Computational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
Computational Methods for Differential Equationsendaily1Thu, 01 Jul 2021 00:00:00 +0430Thu, 01 Jul 2021 00:00:00 +0430Mixed reproducing kernel-based iterative approach for nonlinear boundary value problems with nonlocal conditions
https://cmde.tabrizu.ac.ir/article_10775.html
In this paper, a mixed reproducing kernel function (RKF) is introduced. The kernel&nbsp;function consists of piecewise polynomial kernels and polynomial kernels. On the&nbsp;basis of the mixed RKF, a new numerical technique is put forward for solving non-linear boundary value problems (BVPs) with nonlocal conditions. Compared with&nbsp;the classical RKF-based methods, our method is simpler since it is unnecessary to&nbsp;convert the original equation to an equivalent equation with homogeneous boundary&nbsp;conditions. Also, it is not required to satisfy the homogeneous boundary conditions&nbsp;for the used RKF. Finally, the higher accuracy of the method is shown via several&nbsp;numerical tests.Second Order Boundary Value Problems of Nonsingular Type on Time Scales
https://cmde.tabrizu.ac.ir/article_12148.html
In this study, existence of positive solutions are considered for second order boundary value problems on any time scales even in the case when y =0 may also be a solution.Numerical analysis of fractional differential equation by TSI-wavelet method
https://cmde.tabrizu.ac.ir/article_10385.html
In this paper, we propose a new numerical algorithm for the approximate solution of&nbsp;non-homogeneous fractional differential equation. Using this algorithm the fractional&nbsp;differential equations are transformed into a system of algebraic linear equations&nbsp;by operational matrices of block-pulse and hybrid functions. Based on our new&nbsp;algorithm, this system of algebraic linear equations can be solved by a proposed (TSI)&nbsp;method. Further, some numerical examples are given to illustrate and establish the&nbsp;accuracy and reliability of the proposed algorithm.PDE-based hyperbolic-parabolic model for image denoising with forward-backward diffusivity
https://cmde.tabrizu.ac.ir/article_12150.html
In the present study, we propose an effective nonlinear anisotropic diffusion-based hyperbolic parabolic model for image denoising and edge detection. The hyperbolic-parabolic model employs a second-order PDEs and have a second-time derivative to time t. This approach is very effective to preserve sharper edges and better-denoised images of noisy images. Our model is well-posed and it has a unique weak solution under certain conditions, which is obtained by using an iterative finite difference explicit scheme. The results are obtained in terms of peak signal to noise ratio (PSNR) as a metric, using an explicit scheme with forward-backward diffusivities.New exact solutions and numerical approximations of the generalized KdV equation
https://cmde.tabrizu.ac.ir/article_10328.html
This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B splines over finite elements. Propagation of a single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L&infin; error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.Solving the forward-backward heat equation with a non-overlapping domain decomposition method based on multiquadric RBF meshfree method
https://cmde.tabrizu.ac.ir/article_10952.html
&lrm;&lrm;&lrm;&lrm;&lrm;In this paper&lrm;, &lrm;we present a numerical technique to deal with the one-dimensional forward-backward heat equations&lrm;.&nbsp;First&lrm;, &lrm;the physical domain is divided into two non-overlapping subdomains resulting in two separate forward and backward subproblems&lrm;, &lrm;and then a meshless method based on multiquadric radial basis functions is employed to treat the spatial variables in each subproblem using the Kansa's method&lrm;. &lrm;We use a time discretization scheme to&lrm;&nbsp;approximate the time derivative by the forward and backward finite difference formulas&lrm;.&nbsp;In order to have adequate boundary conditions for each subproblem&lrm;, &lrm;an initial approximate solution is assumed on the interface boundary&lrm;, &lrm;and the solution is improved by solving the subproblems in an iterative way&lrm;. &lrm;The numerical results show that the proposed method is very useful and computationally efficient in comparison with the previous works&lrm;.Stability and bifurcation of fractional tumor-immune model with time delay
https://cmde.tabrizu.ac.ir/article_10334.html
&lrm;The present study aims are to analyze a delay tumor-immune fractional-order system to describe the rivalry among the immune system and tumor cells. Given that the dynamics of this system depend on the time delay parameter, we examine the impact of time delay on this system to attain better compatibility with actuality. For this purpose, we analytically evaluated the stability of the system&rsquo;s equilibrium points. It is shown that Hopf bifurcation occurs in the fractional system when the delay parameter passes a certain value. Finally, by using numerical simulations, the analytical results were compared to the numerical results to acquire several dynamical behaviors of this system.Eigenvalues of Fractional Sturm-Liouville Problems by Successive method
https://cmde.tabrizu.ac.ir/article_10955.html
In this paper&lrm;, &lrm;we consider a fractional Sturm-Liouville equation of the form&lrm;, &lrm;\begin{equation*}&lrm; -&lrm;{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0&lt;\alpha &lt;1,\quad t\in[0,1]&lrm;, &lrm;\end{equation*}&lrm; &lrm;with Dirichlet boundary conditions&lrm; &lrm;$$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0,\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$&lrm; &lrm;where&lrm;, &lrm;the sign $\circ$ is composition of two operators and $q\in L^2(0,1)$&lrm;, &lrm;is a real-valued potential function&lrm;. &lrm;We use a recursive method based on Picard's successive method to find the solution of this problem&lrm;. &lrm;We prove the method is convergent and show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives&lrm;.Constructing an efficient multi-step iterative scheme for nonlinear system of equations
https://cmde.tabrizu.ac.ir/article_10336.html
The objective of this research is to propose a new multi-step method in tackling a&nbsp;system of nonlinear equations. The constructed iterative scheme achieves a higher&nbsp;rate of convergence whereas only one LU decomposition per cycle is required to&nbsp;proceed. This makes the efficiency index to be high as well in contrast to the existing&nbsp;solvers. The usefulness of the presented approach for tackling differential equations&nbsp;of nonlinear type with partial derivatives is also given.On the Existence and Uniqueness of Positive Solutions for a p-Laplacian Fractional Boundary Value Problem with an Integral Boundary Condition with a parameter
https://cmde.tabrizu.ac.ir/article_10956.html
&lrm;&lrm;The aim of this work is to prove the existence and uniqueness of the positive solutions for a fractional boundary value problem by a parameterized integral boundary condition with p-Laplacian operator&lrm;. &lrm;By using iteration sequence&lrm;, &lrm;existence of two solutions is proved&lrm;. &lrm;Also by applying a fixed point theorem on solid cone&lrm;, &lrm;the result for the uniqueness of positive solution to the problem is obtained&lrm;. &lrm;Two examples are given to confirm our results&lrm;.An efficient technique based on the HAM with Green's function for a class of nonlocal elliptic boundary value problems
