scholarly journals An inverse source problem for a two terms time-fractional diffusion equation

2021 ◽  
Vol 40 ◽  
pp. 1-15
Author(s):  
Fatima Dib ◽  
Mokhtar Kirane
Keyword(s):  
Boundary Condition ◽  
Fractional Derivatives ◽  
Source Term ◽  
Space Variable ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Integral Type ◽  
Time Fractional Diffusion Equation

In this paper, we consider an inverse problem for a linear heat equation involving two time-fractional derivatives, subject to a nonlocal boundary condition. We determine a source term independent of the space variable, and the temperature distribution with an over- determining function of integral type.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

scholarly journals A Two Dimensional Discrete Mollification Operator and the Numerical Solution of an Inverse Source Problem

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 89 ◽  
Author(s):  
Manuel Echeverry ◽  
Carlos Mejía
Keyword(s):  
Inverse Problem ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Two Dimensional ◽  
Regularization Procedure ◽  
Numerical Examples ◽  
Convergence Results ◽  
Time Fractional Diffusion Equation ◽  
Operator Convergence

We consider a two-dimensional time fractional diffusion equation and address the important inverse problem consisting of the identification of an ingredient in the source term. The fractional derivative is in the sense of Caputo. The necessary regularization procedure is provided by a two-dimensional discrete mollification operator. Convergence results and illustrative numerical examples are included.

Inverse source problem for a distributed-order time fractional diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Lele Yuan ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order ◽  
Regularized Solution

AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

2015 ◽  
Vol 8 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Zhousheng Ruan ◽  
Zhijian Yang ◽  
Xiliang Lu
Keyword(s):  
Diffusion Equation ◽  
Optimization Problem ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Final Time ◽  
Elastic Net Regularization ◽  
Semismooth Newton ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

scholarly journals Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation

2020 ◽  
Vol 23 (6) ◽  
pp. 1702-1711
Author(s):  
Marian Slodička
Keyword(s):  
Source Term ◽  
Linear Time ◽  
Fractional Diffusion ◽  
Time Measurement ◽  
Uniqueness Result ◽  
Inverse Source Problem ◽  
Final Time ◽  
Fractional Diffusion Equations ◽  
Time Fractional Diffusion Equation ◽  
Uniqueness Of A Solution

Abstract We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time-dependent coefficients. We consider source term in a separated form h(t)f (x). The unknown source f (x) is recovered from the final time measurement u (x, T). A new uniqueness result is formulated in Theorem 3.1 under the assumption that h ∈ C ([0, T]) and 0 ≢ h ≥ 0. No monotonicity in time for h(t) and for coefficients of the differential operator is required.

scholarly journals Spectrum curves for a discrete Sturm–Liouville problem with one integral boundary condition

2019 ◽  
Vol 24 (5) ◽  
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Function Method ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Trapezoidal Rule ◽  
Integral Type ◽  
Integral Boundary ◽  
Sturm Liouville

This paper presents new results on the spectrum on complex plane for discrete Sturm–Liouville problem with one integral type nonlocal boundary condition depending on three parameters: γ, ξ1 and ξ2. The integral condition is approximated by the trapezoidal rule. The dependence on parameter γ is investigated by using characteristic function method and analysing spectrum curves which gives qualitative view of the spectrum for fixed ξ1 = m1 / n and ξ2 = m2 / n, where n is discretisation parameter. Some properties of the spectrum curves are formulated and illustrated in figures for various ξ1 and ξ2. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).

scholarly journals Investigation of the spectrum of the Sturm–Liouville problem with a nonlocal integral condition

2013 ◽  
Vol 54 ◽  
pp. 67-72
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Integral Condition ◽  
Characteristic Functions ◽  
Integral Type ◽  
Complex Characteristic ◽  
Nonlocal Integral Condition ◽  
Sturm Liouville

This paper presents some new results on the spectrum for the second order dif-ferential problem with one integral type nonlocal boundary condition (NBC). We investigate how the spectrum of this problem depends on the integral nonlocal boundary condition pa-rameters γ, ξ and the symmetric interval in the integral. Some new results are given on the complex spectra of this problem. Many results are presented as graphs of real and complex characteristic functions.

scholarly journals Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition

2021 ◽  
Vol 26 (5) ◽  
pp. 969-991
Author(s):  
Artūras Štikonas ◽  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Arbitrary Order ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Classical Type ◽  
Integral Type ◽  
Sturm Liouville ◽  
Type Boundary

In this study, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. We investigate solutions of special initial value problem and find asymptotic formulas of arbitrary order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic formulas of arbitrary order. We apply the obtained results to the problem with integral-type nonlocal boundary condition.

scholarly journals Investigation of the Sturm–Liouville problems with integral boundary condition

2011 ◽  
Vol 52 ◽  
pp. 297-302
Author(s):  
Agnė Skučaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Trapezoidal Rule ◽  
Integral Conditions ◽  
Integral Type ◽  
Integral Boundary ◽  
Complex Eigenvalues ◽  
Sturm Liouville

This paper presents some new results on the spectrum of a complex plane for the second order Finite-Difference Scheme with one integral type nonlocal boundary condition (NBC). We analyze how complex eigenvalues of these problems depend on the parameters of the integral NBC. The integral conditions are approximated by the trapezoidal rule or by Simpson’s rule.

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