An Inverse Source Non-local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

Abstract

In the present work, we discuss a unique solvability of an inversesource problem with integral transmitting condition for time-fractional mixed type equation in a rectangular domain, where the unknown source term depends on space variable only.The method of solution based on a series expansion using biorthogonal basis of space corresponding to a nonself-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. Influence of transmitting condition on the solvability of the problem is shown as well. Precisely, two different cases were considered; a case of full integral form (0 < γ < 1) and a special case (γ = 1) of transmitting condition.In order to simplify the bulky expressions appearing in the proof of the main result, we have established a new property of the recently introduced Mittag-Leffler type function of two variables (see Lemma 2.1).

A Fractional Tikhonov Regularisation Method for Finding Source Terms in a Time-Fractional Radial Heat Equation

“…and then estimate each term separately. Analogously to the evaluation of the term I 1 in the proof of Theorem 4.1, we obtain 12) and the application of Lemma 4.6 and (4.12) leads to the estimate…”

“…Thus Wang et al [30] applied reproducing kernel space method to solve an inverse space-dependent source problem, Wei and Wang [31] used a modified quasi-boundary value method, Zhang and Xu [37] employed the Cauchy data at one end, Tatar et al [26,27] considered it for a spacetime fractional diffusion equation and investigated a nonlocal inverse source problem, Cheng et al [4] used a spectral method to determine an unknown heat source term from the final temperature history in the radial domain and provided logarithmic-type error estimates for regularised solutions, Xiong and Ma [33] discussed a backward ill-posed problem for an axis-symmetric fractional diffusion equation. For other relevant results, the reader is referred to [3,9,12,14,28,[34][35][36].…”

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“…In [25], a hyperbolic-parabolic system, in relation to pulse combustion, is investigated. Mixed type fractional differential equations are studied in many works by scientists-particularly in [26][27][28][29][30][31][32][33][34][35].…”

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“…The inverse source problem for time-fractional diffusion equation is widely studied. Thus Zhang et al [44] found the solution of an inverse space-dependent source problem from the Cauchy data at x = 0, Sakamoto et al [28] established a stability estimate for identification of a time-dependent source from the measurements at an interior point, Wang et al [35] used a reproducing kernel space method to study an inverse spacedependent source problem, Wei and Wang [38] identified the space-dependent source term by a modified quasi-boundary value method, Wei et al [36] used a conjugate gradient algorithm to determine a time-dependent source term, Fujishiro and Kian [9] established time-dependent factors of a zero order term coefficient from the solution at an interior point, Karimov et al [14] used a bi-orthogonal basis to show the unique solvability of an inverse-space source problem in rectangular domain, Liu and Zhang [20] employed fixedpoint iterations to reconstruct a time-dependent source from a single point data, Yan and Zhou [40,41] developed an adaptive multi-fidelity approach to Bayesian inverse and to nonlinear inverse problems.…”