https://cmde.tabrizu.ac.ir/article_10757.html
In this paper, we propose an efficient technique-based optimal homotopy analysis method with Green&rsquo;s function technique for the approximate solutions of nonlocal elliptic boundary value problems. We first transform the nonlocal boundary value problems into the equivalent integral equations. We then apply the optimal homotopy analysis method for the approximate solution of the considered problems. Several examples are considered to compare the results with the existing technique. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. We also provide the convergence and the error estimation of the proposed method.Meshless approach for pricing Islamic Ijarah under stochastic interest rate models
https://cmde.tabrizu.ac.ir/article_11204.html
Nowadays, the fixed interest rate financing method is commonly used in the capitalist financial system and in a wide range of financial liability instruments, the most important of which is bond. In the Islamic financial system, using these instruments is considered as usury and has been prohibited. In fact, Islamic law, Shari&rsquo;ah, forbids Muslims from receiving or paying off the Riba. Therefore, using customary financial instruments such as bond is not acceptable or applicable in countries which have a majority of Muslim citizens. In this paper, we introduce one financial instrument, Sukuk, as a securities-based asset under stochastic income. These securities can be traded in secondary markets based on the Shari&rsquo;ah law. To this end, this paper will focus on the most common structure of the Islamic bond, the Ijarah sukuk and its negotiation mechanism. Then, by presenting the short-term stochastic model, we solve fixed interest rate and model the securities-based asset by the stochastic model. Finally, we approximate the resulting model by radial basis function method, as well as utilizing the Matlab softwareFourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term
https://cmde.tabrizu.ac.ir/article_10759.html
&lrm;This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.Laguerre collocation method for solving Lane-Emden type equations
https://cmde.tabrizu.ac.ir/article_12107.html
In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.VIM-Pad´e technique for solving nonlinear and delay initial value problems
https://cmde.tabrizu.ac.ir/article_10758.html
In this work, we employ a combination of variational iteration method (VIM) and Pad&acute;e approximation method, called the VIM-Pad&acute;e technique, to solve some nonlinear initial value problems and a delay differential equation (DDE). Some examples are provided to illustrate the capability and reliability of the technique. The obtained results by using the VIM are compared to the results of this technique. This comparison shows that VIM-Pad&acute;e technique is more effective than VIM and yields faster convergence compared to the VIM.Analytical Fuzzy Solution of the Ventricular Pressure Equation and Prediction of the Blood Pressure
https://cmde.tabrizu.ac.ir/article_12108.html
The cardiovascular system is an extremely intelligent and dynamic system which adjusts its performance depending on the individual's physical and environmental conditions. Some of these physical and environmental conditions may create slight disruptions in the cardiovascular system leading to a variety of diseases. Since prevention has always been preferable to treatment, this paper examined the Instantaneous Pressure-Volume Relation (IPVR) and also the pressure of the artery root. The fuzzy mathematics as a powerful tool is used to evaluate and predict the status of an individual's blood pressure. The arterial pressure is modeled as a first-order fuzzy differential equation and an analytical solution for this equation is obtained and an example show the behavior of the solution. The risk factors using fuzzy rules are assessed, which help diagnose the status of individual's blood pressure. Using the outcome by drawing the individual's attention to these risk factors, the individual's health is improved. Moreover, in this study adaptive neuro-fuzzy inference system (ANFIS) models is evaluated to predict the status of an individual's blood pressure on the basis of the inputs.Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations
https://cmde.tabrizu.ac.ir/article_10760.html
In this paper, we first construct Alpert wavelet system and propose a computational method for solving a fractional nonlinear Fredholm integro-differential equation. Then we create an operational matrix of fractional integration and use it to simplify the equation to a system of algebraic equations. By using Newtons iterative method, this system is solved and the solution of the fractional nonlinear Fredholm integro-differential equations is achieved. Thresholding parameter is used to increase the sparsity of matrix coefficients and the speed of computations. Finally, the method is demonstrated by examples, and then compared results with CAS wavelet method show that our proposed method is more effective and accurate.Extending a new two-grid waveform relaxation on a spatial finite element discretization
https://cmde.tabrizu.ac.ir/article_12110.html
In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.An accurate method for nonlinear local fractional Wave-Like equations with variable coefficients
https://cmde.tabrizu.ac.ir/article_10935.html
The basic motivation of the present study is to apply the local fractional Sumudu variational iteration method (LFSVIM) for solving nonlinear wave-like equations with variable coefficients and within local fractional derivatives. The derivatives operators are taken in the local fractional sense. The results show that the LFSVIM is an appropriate method to find non-differentiable solutions for similar problems. The results of the solved examples showed the flexibility of applying this method and its ability to reach accurate results even with these new differential equations.A new methodology to estimate constant elasticity of variance
https://cmde.tabrizu.ac.ir/article_12147.html
This paper introduces a novel method for estimation of the parameters of the constant elasticity of variance model. To do this, the likelihood function will be constructed based on the approximate density function. Then, to estimate the parameters, some optimization algorithms will be appliedSensitivity analytic and synchronization of a new fractional-order financial system
https://cmde.tabrizu.ac.ir/article_10774.html
In this paper, we present a new fractional-order financial system (FOFS) with the new parameters. We study the synchronization for commensurate order of the fractional-order financial system with disturbance observer (FOFSDO) on the new parameters. Also, the sensitivity analysis of the synchronization error was investigated by using the feedback control technique for the FOFSDO. The stability of the used method demonstrates by Lyapunov stability theorem. Numerical simulations are presented to ensure the validity and influence of the target feedback control design in the presence of extrinsic bounded unknown disturbance.Toward a new understanding of cohomological method for fractional partial differential equations
