Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Abstract

This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order timefractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t)) −1 . Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.MSC 2010 : Primary 35R11; Secondary 35B40, 35S11, 44A10

Time-fractional diffusion equation in the fractional Sobolev spaces

“…in the monographs [5], [13], [10], and [23]. We mention here also the papers [3], [14], [16]- [19], [25], where some recent developments regarding the partial fractional differential equations are presented.…”

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

“…Applications of the results are therefore presented subsequently, including analytic solutions of fractional distributed order relaxation processes and time domain impulse response of fractional distributed order operators in new simple series expansions. Due to the physical implications and importance of these two problems as well as omnipresence of fractional (distributed order) differential equations, numerical schemes for these kinds of equations and analytical exact and approximate expressions of their solutions that are simple and easy to compute are interesting to scientists . In addition to that, new alternative series expansions for some special functions are obtained in this paper.…”

“…Due to the physical implications and importance of these two problems [9][10][11][12][13] as well as omnipresence of fractional (distributed order) differential equations, 14 numerical schemes for these kinds of equations 15 and analytical exact and approximate expressions of their solutions that are simple and easy to compute are interesting to scientists. [16][17][18][19][20] In addition to that, new alternative series expansions for some special functions are obtained in this paper. The provided numerical simulations suggest that the obtained solutions and expressions are convergent and valid.…”

Fractional transient thermal mixed boundary value problem of distributed order

“…and was developed by other researchers for different types of differential equations (for example, see other works [29][30][31][32][33][34][35][36][37][38][39] ). In this definition, C t D 0 + is the Caputo fractional derivative of order given by 40,41…”