Numerical Solution for Time-Fractional Murray Reaction-Diffusion Equations via Reduced Differential Transform Method

Abstract

Bu çalışmada mühendislik ve fen bilimlerinde ortaya çıkan zaman-kesirli diferansiyel denklemin yarı analitik ve sayısal çözümleri indirgenmiş diferansiyel dönüşüm metodu kullanılarak incelenmiştir. Öncelikle kesirli mertebeden türevlerin tanımı ve önemli özellikler verilmiştir. Daha sonra Caputo kesirli türev tanımı kullanılarak indirgenmiş diferansiyel metodu sunulmuştur. Son olarak, kesirli mertebeden Murray diferansiyel denkleminin yarı analitik ve sayısal çözümleri İndirgenmiş diferansiyel dönüşüm metodu kullanılarak elde edilmiştir. Elde edilen çözümler tablo ve grafik üzerinde gösterilerek karşılaştırılma yapılmıştır.

Keywords

• Time-fractional differential equations
• reduced differential transform methods
• murray equations
• caputo fractional derivative

References

1. Zhou J.K., (1986) Differential transformation and its applications for electrical circuits, Huazhong University Press. Wuhan, China
2. Adomian G., (1994) Solving frontier problems of physics: The Decomposition Method, Kluwer Academic Publishers, Boston, Usa
3. Chen C.K., Ho S.H., (1996) Application of differential transformation to eigenvalue problems, Applied Mathematics and Computation, 79(2-3), 173-188. https://doi.org/10.1016/0096-3003(95)00253-7.
4. He J.H., (1999) Variational iteration method-a kind of non-lineer analytical technique: some examples, International Journal of Non-Linear Mechanics, 34(4), 699-708. https://doi.org/10.1016/S0020-7462(98)00048-1.
5. Chen C.K., Ho S.H., (1999) Solving partial differential equations by two dimensional differential transform method, Applied Mathematics and Computation, 106(2-3), 171-179. https://doi.org/10.1016/S0096-3003(98)10115-7. Ayaz F., (2003) On the two-dimensional differential transform method, Applied Mathematics and computation, 143(2-3), 361-374. https://doi.org/10.1016/S0096-3003(02)00368-5.
6. Keskin Y., Oturanç G., (2009) Reduced differential transform method for partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 10(6), 741-749. https://doi.org/10.1515/IJNSNS.2009.10.6.741.
7. Gupta P.K., (2011) Approximate analytical solutions of fractional Benney-lin equation by reduced differential transform method and the homotopy perturbation method. Computers and Mathematics with Applications, 61(9), 2829-2842. https://doi.org/10.1016/j.camwa.2011.03.057.
8. Srivastava V.K., (2014) Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method, Egyptian Journal of Basic and Applied Sciences, 1(1):60-66. https://doi.org/10.1016/j.ejbas.2014.01.002.

9. Bhrawy A.H., Doha E.H., Abdelkawy M.A., Van Gorder R.A., (2016) Jacobi gauss lobatto collocation method for solving nonlinear reaction-diffusion equations subject to dirichlet boundary conditions, Applied Mathematical Modelling, 40(3), 1703-1716. https://doi.org/10.1016/j.apm.2015.09.009.
10. Murray J.D., (1977) Nonlinear differential equation models in biology, Clarendon Press, Oxford, England
11. Murray J.D., (1989) Mathematical biology, Springer, Berlin, Germany
12. Luckho Y., Gorenflo R., (1999) An Operational Method for Solving Fractional Differential Equations with The Caputo Derivatives, Acta Mathematica Vietnamica, Vol. 24, 207-233.
13. Oldham K.B., Spainer J., (1974) The Fractional calculus: theory and applications of differentiation and integration to arbitrary order, Academic Press, California, USA
14. Podlubny I., (1999) Fractional differential equations, Academic Press, San Diego, USA
15. Caputo M., (1976) Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal of The Royal Astronomical Society, 13(5), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.
16. Kılbas A.A., Srivastava H.M., Trujillo J.J., (2006) Theory and applications of fractional differential equations, Vol. 204, Elsevier, Amsterdam, Holland
17. Arshad M., Lu D., Wang J., (2017) (N+1)-dimensional fractional reduced differential transform method for fractional order partial differential equations, Commun Nonlinear Sci Numer Simulat, 48, 509-519. https://doi.org/10.1016/j.cnsns.2017.01.018.
18. Momani S., Odibat Z., Erturk V.S., (2007) Generalized differential transform method for solving a space and time- fractional diffusion-wave equation, Physics Letters A., 370(5-6), 379-387. https://doi.org/10.1016/j.physleta.2007.05.083.
19. Keskin Y., (2010) Kısmi türevli diferansiyel denklemlerin indirgenmiş diferansiyel dönüşüm yöntemiyle çözülmesi, Ph.D. Thesis, Selcuk University, Konya, TR
20. Srivastava V.K., Awasthi M.K., Tamsir M., (2013) RDTM solution of Caputo time fractional-order hyperbolic telegraph equation, AIP Advances, Vol.3., https://doi.org/10.1063/1.4799548.