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A numerical method for solving a class of variable-order differential equations using Hosoya Polynomial of simple paths

Salati Saha, Jafari Hossein, Matinfar Mashallah

Journal of Applied Mathematics and Computational Mechanics

2024

Vol. 23, nr 1

97-108

In this paper, we use the operational matrices (OMs) and collocation method (CM) to obtain a numerical solution for a class of variable-order differential equations (VO-DEs). The fractional derivatives and the VO-derivatives are in the Caputo sense. The operational matrices are computed based on the Hosoya polynomials (HPs) of simple paths. Firstly, we assume the unknown function as a finite series by using the Hosoya polynomials as the basis functions. To obtain the unknown coefficients of this approximation, we computed the operational matrices of all terms of the main equations. Then, by using the operational matrix and collocation points, the governing equations are converted to a set of algebraic equations. Finally, an approximate solution is obtained by solving those algebraic equations.

Numerical solution of two-dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method

Ishola Christie Yemisi, Taiwo Omotayo Adebayo, Adebisi Ajimot Folasade, Peter Olumuyiwa James

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 1

29--40

This paper presents the Chebyshev Integral Operational Matrix Method (CIOMM) for the numerical solution of two-dimensional Fredholm Integro-Differential Equations (2D-FIDEs). The process of the method is obtaining the operational matrix of integration by evaluating a 2D integral of 2D Chebyshev polynomial basis functions and assuming approximate solutions of the 2D-FIDEs as a truncated 2D Chebyshev series. This leads to a system of linear algebraic equations which are solved to obtain the values of the unknown constants using Maple 18. Some numerical problems are solved to illustrate the practicability of the method.

Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets

Khajehnasiri Amir Ahmad, Ezzati R., Kermani M.

Journal of Applied Analysis

2021

Vol. 27, nr 2

239--257

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.