An inverse source problem in a semilinear time-fractional diffusion equation

Slodička, Marián; Šišková, Katarína

doi:10.1016/j.camwa.2016.07.029

Cited by 30 publications

(17 citation statements)

References 21 publications

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“…in Q T . (46) When discussing the uniqueness of a solution, we get again from Corollary 3.1 inequality (39) where β 1 = max{β 1 , β 2 } in (8). We conclude the following theorem.…”

Section: A∇u(t) • Nmentioning

confidence: 60%

Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

Bockstal

2021

Adv Differ Equ

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In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

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“…in Q T . (46) When discussing the uniqueness of a solution, we get again from Corollary 3.1 inequality (39) where β 1 = max{β 1 , β 2 } in (8). We conclude the following theorem.…”

Section: A∇u(t) • Nmentioning

confidence: 60%

Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

Bockstal

2021

Adv Differ Equ

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“…[17,6,53,50,37,52,49,41]. Identification of the time-dependent part of the source term in a fractional diffusion equation assuming various boundary conditions and additional measurements has been studied in [36,51,10,38]. The well-posedness of the problem in a one-dimensional setting with nonlocal boundary conditions, F = 0 and a measurement in the form of integral over the domain is studied in [6].…”

Section: Introductionmentioning

confidence: 99%

“…The Tikhonov regularization method is applied on the problem considering a simple fractional diffusion equation in the one-dimensional case in [49]. In [38] an ISP for a semilinear time-fractional diffusion equation with a solely time-dependent unknown source was studied. The existence and uniqueness of a weak solution for the ISP were proved.…”

Section: Introductionmentioning

confidence: 99%

“…The corresponding error estimates are always restrictive since often the solution does not have the required regularity. The Rothe's method in [38] was based on a compatibility condition between the initial data, sources and boundary condition, and so the corresponding error estimates are also restrictive. More recently, the authors in [8,43] analyzed the L1-formula on graded time grids by taking into account the initial singularity in the Caputo's fractional subdiffusion problems.…”

Section: Introductionmentioning

confidence: 99%

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On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Hendy

Bockstal

2021

J Sci Comput

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An inverse source problem for a non-automonous time fractional diffusion equation of order (0 < β < 1) is considered in a bounded Lipschitz domain in R d . The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe's method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from t = 0. Rothe's method on a uniform grid addresses the existence of a such a solution (non-smooth with t γ term where 1 > γ > β ) under low regularity assumptions, whilst Rothe's method over graded grids has the advantage to cope better with the behaviour at t = 0 (also here t β is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

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“…19 Existence, uniqueness, and regularity of the weak solution for a semilinear time-fractional diffusion equation is addressed in Slodicka and Siskova. 20 Determination of the spatial and temporal component of source term for the space-time-fractional diffusion equation is considered in Ali et al, 21,22 respectively. Generalized Tikhonov regularization method is used to construct the solution of the inverse source problem for time-fractional diffusion equation.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Some inverse problems for time‐fractional diffusion equation with nonlocal Samarskii‐Ionkin type condition

Ali

Aziz

2020

Math Methods in App Sciences

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Two inverse problems for time-fractional diffusion equation having a family of nonlocal boundary conditions are discussed. In first inverse problem, initial distribution is determined provided that the data at final temperature t = T is given.Second inverse problem addresses the recovery of temporal component of source term whenever total energy of the system is known. A bi-orthogonal system of functions is used to write the series solution by Fourier's method. The classical nature of the solution of both inverse problems is established by using the estimates of Mittag-Leffler function and by imposing some regularity conditions on given datum.

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An inverse source problem in a semilinear time-fractional diffusion equation

Cited by 30 publications

References 21 publications

Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

Some inverse problems for time‐fractional diffusion equation with nonlocal Samarskii‐Ionkin type condition

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