THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Aydin Secer

,

Melih Cinar

A JACOBI WAVELET COLLOCATION METHOD FOR FRACTIONAL FISHER'S EQUATION IN TIME

ABSTRACT
In this study, the Jacobi wavelet collocation method is studied to derive a solution of the time-fractional Fisher's equation in Caputo sense. Jacobi wavelets can be considered as a generalization of the wavelets since the Gegenbauer, and thus also Chebyshev and Legendre polynomials are a special type of the Jacobi polynomials. So, more accurate and fast convergence solutions can be possible for some kind of problems thanks to Jacobi wavelets. After applying the proposed method to the considered equation and discretizing the equation at the collocation points, an algebraic equation system is derived and solving the equation system is quite sim­ple rather than solving a non-linear PDE. The obtained values of our method are checked against the other numerical and analytic solution of considered equation in the literature and the results are visualized by using graphics and tables so as to reveal whether the method is effectiveness or not. The obtained results evince that the wavelet method is quite proper because of its simple algorithm, high accuracy and less CPU time for solving the considered equation.
KEYWORDS
time-fractional Fisher’s equation, collocation method, fractional differential equation, Jacobi wavelet
PAPER SUBMITTED: 2020-03-06
PAPER REVISED: 2020-05-23
PAPER ACCEPTED: 2020-06-01
PUBLISHED ONLINE: 2020-10-25
DOI REFERENCE: https://doi.org/10.2298/TSCI20S1119S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Supplement 1, PAGES [S119 - S129]
REFERENCES
  1. Zhang, Y. A Finite Difference Method for Fractional Partial Differential Equation, Applied Mathematics and Computation, 215 (2009), 2, pp. 524-529
  2. Li, C., Zeng, F., Finite Difference Methods for Fractional Differential Equations, Int. Jour. of Bifurcation and Chaos, 22 (2012), 4, 1230014
  3. Momani, S., Odibat, Z., Analytical Solution of a Time-Fractional Navier-Stokes Equation by Adomian Decomposition Method, Applied Mathematics and Computation, 177 (2006), 2, pp. 488-494
  4. Hu, Y., et al., Analytical Solution of the Linear Fractional Differential Equation by Adomian Decomposition Method, Journal of Computational and Applied Mathematics, 215 (2008), 1, pp. 220-229
  5. Abdulaziz, O., et al., Application of Homotopy-Perturbation Method to Fractional IVP, Journal of Computational and Applied Mathematics, 216 (2008), 2, pp. 574-584
  6. Hosseinnia, S. H., et al., Using an Enhanced Homotopy Perturbation Method in Fractional Differential Equations Via Deforming the Linear Part, Computers and Mathematics with Applications, 56 (2008), 12, pp. 3138-3149
  7. Podlubny, I., An Introduction Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, in: Fractional Derivatives and Integrals, In Fractional Differential Equations, New York, USA, 1st ed., Academic Press, 1998
  8. Jafari, H., et al., A New Approach for Solving a System of Fractional Partial Differential Equations, Computers and Mathematics with Applications, 66 (2013), 5, pp. 838-843
  9. Partohaghighi, M., et al., Ficitious Time Integration Method for Solving the Time Fractional Gas Dynamics Equation, Thermal Science, 23 (2019), 6, pp. 2009-2016
  10. Akgul, A., et al., New Method for Investigating the Density-Dependent Diffusion Nagumo Equation, Thermal Science, 22 (2018), Suppl. 1, pp. S143-S152
  11. Secer, A., Ozdemir, N., An Effective Computational Approach Based on Gegenbauer Wavelets for Solving the Time-Fractional Kdv-Burgers-Kuramoto Equation, Advances in Difference Equations, 2019 (2019), 386
  12. Zhang, X., et al., A Fully Discrete Local Discontinuous Galerkin Method for 1-D Time-Fractional Fisher's Equation, International Journal of Computer Mathematics, 91 (2014), 9, pp. 2021-2038
  13. Al Qurashi, M. M., et al., A New Iterative Algorithm on the Time-Fractional Fisher Equation: Residual Power Series Method, Advances in Mechanical Engineering, 9 (2017), 9, pp. 1-8.
  14. Demir, A., et al., An Approximate Solution of the Time-Fractional Fisher Equation with Small Delay by Residual Power Series Method, Mathematical Problems in Engineering, 2018 (2018), ID 9471910
  15. Zhang, X., Liu, J., An Analytic Study on Time-Fractional Fisher Equation Using Homotopy Perturbation Method, Walailak Journal of Science and Technology, 11 (2014), 11, pp. 975-985
  16. Vanani, S., Aminataei, A., On the Numerical Solution of Fractional Partial Differential Equations, Mathematical and Computational Applications, 17 (2012), 2, pp. 140-151
  17. Fisher, R. A., The Wave of Advance of Advantageous Genes, Annals of Human Genetics, 7 (1937), 4, pp. 299-398
  18. Lepik, U., Solving PDE with the Aid of 2-D Haar Wavelets, Computers and Mathematics with Applications, 61 (2011), 7, pp. 1873-1879
  19. Hariharan, G., et al., Haar Wavelet Method for Solving Fisher's Equation, Applied Mat. and Computation, 211 (2009), 2, pp. 284-292
  20. Rong, L. J., Chang, P., Jacobi Wavelet Operational Matrix of Fractional Integration for Solving Fractional Integro-Differential Equation, In Journal of Physics: Conference Series, 693 (2016), 012002
  21. Zaky, M. A., et al., A New Operational Matrix Based on Jacobi Wavelets for a Class of Variable-Order Fractional Differential Equations, Proceedings of the Romanian Academy Series A- Mathematics Physics Technical Sciences Information Science, 18 (2017), 4, pp. 315-322
  22. Abbassa, N., et al., Regularized Jacobi Wavelets Kernel for Support Vector Machines, Statistics, Optimization and Information Computing, 7 (2019), 4, pp. 669-685
  23. Chi-Hsu, W., On the Generalization of Block Pulse Operational Matrices for Fractional and Operational Calculus. Journal of the Franklin Institute, 315 (1983), 2, pp. 91-102
  24. Hale, J. K., Functional Differential Equations, Springer, New York, USA, 1971, pp. 9-22
  25. Moncayo, M., Yanez, R. J., An Application of the Jacobi Summability to the Wavelet Approximation, Integral Transforms and Special Functions, 19 (2008), 4, pp. 249-258
  26. Yang, Y., et al., Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations, Acta Mathematica Scientia, 34 (2014), 3, pp. 673-690
  27. Kilicman, A., Al Zhour, Z. A. A., Kronecker Operational Matrices for Fractional Calculus and Some Applications, Applied Mathematics and Computation, 187 (2007), 1, 250-265
  28. Singh, P., Sharma, D., Comparative Study of Homotopy Perturbation Transformation with Homotopy Perturbation Elzaki Transform Method for Solving Non-Linear Fractional PDE, Non-Linear Engineering, 9 (2020), 1, pp. 60-71
  29. Chandraker, V., et al., A Numerical Treatment of Fisher Equation, In Procedia Engineering, 127 (2015), Dec., pp. 1256-1262
PDF VERSION [DOWNLOAD]
A Jacobi wavelet collocation method for fractional fisher's equation in time

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence