University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 A numerical method based on the moving mesh for the solving of a mathematical model of the avascular tumor growth 327 346 10353 10.22034/cmde.2020.31455.1472 EN Mina Bagherpoorfard Department of applied mathematics, Ferdowsi university of Mashhad, Mashhad. Iran. Ali Reza Soheili Department of applied mathematics, Ferdowsi university of Mashhad, Mashhad, Iran. Journal Article 2019 01 10 Using adaptive mesh methods is one of the strategies to improve numerical solutions in time dependent partial differential equations. The moving mesh method is an adaptive mesh method, which, firstly does not need an increase in the number of mesh points, secondly reduces the concentration of points in the steady areas of the solutions that do not need a high degree of accuracy, and finally places the points in the areas, where a high degree of accuracy is needed. In this paper, we improved the numerical solutions for a three-phase model of avascular tumor growth by using the moving mesh method. The physical formulation of this model uses reaction-diffusion dynamics with the mass conservation law and appears in the format of the nonlinear system of partial differential equations based on the continuous density of three proliferating, quiescent, and necrotic cell categorizations. Our numerical results show more accurate numerical solutions, as compared to the corresponding fixed mesh method. Moreover, this method leads to the higher order of numerical convergence. Adaptive Moving Mesh Tumor Growth Avascular Tumor Growth Mathematical Modeling https://cmde.tabrizu.ac.ir/article_10353_c30e275526321601a1a6fc5896291675.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Necessary and sufficient conditions for M-stationarity of nonsmooth optimization problems with vanishing constraints 347 357 10352 10.22034/cmde.2020.30733.1459 EN Hadis Mokhtavayi Department of Mathematics, Payam Noor University, P. O. Box 19395-3697, Tehran, Iran. Aghileh Heidari Department of Mathematics, Payam Noor University, P. O. Box 19395-3697, Tehran, Iran. Nader Kanzi Department of Mathematics, Payam Noor University, P. O. Box 19395-3697, Tehran, Iran. Journal Article 2018 12 05 We consider a nonsmooth optimization problem with a feasible set defined by vanishing constraints. First, we introduce a constraint qualification for the problem, named NNAMCQ. Then, NNAMCQ is applied to obtain a necessary M-stationary condition. Finally, we present a sufficient condition for M-stationarity, under generalized convexity assumption. Our results are formulated in terms of Mordukhovich subdifferential. Stationary conditions Vanishing constraints Nonsmooth optimization Constraint qualification https://cmde.tabrizu.ac.ir/article_10352_0b6a2959c2b43ba7461b1c2201c089da.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Numerical investigation based on a local meshless radial point interpolation for solving coupled nonlinear reaction-diffusion system 358 374 10322 10.22034/cmde.2019.30396.1450 EN Elyas Shivanian Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran. Ahmad Jafarabadi Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran. Journal Article 2018 11 15 In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction-diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The effect of parameters and conditions are studied by considering the well known Brusselator model. Two test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme. Turing systems Brusselator model Spectral meshless radial point interpolation (SMRPI) method Radial basis function, Finite difference method https://cmde.tabrizu.ac.ir/article_10322_fb26a63e2c917e8049427f3c3ecd2a63.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Partial eigenvalue assignment of descriptor fractional discrete-time linear systems by parametric state feedback 375 392 10325 10.22034/cmde.2020.33660.1544 EN Sakineh Bigom Mirassadi Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran. Hojjat Ahsani Tehrani Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran. Journal Article 2019 05 23 In this paper, we present a nonlinear parametric method to stabilize descriptor fractional discrete time linear system practically. Parametric methods with the free parameters can be adjusted to obtain better performance responses like minimum norm in state feedback. The aim is assigning desirable eigenvalues to obtain satisfactory responses by forward state feedback and forward and propositional state feedback in new systems with large matrices. However, finding the solution to nonlinear parametric equations makes some errors. In partial eigenvalue assignment, just a part of the open-loop spectrum of the standard linear systems is reassigned, while leaving the rest of the spectrum invariant. The size of matrices, state, and input vectors are decreased and the stability is kept. At the end, summary and conclusions are proposed and the convergence of state vectors in the descriptor fractional discrete-time system to zero is also shown by figures in a numerical example. Our method is also compared with another method with one of orthogonality relations in our article and example. Descriptor fractional discrete-time Nonlinear equations Parametric state feedback Partial eigenvalue assignment https://cmde.tabrizu.ac.ir/article_10325_664a2b3731a01c3a36e6f95c75d95b79.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Solving some stochastic differential equation using Dirichlet distributions 393 398 10324 10.22034/cmde.2019.32914.1533 EN Hawre Hadad Department of Statistics, Sciences and Research branch, Islamic Azad University, Tehran, Iran. Hazhir Homei Department of Statistic University of Tabriz P.O.Box 51666-16471 Tabriz, Iran. Mohammad Hassan Behzadi Department of Statistics, Sciences and Research branch, Islamic Azad University, Tehran, Iran. Rahman Farnoosh School of Mathematics ,Iran University of Science and Technology, Tehran, Iran. Journal Article 2019 04 28 Stochastic linear combinations of some random vectors are studied where the distribution of the random vectors and the joint distribution of their coefficients have Dirichlet distributions. A method is provided for calculating the distribution of these combinations which has been studied before. Our main result is the same as but from a different point of view. Stochastic Linear Combination Dependent Components Lifetime https://cmde.tabrizu.ac.ir/article_10324_cc28d04aa0f66ed68800b2c95ebd51c7.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Hyperbolic Ricci-Bourguignon flow 399 409 10327 10.22034/cmde.2020.34205.1566 EN Shahroud Azami Department of pure Mathematics, Faculty of Sciences Imam Khomeini International University, Qazvin, Iran. Journal Article 2019 06 23 In this paper, we consider the hyperbolic Ricci-Bourguignon flow on a compact manifold M and show that this flow has a unique solution on short-time with imposing on initial conditions. After then, we find evolution equations for Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of M under this flow. In the final section, we give some examples of this flow on some compact manifolds. Geometric flow Hyperbolic equation Strictly hyperbolicity https://cmde.tabrizu.ac.ir/article_10327_4eed60752e4234f9632a77115adc98e9.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 A combining method for the approximate solution of spatial segregation limit of reaction-diffusion systems 410 426 10351 10.22034/cmde.2020.29291.1412 EN Maryam Dehghan Department of Mathematics, Faculty of Sciences, Persian Gulf University, Bushehr 75169, Iran. Saeed Karimi Jafarbigloo Department of Mathematics, Faculty of Sciences, Persian Gulf University, Bushehr 75169, Iran. Journal Article 2018 09 11 In this paper, we concern ourselves with the study of a class of stationary states for reaction-diffusion systems with densities having disjoint supports. Major contribution of this work is computing the numerical solution of problem as the rate of interaction between two different species tend to infinity. The main difficulty is the nonlinearity nature of problem. To do so, an efficient iterative method is proposed by hybrid of the radial basis function (RBF) collocation and finite difference (FD) methods to approximate the solution. Numerical results with good accuracies are achieved where the shape parameter is carefully selected. Finally, some numerical examples are given to illustrate the good performance of the method. Free boundary problems Two-phase membrane One phase obstacle problem Segregation finite difference method Multiquadric radial basis functions https://cmde.tabrizu.ac.ir/article_10351_eb1d75eb88216a96cbd77deb15a0c1f1.