A Numerical Method with Shifted Chebyshev Polynomials for a Set of Variable Order Fractional Partial Differential Equations

2020 ◽  
Vol 7 (1) ◽  
pp. 57-69
Author(s):  
Xing Zhang ◽  
◽  
Hong Sun ◽  
Yi-Ming Chen ◽  
Lei Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Chebyshev Polynomials ◽  
Variable Order ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

scholarly journals A fast numerical method for fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
S. Mockary ◽  
E. Babolian ◽  
A. R. Vahidi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

scholarly journals Legendre Polynomials Operational Matrix Method for Solving Fractional Partial Differential Equations with Variable Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yongqiang Yang ◽  
Yunpeng Ma ◽  
Lifeng Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Legendre Polynomials ◽  
Linear Equations ◽  
Operational Matrix ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Operational Matrix Method

A numerical method for solving a class of fractional partial differential equations with variable coefficients based on Legendre polynomials is proposed. A fractional order operational matrix of Legendre polynomials is also derived. The initial equations are transformed into the products of several matrixes by using the operational matrix. A system of linear equations is obtained by dispersing the coefficients and the products of matrixes. Only a small number of Legendre polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.

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