Analytical solutions of linear fractional partial differential equations using fractional Fourier transform

2021 ◽  
Vol 385 ◽  
pp. 113202
Author(s):  
Teekam Chand Mahor ◽  
Rajshree Mishra ◽  
Renu Jain
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Fractional Fourier Transform ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

scholarly journals A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ji Juan-Juan ◽  
Guo Ye-Cai ◽  
Zhang Lan-Fang ◽  
Zhang Chao-Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Table Lookup ◽  
Exact Analytical Solutions ◽  
Function Solution

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

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.

scholarly journals A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Mehmet Ali Akinlar ◽  
Muhammet Kurulay
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Solution Technique ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Vector Type ◽  
Novel Method ◽  
Fractional Differential ◽  
Finite Sum

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.

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