https://cmde.tabrizu.ac.ir/article_12149.html
One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation&lrm; &lrm;$ \psi^{\alpha} f=g $&lrm;, &lrm;on&lrm; &lrm;$ \mathbb{T}^n $&lrm;. &lrm;We prove that if&lrm; &lrm;$ f $&lrm; &lrm;is not analytic&lrm;, &lrm;then fractional integro-differential equation&lrm; &lrm;$ I_t^{1-\alpha} D_x^{\alpha}u(x,t)+i I_x^{1-\alpha} D_t^{\alpha}u(x,t)=f(t) $&lrm; &lrm;has no solution in&lrm; &lrm;$ C^1(B) $ with $0&lt; \alpha \leq 1$&lrm;. &lrm;ٌ&lrm;W&lrm;e &lrm;also&lrm; obtain &lrm;solutions &lrm;for&lrm; the space-time fractional heat &lrm;equations&lrm; on&lrm; &lrm;$ \mathbb{S}^1 $&lrm; &lrm;and &lrm;$ \mathbb{T}^n $&lrm;. &lrm;At the end of this article&lrm;, &lrm;there are examples of fractional partial differential equations and a fractional integral equation together with their solutions&lrm;.Solving of partial differential equations with distributed order in time using fractional-order Bernoulli-Legendre functions
https://cmde.tabrizu.ac.ir/article_10773.html
In this paper, an efficient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation
https://cmde.tabrizu.ac.ir/article_12151.html
A radial basis functions (RBFs) method for solving nonlinear time- and space-fractional Fokker-Planck equation is presented. The time-fractional derivative is Caputo type and the space-fractional derivative is Caputo or Riemann-Liouville type. The Caputo and Riemann-Liouville fractional derivatives of RBFs are utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step the time-fractional derivative is approximated by the high order formulas introduced in \cite{CaoLiChen} and, a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Two numerical examples are included to demonstrate the applicability, accuracy and stability of the method. Numerical experiments show that the experimental order of convergence is $4-\alpha$, $\alpha$ is the order of time derivative.Multiple solutions for a fourth-order elliptic equation involving singularity
https://cmde.tabrizu.ac.ir/article_10936.html
Here, we consider a fourth-order elliptic problem involving singularity and p(x)- biharmonic operator. Using Hardy&rsquo;s inequality, S+-condition, and Palais-Smale condition, the existence of weak solutions in a bounded domain in RN is proved. Finally, we percent some examples.Dynamical behaviours of Bazykin-Berezovskaya model with fractional-order and its discretization
https://cmde.tabrizu.ac.ir/article_12152.html
&lrm;This paper is devoted to study dynamical behaviours of the fractional-order Bazykin-Berezovskaya model and its discretization&lrm;. &lrm;The fractional derivative has been described in the Caputo sense&lrm;. &lrm;We show that the discretized system&lrm;, &lrm;exhibits more complicated dynamical behaviours than its corresponding fractional-order model&lrm;. &lrm;Specially&lrm;, &lrm;in the discretized model Neimark-Sacker and flip bifurcations and also chaos phenomena will happen&lrm;. &lrm;In the final part&lrm;, &lrm;some numerical simulation verify the analytical results&lrm;.The numerical approximation for the solution of linear and nonlinear integral equations of the second kind by interpolating moving least squares
https://cmde.tabrizu.ac.ir/article_10934.html
In this paper, the interpolating moving least-squares (IMLS) method is discussed. The interpolating moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for the numerical solution of Volterra&ndash;Fredholm integral equations, and finally some examples are given to show the accuracy and applicability of the method.Finite Volume Element Approximation For Time-dependent Convection-Diffusion-ReactionEquations With Memory
https://cmde.tabrizu.ac.ir/article_12153.html
Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated.\ For the spatially discrete scheme, optimal order error estimates in $L^{2},$ $H^{1},\ $ and \ $W^{1,p\ }$ norms for $2\leq p &lt;\infty ,$ are obtained. Inthis paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.Bernoulli wavelet method for numerical solutions of system of fuzzy integral equations
https://cmde.tabrizu.ac.ir/article_12839.html
In this paper, we have proposed an efficient numerical method to solve a system linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM). Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First, we introduce properties of Bernoulli wavelets then we used it to transform the integral equations to the system of algebraic equations, the error estimates of the proposed method are given and compared by solving some numerical examples.Non-uniform L1/DG method for one-dimensional time-fractional convection equation
https://cmde.tabrizu.ac.ir/article_12171.html
In this paper, we present an efficient numerical method to solve a one-dimensional time-fractional convection equation whose solution has a certain weak regularity at the starting time, where the time-fractional derivative in the Caputo sense with order in (0,1) is discretized by the L1 finite difference method on non-uniform meshes and the spatial derivative by the discontinuous Galerkin (DG) finite element method. The stability and convergence of the method are analyzed. Numerical experiments are provided to confirm the theoretical results.Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations
https://cmde.tabrizu.ac.ir/article_10933.html
This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method.The Symmetry Analysis and Analytical Studies of the Rotational Green-Naghdi (R-GN) Equation
https://cmde.tabrizu.ac.ir/article_12172.html
The simplified phenomenological model of long-crested shallow-water wave propagations is considered without/with the Coriolis effect. Symmetry analysis is taken into consideration to obtain exact solutions. Both classical wave transformation and transformations are obtained with symmetries and solvable equations are kept thanks to these transformations. Additionally, the exact solutions are obtained via various methods which are ansatz based methods. The obtained results have a major role in the literature so that the considered equation is seen in a large scale of applications in the area of geophysical.Optimal control of double delayed HIV-1 infection model of fighting a virus with another virus
https://cmde.tabrizu.ac.ir/article_12109.html
A double delayed- HIV-1 infection model with optimal controls is taken into account. The proposed model consists of double-time delays and the following five compartments: uninfected cells CD4+ T cells, infected CD4+ T cells, double infected CD4+ T cells, human immunodeficiency virus, and recombinant virus. Further, the optimal controls functions are introduced into the model. Objective functional is constituted which aims to (i) minimize the infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood Then, the existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically using an iterative method with Runge-Kutta fourth-order scheme.An infinite number of nonnegative solutions for iterative system of singular fractional order boundary value problems
https://cmde.tabrizu.ac.ir/article_12173.html
