The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function
DOI:
https://doi.org/10.26713/cma.v10i4.1290Keywords:
Caputo fractional derivative, Space-fractional diffusion equation, Mittag-Leffler function, Initial-boundary-value problems, Spectral methodAbstract
In this research, we determine the analytic solution of initial boundary value problem including time-space fractional differential equation with Dirichlet boundary conditions in one dimension. By using separation 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 including fractional derivative in Caputo sense. A new inner product with weighted function is defined to obtain coefficients in the Fourier series.
Downloads
References
M. A. Bayrak and A. Demir, A new approach for space-time fractional partial differential equations by Residual power series method, Applied Mathematics and Computation 336 (2013), 215 – 230, DOI: 10.1016/j.amc.2018.04.032.
M. A. Bayrak and A. Demir, Inverse problem for determination of an unknown coefficient in the time fractional diffusion equation, Communications in Mathematics and Applications 9 (2018), 229 – 237, DOI: 10.26713/cma.v9i2.722.
A. Demir, M. A. Bayrak and E. Ozbilge, A new approach for the Approximate Analytical solution of space time fractional differential equations by the homotopy analysis method, Advances in Mathematical 2019 (2019), Article ID 5602565, DOI: 10.1155/2019/5602565.
A. Demir, S. Erman, B. Ozgur and E. Korkmaz, Analysis of fractional partial differential equations by Taylor series expansion, Boundary Value Problems 2013 (2013), 68, DOI: 10.1186/1687-2770-2013-68.
A. Demir, F. Kanca and E. Ozbilge, Numerical solution and distinguishability in time fractional parabolic equation, Boundary Value Problems 2015 (2015), 142, DOI: 10.1186/s13661-015-0405-6.
A. Demir and E. Ozbilge, Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions Boundary Value Problems 2014 (2014), 134, DOI: 10.1186/1687-2770-2014-134.
S. Erman and A. Demir, A novel approach for the stability analysis of state dependent differential equation, Communications in Mathematics and Applications 7 (2016), 105 – 113, DOI: 10.26713/cma.v7i2.373.
F. Huang and F. Liu, The time-fractional diffusion equation and fractional advection-dispersion equation, The ANZIAM Journal 46 (2005), 317 – 330, DOI: 10.1017/S1446181100008282.
Y. Luchko, Initial-boundary-value problems for the one dimensional time-fractional diffusion equation, Fractional Calculus and Applied Analysis 15 (2012), 141 – 160, DOI: 10.2478/s13540-012-0010-7.
Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, Journal of Mathematical Analysis and Applications 74(2) (2011), 538 – 548, DOI: 10.1016/j.jmaa.2010.08.048.
S. Momani ad Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons and Fractals 31(5) (2007), 1248 – 1255, DOI: 10.1016/j.chaos.2005.10.068.
B. Ozgur and A. Demir, Some stability charts of a neural field model of two neural populations, Communications in Mathematics and Applications 7 (2016), 159 – 166, DOI: 10.26713/cma.v7i2.481.
Å. PÅ‚ociniczak, Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Communications in Nonlinear Science and Numerical Simulation 24(1) (2015), 169 – 183, DOI: 10.1016/j.cnsns.2015.01.005.
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
J. J. Nieto and R. Rodríguez-López, Fractional Differential Equations Theory, Methods and Applications, MDPI, Basel (2019), DOI: 10.3390/books978-3-03921-733-5.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
