scholarly journals Stable numerical results to a class of time-space fractional partial differential equations via spectral method

2020 ◽  
Vol 25 ◽  
pp. 39-48 ◽  
Author(s):  
Kamal Shah ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spectral Method ◽  
Numerical Results ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Time Space

B-spline wavelet operational method for numerical solution of time-space fractional partial differential equations

2017 ◽  
Vol 15 (04) ◽  
pp. 1750034 ◽  
Author(s):  
Zeynab Kargar ◽  
Habibollah Saeedi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Results ◽  
Scaling Functions ◽  
Operational Matrices ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
B Spline ◽  
Time Space ◽  
Linear B

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

scholarly journals Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Weishi Yin ◽  
Fei Xu ◽  
Weipeng Zhang ◽  
Yixian Gao
Keyword(s):  
Asymptotic Expansion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Asymptotic Expansion Of Solutions

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

Extrapolating for attaining high precision solutions for fractional partial differential equations

2018 ◽  
Vol 21 (6) ◽  
pp. 1506-1523 ◽  
Author(s):  
Fernanda Simões Patrício ◽  
Miguel Patrício ◽  
Higinio Ramos
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Precision ◽  
Euler Method ◽  
Accurate Solution ◽  
Numerical Results ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Temporal Integrator ◽  
Present Technique

Abstract This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.

Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

2018 ◽  
Vol 21 (2) ◽  
pp. 312-335 ◽  
Author(s):  
Xiao-Li Ding ◽  
Juan J. Nieto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Laplacian Operator ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Time Space ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

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