Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

Abstract

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

Identification of a Time-Dependent Source Term in a Nonlocal Problem for Time Fractional Diffusion Equation

“…For a TFDe inverse problem of identification a time-dependent source term from temperature measurement at the selected point in the spatial domain was considered in [5,19] and determining a time-dependent source term from integral type overdetermination condition was studied in [2,3,16,18]. The initial and boundary data identification from final measurements in the initial boundary value problem for time-fractional heat equation was studied in [1,21] and [9], respectively.…”

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

“…At the initial time, the typical solutions of such problems have a key factor that must be considered (as in [6]); this factor is weak singularity which significantly complicates analysis. Now, many time-fractional initial-boundary value problems with Robin boundary conditions are widely used in the research fields of heat equation, biomathematics, and so on [7][8][9]. That is the main reason why this type of boundary condition is considered in this paper.…”