In this paper, we consider the iterative system of singular Rimean-Liouville fractional order boundary value problems with RiemannStieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii&rsquo;s cone fixed point theorem in a Banach space, we derive sufficent conditions for the existence of infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of unique nonnegative solution to the addressed problem by fixed point theorem in a complete metric space. As an application, we present an example to illustrate the main results.Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems
https://cmde.tabrizu.ac.ir/article_12113.html
This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and both the results are presented in tables and graphs. The numerical results are compared with findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.An approximation to the solution of one-dimensional hyperbolic telegraph equation based on the collocation of quadratic b-spline functions
https://cmde.tabrizu.ac.ir/article_12204.html
In this work, collocation method based on B-spline functions is used to obtained a numerical solution for one-dimensional hyperbolic telegraph equation. The proposed method is consists of two main steps. As first step, by using finite difference scheme for time variable, partial differential equation is converted to an ordinary differential equation by space variable. In the next step, for solving this equation collocation method is used. In the analysis section of the proposed method, the convergence of the method is studied. Also, some numerical results are given to demonstrate the validity and applicability of the presented technique. The L&infin;, L2 and Root-Mean-Square(RMS) in the solutions show the efficiency of the method computationally.Solving free boundary problem for an initial cell layer in multispecies biofilm formation by Newton-Raphson method
https://cmde.tabrizu.ac.ir/article_10908.html
The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.Cubic B-spline collocation method on non-uniform mesh for solving non-linear parabolic partial differential equation
https://cmde.tabrizu.ac.ir/article_12205.html
In this paper, an approximate solution of non-linear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme is shown by numerical experiments. We have compared the approximate solutions with that in the literature.Conformable double Laplace transform method for solving conformable fractional partial differential equations
https://cmde.tabrizu.ac.ir/article_12112.html
In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional differential in mathematical physics. The results obtained show that the proposed method is efficient, reliable, and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.An Exponential Cubic B-spline Algorithm for the Nonlinear Coupled Burgers' Equation
https://cmde.tabrizu.ac.ir/article_12223.html
The collocation method based on the exponential cubic B-splines (ECB-splines) together with the Crank-Nicolson formula is used to solve nonlinear coupled Burgers' equation (CBE). This method is tested by studying three different problems. The proposed scheme is compared with some existing methods. \textbf{It produced accurate results }with the suitable selection of the free parameter of the ECB-spline function. It produces accurate results. Stability of the fully discretized CBE is investigated by the Von Neumann analysis.Controllability and observability of linear impulsive differential algebraic system with Caputo fractional derivative
https://cmde.tabrizu.ac.ir/article_12206.html
Linear impulsive fractional differential-algebraic systems (LIFDAS) in a finite-dimensional space are studied. We obtain the solution of LIFDAS. Using Gramian matrices, necessary and sufficient conditions for controllability and observability of time-varying LIFDAS are established. We acquired criterion for time-invariant LIFDAS in the form of rank conditions. The results are more generalized than the results that are obtained for various differential-algebraic systems without impulsesBounding error of calculating the matrix functions
https://cmde.tabrizu.ac.ir/article_12207.html
Matrix functions play important roles in various branches of science and engineering. In numerical computations and physical measurements there are several sources of error which significantly affect the main results obtained from solving the problems. This effect also influences the matrix computations. In this paper, we propose some approaches to enclose the matrix functions. We then present some analytical arguments to ensure that the obtained enclosures contain the exact result. Numerical experiments are given to illustrate the performance and effectiveness of the proposed approaches.Design of Normal distribution-based algorithm for solving systems of nonlinear equations
https://cmde.tabrizu.ac.ir/article_12208.html
In this paper, a completely new statistical based approach is developed for solving the system of nonlinear equations. The developed approach utilizes the characteristics of the normal distribution to search the solution space. The normal distribution is generally introduced by two parameters, i.e., mean and standard deviation. In the developed algorithm, large values of standard deviation enable the algorithm to escape from a local optimum, and small values of standard deviation help the algorithm to find the global optimum. In the following, six benchmark tests and thirteen benchmark case problems are investigated to evaluate the performance of the Normal Distribution-based Algorithm (NDA). The obtained statistical results of NDA are compared with those of PSO, ICA, CS, and ACO. Based on the obtained results, NDA is the least time-consuming algorithm that gets high-quality solutions. Furthermore, few input parameters and simple structure introduce NDA as a user friendly and easy-to-understand algorithm.Qualitative analysis of fractional differential equations with $\psi$-Hilfer fractional derivative
https://cmde.tabrizu.ac.ir/article_12209.html
We discuss successive approximation techniques for the investigation of solutions of fractional differential equations with $\psi$-Hilfer fractional derivative. Next, we present the continuous dependence of a solution for the given Cauchy-type problem.Existence of solution for nonlinear integral inclusions
https://cmde.tabrizu.ac.ir/article_12210.html
In this paper, we prove the existence of solution of two nonlinear integral inclusions by using generalization of Krasnoselskii fixed point theorem for set-valued mappings. As an application we prove the existence of solution of the boundary valued problem of ordinary differential inclusion.The monotonicity and convexity of the period function for a class of symmetric Newtonian systems of degree 8
https://cmde.tabrizu.ac.ir/article_12211.html
In this paper, we study the monotonicity and convexity of the period function associated with centers of a specific class of symmetric Newtonian systems of degree 8. In this regard, we prove that if the period annulus surrounds only one elementary center, then the corresponding period function is monotone; but, for the other cases, the period function has exactly one critical point. We also prove that in all cases, the period function is convex.New midpoint type inequalities for generalized fractional integral
https://cmde.tabrizu.ac.ir/article_12212.html
