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

Li, Zhiyuan; Luchko, Yuri; Yamamoto, Mahito

doi:10.2478/s13540-014-0217-x

Cited by 86 publications

(77 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Introductionmentioning

confidence: 98%

Time-fractional diffusion equation in the fractional Sobolev spaces

Gorenflo

Luchko

Yamamoto

2015

FCAA

Self Cite

214

199

View full text Add to dashboard Cite

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of smooth functions that appear to be too narrow for several important applications. In this paper, we propose a definition of the Caputo derivative on a finite interval in the fractional Sobolev spaces and investigate it from the operator theoretic viewpoint. In particular, some important equivalences of the norms related to the fractional integration and differentiation operators in the fractional Sobolev spaces are given. These results are then applied for proving the maximal regularity of the solutions to some initial-boundary-value problems for the time-fractional diffusion equation with the Caputo derivative in the fractional Sobolev spaces. MSC 2010 : Primary 26A33; Secondary 35C05, 35E05, 35L05, 45K05, 60E99 Key Words and Phrases: Riemann-Liouville integral, Caputo fractional derivative, fractional Sobolev spaces, norm equivalences, fractional diffusion equation in Sobolev spaces, norm estimates of the solutions, initialboundary-value problems, weak solution, existence and uniqueness results c 2015 Diogenes Co., Sofia pp. 799-820 ,

show abstract

Section: Introductionmentioning

confidence: 98%

Time-fractional diffusion equation in the fractional Sobolev spaces

Gorenflo

Luchko

Yamamoto

2015

FCAA

Self Cite

214

199

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Taghavian

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.

show abstract

“…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…”

Section: Introductionmentioning

confidence: 99%

Fractional transient thermal mixed boundary value problem of distributed order

Masomi

Ansari

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, we introduce the fractional transient thermal mixed boundary value problem of distributed order. We employ the Cagniard-de Hoop and the Wiener-Hopf techniques to solve this problem. The Laplace and Fourier transforms are used to present the formal solution of this problem in terms of some special functions. The numerical simulation of a model example is also given for the heat flow through the foundation of building. KEYWORDS boundary value problem, Cagniard-de Hoop method, distributed order, Fourier transform, Laplace transform, Wiener-Hopf method 6726

show abstract

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

Cited by 86 publications

References 15 publications

Time-fractional diffusion equation in the fractional Sobolev spaces

Time-fractional diffusion equation in the fractional Sobolev spaces

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

Fractional transient thermal mixed boundary value problem of distributed order

Contact Info

Product

Resources

About