Time Fractional Equation with Non-homogenous Dirichlet Boundary Conditions

Abstract

In this research, we discuss the construction of analytic solution of non-homogenous initial boundary value problem including PDEs of fractional order. Since non-homogenous initial boundary value problem involves Caputo fractional order derivative, it has classical initial and boundary conditions. By means of separation of variables method and the inner product defined on L^2 [0,l], 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 used in this study. Illustrative example presents the applicability and influence of separation of variables method on fractional mathematical problems.

Keywords

• Caputo fractional derivative
• Time-fractional diffusion equation
• Mittag-Leffler function
• Initial-boundary-value problems
• Spectral method

References

1. A. Demir, M. A. Bayrak and E. Ozbilge, “New approaches for the solution of space-time fractional Schrödinger equation,” Advances in Difference Equation, vol. 2020, no.133, 2020.
2. A. Demir and M. A. Bayrak, “A New Approach for the Solution of Space-TimeFractional Order Heat-Like Partial Differential Equations by Residual Power Series Method” Communications in Mathematics and Applications, vol. 10, no. 3, pp. 585–597, 2019.
3. 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 Physics, vol. 2019, Article ID 5602565, 2019.
4. A. Demir, M. A. Bayrak and E. Ozbilge, “An Approximate Solution of the Time-Fractional FisherEquation with Small Delay by Residual Power Series Method”, Mathematical Problems in Engineering, vol. 2018, Article ID 9471910, 2018.
5. S. Cetinkaya, A. Demir and H. Kodal Sevindir, “The analytic solution of initial boundary value problem including time-fractional diffusion equation,” Facta Universitatis Ser. Math. Inform, vol. 35, no. 1, pp. 243-252, 2020.
6. S. Cetinkaya, A. Demir, and H. Kodal Sevindir, “The analytic solution of sequential space-time fractional diffusion equation including periodic boundary conditions,” Journal of Mathematical Analysis, vol. 11, no.1, pp. 17-26, 2020.
7. S. Cetinkaya and A. Demir, “The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function,” Communications in Mathematics and Applications, vol. 10, no. 4, pp. 865-873, 2019.
8. S. Cetinkaya, A. Demir, and H. Kodal Sevindir, “The Analytic Solution of Initial Periodic Boundary Value Problem Including Sequential Time Fractional Diffusion Equation,” Communications in Mathematics and Applications, vol. 11, no. 1, pp. 173-179, 2020.

9. S. Cetinkaya and A. Demir, “Time Fractional Diffusion Equation with Periodic Boundary Conditions,” Konuralp Journal of Mathematics, vol. 8, no. 2, pp. 337-342, 2020.
10. A. A. Kilbas, H. M. Srivastava and J. J. Trujıllo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.
11. I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.