Computational Methods for Differential Equations
https://cmde.tabrizu.ac.ir/
Computational Methods for Differential Equationsendaily1Sat, 01 Jan 2022 00:00:00 +0330Sat, 01 Jan 2022 00:00:00 +0330Qualitative analysis of fractional differential equations with ψ-Hilfer fractional derivative
https://cmde.tabrizu.ac.ir/article_12209.html
In this paper, we investigate the solutions of a class of &psi;-Hilfer fractional differential equations with the initial values in the sense of &psi;-fractional integral by using the successive approximation techniques. Next, the continuous dependence of a solution for the given Cauchy-type problem is presented.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) to approximate the Caputo-Fabrizio fractional derivative of order &alpha;, (0 &lt; &alpha; &lt; 1) is developed. It is established by means of the quadratic interpolation approximation using three points (tj&minus;2,y(tj&minus;2)),(tj&minus;1,y(tj&minus;1)), and (tj, y(tj)) on each interval [tj&minus;1,tj] for (j &ge; 2), while the linear interpolation approximation are applied on the first interval [t0,t1]. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for y(t). Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of the CF2 formula. 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.Cubic B-spline collocation method on a non-uniform mesh for solving nonlinear parabolic partial differential equation
https://cmde.tabrizu.ac.ir/article_12205.html
In this paper, an approximate solution of a nonlinear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary conditions 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 are shown by numerical experiments. We have compared the approximate solutions with that in the 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 an interval) containing all these errors. The described method can also include data uncertainties in the intervals obtained.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. In the end, illustrative examples are included to demonstrate the validity and effectiveness of the presented approach.&nbsp;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 is 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.&nbsp;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 systems and the integer-order hyperchaotic system has been investigated. Due to increased complexity and the 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 nonlinear&nbsp;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 the field of secure communication. The results have been compared with published literature.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 an 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. The L2 convergence analysis of the numerical method is analyzed. Numerical results for different examples are examined to verify the accuracy of the spectral method and justification the theoretical analysis and to compare with other existing methods in the literatures.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 symmetric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional-differential equations with proportional delays. In this method, we control the absolute errors and the condition number of the system matrix through the program prepared with Maple 18.0 by increasing the number of collocation points that have a direct effect on the defined shape parameter. Also, we present the tables of the rate of the convergence (ROC) to investigate and show the convergence rate of this method compared to the RBF method with constant shape parameter. 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).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(G0/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.&nbsp;Optimal control of satellite attitude and its stability based on quaternion parameters
https://cmde.tabrizu.ac.ir/article_12509.html
This paper proposes an optimal control method for the chaotic attitude of the satellite when it is exposed to external disturbances. When there is no control over the satellite, its chaotic attitude is investigated using Lyapunov exponents (LEs), Poincare diagrams, and bifurcation diagrams. In order to overcome the problem of singularity in the great maneuvers of satellite, we consider the kinematic equations based on quaternion parameters instead of Euler angles, and obtain control functions by using the Pontryagin maximum principle (PMP). These functions are able to reach the satellite attitude to its equilibrium point. Also the asymptotic stability of these control functions is investigated by Lyapunov&rsquo;s stability theorem. Some simulation results are given to visualize the effectiveness and feasibility of the proposed method.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 ( G'/G2)-expansion function methods in constructing various optical soliton and other solutions to the (2+1)-dimensional hyperbolic nonlinear Schrodinger&rsquo;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.&nbsp;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 study the existence and Hyers-Ulam stability of random impulsive stochastic functional integrodifferential equations with finite delays. Firstly, we prove the existence of mild solutions to the equations by using Banach fixed point theorem. In the later case we explore the Hyers Ulam stability results under the Lipschitz condition on a bounded and closed interval.&nbsp;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 the 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 impulses.&nbsp;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.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 &rdquo;speckle&rdquo;. 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 (Catte&rsquo;s pde model).&nbsp;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.Bounding 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.&nbsp;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.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.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.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.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;Exact solutions of the space time-fractional Klein-Gordon equation with cubic nonlinearities using some methods
https://cmde.tabrizu.ac.ir/article_13045.html
Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this work, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extacted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.Local Fractal Fourier Transform and Applications
https://cmde.tabrizu.ac.ir/article_13049.html
In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta are also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when $\alpha=1$, we obtain the same results in the standard calculus.A numerical solution of two-dimensional hyperbolic telegraph equation based on moving least square meshless method and radial basis functions
https://cmde.tabrizu.ac.ir/article_13341.html
In this research, linear combination of moving least square (MLS) and local radial basis functions(LRBFs)is considered within the framework of meshless method to solve two-dimensional hyperbolic telegraph equation.Besides, differential quadrature method (DQM) is employed to discretize temporal derivatives. Furthermore, a control parameter is introduced and optimized to achieve minimum errors via an experimental approach.Illustrative examples are provided to demonstrate applicability and efficiency of the method. The results prove the superiority of this method overusing MLS and LRBF individually.A novel local meshless scheme based on the radial basis function for pricing multi-asset options
https://cmde.tabrizu.ac.ir/article_13342.html
&lrm;A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model&lrm;. &lrm;The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction&lrm;. &lrm;Therefore&lrm;, &lrm;it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation&lrm;. &lrm;We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach&lrm;. &lrm;Moreover&lrm;, &lrm;based on a review performed in the current study&lrm;, &lrm;the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced&lrm;. &lrm;As seen in numerical experiments&lrm;, &lrm;it has a low computational cost and generates higher accuracy&lrm;. &lrm;Finally&lrm;, &lrm;the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector&lrm;, &lrm;as well as in other fields of engineering and science&lrm;. &lrm;Finally&lrm;, &lrm;we conclude that the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector&lrm;, &lrm;as well as in other fields of engineering and science&lrm;.Lie symmetries, exact solutions, and conservation laws of the nonlinear time-fractional Benjamin-Ono equation
https://cmde.tabrizu.ac.ir/article_13343.html
In this work, we use the symmetry of the Lie group analysis as one of the powerful tools which that deals with the wide class of fractional order differential equation in the Riemann-Liouville concept. We employ the classical Lie symmetries to obtain similarity reductions of nonlinear time-fractional Benjamin-Ono equation and then, we find the related exact solutions for the derived generators. Finally, according to the Lie symmetry generators obtained, we construct conservation laws for related classical vector fields of time-fractional Benjamin-Ono equation.Backward bifurcation in a two strain model of heroin addiction
https://cmde.tabrizu.ac.ir/article_13344.html
Among the various causes of heroin addiction, the use of &lrm;prescription &lrm;opioids&lrm; is one of the main reasons. In this article, we introduce and analyze a two &lrm;strain&lrm; epidemic model with super infection for modeling the effect of &lrm;prescrib&lrm;ed opioids abuse on heroin &lrm;addiction.&lrm; &lrm;Our &lrm;model &lrm;contains &lrm;the &lrm;effect &lrm;of &lrm;relapse &lrm;of &lrm;individuals &lrm;under &lrm;treatment/rehabilitation&lrm; &lrm;to drug abuse in each &lrm;strain.&lrm; &lrm;We &lrm;extract&lrm; the basic reproductive &lrm;ratio, &lrm;the&lrm; invasion numbers&lrm;, &lrm;and study the occurrence of backward bifurcation in strain &lrm;domi&lrm;nance equilibria, i.e., boundary &lrm;equilibria. &lrm;Also, &lrm;we &lrm;study &lrm;both&lrm; &lrm;&lrm;local and global stability of DFE and boundary equilibria &lrm;under suitable conditions&lrm;.&lrm; &lrm;Furthermore, we study the &lrm;existence of the coexistence equilibrium point&lrm;. We prove that when &lrm;$&lrm;R_0&lt;1&lrm;$&lrm;, the coexistence equilibrium point can exist, i.e., backward bifurcation &lrm;occurs&lrm; in coexistence equilibria. &lrm;Finally, we use numerical simulation to describe the obtained analytical results.&lrm;On exact solutions of the generalized Pochhammer-Chree equation
https://cmde.tabrizu.ac.ir/article_13345.html
In the current study, we consider the generalized Pochhammer-Chree equation with term of order $n$. Based on the (1/G')-expansion method and with the aid of symbolic computation, we construct some distinct exact solutions for this nonlinear model. Various exact solutions are produced to the studied equation including singular solutions, periodic wave solutions. In addition to 2D, 3D and contour plots are graphed for all obtaining solutions via choosing the suitable values for the involved parameters. All gained solutions verify the governing equation.Hybrid shrinking projection extragradient-like algorithms for equilibrium and fixed point problems
https://cmde.tabrizu.ac.ir/article_13350.html