In this paper, we first establish two new identities for differentiable function involving generalized fractional integrals. Then, by utilizing these equalities, we obtain some midpoint type inequalities involving generalized fractional integrals for mappings whose derivatives in absolute values are convex. We also give several results as special cases of our main results.Synchronization between Integer and Fractional Chaotic Systems Via. Tracking Control and Non Linear Control With Application
https://cmde.tabrizu.ac.ir/article_12213.html
In this paper the synchronization between complex fractional order chaotic system and integer order hyper chaotic system has been investigated. Due to increased complexity and presence of additional variables, it seems to be very interesting and can be associated with real life problems. Based on the idea of tracking control and non linear control, we have designed the controllers to obtain the synchronization between the chaotic systems. To establish the efficacy of the methods computations have been carried out. Excellent agreement between the analytical and computational studies has been observed. The achieved synchronization is illustrated in field of secure communication. The results have been compared with published literature.An interval version of the Kuntzmann-Butcher method for solving the initial value problem
https://cmde.tabrizu.ac.ir/article_12214.html
The Kutzmann-Butcher method is the unique implicit four-stage Runge-Kutta method of order 8. In many problems in ordinary differential equations this method realized in floating-point arithmetic gives quite good approximations to the exact solutions, but the results obtained do not contain any information on rounding errors, representation errors and the error of the method. Thus, we describe an interval version of this method, which realized in floating-point interval arithmetic gives approximations (enclosures in the form of interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.Existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays
https://cmde.tabrizu.ac.ir/article_12215.html
In this article, we concentrate on the existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Initially, the existence of the mild solutions to the equations by utilizing Banach fixed point theorem is demonstrated. In the later case we explore the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval.A numerical technique for solving nonlinear fractional stochastic integro-differential equations with n-dimensional Wiener process
https://cmde.tabrizu.ac.ir/article_12216.html
This paper deals with the numerical solution of nonlinear fractional stochastic integro-differential equations with the n-dimensional Wiener process. A new computational method is employed to approximate the solution of the considered problem. This technique is based on the modified hat functions, the Caputo derivative and a suitable numerical integration rule. Error estimate of the method is investigated in detail. At the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.New optical soliton solutions for the thin-film ferroelectric materials equation instead of the numerical solution
https://cmde.tabrizu.ac.ir/article_12217.html
In this article, we will implement the (G&prime;/G)-expansion method which is used for the first time to obtain new optical soliton solutions of the thin-film ferroelectric materials equation (TFFME). Also, the numerical solutions of the suggested equation according to the variational iteration method (VIM) are demonstrated effectively. A comparison between the achieved exact and numerical solutions has been established successfully.Some new soliton solutions for the nonlinear the fifth-order integrable equations
https://cmde.tabrizu.ac.ir/article_12218.html
In this work, we established some exact solutions for the (1 + 1)-dimensional and (2 + 1)-dimensional fifth-order integrable equations ((1+1)D and (2+1)D FOIEs) which is considered based on the improved tanh(ϕ(&xi;)/2)-expansion method (IThEM), by utilizing Maple software. We obtained new periodic solitary wave solutions. The obtained solutions include soliton, periodic, kink, kink-singular wave solutions. Comparing our new results with Wazwaz results, namely, Hereman-Nuseri method [2, 3] show that our results give the further solutions. Many other such types of nonlinear equations arising in uid dynamics, plasma physics and nonlinear physics.An Anomalous Diffusion Approach for Speckle Noise Reduction in Medical Ultrasound Images
https://cmde.tabrizu.ac.ir/article_12219.html
Medical ultrasound images are usually degraded by a specific type of noise, called "speckle". The presence of speckle noise in medical ultrasound images will reduce the image quality and affect the effective information, which can potentially cause a misdiagnosis. Therefore, medical image enhancement processing has been extensively studied and several denoising approaches have been introduced and developed. In the current work, a robust fractional partial differential equation (FPDE) model based on the anomalous diffusion theory is proposed and used for medical ultrasound image enhancement. An efficient computational approach based on a combination of a time integration scheme and localized meshless method in a domain decomposition framework is performed to deal with the model. {In order to evaluate the performance of the proposed de-speckling approach, it is used for speckle noise reduction of a synthetic ultrasound image degraded by different levels of speckle noise. The results indicate the superiority of the proposed approach in comparison with classical anisotropic diffusion denoising model (Catt$\acute{e}$'s pde model).}Modulation instability analysis, optical solitons and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schrodinger's equation
https://cmde.tabrizu.ac.ir/article_12220.html
The current study utilizes the extended sinh-Gordon equation expansion and ($\frac{G^{\prime}}{G^2}$)-expansion function methods in constructing various optical soliton and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schr${\ddot o}$dinger's equation which describes the elevation of water wave surface for slowly modulated wave trains in deep water in hydrodynamics. We secure different kinds of solutions like optical dark, bright, singular, combo solitons as well as hyperbolic and trigonometric functions solutions. Moreover, singular periodic wave solutions are recovered and the constraint conditions which provide the guarantee to the soliton solutions are also reported. In order to shed more light on these novel solutions, graphical features 3D, 2D and contour with some suitable choice of parameter values have been depicted. We also discuss the stability analysis of the studied nonlinear model with aid of modulation instability analysis.Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials
https://cmde.tabrizu.ac.ir/article_12221.html
This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package.A new numerical fractional differentiation formula to approximate the Caputo-Fabrizio fractional derivative: error analysis and stability
https://cmde.tabrizu.ac.ir/article_12222.html
