A NONSTANDARD FINITE DIFFERENCE SCHEME FOR TWO-SIDED SPACE-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250079 ◽  
Author(s):  
SHAHER MOMANI ◽  
ABDULLAH ABU RQAYIQ ◽  
DUMITRU BALEANU
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Boundary ◽  
Variable Coefficients ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Left Handed ◽  
Nonstandard Finite Difference Scheme ◽  
Initial Boundary Value

In this paper, we apply the Mickens nonstandard discretization method to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and thereby increase the accuracy of the solutions. We examine the case when a left-handed and a right-handed fractional spatial derivative may be present in the partial differential equation. Two numerical examples using this method are presented and compared successfully with the exact analytical solutions.

scholarly journals Approximation Solution of Fractional Partial Differential Equations by Neural Networks

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Adel A. S. Almarashi
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Boundary ◽  
Variable Coefficients ◽  
Finite Domain ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Approximation Solution ◽  
Initial Boundary Value

Neural networks with radial basis functions method are used to solve a class of initial boundary value of fractional partial differential equations with variable coefficients on a finite domain. It takes the case where a left-handed or right-handed fractional spatial derivative may be present in the partial differential equations. Convergence of this method will be discussed in the paper. A numerical example using neural networks RBF method for a two-sided fractional PDE also will be presented and compared with other methods.

scholarly journals Exact, approximate solutions and error bounds for coupled implicit systems of partial differential equations

1992 ◽  
Vol 15 (4) ◽  
pp. 663-672
Author(s):  
Lucas Jódar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Value Systems ◽  
Finite Domain ◽  
Partial Differential ◽  
Implicit Systems ◽  
Initial Boundary Value ◽  
The Given

In this paper coupled implicit initial-boundary value systems of second order partial differential equations are considered. Given a finite domain and an admissible errorϵan analytic approximate solution whose error is upper bounded byϵin the given domain is constructed in terms of the data.

scholarly journals Novel solution methods for initial boundary value problems of fractional order with conformable differentiation

2017 ◽  
Vol 8 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Mehmet Yavuz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Initial Boundary Value

In this work, we develop a formulation for the approximate-analytical solution of fractional partial differential equations (PDEs) by using conformable fractional derivative. Firstly, we redefine the conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we solve some initial boundary value problems (IBVP) by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of the IBVP. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions for the IBVP of fractional order in the conformable sense.  

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).

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