An inverse source problem for a two-parameter anomalous diffusion with local time datum

Abstract

We determine the space-dependent source term for a two-parameter fractional diffusion problem subject to nonlocal non-self-adjoint boundary conditions and two local time-distinct datum. A bi-orthogonal pair of bases is used to construct a series representation of the solution and the source term. The two local time conditions spare us from measuring the fractional integral initial conditions commonly associated with fractional derivatives. On the other hand, they lead to delicate 2 × 2 linear systems for the Fourier coefficients of the source term and of the fractional integral of the solution at t = 0. The asymptotic behavior and estimates of the generalized Mittag-Leffler function are used to establish the solvability of these linear systems, and to obtain sufficient conditions for the existence of our construction. Analytical and numerical examples are provided.

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

“…The Mittag-Leffler-type function e ,1 (t; ) is positive decreasing function (see lemma 2 of Furati et al 27 ), ie,…”

“…25 For = 0, the ISP with Riemann-Liouville fractional derivative, ie, = 0, was considered in Kirane et al, 26 and the ISP for the Hilfer fractional derivative was considered by Furati et al in previous study. 27 Here, we are interested in the nontrivial case when ≠ 0. The over-specified data at 2 local time allow us to avoid the initial condition with fractional integral, 27 usually associated with Hilfer fractional derivative.…”

“…27 Here, we are interested in the nontrivial case when ≠ 0. The over-specified data at 2 local time allow us to avoid the initial condition with fractional integral, 27 usually associated with Hilfer fractional derivative. For = 1, ie, with Caputo fractional derivative in (1), the inverse problem of determination of a space-dependent source term was considered in Ali et al 28 The inverse problem considered here is a generalization of the inverse problem considered in Ali et al 28 A regular solution of the ISP is a pair of functions 1].…”

“…The ISP of determination of a space-dependent source term from final temperature distribution was considered by previous studies. 26,27,29,30 The ISPs for space time fractional diffusion equations are considered in 3 studies. [31][32][33] The determination of a source term using maximum principle technique for nonlinear time fractional diffusion equation is considered in Luchko et al, 34 where as inverse problem of recovering nonlinear boundary conditions for a time fractional differential equation was considered by Rundell et al 35 For a nonlinear time fractional diffusion equation, an inverse coefficient problem was considered in Tatar et al 36 The ISP of recovering a space-dependent source term in 2-dimensional space for a time fractional diffusion equation with nonlocal boundary conditions is considered in Malik et al 37 A topical review on inverse problems related to fractional diffusion equations is provided in Jin et al 38 The inverse problems related to several diffusion equations involving time and space fractional derivatives have been reported (for more details, see Jin et al 38 ).…”

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

“…34 Wei et al 35 proved unique identification of a space-dependent source term for TFDE. A multiple-scale radial basis function method is applied to solve direct and inverse Cauchy problem by Liu et al 36 There has been growing interest in studying ISPs related to FODEs, see, for example, Kirane et al 37,38 and Furati et al 39 For a TFDE, Al-Jamal 40 considered recovery of the initial condition whenever over-specified data are obtained from interior of the spatial domain. Direct and inverse problems for a family of TFDEs with nonlocal boundary conditions are proved to be well-posed in Ali and Malik.…”