Based on the extragradient-like method combined with shrinking projection, we propose two algorithms, the first algorithm is obtained using sequential computation of extragradientlike method and the second algorithm is obtained using parallel computation of extragradient-like method, to find a common point of the set of fixed points of nonexpansive mapping and the solution set of the equilibrium problem of a bifunction given as a sum of finite number of H&uml;older continuous bifunctions. The convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for the bifunction and its summands.Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation
https://cmde.tabrizu.ac.ir/article_13735.html
In this study, one explicit and one implicit finite differencescheme is introduced for the numerical solution of time-fractionalRiesz space diffusion equation. The time derivative is approximatedby the standard Gr\"{u}nwald Letnikov formula of order one, whilethe Riesz space derivative is discretized by Fourier transform-basedalgorithm of order four. The stability and convergence of theproposed methods are studied. It is proved that the implicit schemeis unconditionally stable, while the explicit scheme is stableconditionally. Some examples are solved to illustrate the efficiencyand accuracy of the proposed methods. Numerical results confirm thatthe accuracy of present schemes is of order one.A Study on Homotopy Analysis Method and Clique Polynomial Method
https://cmde.tabrizu.ac.ir/article_13736.html
This paper generated the novel approach called the Clique polynomial method (CPM) using the Clique polynomials raised in graph theory. Non-linear initial value problems are converted into non-linear algebraic equations by discretion with suitable grid points in the current approach. We solved highly non-linear initial problems using the (HAM) Homotopy analysis method and CPM. Obtained results reveal that the present technique is better than HAM that is discussed through tables and simulations. Convergence analyses are reflected in terms of the theorem.An effective technique for the conformable space-time fractional cubic-quartic nonlinear Schrodinger equation with different laws of nonlinearity
https://cmde.tabrizu.ac.ir/article_13737.html
In the present study, we investigate the conformable space-time fractional cubic-quartic nonlinear Schrodinger equation with three different laws of nonlinearity namely, parabolic law, quadratic-cubic law, and weak non-local law.This model governs the propagation of solitons through nonlinear optical fibers. An effective approach namely, the exp(-Pi(xi))-expansion method is applied to construct some new exact solutions of the governing model. Consequently, the dark, singular, rational and periodic solitary wave solutions are successfully revealed. The comparisons with other results are also presented. In addition, the dynamical structures of obtained solutions are presented through 3D and 2D plots.Asymptotic method for solution of oscillatory fractional derivative
https://cmde.tabrizu.ac.ir/article_13860.html
In the paper an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of solutions of the relative small parameter are constructed. The results are illustrated on the concrete example, where the solution differs from the analytical solution by 10^-2 order.Application of variation of parameter's method for hydrothermal analysis on MHD squeezing nanofluid flow in parallel plates
https://cmde.tabrizu.ac.ir/article_13865.html
In this paper, the transport of flow and heat transfer through parallel plates arranged horizontally against each other is studied. The mechanics of fluid transport and heat transfer is formulated utilizing systems of coupled higher-order numerical model. This governing transport model is investigated via applying the variation of parameter&rsquo;s method. Result obtained from the analytical study is reported graphically. It is observed from generated result that the velocity profile and thermal profile drop by increasing the squeeze parameter. Drop in flow is due to limitations in velocity as plates are close to each other. Also, thermal transfer due to flow pattern causes decreasing boundary layer thickness at the thermal layer, consequently drop in thermal profile. The analytical result obtained from this study is compared with study in literature for simplified case, this shows good agreement. Results obtained may therefore provide useful insight to practical applications including food processing, lubrication, and polymer processing industries amongst other relevant applications.Obtaining soliton solutions of equations combined with the Burgers and equal width wave equations using a novel method
https://cmde.tabrizu.ac.ir/article_13869.html
In the present paper, a modified simple equation method is used to obtain exact solutions of the equal width wave Burgers and modified equal width wave Burgers equations. By giving specific values to the parameters, particular solutions are obtained and the plots of solutions are drawn. It shows that the proposed method can be easily generalized to solve a variety of non-linear equations by implementing a robust and straightforward algorithm without the need for any tools.The Generalized Conformable Derivative for 4$\alpha$-Order Sturm-Liouville Problems
https://cmde.tabrizu.ac.ir/article_13871.html
In this paper, we discuss on the new generalized fractional operator. This operator similarly conformable derivative satisfies properties such as the sum, product/quotient and chain rule. Laplace transform is defined in this case, and some of its properties are stated. In the following, the Sturm-Liouville problems are investigated, and also eigenvalues and eigenfunctions are obtained.Approximate price of the option under discretization by applying fractional quadratic interpolation
