On self-similar solutions of time and space fractional sub-diffusion equations

Fatma  Al-Musalhi; Erkinjon  Karimov

doi:10.11121/ijocta.2021.1065

Authors

Fatma Al-Musalhi Center for Preparatory Studies, Sultan Qaboos University, Oman https://orcid.org/0000-0003-1108-3879
Erkinjon Karimov Ferghana State University, Uzbekistan https://orcid.org/0000-0003-4443-6300

DOI:

https://doi.org/10.11121/ijocta.2021.1065

Keywords:

Self-similar solution, Erdelyi-Kober fractional derivative, Hilfer derivatives, Hyper-Bessel operator, Successive iteration method

Abstract

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

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Author Biographies

Fatma Al-Musalhi, Center for Preparatory Studies, Sultan Qaboos University, Oman

is an instructor at the Center for Preparatory Studies, Sultan Qaboos University (SQU) in Oman and also a researcher. She received the master's and Ph.D. degree in Mathematics at SQU. Her research interests are on Fractional differential equations and mathematical modeling.

Erkinjon Karimov, Ferghana State University, Uzbekistan

is an Associate Professor at the Ferghana State University and Senior Researcher at the V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences (part-time). In 1998 he has got his Master degree from Fergana State University, then in 2006 he has got his PhD and in 2020 DSc from the V.I. Romanovskiy Institute of Mathematics.

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