Mohammad Aslefallah¹, Şuayip Yüzbaşi², Saeid Abbasbandy^1,*

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.2, pp. 1687-1706, 2023, DOI:10.32604/cmes.2023.025647

Abstract In this work, the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus (COVID-19). The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics, namely, susceptible (S), infected (I), treatment (T), and recovered (R). The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points. To indicate the usefulness of this method, we employ it in some cases. For error analysis of the method, the… More > Graphic Abstract

Yanxin Wang^{1, *}, Li Zhu¹, Zhi Wang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.2, pp. 339-350, 2019, DOI:10.31614/cmes.2019.04575

Abstract An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper. The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented. Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. And the convergence of the Euler wavelets basis is given. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy. More >

A.H. Bhrawy^1,2, E.H. Doha³, S.S. Ezz-Eldien⁴, M.A. Abdelkawy²

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.3, pp. 185-209, 2015, DOI:10.3970/cmes.2015.104.185

Abstract In this paper, a high accurate numerical approach is investigated for solving the time-fractional linear and nonlinear Korteweg-de Vries (KdV) equations. These equations are the most appropriate and desirable definition for physical modeling. The spectral collocation method and the operational matrix of fractional derivatives are used together with the help of the Gauss-quadrature formula in order to reduce such problem into a problem consists of solving a system of algebraic equations which greatly simplifying the problem. Our approach is based on the shifted Jacobi polynomials and the fractional derivative is described in the sense of Caputo. In addition, the presented… More >

Jinsheng Wang¹, Liqing Liu², Yiming Chen², Lechun Liu², Dayan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.1, pp. 69-85, 2015, DOI:10.3970/cmes.2015.104.069

Abstract The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials. The fractional derivative is described in the Caputo sense. Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable. By solving the algebraic equations, the numerical solutions are acquired. The method in general is easy to implement and yields good results. Numerical examples are provided to demonstrate the validity and applicability of the method. More >

Baofeng Li ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.99, No.2, pp. 105-122, 2014, DOI:10.3970/cmes.2014.099.105

Abstract In this paper, operational matrix method based on the generalized hat functions is introduced for the approximate solutions of linear and nonlinear fractional integro-differential equations. The fractional order generalized hat functions operational matrix of integration is also introduced. The linear and nonlinear fractional integro-differential equations are transformed into a system of algebraic equations. In addition, the method is presented with error analysis. Numerical examples are included to demonstrate the validity and applicability of the approach. More >

Yiming Chen¹, Xiaoning Han¹, Lechun Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.5, pp. 391-405, 2014, DOI:10.3970/cmes.2014.097.391

Abstract In this paper, a class of linear system of fractional differential equations is considered. It has been solved by operational matrix of Haar wavelet method which converts the problem into algebraic equations. Moreover the convergence of the method is studied, and three numerical examples are provided to demonstrate the accuracy and efficiency. More >

Yiming Chen¹, Liqing Liu¹, Xuan Li¹ and Yannan Sun¹

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.1, pp. 81-100, 2014, DOI:10.3970/cmes.2014.097.081

Abstract In this paper, Bernstein polynomials method is proposed for the numerical solution of a class of variable order time fractional diffusion equation. Coimbra variable order fractional operator is adopted, as it is the most appropriate and desirable definition for physical modeling. The Coimbra variable order fractional operator can also be regarded as a Caputo-type definition. The main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of Bernstein polynomials. With the operational matrixes, the equation is transformed into the products of several dependent matrixes which can also be viewed as the system of… More >

W.M. Abd- Elhameed^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.101, No.3, pp. 159-185, 2014, DOI:10.3970/cmes.2014.101.159

Abstract This paper is concerned with developing two new algorithms for direct solutions of linear and nonlinear sixth-order two point boundary value problems. These algorithms are based on the application of the two spectral methods namely, collocation and Petrov-Galerkin methods. The suggested algorithms are completely new and they depend on introducing a novel operational matrix of derivatives which is expressed in terms of the well-known harmonic numbers. The basic idea for the suggested algorithms rely on reducing the linear or nonlinear sixth-order boundary value problem governed by its boundary conditions to a system of linear or nonlinear algebraic equations which can… More >

Mingxu Yi¹, Jun Huang¹, Lifeng Wang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.5, pp. 361-377, 2013, DOI:10.3970/cmes.2013.096.361

Abstract In this paper, operational matrix method based upon the Bernstein polynomials is proposed to solve the variable order fractional integro-differential equations in the Caputo derivative sense. We derive the Bernstein polynomials operational matrix of fractional order integration and introduce the product operational matrix of Bernstein polynomials. A truncated the Bernstein polynomials series together with the polynomials operational matrix are utilized to reduce the variable order fractional integro-differential equations to a system of algebraic equations. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Some examples are included to demonstrate the validity and applicability of the… More >

Yunpeng Ma¹, Lifeng Wang¹, Zhijun Meng¹

CMES-Computer Modeling in Engineering & Sciences, Vol.96, No.1, pp. 31-47, 2013, DOI:10.3970/cmes.2013.096.031

Abstract In this paper, we propose a numerical algorithm for solving linear and nonlinear fractional integro-differential equations based on our constructed fractional order generalized block pulse functions operational matrix of integration. The linear and nonlinear fractional integro-differential equations are transformed into a system of algebraic equations by the matrix and these algebraic equations are solved through known computational methods. Further some numerical examples are shown to illustrate the accuracy and reliability of the proposed approach. Moreover, comparing the methodology with the known technique shows that our approach is more efficient and more convenient. More >