https://cmde.tabrizu.ac.ir/article_13873.html
&lrm;The time-fractional Black-Scholes model (TFBSM) governing European options in which the temporal derivative&lrm; &lrm;is focused on the Caputo fractional derivative with $0&lt;\beta \leq 1$ is considered in this article&lrm;.&lrm;Approximating financial options with respect to their hereditary characteristics can be well&lrm; &lrm;understood and explained due to its outstanding memory effect current in fractional derivatives&lrm;.&lrm;Compelled by the stated cause&lrm;, &lrm;It is important to find reasonably accurate and successful numerical methods when approaching fractional differential equations&lrm;.&lrm;The simulation model given here is developed in two ways&lrm;: &lrm;one&lrm;, &lrm;the semi-discrete is produced in the time using a quadratic interpolation with the order of precision $\tau^{3-\alpha}$ \textcolor{red}{ in the case of smooth solution}&lrm;, &lrm;and subsequently&lrm;, &lrm;the unconditional stability and convergence order are investigated&lrm;.&lrm;The spatial derivative variables are simulated using the collocation approach based on a Legendre basis for the designed full-discrete scheme&lrm;.&lrm;Last&lrm;, &lrm;we employ various test problems to demonstrate the suggested design's high precision&lrm;.&lrm;Moreover&lrm;, &lrm;the obtained results are compared to those obtained using other methodologies&lrm;, &lrm;demonstrating that the proposed technique is highly accurate and practicable&lrm;.A Numerical Scheme for Solving Time-Fractional Bessel Differential Equations
https://cmde.tabrizu.ac.ir/article_13874.html
The object of this paper devotes on offering an indirect schemebased on time-fractional Bernoulli functions in the sense ofRieman-Liouville fractional derivative which ends up to the highcredit of the obtained approximate fractional Bessel solutions. Inthis paper, the operational matrices of fractional Rieman-Liouvilleintegration for Bernoulli polynomials are introduced. Utilizingthese operational matrices along with the properties of Bernoullipolynomials and the least squares method, the fractional Besseldifferential equation converts into a nonlinear system of algebraic.To solve these nonlinear algebraic equations which are a prominentproblem, there is a need to employ Newton's iterative method. Inorder to elaborate the study, the synergy of the proposed method isinvestigated and then the accuracy and the efficiency of the methodare clearly evaluated by presenting numerical resultsExistence and properties of positive solutions for Caputo fractional difference equation and Applications
https://cmde.tabrizu.ac.ir/article_13909.html
&lrm;&lrm;This paper deals with a typical Caputo fractional&lrm;&lrm;difference equation&lrm;. &lrm;This equation appears in important applications such as modelling of medicine distributing throughout the body via injection and equation for general population growth&lrm;. &lrm;We use fixed point theory of concave operators in specific normed spaces to find a parameter interval for which the unique positive solution exists&lrm;. &lrm;Some properties of positive solutions are studied and illustrative examples are given&lrm;.Sliding mode control of a class of uncertain nonlinear fractional order time-varying delayed systems based on Razumikhin approach
https://cmde.tabrizu.ac.ir/article_13921.html
Within the current paper, we design a sliding-based control law to stabilizea set of systems that are nonlinear, having fractional order, and involve delay, perturba-tion, and uncertainty. A control law based sliding mode is considered in such a way thatthe variables of the closed loop system reach sliding surface in a limited time and stayon it for later times. Then, using the Razomokhin stability theorem, the stability of thesystems is proved and at the end, a calculation is found to search for useful methods.ERROR ANALYSIS AND KRONECKER IMPLEMENTATION OF CHEBYSHEV SPECTRAL COLLOCATION METHOD FOR SOLVING LINEAR PDES
https://cmde.tabrizu.ac.ir/article_13937.html
Numerical methods have essential role to approximate the so-lutions of Partial Differential Equations (PDEs). Spectral method is oneof the best numerical methods of exponential order with high convergencerate to solve PDEs. In recent decades the Chebyshev Spectral Collocation(CSC) method has been used to approximate solutions of linear PDEs.In this paper, by using linear algebra operators, we implement KroneckerChebyshev Spectral Collocation (KCSC) method for n-order linear PDEs.By statistical tools, we obtain that the Run times of KCSC method haspolynomial growth, but the Run times of CSC method has exponentialgrowth. Moreover, error upper bounds of KCSC and CSC methods arecompared.An epidemic model for drug addiction
https://cmde.tabrizu.ac.ir/article_14146.html
&lrm;The two most common ways to prevent spreading drug addiction are counseling and imprisonment&lrm;. &lrm;In this paper&lrm;, &lrm;we propose and study a model for the spread of drug addiction incorporating the effect of consultation and incarceration of addicted individuals&lrm;. &lrm;We extract the basic reproductive ratio and study the occurrence of backward bifurcation&lrm;. &lrm;Also&lrm;, &lrm;we study the local and global stability of drug-free and endemic equilibria under suitable conditions&lrm;. &lrm;Finally&lrm;, &lrm;we use numerical simulations to illustrate the obtained analytical results&lrm;.On the extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions
https://cmde.tabrizu.ac.ir/article_14147.html
