Existence and uniqueness of the solution for a time-fractional diffusion equation

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
J. Kemppainen
Keyword(s):  
Integral Equation ◽  
Diffusion Equation ◽  
Unique Solution ◽  
Double Layer ◽  
Existence And Uniqueness ◽  
Fractional Diffusion ◽  
Double Layer Potential ◽  
Layer Potential ◽  
Uniqueness Of The Solution ◽  
Time Fractional Diffusion Equation

AbstractIn the paper existence and uniqueness of the solution for a time-fractional diffusion equation on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a fundamental solution of the problem is known, we may seek the solution as the double layer potential. This approach leads to a Volterra integral equation of the second kind associated with a compact operator. Then classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary datum is continuous and satisfies a compatibility condition. This proves that the original problem has a unique solution and the solution is given by the double layer potential.

scholarly journals Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jukka Kemppainen
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Original Problem ◽  
Robin Boundary Condition ◽  
Fractional Diffusion ◽  
Uniqueness Of The Solution ◽  
Time Fractional Diffusion Equation ◽  
Robin Boundary

Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

scholarly journals A Tutorial on the Basic Special Functions of Fractional Calculus

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Probability Theory ◽  
Integral Equation ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Special Function ◽  
Special Functions ◽  
Fractional Diffusion ◽  
Basic Properties ◽  
Fractional Relaxation ◽  
Time Fractional Diffusion Equation

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.

Identification of the diffusion coefficient in a time fractional diffusion equation

2020 ◽  
Vol 28 (2) ◽  
pp. 299-306
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri ◽  
Soheila Bodaghi ◽  
M. Heshmati
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Stability Result ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Cost Functional ◽  
Time Fractional Diffusion Equation ◽  
The Cost ◽  
The Uniqueness Theorem ◽  
Admissible Functions

AbstractIn this paper, we study the existence and uniqueness of a quasi solution to a time fractional diffusion equation related to {{}^{C}D_{t}^{\alpha}u-\nabla\cdot(k(x)\nabla u)=f}, where the function {k=k(x)} is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function k. At the first step of the methodology, we give a stability result corresponding to connectivity of k and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the end, convexity of the introduced cost functional and subsequently the uniqueness theorem of the quasi solution are given.

scholarly journals Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

2021 ◽  
Vol 5 (3) ◽  
pp. 64
Author(s):  
Igor V. Malyk ◽  
Mykola Gorbatenko ◽  
Arun Chaudhary ◽  
Shivani Sharma ◽  
Ravi Shanker Dubey
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Numerical Solution ◽  
Analytical Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Derivative Operator ◽  
Uniqueness Of The Solution ◽  
Time Fractional Diffusion Equation

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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 Analytical solution of non-linear fractional diffusion equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Obaid Alqahtani
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
Approximate Analytical Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution

AbstractIn this paper, we obtain an approximate/analytical solution of nonlinear fractional diffusion equation using the q-homotopy analysis transform method. The existence and uniqueness of the solution for this problem are also derived. Further, the applicability of the model is discussed based on graphical results and numerical examples.

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