scholarly journals ψ-SHIFTED OPERATIONAL MATRIX SCHEME FOR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Shazia Sadiq ◽  
◽  
Mujeeb ur Rehman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Operational Matrix ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Matrix Scheme

scholarly journals New method for solving a class of fractional partial differential equations with applications

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 277-286 ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations ◽  
Spline Polynomial

In this work we suggest a numerical approach based on the B-spline polynomial to obtain the solution of linear fractional partial differential equations. We find the operational matrix for fractional integration and then we convert the main problem into a system of linear algebraic equations by using this matrix. Examples are provided to show the simplicity of our method.

scholarly journals Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

2021 ◽  
Vol 5 (4) ◽  
pp. 208
Author(s):  
Muhammad I. Bhatti ◽  
Md. Habibur Rahman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Initial Conditions ◽  
Operational Matrix ◽  
Basis Sets ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

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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