Existence results of self-similar solutions of the space-fractional diffusion equation involving the generalized Riesz-Caputo fractional derivative
Keywords:
Space-fractional diffusion equation, fixed point theorems, self-similar solutions, generalized Riesz-Caputo fractional derivativeAbstract
In this paper, we have discussed the problem of existence and uniqueness of solutions under the self-similar form to the space-fractional diffusion equation. The space-fractional derivative which will be used is the generalized Riesz-Caputo fractional derivative. Based on the similarity variable η, we have introduced the equation satisfied by the self-similar solutions for the aforementioned problem. To study the existence and uniqueness of self-similar solutions for this problem, we have applied some known fixed point theorems (i.e.~Banach's contraction principle, Schauder's fixed point theorem and the nonlinear alternative of Leray-Schauder type).
References
Aleem, Maryam, et al. "On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz-Caputo operator." Adv. Difference Equ. (2021): Paper No. 303.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Almeida Ricardo. "A Gronwall inequality for a general Caputo fractional operator." Arxiv (2017): arxiv.org/pdf/1705.10079.pdf.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Almeida, Ricardo, Agnieszka B. Malinowska, and Tatiana Odzijewicz. "Fractional differential equations with dependence on the Caputo–Katugampola derivative." J. Comput. Nonlinear Dynam. 11, no. 6 (2016): Paper No. 061017.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Arioua, Yacine, Bilal Basti, and Nouredine Benhamidouche. "Initial value problem for nonlinear implicit fractional differential equations with Katugampola derivative." Appl. Math. E-Notes 19 (2019): 397-412.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Arioua, Yacine, and Maria Titraoui. "Boundary value problem for a coupled system of nonlinear fractional differential equations involving Erdélyi-Kober derivative." Appl. Math. E-Notes 21 (2021): 291-306.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Arioua, Yacine, and Li Ma. "On criteria of existence for nonlinear Katugampola fractional differential equations with p-Laplacian operator." Fract. Differ. Calc. 11, no. 1 (2021): 55-68.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Baleanu, Dumitru I., Octavian G. Mustafa, and Ravi Prakash Agrawal. "On the solution set for a class of sequential fractional differential equations." J. Phys. A 43, no. 38 (2010): Art. No. 385209.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Basti, Bilal, Yacine Arioua, and Nouredine Benhamidouche. "Existence and uniqueness of solutions for nonlinear Katugampola fractional differential equations." J. Math. Appl. 42 (2019): 35-61.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Basti, Bilal, Yacine Arioua, and Nouredine Benhamidouche. "Existence results for nonlinear Katugampola fractional differential equations with an integral condition." Acta Math. Univ. Comenian. (N.S.) 89, no. 2 (2020): 243-260.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Basti, Bilal, and Nouredine Benhamidouche. "Existence results of self-similar solutions to the Caputo-type’s space-fractional heat equation." Surv. Math. Appl. 15 (2020): 153-168.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Boucenna, Djalal, Abdellatif Ben Makhlouf, and Mohamed Ali Hammami. "On Katugampola fractional order derivatives and Darboux problem for differential equations." Cubo 22, no. 1 (2020): 125-136.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Bouchama, Kaouther, Yacine Arioua, and Abdelkrim Merzougui. "The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative." Numer. Algebra Control Optim. 12, no. 3 (2022): 621-636.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Cernea, Aurelian. "On the solutions of a class of fractional hyperbolic integrodifferential inclusions." Int. J. Anal. Appl. 17, no. 6 (2019), 904-916.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Buckwar, Evelyn, and Yurii F. Luchko. "Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations." J. Math. Anal. Appl. 227, no. 1 (1998): 81-97.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Deng, Keng, and Howard A. Levine. "The role of critical exponents in blow-up theorems: the sequel." J. Math. Anal. Appl. 243, no. 1 (2000): 85-126.
