Inverse coefficient problem by fractional Taylor series method

Mine Aylin Bayrak, Ali Demir

Abstract


This study focus on determining the unknown function of time or space in space-time fractional differential equation by fractional Taylor series method. A significant advantage of this method is that over-measured data is not used unlike most inverse problems.

This advantage allows us to determine the unknown function with less error. The presented examples illustrate that the obtained solutions are in a high agreement with the exact solutions of the corresponding inverse problems.


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References


A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier San Diego, CA, USA, 2006.

I. Podlubny, Fractional Differential Equations. Academic Press New York, NY, USA, 1999.

J. Sabatier, O.P. Agarwal and J.A.T. Machado(eds.), Advances in fractional calculus: theoretical developments and applications in physics and engineering, Dordrecht: Springer, 2007.

S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives theory and applications, Amsterdam: Gordon and Breach, 1993.

Z. Odibat, Approximations of fractional integrals and Caputo fractional derivatives, Appl. Math. Comput. 178(2006), 527-533. https://doi.org/10.1016/j.jksus.2017.12.017.

S. Momani and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006), 271-279. https://doi.org/10.1016/j.physleta.2006.02.048.

K. Seki, M. Wojcik and M.Tachiya, Fractional reaction-diffusion equation, J. Chem. Phys. 119(2003), 2165. https://doi.org/10.1063/1.1587126.

R.K. Saxena, A.M. Mathai and H.J. Haubold, Fractional reaction–diffusion equations, Astrophys. Space Sci. 305 (2006), 289-296 DOI 10.1007/s10509-006-9189-6.

S. Abbasbandy, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Lett. A 360 (2006), 109-113. https://doi.org/10.1016/j.physleta.2006.07.065.

A. Atangana, On the new fractional derivative and application to nonlinear Fisher-reaction- diffusion equation, Appl. Math. Comput. 273 (2016), 948-956. https://doi.org/10.1016/j.amc.2015.10.021.

A. El-Ajou, O. Abu Arqub, S. Momani, D. Baleanu and A. Alsaedi, A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. 257(2015), 119-133. https://doi.org/10.1016/j.amc.2014.12.121.

M.A. Bayrak and A. Demir, A new approach for space-time fractional partial differential equations by residual power series method, Appl. Math. Comput. 336(2018), 215-230. https://doi.org/10.1016/j.amc.2018.04.032.

X. Xiangtuan G. Hongbo and L. Xiaohong , An inverse problem for a fractional diffusion equation,Journal of Computational and Applied Mathematics 236 (2012), 4474–4484. https://doi.org/10.1016/j.cam.2012.04.019.

I.I. Mansur and C. Muhammed , Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Applied Mathematical Modelling 40 4 (2016), 891-899. https://doi.org/10.1016/j.apm.2015.12.020.

L. Songshu and F. Lixin , An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation, Mathematical Problems in Engineering 2020 (2020), Article ID 5865971, 13 pages.https://doi.org/10.1016/j.apm.2015.12.020.

L. Zhiyuan C. Xing and L. Gongsheng, An inverse problem in time-fractional diffusion equations with nonlinear boundary condition, Journal of Mathematical Physics 60 (2019), 091502. DOI:10.1063/1.5047074.

I. Jaradat, M. Alquran and R. Abdel-Muhsen, An analytical framework of 2D diffusion, wavelike, telegraph and Burgers models with twofold Caputo derivatives ordering, Nonlinear Dynamics,

(4) (2018), 1911-1922. DOI:10.1007/s11071-018-4297-8.




DOI: https://doi.org/10.52846/ami.v50i2.1677