The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function

Süleyman í‡etinkaya; Ali Demir

doi:10.26713/cma.v10i4.1290

Authors

Süleyman í‡etinkaya Department of Mathematics, Kocaeli University, Kocaeli
Ali Demir Department of Mathematics, Kocaeli University, Kocaeli

DOI:

https://doi.org/10.26713/cma.v10i4.1290

Keywords:

Caputo fractional derivative, Space-fractional diffusion equation, Mittag-Leffler function, Initial-boundary-value problems, Spectral method

Abstract

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.

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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.