One Inverse Problem for the Diffusion-Wave Equation in Bounded Domain

Lopushans’kyi, A.; Lopushanska, Halyna

doi:10.1007/s11253-014-0969-9

Cited by 10 publications

(4 citation statements)

References 8 publications

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“…For example, this method does not work for inverse source problems. Furthermore, Lopushanskyi and Lopushanska [LL14] examined the same model as investigated in [Zh16] for the case α ∈ (0, 2). They employed the Green function to obtain a representation of the solution v. Additionally, they introduced an operator for σ(t) that guarantees the existence and uniqueness of the pair (σ, v).…”

Section: • the Continuous Dependence On The Data;mentioning

confidence: 99%

A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

Serikbaev¹,

Tokmagambetov²

2020

Bul. Kar. Univ. - Math.

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A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation with one fractional Sturm–Liouville operator is considered. In this paper, we prove the existence and uniqueness results of the solutions using by the variable separation method that is to say the Fourier method. We are especially interested in proving the existence and uniqueness of the solutions in the abstract setting of Hilbert spaces. The mentioned results are presented as well as for the Caputo time fractional pseudoparabolic equation. There are many cases in which practical needs lead to problems determining the coefficients or the right side of a differential equation from some available decision data. These are called inverse problems of mathematical physics. Inverse problems arise in various areas of human activity, such as seismology, mineral exploration, biology, medicine, industrial quality control goods, and so on. All these circumstances put the inverse problems among the important problems of modern mathematics.

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Section: • the Continuous Dependence On The Data;mentioning

confidence: 99%

A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

Serikbaev¹,

Tokmagambetov²

2020

Bul. Kar. Univ. - Math.

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“…See also [29] for recovering two coefficients from the Dirichlet-to-Neumann map. Zhang [45] proved the unique recovery of q(t) from lateral Cauchy data; see also [36]. Nonetheless, there seems still no known stability result for the inverse problem, and Theorems 3.1 and 3.2 are first known stability results for the concerned inverse problem.…”

Section: Introductionmentioning

confidence: 99%

Recovery of a Space-Time Dependent Diffusion Coefficient in Subdiffusion: Stability, Approximation and Error Analysis

Jin¹

2021

Preprint

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In this work, we study an inverse problem of recovering a space-time dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian-Caputo fractional derivative of order α ∈ (0, 1) in time. The main technical challenges of both theoretical and numerical analysis lie in the limited smoothing properties due to the fractional differential operator and the high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. Theoretically, we establish two conditional stability results using a novel test function, which leads to a stability bound in L 2 (0, T ; L 2 (Ω)) under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for the recovery of the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal L p regularity, and a nonstandard energy argument. Under the positivity condition, we obtain a standard L 2 (0, T ; L 2 (Ω)) error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

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“…In [9] and [11] authors considered some inverse source problems for time-fractional mixed parabolic-hyperbolic equations. Also in [15] authors investigated an inverse problem of determining diffusion coefficient in the diffusion-wave equation. In this work, we employ a spectral method with Jacobi polynomials as the basis functions.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equation

Nemati¹,

Babaei²

2019

Taiwanese J. Math.

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In this paper, we consider an inverse problem of identifying an unknown time dependent source function in a time-fractional diffusion-wave equation. First, some basic properties of the shifted Jacobi polynomials (SJPs) are presented. Then, the analytical solution of the direct problem is given and used to obtain an approximation of the unknown source function in a series of SJPs. Due to ill-posedness of this inverse problem, the Tikhonov regularization method with Morozov's discrepancy principle criterion is applied to find a stable solution. After that, an error bound is obtained for the approximation of the unknown source function. Finally, some numerical examples are provided to show effectiveness and robustness of the proposed algorithm.

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One Inverse Problem for the Diffusion-Wave Equation in Bounded Domain

Cited by 10 publications

References 8 publications

A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

Recovery of a Space-Time Dependent Diffusion Coefficient in Subdiffusion: Stability, Approximation and Error Analysis

A Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equation

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