In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula)&nbsp;to approximate the Caputo-Fabrizio fractional derivative of order $\alpha,$ $(0&lt;\alpha&lt;1)$ is developed.&nbsp;It is established by means of the quadratic interpolation approximation using&nbsp;three points&nbsp;$ (t_{j-2},y(t_{j-2}))$,&nbsp; $(t_{j-1},y(t_{j-1})) $&nbsp;and&nbsp;$ (t_{j},y(t_{j})) $&nbsp;on each interval&nbsp;$[t_{j-1},t_{j}]$ for&nbsp;$ ( j \geq 2 )$,&nbsp;while the linear interpolation approximation is applied on the first interval&nbsp;$[t_{0},t_{1}]$. As a result, the new formula can be&nbsp;formally viewed as a modification of the classical CF1 formula,&nbsp;which is obtained by the piecewise linear approximation for&nbsp;$y(t)$. Both the computational efficiency and numerical accuracy of&nbsp;the new formula is superior to that of the CF1 formula.&nbsp;The coefficients and truncation errors of this formula are discussed in detail.&nbsp;{Two test example show} the numerical accuracy of the CF2 formula.&nbsp;The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated.Fractional study on heat and mass transfer of MHD Oldroyd-B fluid with ramped velocity and temperature
https://cmde.tabrizu.ac.ir/article_12288.html
This study explores the time-dependent convective flow of MHD Oldroyd-B fluid under the effect of ramped wall velocity and temperature. The flow is confined to an infinite vertical plate embedded in a permeable surface with the impact of heat generation and thermal radiation. Solutions of velocity, temperature, and concentration are derived symmetrically by applying non-dimensional parameters along with Laplace transformation $(LT)$ and numerical inversion algorithm. Graphical results for different physical constraints are produced for the velocity, temperature, and concentration profiles. Velocity and temperature profile decrease by increasing the effective Prandtl number. The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity. Velocity is decreasing for $\kappa$, $M$, $Pr_{reff,}$ and $Sc$ while increasing for $G_{r}$ and $G_{c}$. Temperature is an increasing function of the fractional parameter. Additionally, Atangana-Baleanu $(ABC)$ model is good to explain the dynamics of fluid with better memory effect as compared to other fractional operators.Numerical Method for the Solution of Algebraic Fuzzy Complex Equations
https://cmde.tabrizu.ac.ir/article_12289.html
In this paper, the numerical solution of an algebraic complex fuzzy equation of degree ${n}$, based on the parametric fuzzy numbers, is discussed. The unknown variable and right-hand side of the equation are considered as fuzzy complex numbers, whereas, the coefficients of the equation, are considered to be real crisp numbers. The given method is a numerical method and proposed based on the separation of the real and imaginary parts of the equation and using the parametric forms of the fuzzy numbers in the form of polynomials of degree at most ${m}$. In this case, a system of nonlinear equations achieved. To get the solutions of the system, we used the Gauss-Newton iterative method. We also very briefly explain the conjugate of the solution of such equations. Finally, the efficiency and quality of the given method are tested by applying it to some numerical examples.Numerical solution for solving fractional parabolic partial differential equations
https://cmde.tabrizu.ac.ir/article_12508.html
In this paper, A reliable numerical scheme is developed and reviewed in order to obtain approximate solution of time fractional parabolic partial differential equations. The introduced scheme is based on Legendre tau spectral approximation and the time fractional derivative is employed in the Caputo sense. TheL2convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of spectral method and justification the theoretical analysis, and to compare with other existing methods in the literaturesOptimal control of satellite attitude and its stability based on quaternion parameters
https://cmde.tabrizu.ac.ir/article_12509.html
&lrm;This paper proposes an optimal control method for the chaotic &lrm;attitude of the satellite when it is exposed to external disturbances. When there is no control over the satellite, its chaotic attitude &lrm;is investigated using Lyapunov exponents (LEs)&lrm;, Poincare diagrams, and bifurcation diagrams. &lrm;In order to overcome the problem of singularity in the great maneuvers of satellite, &lrm;we consider the kinematic equations based on quaternion parameters instead of Euler angles, &lrm;and obtain control functions by using the Pontryagin maximum principle (PMP)&lrm;. &lrm;These functions are able to reach the satellite attitude to its equilibrium point. &lrm;Also the asymptotic stability of these control functions is investigated by Lyapunov's stability theorem. &lrm;Some simulation results are given to visualize the effectiveness and feasibility of the proposed method.Qualitative Stability Analysis of a Non-Hyperbolic Equilibrium Point of a Caputo Fractional System
https://cmde.tabrizu.ac.ir/article_12510.html
In this manuscript a center manifold reduction of the flow of a non-hyperbolic equilibrium point on a planar dynamical system with the Caputo derivative is proposed. The stability of the non-hyperbolic equilibrium point is shown to be locally asymptotically stable, under suitable conditions, by using the fractional Lyapunov direct method.Studying the Thermal Analysis of Rectangular Cross Section Porous Fin: A Numerical Approach
https://cmde.tabrizu.ac.ir/article_12511.html
In this work, a direct computational method has been developed for solving the thermal analysis of porous fins with a rectangular cross-section with the aid of Chebyshev polynomials. The method transforms the nonlinear differential equation into a system of nonlinear algebraic equations and then solved using a novel technique. The solution of the system gives the unknown Chebyshev coefficients. An algorithm for solving this nonlinear system is presented. The results are obtained for different values of the variables and a comparison with other methods is made to demonstrate the effectiveness of the method.The interior inverse boundary value problem for the impulsive Sturm-Liouville operator with the spectral boundary conditions
https://cmde.tabrizu.ac.ir/article_12512.html
In this study, we discuss the inverse problem for the Sturm-Liouville operator with the impulse and with the spectral boundary conditions on the finite interval (0, \pi). By taking the Mochizuki-Trooshin's method, we have shown that some information of eigenfunctions at some interior point and parts of two spectra can uniquely determine the potential function q(x) and the boundary conditions.PDTM Approach to Solve Black Scholes Equation for Powered ML-Payoff Function
https://cmde.tabrizu.ac.ir/article_12687.html
In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions, $\max {S^k\ln\big(\frac{S}{K}\big),0\}$ and $\max\{S^k\ln\big(\frac{K}{S}\big),0\}, (k\in \mathbb{R^{+}}\cup \{0\})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM is quite accurate to the closed form solutions.Numerical solution of optimal control problem for economic growth model using RBF collocation method
https://cmde.tabrizu.ac.ir/article_12688.html
