scholarly journals Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time

2021 ◽  
Vol 6 (11) ◽  
pp. 12114-12132
Author(s):  
Shuang-Shuang Zhou ◽  
◽  
Saima Rashid ◽  
Asia Rauf ◽  
Khadija Tul Kubra ◽  
...  
Keyword(s):  
Diffusion Equation ◽  
Spectral Problem ◽  
Direct Problem ◽  
Fractional Derivatives ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value ◽  
The Given

<abstract><p>For a multi-term time-fractional diffusion equation comprising Hilfer fractional derivatives in time variables of different orders between $ 0 $ and $ 1 $, we have studied two problems (direct problem and inverse source problem). The spectral problem under consideration is self-adjoint. The solution to the given direct and inverse source problems is formulated utilizing the spectral problem. For the solution of the given direct problem, we proposed existence, uniqueness, and stability results. The existence, uniqueness, and consistency effects for the solution of the given inverse problem were addressed, as well as an inverse source for recovering space-dependent source term at certain $ T $. For the solution of the challenges, we proposed certain relevant cases.</p></abstract>

scholarly journals Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
The Maximum Principle ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Yuri Luchko ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value ◽  
General Time

AbstractIn this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

scholarly journals On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
D. Goos ◽  
G. Reyero ◽  
S. Roscani ◽  
E. Santillan Marcus
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
The Real ◽  
Positive Semiaxis ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

We consider the time-fractional derivative in the Caputo sense of orderα∈(0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function inR+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit whenα↗1of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems whenα= 1, and the fractional diffusion equation becomes the heat equation.

scholarly journals Maximum principle and its application to multi-index Hadamard fractional diffusion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xueyan Ren ◽  
Guotao Wang ◽  
Zhanbing Bai ◽  
A. A. El-Deeb
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Classical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Multi Index ◽  
Initial Boundary Value

AbstractThis study establishes some new maximum principle which will help to investigate an IBVP for multi-index Hadamard fractional diffusion equation. With the help of the new maximum principle, this paper ensures that the focused multi-index Hadamard fractional diffusion equation possesses at most one classical solution and that the solution depends continuously on its initial boundary value conditions.

Inverse coefficient problem for the time-fractional diffusion equation

2021 ◽  
Vol 9 (1) ◽  
pp. 44-54
Author(s):  
D. K. Durdiev ◽  
Keyword(s):  
Inverse Problem ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Direct Problem ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Unknown Coefficient ◽  
Single Observation ◽  
Uniqueness Results ◽  
Time Fractional Diffusion Equation

We study the inverse problem of determining the time depending reaction diffu- sion coefficient in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is used and properties of this solution are investigated. The fundamental solution contains the Fox’s H− functions widely used in fractional calculus. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient which will be used in study inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.

Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach

2019 ◽  
Vol 27 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Xiong Bin Yan ◽  
Ting Wei
Keyword(s):  
Diffusion Equation ◽  
Tikhonov Regularization ◽  
Direct Problem ◽  
Fractional Diffusion ◽  
Adjoint Problem ◽  
Fractional Diffusion Equation ◽  
Gradient Algorithm ◽  
Tikhonov Regularization Method ◽  
Series Expression ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

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