This paper is devoted to prove the existence of extremal solutions for multi-term nonlinearfractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputotype and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishinga new comparison theorem and applying the monotone iterative technique, we show the existence ofextremal solutions. The method is a constructive method that yields monotone sequences that convergeto the extremal solutions. As an application, some examples are presented to illustrate the main results.A Novel Perspective for Simulations of the MEW Equation By Trigonometric Cubic B-spline Collocation Method Based on Rubin-Graves Type Linearization
https://cmde.tabrizu.ac.ir/article_14148.html
In the present study, the Modified Equal Width (MEW) wave equation is going to be solved numerically by presenting a new technique based on collocation finite element method in which trigonometric cubic B-splines are used as approximate functions. In order to support the present study, three test problems; namely, the motion of a single solitary wave, interaction of two solitary waves and the birth of solitons are studied. The newly obtained results are compared with some of the other published numerical solutions available in the literature. The accuracy of the proposed method is discussed by computing the numerical conserved laws as well as the error norms L₂ and L_{&infin;}.A two-step method adaptive with memory with eighth-order for solving nonlinear equations and its dynamic
https://cmde.tabrizu.ac.ir/article_14153.html
In this work, we have constructed the without-memory two-step method with four convergence degrees by entering the maximum self-accelerator parameter(three parameters). Then, using Newton&rsquo;s interpolation, a with-memory method with a convergence order of 7.53 is constructed. Using the information of all the steps, we will improve the convergence order by one hundred percent,and we will introduce our method with convergence order 8. Numerical examples demonstrate the exceptional convergence speed of the proposed method and conﬁrm theoretical results. Finally, we have presented the dynamics of the adaptive method and other without-memory methods for complex polynomials of degrees two, three, and four. The basins of attraction of existing with-memorymethods are present and compared to illustrate their performance.Numerical solution of the hyperbolic telegraph equation using cubic B-spline-based differential quadrature of high accuracy
https://cmde.tabrizu.ac.ir/article_14155.html
By constructing a newly modified cubic B-splines having the optimal accuracy order four, we propose a numerical scheme for solving the hyperbolic telegraph equation using a differential quadrature method.The spatial derivatives are approximated by the differential quadrature whose weight coefficients are computed using the newly modified cubic B-splines. Our modified cubic B-splines retain the tridiagonal structure and achieve the fourth order convergence rate. The solution of the associated ODEs is advanced in the time domain by the SSPRK scheme. The stability of the method is analyzed using the discretization matrix. Our numerical experiments demonstrate the better performance of our proposed scheme over several known numerical schemes reported in the literature.Numerical solution of Drinfel’d–Sokolov system with the Haar wavelets method
https://cmde.tabrizu.ac.ir/article_14157.html
In this article, we use the Haar wavelets (HWs) method to numerically solve the nonlinear Drinfel&rsquo;d&ndash;Sokolov (DS) system. For this purpose, we use an approximation of functions with the help of HWs, and we approximate spatial derivatives using this method. In this regard, to linearize the nonlinear terms of the equations, we use the quasilinearization technique. In the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided.An optimal B-spline collocation technique for numerical simulation of viscous coupled Burgers' equation
https://cmde.tabrizu.ac.ir/article_14181.html
In this paper, an optimal cubic B-spline collocation method is applied to solve the viscous coupledBurgers&rsquo; equation, which helps in modeling the polydispersive sedimentation. As it is not possible to obtain optimal order of convergence with the standard collocation method, so to overcome this, posteriori corrections are made in cubic B-spline interpolant and its higher-order derivatives. This optimal cubic B-spline collocation method is used for space integration and for time-domain integration, the Crank-Nicolson scheme is applied along with the quasilinearization process to deal with the nonlinear terms in the equations. Von-Neumann stability analysis is carried out to discuss the stability of the technique. Few test problems are solved numerically along with the calculation of L2, L1 error norms as well as the order of convergence. The obtained results are compared with those available in the literature, which show the improvement in results over the standard collocation method and many other existing techniques also.A Bernoulli Tau method for numerical solution of feedback Nash differential games with an error estimation
https://cmde.tabrizu.ac.ir/article_14182.html
In the present study, an efficient combination of the Tau method with the Bernoulli polynomials is proposed for computing the Feedback Nash equilibrium in differential games over a finite horizon. By this approach, the system of Hamilton-Jacobi-Bellman equations of a differential game derived from Bellman's optimality principle is transferred to a nonlinear system of algebraic equations solvable by using Newton's iteration method. Some illustrative examples are provided to show the accuracy and efficiency of the proposed numerical method.