In the current paper, for the economic growth model, an efficient numerical approach on arbitrary collocation points is described according to Radial Basis Functions (RBFs) interpolation to approximate the solutions of optimal control problem. The proposed method is based on parametrizing the solutions with any arbitrary global RBF and transforming the optimal control problem into a constrained optimization problem using arbitrary collocation points. The superiority of the method is its flexibility to select between different RBF functions for the interpolation and also parametrization an extensive range of arbitrary nodes. The Lagrange&nbsp;multipliers method is employed to convert the constrained optimization problem into a system of algebraic equations. Numerical results approve the accuracy and performance of the presented method for solving optimal control problems in the economic growth model.Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrodinger equation by beta-fractional derivative
https://cmde.tabrizu.ac.ir/article_12689.html
In this article, a new version of the trial equation method is suggested. This method allows new&nbsp;exact solutions of the nonlinear partial differential equations. The developed method is applied to&nbsp;unstable nonlinear fractional-order Schr&uml;odinger equation in fractional time derivative form&nbsp;of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the&nbsp;new powerful expansion approach using by beta-fractional derivatives which are used to get many&nbsp;solitary wave solutions by changing various parameters. New exact solutions are expressed with&nbsp;rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions,&nbsp;dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr&uml;odinger&nbsp;equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact&nbsp;solution are given with two and three dimensional graphs.New analytical methods for solving a class of conformable fractional differential equations by fractional Laplace transform
https://cmde.tabrizu.ac.ir/article_12690.html
In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDE) and some more results about Laplace transform introduced by Abdeljawad \cite{abdeljawad2015conformable} are investigated. The Laplace transform method is developed to get the exact solution of conformable fractional differential equations. The aim of this paper is to convert the conformable fractional differential equations into ordinary differential equations (ODE), this is done by using the fractional Laplace transformation of $(\alpha+\beta)$ order.A meshless technique based on the radial basis functions for solving systems of partial differential equations
https://cmde.tabrizu.ac.ir/article_12706.html
The radial basis functions (RBFs) methods were first developed by Kansa to approximate partial differential equations (PDEs). The RBFs method is being truly meshfree becomes quite appealing, owing to the presence of distance function, straight-forward implementation, and ease of programming in higher dimensions. Another considerable advantage is the presence of a tunable free shape parameter, contained in most of the RBFs that control the accuracy of the RBFs method. Here, the solution of the two dimensional system of nonlinear partial differential equations is examined numerically by a Global Radial Basis Functions Collocation Method (GRBFCM). It can work on a set of random or uniform nodes with no need for element connectivity of input data. For the time-dependent partial differential equations, a system of ordinary differential equations (ODEs) is derived from this scheme. Like some other numerical methods, a comparison between numerical results with analytical solutions is implemented confirming the efficiency, accuracy, and simple performance of the suggested method.Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation
https://cmde.tabrizu.ac.ir/article_12707.html
&lrm; In this paper option pricing is given via stochastic analysis and invariant subspace method. Finally numerical solutions is driven and shown via diagram. The considered model is one of the most well known non-linear time series model in which the switching mechanism is controlled by an unobservable state variable that follows a first-order Markov chain. Some analytical solutions for option pricing are given under our considered model. Then numerical solutions are presented via finite difference method.An efficient approximate solution of Riesz fractional advection-diffusion equation
https://cmde.tabrizu.ac.ir/article_12721.html
The Riesz fractional advection-diffusion is a result of the mechanics of chaotic dynamics. It's of preponderant importance to solve this equation numerically. Moreover, the utilization of Chebyshev polynomials as a base in several mathematical equations shows the exponential rate of convergence. To this approach, we transform the interval of state space into the interval [-1,1] * [-1,1] Then, we use the operational matrix to discretize fractional operators. Applying the resulting discretization, we obtain a linear system of equations, which leads to the numerical solution. Examples show the effectiveness of the method.Exact solutions and numerical simulation for Bakstein-Howison model
https://cmde.tabrizu.ac.ir/article_12764.html
In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact stochastic analysis and Lie group analysis are applied to find exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.An adaptive Monte Carlo algorithm for European and American options
https://cmde.tabrizu.ac.ir/article_12765.html
Abstract. In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different size and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm.Application of fuzzy systems on the numerical solution of the elliptic PDE-constrained optimal control problems
https://cmde.tabrizu.ac.ir/article_12766.html
&lrm;This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an optimal control problem governed by Poisson's differential equation&lrm;. The considered problem may or may not be accompanied by a control box constraint&lrm;. &lrm;The first-order necessary optimality conditions have been derived, which may contain a variational inequality in function space&lrm;. &lrm;In the presented method&lrm;, &lrm;the obtained optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized optimality conditions&lrm;. &lrm;The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve it&lrm;. A fuzzy-system-based semi-smooth Newton method has also been introduced&lrm; &lrm;to deal with the obtained system&lrm;. &lrm;Solving optimality systems by the presented method gets us unknown fuzzy quantities on the state and control fuzzy expansions&lrm;. &lrm;Finally&lrm;, &lrm;some test problems&lrm; &lrm;have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique&lrm;.The Convergence of exponential Euler method for weighted fractional stochastic equations
https://cmde.tabrizu.ac.ir/article_12794.html
In this paper, we propose an exponential Euler method to&nbsp;approximate the solution of a stochastic functional differential equation&nbsp;driven by weighted fractional Brownian motion B{a,b} under some&nbsp;assumptions on a and b. We obtain also the convergence rate of the method to&nbsp;the true solution after proving an L2 -maximal bound for the stochastic&nbsp;ntegrals in this case.Mean-square stability of a constructed Third-order stochastic Runge--Kutta schemes for general stochastic differential equations
https://cmde.tabrizu.ac.ir/article_12795.html
In this paper, we are interested in construction of an explicit third-order stochastic Runge&ndash;Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the It&circ;o-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered with a large number of equations and could find to derive four families for SRK3 schemes. Also we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.Shifted Jacobi collocation method for Volterra-Fredholm integral equation
