scholarly journals A solution of linear and non-linear partial differential equation of fractional order

2021 ◽  
Author(s):  
Raju Muneshwar ◽  
K. L. Bondar ◽  
Y. H. Shirole
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Series Solution ◽  
Solution Method ◽  
Linear Partial Differential Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear ◽  
Time Variables ◽  
Better Than

We know that the solution of partial differential equations by analytical method is better than the solution by approximate or series solution method. In this paper, we discuss the solution of linear and non-linear fractional partial differential equations involving derivatives with respect to time or space variables by converting them into the partial differential equations of integer order. Also we develop an analytical formulation to solve such fractional partial differential equations. Moreover, we discuss the method to solve the fractional partial differential equations in space as well as time variables simultaneously with the help of some examples

scholarly journals Invariant subspace method: A tool for solving fractional partial differential equations

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Caputo Derivative ◽  
Linear Time ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Non Linear

AbstractIn this paper invariant subspace method has been employed for solving linear and non-linear time and space fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the effectiveness and applicability of the method.

scholarly journals A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

2021 ◽  
Vol 9 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Zeyad Al-Zhour
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Hyperbolic System ◽  
Series Solution ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

scholarly journals New exact solutions of fractional Hirota-Satsuma coupled Korteweg-de Vries equations

2015 ◽  
Vol 19 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
Lian-Xiang Cui ◽  
Li-Mei Yan ◽  
Yan-Qin Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Function Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
New Exact Solutions ◽  
Non Linear ◽  
De Vries

An improved extended tg-function method, which combines the fractional complex transform and the extended tanh-function method, is applied to find exact solutions of non-linear fractional partial differential equations. Generalized Hirota-Satsuma coupled Korteweg-de Vries equations are used as an example to elucidate the effectiveness and simplicity of the method.

scholarly journals Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method

2015 ◽  
Vol 19 (4) ◽  
pp. 1239-1244 ◽  
Author(s):  
Hong-Cai Ma ◽  
Dan-Dan Yao ◽  
Xiao-Fang Peng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Space Time ◽  
Equation Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Non Linear

This paper studies the space-time fractional Whitham-Broer-Kaup equations by the existed fractional sub-equation method, and exact solutions are obtained.

A modified method for solving non-linear time and space fractional partial differential equations

2019 ◽  
Vol 36 (7) ◽  
pp. 2162-2178
Author(s):  
Umer Saeed ◽  
Muhammad Umair
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Time ◽  
Differential Quadrature Method ◽  
Fractional Partial Differential Equations ◽  
Lagrange Polynomial ◽  
Time And Space ◽  
Partial Differential ◽  
Content Type ◽  
Non Linear

Purpose The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain. Design/methodology/approach The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations. Findings The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed. Originality/value Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

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