THE ANALYTIC SOLUTION OF INITIAL BOUNDARY VALUE PROBLEM INCLUDING TIME FRACTIONAL DIFFUSION EQUATION

Süleyman Çetinkaya, Ali Demir, Hülya Kodal Sevindir

DOI Number
https://doi.org/10.22190/FUMI2001243C
First page
243
Last page
252

Abstract


The motivation of this study is to determine the analytic solution of initial boundary value problem including time fractional differential equation with Neumann boundary conditions in one dimension. By making use of seperation of variables, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem.


Keywords

Caputo fractional derivative, space-fractional diffusion equation, Mittag-Leffler function, initial-boundary-value problems, spectral method.

Full Text:

PDF

References


bibitem{1} {sc P. Agarwal, G. V. Milovanovic {rm and} S. K. Nisar}: textit{A Fractional Integral Operator Involving the Mittag-Leffler Type with Four Parameters}. Facta Universitatis, Series: Mathematics and Informatics {bf 30 5} (2015), 597--605.

bibitem{2} {sc M. A. Bayrak {rm and} A. Demir}: textit{A new approach for space-time fractional partial dierential equations by Residual power series method}. Appl. Math. And Comput. {bf 336} (2013), 215--230.

bibitem{3} {sc M. A. Bayrak {rm and} A. Demir}: textit{Inverse Problem for Determination of An Unknown Coefficient in the Time Fractional Diffusion Equation}. Communications in Mathematics and Applications {bf 9} (2018), 229--237.

bibitem{4} {sc A. Benlabbes, M. Benbachir {rm and} M. Lakrib}: textit{Boundary value problems for nonlinear fractional differential equations}. Facta Universitatis, Series: Mathematics and Informatics {bf 30 2} (2015), 157--168.

bibitem{5} {sc A. Demir, M. A. Bayrak {rm and} E. Ozbilge}: textit{A new approach for the Approximate Analytical solution of space time fractional differential equations by the homotopy analysis method}. Advances in mathematichal {bf 2019} (2019), articleID 5602565.

bibitem{6} {sc A. Demir, S. Erman, B. Özgür {rm and} E. Korkmaz}: textit{Analysis of fractional partial differential equations by Taylor series expansion}. Boundary Value Problems {bf 2013 68} (2013).

bibitem{7} {sc A. Demir, F. Kanca {rm and} E. Özbilge}: textit{Numerical solution and distinguishability in time fractional parabolic equation}. Boundary Value Problems {bf 2015 142} (2015).

bibitem{8} {sc A. Demir {rm and} E. Özbilge}: textit{Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions}. Boundary Value Problems {bf 2014 134} (2014).

bibitem{9} {sc S. Erman {rm and} A. Demir}: textit{A Novel Approach for the Stability Analysis of State Dependent Differential Equation}. Communications in Mathematics and Applications {bf 7} (2016), 105--113.

bibitem{10} {sc M. Houas {rm and} M. Bezziou}: textit{Existence and Stability Results for Fractional Differential Equations with Two Caputo Fractional Derivatives}. Facta Universitatis, Series: Mathematics and Informatics {bf 34 2} (2019), 341--357.

bibitem{11} {sc F. Huang {rm and} F. Liu}: textit{The time-fractional diffusion equation and fractional advection-dispersion equation}. The ANZIAM Journal {bf 46 3} (2005), 317--330.

bibitem{12} {sc Y. Luchko}: textit{Initial boundary value problems for the one dimensional time-fractional diffusion equation}. Fractional Calculus and Applied Analysis {bf 15} (2012), 141--160.

bibitem{13} {sc A. A. Kilbas, H. M. Srivastava {rm and} J. J. Trujillo}: textit{Theory and Applications of Fractional Differential Equations}. Elsevier, Amsterdam, 2006.

bibitem{14} {sc Y. Luchko}: textit{Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation}. Journal of Mathematical Analysis and Applications {bf 74 2} (2011), 538--548.

bibitem{15} {sc S. Momani {rm and} Z. Odibat}: textit{Numerical comparison of methods for solving linear differential equations of fractional order}. Chaos Solitons and Fractals {bf 31 5} (2007), 1248--1255.

bibitem{16} {sc B. Özgür {rm and} A. Demir}: textit{Some Stability Charts of A Neural Field Model of Two Neural Populations}. Communications in Mathematics and Applications {bf 7} (2016), 159--166.

bibitem{17} {sc L. Plociniczak}: textit{Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications}. Commun. Nonlinear Sci. Numer. Simul. {bf 24 1} (2015), 169--183.

bibitem{18} {sc K. V. Zhukovsky {rm and} H. M. Srivastava}: textit{Analytical solutions for heat diffusion beyond Fourier law}. Applied Mathematics and Computation {bf 293} (2017), 423--437.

bibitem{19} {sc L. Mahto, S. Abbas, M. Hafayed {rm and} H. M. Srivastava}: textit{Approximate Controllability of Sub-Diffusion Equation with Impulsive Conditio}. Mathematics {bf 7 190} (2019), 1--16.

bibitem{20} {sc X. J. Yang, H. M. Srivastava, D. F. M. Torres {rm and} A. Debbouche}: textit{General Fractional-order Anomalous Diffusion with Non-singuler Power-Law Kernel}. Thermal Science {bf 21 1} (2017), 1--9.

bibitem{21} {sc I. Podlubny}: textit{Fractional Differential Equations}. Academic Press, San Diego, 1999.




DOI: https://doi.org/10.22190/FUMI2001243C

Refbacks





© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)