https://cmde.tabrizu.ac.ir/article_12796.html
In this paper, we evaluate the approximate numerical solution for the Volterra-Fredholm integral equation (V-FIE) using the shifted Jacobi collocation (SJC) method. This method depends on the operational matrices. We present some properties of the shifted Jacobi polynomials. These properties together with the shifted Jacobi polynomials transform the Volterra-Fredholm integral equation into a system of algebraic equations in the expansion coefficients of the solution. We discuss the convergence and error analysis of the shifted Jacobi polynomials in detail. The efficiency of this method is verified through numerical examples and compared with others.A Robust computational method for singularly perturbed delay parabolic convection-diffusion equations arising in the modeling of neuronal variability
https://cmde.tabrizu.ac.ir/article_12797.html
In this study, a robust computational method involving exponential cubic spline for solving singularly perturbed parabolic convection-diffusion equations arising in the modeling of neuronal variability has been presented. Some numerical examples are considered to validate the theoretical findings. The proposed scheme is shown to be an &epsilon;-uniformly convergent accuracy of order O(&Delta;t+h^2 ).A Numerical Method For Solving Fractional Optimal Control Problems Using The Operational Matrix Of Mott Polynomials
https://cmde.tabrizu.ac.ir/article_12798.html
&lrm;This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs) based on numerical polynomial approximation&lrm;. &lrm;The fractional derivative in the dynamic system is described in the Caputo sense&lrm;. &lrm;We used the approach in order to approximate the state and control functions by the Mott polynomials (M-polynomials)&lrm;. &lrm;We introduced the operational matrix of fractional Riemann-Liouville integration and apply it to approximate the fractional derivative of the basis&lrm;. &lrm;We investigated the convergence of the new method and some examples are included to demonstrate the validity and applicability of the proposed method&lrm;.Exact solutions of Diffusion Equation on sphere
https://cmde.tabrizu.ac.ir/article_12827.html
&lrm;We examine the diffusion&lrm; &lrm;equation on the sphere&lrm;. &lrm;In this sense&lrm;, &lrm;we answer question of the symmetry classification&lrm;. &lrm;We provide the algebra of symmetry and build&lrm; &lrm;the optimal system of Lie subalgebras&lrm;. &lrm;We prove for one-dimensional optimal systems of Eq&lrm;.(4), &lrm;space is expanding Ricci solitons&lrm;. &lrm;Reductions of similarities related to subalgebras are classified&lrm;, &lrm;and some exact invariant solutions of the diffusion&lrm; &lrm;equation on the sphere are presented&lrm;.Numerical investigation of the generalized Burgers-Huxley equation using combination of multiquadric quasi-interpolation and method of lines
https://cmde.tabrizu.ac.ir/article_12831.html
In this article, an efficient method for approximate the solution of the generalized Burgers-Huxley (gB-H) equation using multiquadric quasi-interpolation approach is considered. This method consists of two phases. First, the spatial derivatives are evaluated by MQ quasi-interpolation, So the gB-H equation is reduces to a nonlinear system of ordinary differential equations. In phase two, the obtained system is solved by using ODE solvers. Numerical examples demonstrate the validity and applicability of the method.Collocation method based on radial basis functions via symmetric variable shape parameter for solving a particular class of delay differential equations
https://cmde.tabrizu.ac.ir/article_12954.html
In this article, we use the collocation method based on the radial basis functions with sym- metric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional- differential equations with proportional delays. We used Gaussian radial basis functions with SVSP. Using non uniform collocation points, we achieved a system and solving this system yielded the prob- lem solutions. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs). Finally, numerical rate of convergence analysis of the numerical approximation was also obtained. It is observed that by comparing be- tween the obtained ROC values of error norms by the SVSP and CSP method, SVSP results were considered acceptable.Regularized Prabhakar Derivative Applications to Partial Differential Equations
https://cmde.tabrizu.ac.ir/article_12975.html
Prabhakar fractional operator was applied recently for studying the dynamics&nbsp;of complex systems from several branches of sciences and engineering. In&nbsp;this manuscript we discuss the regularized Prabhakar derivative applied to&nbsp;fractional partial differential equations using the Sumudu homotopy analysis&nbsp;method(PSHAM). Three illustrative examples are investigated to confirm our&nbsp;main results.Combining the reproducing kernel method with a practical technique to solve the system of nonlinear singularly perturbed boundary value problems
https://cmde.tabrizu.ac.ir/article_12976.html
In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision.Uniformly convergent fitted operator method for singularly perturbed delay differential equations
https://cmde.tabrizu.ac.ir/article_12977.html
This paper deals with numerical treatment of singularly perturbed delay differential equations having delay on first derivative term. The solution of the considered problem exhibits boundary layer behaviour on left or right side of the domain depending on the sign of the convective&nbsp;term. The term with the delay is approximated using Taylor series approximation, resulting to asymptotically equivalent singularly perturbed boundary value problem. Uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of&nbsp;the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.A simulation study of the COVID-19 pandemic based on the Ornstein-Uhlenbeck processes
https://cmde.tabrizu.ac.ir/article_12978.html
&lrm;&lrm;The rapid spread of &lrm;coronavirus &lrm;disease&lrm; (&lrm;COVID-19) &lrm;has&lrm;&lrm;&lrm; increased the attention to the mathematical modeling of spreading the disease in the &lrm;world.&lrm; &lrm;The behavior of spreading &lrm;is &lrm;not &lrm;deterministic&lrm; &lrm;in &lrm;the &lrm;last &lrm;year&lrm;. The purpose of this paper is to presents a stochastic differential equation for modeling the data sets of the COVID-19 involving &lrm;infected&lrm;, recovered, and death cases. &lrm;At &lrm;first, &lrm;the &lrm;time &lrm;series&lrm; of the covid-19 &lrm;is modeling with the Ornstein-Uhlenbeck process and then using the Ito lemma and Euler approximation the analytical and numerical simulations for &lrm;the stochastic &lrm;differential equation are &lrm;achieved.&lrm;&lrm; Parameters estimation is done using the maximum &lrm;likelihood estimator. Finally, numerical simulations are performed using reported data by &lrm;the world health &lrm;organization&lrm; for case studies of Italy and Iran. The numerical simulations and root mean square error criteria confirm the &lrm;accuracy and &lrm;efficiency of the findings of the present &lrm;study.&lrm;&lrm;&lrm;