##plugins.generic.googleScholarLinks.settings.viewInGS##
El-Shahed, Moustafa. "Positive solutions for boundary value problem of nonlinear fractional differential equation." Abstr. Appl. Anal. (2007): Art. ID 10368.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Al Musalhi, Fatma S. K., and Erkinjon Tulkinovich Karimov. "On self-similar solutions of time and space fractional sub-diffusion equations." Int. J. Optim. Control. Theor. Appl. IJOCTA 11, no. 3 (2021): 16-27.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Hale, Jack K., and Sjoerd M. Verduyn Lunel. Introduction to functional differential equations. Vol. 99 of Applied Mathematical Sciences. New York: Springer-Verlag, 1993.
##plugins.generic.googleScholarLinks.settings.viewInGS##
He, Ji-Huan. "Approximate analytical solution for seepage flow with fractional derivatives in porous media." Comput. Methods Appl. Mech. Engrg. 167, no. 1-2 (1998): 57-68
##plugins.generic.googleScholarLinks.settings.viewInGS##
Granas, Andrzej and James Dugundji. Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Guo, Tian Liang, and Kanjian Zhang. "Impulsive fractional partial differential equations." Appl. Math. Comput. 257 (2015): 581-590.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Karimov, Erkinjon Tulkinovich. "Frankl-type problem for a mixed type equation with the Caputo fractional derivative." Lobachevskii J. Math. 41, no. 9 (2020): 1829-1836.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Katugampola, Udita N. "New approach to a generalized fractional integral." Appl. Math. Comput. 218, no. 3 (2011): 860-865
##plugins.generic.googleScholarLinks.settings.viewInGS##
Katugampola, Udita N. "A new approach to generalized fractional derivatives." Bull. Math. Anal. Appl. 6, no. 4 (2014): 1-15.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Kilbas, Anatoly Aleksandrovich. "Partial fractional differential equations and some of their applications." Analysis (Munich) 30, no. 1 (2010): 35-66.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Kilbas, Anatoly Aleksandrovich, Hari Mohan Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Differential Equations. Vol. 204 of North-Holland Mathematics Studies. Amsterdam: Elsevier Science B.V, 2006.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Luchko, Yurii, and Rudolf Gorenflo. "Scale-invariant solutions of a partial differential equation of fractional order." Fract. Calc. Appl. Anal. 1, no. 1 (1998): 63-78.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Liu, Jun Yi, and Ming Yu Xu. "An exact solution to the moving boundary problem with fractional anomalous diffusion in drug release devices." ZAMM Z. Angew. Math. Mech. 84, no. 1 (2004): 22-28.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Mainardi, Francesco. "The time fractional diffusion-wave equation." Izv. Vyssh. Uchebn. Zaved. Radiofiz. 38, no. 1-2 (1995): 20-36; reprinted in Radiophys. and Quantum Electronics 38, no. 1-2 (1996): 13–24.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Marasi, Hamid Reza, Hojjat Afshari, and Chengbo Zhai. "Some existence and uniqueness results for nonlinear fractional partial differential equations." Rocky Mountain J. Math. 47, no. 2 (2017): 571-585.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Miller, Kenneth S., and Bertram Ross. An introduction to the fractional calculus and fractional differential equations New York: John Wiley & sons Inc., 1993.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Oliveira, Daniela dos Santos, and Edmundo Capelas de Oliveira. "On a Caputotype fractional derivative." Adv. Pure Appl. Math. 10, no. 2 (2019): 81-91.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Podlubny, Igor. Fractional differential equations, Vol. 198 of Mathematics in Science and Enginnering. San Diego: Academic Press, 1999. Cited on 50.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Titraoui, Maria, and Yacine Arioua. "Boundary value problem for nonlinear fractional differential equations involving Erdélyi-Kober derivative on unbounded domain." An. Univ. Craiova Ser. Mat. Inform. 48, no. 1 (2021): 18-31.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Wydawnictwo Naukowe Uniwersytetu Pedagogicznego
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