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Global dynamics and numerical bifurcation of a bioeconomic system 427 445 10332 10.22034/cmde.2020.29491.1421 EN Zeynab Lajmiri Sama technical and vocational training college, Islamic Azad Univercity Izeh Branch, Izeh, Iran. Iman Orak Sama technical and vocational training college, Islamic Azad Univercity Izeh Branch, Izeh, Iran. Reza Fereidooni Reserch and development manager of oxin steel company of khozestan, Iran. Journal Article 2018 09 23 A predator-prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. Theoretical results showed that the harvesting system undergoes multiple bifurcations, including fold, supercritical Hopf, Bogdanov-Takens and cusp bifurcations. We determine stability and dynamical behaviors of the equilibrium of this system. Numerical simulation results are given to support our theoretical results. Hopf Bifurcation Bogdanov-Takens bifurcation Dynamical behavior Cusp bifurcations https://cmde.tabrizu.ac.ir/article_10332_935c51f0a6ec1c419007b446bea20387.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 A new numerical Bernoulli polynomial method for solving fractional optimal control problems with vector components 446 466 10330 10.22034/cmde.2020.34992.1598 EN Vahid Taherpour Department of Mathematics, Khorram Abad Branch, Islamic Azad University, Khorram Abad, Iran Mojtaba Nazari Department of Mathematics, Khorram Abad Branch, Islamic Azad University, Khorram Abad, Iran Ali Nemati Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, Ardabil, Iran Journal Article 2019 08 06 In this paper, a numerical method is developed and analyzed for solving a class of fractional optimal control problems (FOCPs) with vector state and control functions using polynomial approximation. The fractional derivative is considered in the Caputo sense. To implement the proposed numerical procedure, the Ritz spectral method with Bernoulli polynomials basis is applied. By applying the Bernoulli polynomials and using the numerical estimation of the unknown functions, the FOCP is reduced to solve a system of algebraic equations. By rigorous proofs, the convergence of the numerical method is derived for the given FOCP. Moreover, a new fractional operational matrix compatible with the proposed spectral method is formed to ease the complexity in the numerical computations. At last, several test problems are provided to show the applicability and effectiveness of the proposed scheme numerically. Fractional derivative Optimal control problem Bernoulli operational matrix Spectral Ritz method Convergence https://cmde.tabrizu.ac.ir/article_10330_cab7ffd5a82352b18833673d5b555aaa.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Analytical solution for descriptor system in partial differential equations 467 479 12686 10.22034/cmde.2021.42195.1824 EN Svetlana Petrovna Zubova Department of mathematical Analysis, Faculty of Mathematics,\\Voronezh State University, Voronezh, Russia. Abdulftah Hosni Mohamad Department of mathematical Analysis, Faculty of Mathematics,\\Voronezh State University, Voronezh, Russia. Journal Article 2020 10 09 We consider a first-order partial differential equation with constant irreversible coefficients in a Banach space in the regular case. The equation is split into equations in subspaces, in which non-degenerate subsystems are obtained. We obtain an analytical solution of each system with Showalter-type conditions. Finally, an example is given to illustrate the theoretical<br /> results. Banach Space descriptor system Differential algebraic equations 0-normal eigenvalue Showalter-type conditions https://cmde.tabrizu.ac.ir/article_12686_425b7d32c369dafd54ba4f58a734c16b.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Solving a class of fractional optimal control problems via a new efficient and accurate method 480 492 10329 10.22034/cmde.2020.35875.1620 EN Samaneh Soradi-Zeid Faculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran Journal Article 2019 10 15 The present paper aims to get through a class of fractional optimal control problems (FOCPs). Furthermore, the fractional derivative portrayed in the Caputo sense through the dynamics of the system as fractional differential equation (FDE). Getting through the solution, firstly the FOCP is transformed into a functional optimization problem. Then, by using known formulas for computing fractional derivatives of Legendre wavelets (LWs), this problem has been reduce to an equivalent system of algebraic equations. In the next step, we can simply solved this algebraic system. In the end, some examples are given to bring about the validity and applicability of this technique and the convergence accuracy. Fractional optimal control problem Fractional integrals Fractional derivatives Legendre wavelets Lagrange multipliers method https://cmde.tabrizu.ac.ir/article_10329_520124cea04625086027797c8505b9d3.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 On the numerical treatment and analysis of Hammerstein integral equation 493 510 10331 10.22034/cmde.2019.29825.1435 EN Maryam Derakhshan epartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran. Mohammad Zarebnia epartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran. Journal Article 2018 10 15 In this paper, we study the quadratic rules for the numerical solution of Hammerstein integral equation based on spline quasi-interpolant. Also the convergence analysis of the methods are given. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part. SPline Quasi-interpolant Quadrature Hammerstein Convergence https://cmde.tabrizu.ac.ir/article_10331_816a1327f4cbe60319bde09568c7ce14.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 ‎Solving brachistochrone problem via scaling functions of Daubechies wavelets 511 522 10333 10.22034/cmde.2020.34778.1588 EN Azad Kasnazani Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran. Amjad AliPanah Department of Applied Mathematics, University of Kurdistan, Sanandaj, Iran. Journal Article 2019 07 22 In this paper, we proposed an effective method based on the scaling function of Daubechies wavelets for the solution of the brachistochrone problem. An analytic technique for solving the integral of Daubechies scaling functions on dyadic intervals is investigated and these integrals are used to reduce the brachistochrone problem into algebraic equations. The error estimate for the brachistochrone problem is proposed and the numerical results are given to verify the effectiveness of our method. Daubechies wavelets scaling function brachistochrone problem Error analysis numerical results https://cmde.tabrizu.ac.ir/article_10333_25aa5d83406bfd2b13f545e6bfc2efd3.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 A compact difference scheme for time-fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options 523 552 10335 10.22034/cmde.2020.36000.1623 EN Maryam Rezaei Mirarkolaei Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran. Ahmadreza Yazdanian Faculty of Finance Sciences Kharazmi University, Tehran, Iran. Seyed Mahdi Mahmoudi Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran Ali Ashrafi Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran Journal Article 2019 10 08 ‎The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory effect in the financial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield supposed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution; hence, we numerically price the American option by using a compact difference scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as α = 1 and β = 0. Moreover, we demonstrate that the introduced difference scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the efficiency and accuracy of the introduced difference scheme. CEV model Time-dependent parameters Option pricing American option Fractional BlackScholes equation Compact difference scheme https://cmde.tabrizu.ac.ir/article_10335_264b729cd71d9f412f817b6d5634349e.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 A Laguerre approach for solving of the systems of linear differential equations and residual improvement 553 576 10754 10.22034/cmde.2020.34871.1591 EN Suayip Yuzbasi Department of Mathematics, Faculty of Science, Akdeniz University, TR 07058 Antalya, Turkey. Gamze Yildirim Department of Mathematics, Faculty of Science, Akdeniz University, TR 07058 Antalya, Turkey. Journal Article 2019 07 26 In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear differential equations with variable coefficients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives, and the solutions in matrix form. Secondly, the system of linear differential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equations is obtained. By solving the system of the obtained new algebraic equation, the coefficients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is offered and approximate solutions are improved. Finally, the presented method and error estimation technique are demonstrated with the help of numerical examples. The results of the proposed method are compared with the results of other methods Collocation method Collocation points Laguerre collocation method Laguerre polynomials Systems of linear differential equations https://cmde.tabrizu.ac.ir/article_10754_4483e85df27b4d22c0556be887767d7a.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Analysis of time delay model for drug therapy on HIV dynamics 577 588 10756 10.22034/cmde.2020.34812.1589 EN Vinoth Sivakumar Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science,India. Dumitru Baleanu Cankaya University, Turkey. Institute of Space Sciences, Romania. Jayakumar Thippan Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science,India. 0000-0002-5276-6775 Prasantha Bharathi Dhandapani Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science ,India. Journal Article 2019 07 23 We present and investigate the delayed model of HIV infection for drug therapy. The stability of the equilibrium states, disease free and infected equilibrium states are derived and the existence of Hopf bifurcation analysis is studied. We show that the system is asymptotically stable and the stability is lost in a range due to length of the delay, then Hopf bifurcation occurs when τ exceeds the critical value. At last numerical simulations are provided to verify the theoretical results. HIV infection Stability Hopf Bifurcation time delay https://cmde.tabrizu.ac.ir/article_10756_e2ec139da13432d8a198743857fbd386.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Existence of solutions for a class of critical Kirchhoff type problems involving Caffarelli-Kohn-Nirenberg inequalities 589 603 10607 10.22034/cmde.2020.32848.1526 EN Nguyen Thanh Chung Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam. Journal Article 2019 04 14 In this paper, we study the existence of a nontrival weak solution for a class of Kirchhoff type problems with singular potentials and critical exponents. The proofs are essentially based on an appropriated truncated argument, Caffarelli-Kohn-Nirenberg inequalities, combined with a variant of the concentration compactness principle. We also get a priori estimates of the obtained solution Kirchhoff type problems Caffarelli-Kohn-Nirenberg inequalities Critical exponents Mountain pass theorem https://cmde.tabrizu.ac.ir/article_10607_7e754b671a24766e874bbd6414d28cb7.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Stability in distribution of neutral stochastic functional differential equations with infinite delay 604 622 10643 10.22034/cmde.2020.32804.1525 EN Hussein Asker Department of Mathematics, Faculty of Computer Science and Mathematics, Kufa University, Al-Najaf, Iraq. Journal Article 2019 04 11 In this paper, we investigate stability in distribution of neutral stochastic functional differential equations with infinite delay (NSFDEwID) at the state space Cr. We drive a sufficient strong monotone condition for the existence and uniqueness of the global solutions of NSFDEwID in the state space Cr. We also address the stability of the solution map xt and illustrate the theory with an example Neutral stochastic functional differential equations Infinite delay Solution map Stability in distribution https://cmde.tabrizu.ac.ir/article_10643_7c762f6191c3073e1dcf6c81c074beab.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Local existence and blow up of solutions for a logarithmic nonlinear viscoelastic wave equation with delay 623 636 10753 10.22034/cmde.2020.35546.1608 EN Erhan Pişkin Dicle University, Department of Mathematics, Diyarbakir, Turkey. Hazal Yuksekkaya Dicle University, Department of Mathematics, Diyarbakir, Turkey. Journal Article 2019 09 09 In this work, we consider a logarithmic nonlinear viscoelastic wave equation with a delay term in a bounded domain. We obtain the local existence of the solution by using the Faedo-Galerkin approximation. Then, under suitable conditions, we prove the blow up of solutions in finite time. Local existence Blow-up Logarithmic nonlinearity Delay term https://cmde.tabrizu.ac.ir/article_10753_17541edaeca34daeb0b2eafaedcf445e.pdf
University of Tabriz Computational Methods for Differential Equations 2345-3982 9 2 2021 04 01 Bounds of Riemann-Liouville fractional integral operators 637 648 10752 10.22034/cmde.2020.32653.1516 EN Ghulam Farid Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, Pakistan. Journal Article 2019 03 25 Fractional integral operators play an important role in generalizations and extensions of various subjects of sciences and engineering. This research is the study of bounds of Riemann-Liouville fractional integrals via (h − m)-convex functions. The author succeeded to find upper bounds of the sum of left and right fractional integrals for (h − m)-convex function as well as for functions which are deducible from aforementioned function (as comprise in Remark 1.2). By using (h − m) convexity of |f ′ | a modulus inequality is established for bounds of Riemann-Liouville fractional integrals. Moreover, a Hadamard type inequality is obtained by imposing an additional condition. Several special cases of the results of this research are identified.  Convex function (h − m)-convex function Riemann-Liouville fractional integral operators Bounds https://cmde.tabrizu.ac.ir/article_10752_c4ec191093d776940f3007fc1d6f70fe.pdf