A Numerical Method for Solving the Inverse Source Problem of the Space-Time Fractional Diffusion Equation

2021 ◽  
Vol 10 (04) ◽  
pp. 996-1002
Author(s):  
柔姿 段
Keyword(s):  
Diffusion Equation ◽  
Numerical Method ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation

Inverse source problem for a space-time fractional diffusion equation

2018 ◽  
Vol 21 (3) ◽  
pp. 844-863 ◽  
Author(s):  
Muhammad Ali ◽  
Sara Aziz ◽  
Salman A. Malik
Keyword(s):  
Diffusion Equation ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Eigenfunction Expansion Method ◽  
Special Cases ◽  
Time Fractional Diffusion Equation

Abstract For a space-time fractional diffusion equation, an inverse problem of determination of a space dependent source term along with the solution is considered. The fractional derivatives in time and space are defined in the sense of Caputo. Due to an over-specified data at final time say T, we proved that there exists a unique solution of the inverse source problem. We use the eigenfunction expansion method to prove our main results. Several special cases of space-time fractional diffusion equations are discussed and results are interpolated from generalized results. Some examples are provided.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

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