Generalized fraction evolution equations with fractional Gross Laplacian

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Samah Horrigue ◽  
Habib Ouerdiane ◽  
Imen Salhi
Keyword(s):  
Diffusion Equation ◽  
Laplace Transform ◽  
Explicit Solution ◽  
Evolution Equations ◽  
Fractional Diffusion ◽  
Generalized Function ◽  
Initial Condition ◽  
Infinite Dimensions ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we define and consider the fractional Gross Laplacian which is characterized by the Laplace transform. As application, we study the generalized Riemann-Liouville time fractional diffusion equation in infinite dimensions. We show that the explicit solution is given as the convolution between the initial condition and a generalized function related to the Mittag-Leffler function.

scholarly journals Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator

2021 ◽  
Vol 24 (6) ◽  
pp. 1899-1918
Author(s):  
Anwar Ahmad ◽  
Muhammad Ali ◽  
Salman A. Malik
Keyword(s):  
Inverse Problems ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fractional Diffusion ◽  
Time Dependent ◽  
Source Terms ◽  
Special Cases ◽  
Time Fractional Diffusion Equation

Abstract Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives named after Riemann-Liouville, Caputo and Hilfer are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are established. The results obtained here generalize several known results.

scholarly journals Fast O(N) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation

2020 ◽  
Vol 23 (02) ◽  
pp. 110-123
Author(s):  
Fouad Mohammad Salama ◽  
Norhashidah Hj. Mohd Ali ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Hybrid Method ◽  
Finite Difference Scheme ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Time Fractional Diffusion Equation ◽  
Memory Complexity

It is time-memory consuming when numerically solving time fractional partial differential equations, as it requires O ( N 2 ) computational cost and O ( M N ) memory complexity with finite difference methods, where, N and M are the total number of time steps and spatial grid points, respectively. To surmount this issue, we develop an efficient hybrid method with O ( N ) computational cost and O ( M ) memory complexity in solving two-dimensional time fractional diffusion equation. The presented method is based on the Laplace transform method and a finite difference scheme. The stability and convergence of the proposed method are analyzed rigorously by the means of the Fourier method. A comparative study drawn from numerical experiments shows that the hybrid method is accurate and reduces the computational cost, memory requirement as well as the CPU time effectively compared to a standard finite difference scheme.

Solving a backward problem for a distributed-order time fractional diffusion equation by a new adjoint technique

2020 ◽  
Vol 28 (4) ◽  
pp. 471-488
Author(s):  
Lele Yuan ◽  
Xiaoliang Cheng ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Convergence Rates ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Initial Condition ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order

AbstractThis paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series representation of the regularized solution, we give convergence rates under an a-priori and an a-posteriori regularization parameter choice rule. With a new adjoint technique to compute the gradient of the functional, the conjugate gradient method is applied to reconstruct the initial condition. Numerical examples in one- and two-dimensional cases illustrate the effectiveness of the proposed method.

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