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 Solving systems of multi-term fractional PDEs: Invariant subspace approach

2019 ◽  
Vol 10 (01) ◽  
pp. 1941010 ◽  
Author(s):  
Sangita Choudhary ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Time And Space ◽  
Partial Differential ◽  
Fractional Pdes

In the present paper, invariant subspace method has been extended for solving systems of multi-term fractional partial differential equations (FPDEs) involving both time and space fractional derivatives. Further, the method has also been employed for solving multi-term fractional PDEs in [Formula: see text] dimensions. A diverse set of examples is solved to illustrate the method.

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.

The Invariant Subspace Method for Solving Fractional Partial Differential Equations

2020 ◽  
pp. 267-282
Author(s):  
Mohamed Soror Abdel Latif ◽  
Abass Hassan Abdel Kader
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Subspace ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Subspace Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Balance Principle ◽  
New Exact Solutions

In this chapter, the authors discuss the effectiveness of the invariant subspace method (ISM) for solving fractional partial differential equations. For this purpose, they have chosen a nonlinear time fractional partial differential equation (PDE) with variable coefficients to be investigated through this method. One-, two-, and three-dimensional invariant subspace classifications have been performed for this equation. Some new exact solutions have been obtained using the ISM. Also, the authors give a comparison between this method and the homogeneous balance principle (HBP).

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 A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

2020 ◽  
Vol 5 (2) ◽  
pp. 35-48 ◽  
Author(s):  
Kamal Ait Touchent ◽  
Zakia Hammouch ◽  
Toufik Mekkaoui
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Invariant Subspace ◽  
Fractional Derivatives ◽  
Subspace Method ◽  
Partial Differential ◽  
Singular Kernels ◽  
Fractional Differential

AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

scholarly journals Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Mehnaz Shakeel ◽  
Iltaf Hussain ◽  
Hijaz Ahmad ◽  
Imtiaz Ahmad ◽  
Phatiphat Thounthong ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Caputo Derivative ◽  
Time Derivative ◽  
Order Derivative ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Central Difference ◽  
Definition Of

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

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