Equation Including Local Fractional Derivative and Neumann Boundary Conditions

Abstract

The aim of this study to discuss the construction of the solution of fractional partial differential equations (FPDEs) with initial and boundary conditions. Since the homogenous initial boundary value problem involves local fractional-order derivative, it has classical initial and boundary conditions. By means of the separation of variables method (SVM) and the inner product on L^2\left[0,l\right], we construct the solution in this series form in terms of eigenfunctions of related Sturm-Liouville problem. An illustrative example presents the applicability and influence of the separation of variables method on fractional mathematical problems.

Keywords

• Local Fractional Derivative
• Neumann boundary conditions
• Spectral method
• Separation of variables

References

1. [1] Dumitru B., Arran F., Akgül A., 2020. On a Fractional Operator Combining Proportional and Classical Differintegrals. Mathematics, 8(360). doi:10.3390/math8030360
2. [2] Bisquert J., 2005. Interpretation of A Fractional Diffusion Equation with Nonconserved Probability Density in Terms of Experimental Systems with Trapping or Recombination. Physical Review E, 72. doi: 10.1103/PhysRevE.72.011109
3. [3] Ndolane S., 2019. Solutions of Fractional Diffusion Equations and Cattaneo-Hristov Diffusion Model. International Journal of Analysis and Applications, 17(2), pp. 191-207. doi: 10.28924/2291-8639-17-2019-191
4. [4] Aguilar J. F. G., Hernández M. M., 2014. Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative. Abstract and Applied Analysis. 2014 doi: 10.1155/2014/283019
5. [5] Naber M., 2004. Distributed order fractional sub-diffusion. Fractals, 12(1), pp. 23-32. doi: 10.1142/S0218348X04002410
6. [6] Nadal E., Abisset C. E., Cueto E., Chinesta F., 2018. On the Physical Interpretation of Fractional Diffusion. Comptes Rendus Mecanique, 346, pp. 581-589. doi: 10.1016/j.crme.2018.04.004
7. [7] Zhang W., Yi M., 2016. Sturm-Liouville Problem and Numerical Method of Fractional Diffusion Equation on Fractals. Advances in Difference Equations, 2016:217. doi: 10.1186/s13662-016-0945-9
8. [8] Cetinkaya S., Demir A., Kodal Sevindir H., 2020. The Analytic Solution of Initial Boundary Value Problem Including Time-fractional Diffusion Equation. Facta Universitatis Ser. Math. Inform, 35(1), pp. 243-252.

9. [9] Cetinkaya S., Demir A., Kodal Sevindir H., 2020. The Analytic Solution of Sequential Space-time Fractional Diffusion Equation Including Periodic Boundary Conditions. Journal of Mathematical Analysis, 11(1), pp. 17-26.
10. [10] Cetinkaya S., Demir A., 2019. The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function. Communications in Mathematics and Applications, 10(4), pp. 865-873.
11. [11] Cetinkaya S., Demir A., Kodal Sevindir H., 2020. The Analytic Solution of Initial Periodic Boundary Value Problem Including Sequential Time Fractional Diffusion Equation. Communications in Mathematics and Applications, 11(1), pp. 173-179.
12. [12] Cetinkaya S., Demir A., Time Fractional Equation Including Non-homogenous Dirichlet Boundary Conditions. Sakarya University Journal of Science (Accepted Paper).
13. [13] Cetinkaya S., Demir A., Sequential Space Fractional Diffusion Equation's solutions via New Inner Product. Asian-European Journal of Mathematics (Accepted Paper). doi: 10.1142/S1793557121501217
14. [14] Cetinkaya S., Demir A., Time Fractional Diffusion Equation with Periodic Boundary Conditions. Konuralp Journal of Mathematics, 8(2), pp. 337-342.