On the Solution of Mathematical Model Including Space-Time Fractional Diffusion Equation in Conformable Derivative, Via Weighted Inner Product

Öz

This research aims to accomplish an analytic solution to mathematical models involving space-time fractional differential equations in the conformable sense in series form through the weighted inner product and separation of variables method. The main advantage of this method is that various linear problems of any kind of differential equations can be solved by using this method. First, the corresponding eigenfunctions are established by solving the Sturm-Liouville eigenvalue problem. Secondly, the coefficients of the eigenfunctions are determined by employing weighted inner product and initial condition. Thirdly, the analytic solution to the problem is constructed in the series form. Finally, an illustrative example is presented to show how this method is implemented for fractional problems and exhibit its effectiveness and accuracy.

Anahtar Kelimeler

• Conformable fractional derivative
• Initial-boundary-value problems
• Space-time fractional differential equation
• Spectral method

Kaynakça

1. [1] Cetinkaya S., Demir A., 2021. Numerical Solutions of Nonlinear Fractional Differential Equations via Laplace Transform. Facta Universitatis Ser. Math. Inform, 36(2), pp. 249-257.
2. [2] Cetinkaya S., Demir A., Baleanu D., 2021. Analysis of Fractional Fokker-Planck Equation with Caputo and Caputo-Fabrizio derivatives. Annals of the University of Craiova, Mathematics and Computer Science Series, 48(2), pp. 334-348.
3. [3] Cetinkaya S., Demir A., 2021. On the Solution of Bratu’s Initial Value Problem in the Liouville-Caputo Sense by ARA Transform and Decomposition Method. Comptes rendus de l'Academie bulgare des Sciences, 74(12), pp. 1729-1738.
4. [4] Kodal Sevindir H., Cetinkaya S., Demir A., 2021. On Effects of a New Method for Fractional Initial Value Problems. Advances in Mathematical Physics, 2021, Article ID 7606442.
5. [5] Cetinkaya S., Demir A., Kodal Sevindir, H., 2021. Solution of Space-Time Fractional Problem by Shehu Variational Iteration Method. Advances in Mathematical Physics, 2021, Article ID 5528928.
6. [6] Cetinkaya S., Demir A., 2021. On Solutions of Hybrid Time Fractional Heat Problem. Bulletin of the Institute of Mathematics Academia Sinica New Series, 16(1), pp. 49-62.
7. [7] Podlubny I., 1999. Fractional Differential Equations, Academic Press.
8. [8] Kilbas A. A., Srivastava H. M., Trujillo J. J., 2006. Theory and Applications of Fractional Differential Equations, Elsevier.

9. [9] Miller K. S., 1993. An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons.
10. [10] Oldham K., Spanier J., 1974. The Fractional Calculus, Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press.
11. [11] Gao W., Günerhan H., Başkonuş H.M., 2020. Analytical and approximate solutions of an epidemic system of HIV/AIDS transmission. Alexandria Engineering Journal, 59(5), pp. 3197-3211.
12. [12] Srivastava H.M., Günerhan H., 2019. Analytical and approximate solutions of fractional-order susceptible-infected-recovered epidemic model of childhood disease. Mathematical Methods in the Applied Sciences, 42, pp. 935–941.
13. [13] Dutta H., Günerhan H., Ali K.K, Yilmazer R., 2020. Exact Soliton Solutions to the Cubic-Quartic Non-linear Schrödinger Equation With Conformable Derivative. Frontiers in Physics, 8, pp. 1-7.
14. [14] Khalil R., Al Horani M., Yousef A., Sababheh M., 2014. A new definition of fractional derivative. J. Comput. Appl. Math., 264, pp. 65-70.
15. [15] Abdeljawad T., On conformable fractional calculus. J. Comput. Appl. Math., 279, pp. 57-66.
16. [16] Abu Hammad M., Khalil R., 2014. Conformable fractional heat differential equation. International Journal of Pure and Applied Mathematics, 94(2), pp. 215-221.