A Method of Finding Source Function for Inverse Diffusion Problem with Time-Fractional Derivative

Gülkaç, Vildan

doi:10.1155/2016/6470949

Cited by 2 publications

(2 citation statements)

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“…Solving the homogeneous equation corresponding to (10) using separation of variables leads to the Legendre equation (8). According to Theorem 7, the Legendre polynomials form a complete orthogonal system in [−1, 1], hence the solution set {U (t, x), h(x)} and the given data v(x), w(x) can be represented in a form of Fourier-Legendre series as follow:…”

Section: Existence and Convergence Of The Solutionmentioning

confidence: 99%

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Inverse source problems for degenerate time-fractional PDE

Al-Salti,

Karimov

2019

Preprint

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In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involve a generalized Mittag-Leffler type function. Moreover, we have established a new estimate for this generalized Mittag-Leffler type function. The obtained results are illustrated by providing example solutions using certain given data at the initial and final time.

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Section: Existence and Convergence Of The Solutionmentioning

confidence: 99%

“…Solutions to inverse problems can be obtained through analytical and numerical methods as well. Analytical Methods include, for example, spectral method [7], the homotopy perturbation method [8] and regularization method [9]. For numerical methods, one may see, for example, [10], [11].…”

Section: Introductionmentioning

confidence: 99%