Application of the Combined Aboodh and Reduced Differential Transform Methods to the Fisher's Type Equations
A. A. Oyewumi *
Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
R. A. Oderinu
Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The Fishers' equation has relevant application in the studies of gas dynamics, uid dynamic models, applied physics and mathematics felds. This equation models the interaction between the convection effect, reaction mechanism and the diffusion transport. In this work the nonlinear time-dependent Fisher's type equations are solved. The combined Aboodh and Reduced differential transform methods are employed to solve two cases of the nonlinear partial differential equations. The solutions obtained by this method reveal that the proposed method is well capable of handling such problems. It is also evident from the results that the method is
highly effcient and accurate.
Keywords: Fisher's equation, partial differential equation, Aboodh transform and reduced differential transform method
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References
Fisher RA. The wave of advance of advantageous genes, Annals of Eu- genics. 1937;7:335-369.
Tan Y, Xu H, Liao SJ. Explicit series solution of travelling waves with a front of Fisher equation. Chaos, Solitons and Fractals. 2007;31(2):462- 472.
Ismail HNA, Raslan K, Rabboh AA. Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations, Applied mathemat- ics and Computational. 2004;159(1):291-301.
Wazwaz A. The extended tanh method for abundant solitary wave so- lutions of nonlinear wave equations. Applied Mathematics and Compu- tation. 2007;187(2):1131-1142.
Britton NF. Reaction diusion equations and their applications to bi- ology, Academic Press, London, UK; 1986.
Khattak AJ. A computational meshless method for the general- ized Burger's- Huxley equation, Applied Mathematics Modelling. 2009;33(9):3718-3729.
Frank DA. Diusion and Heat Exchange in Chemical Kinetics, Prince- ton University Press, Princeton, NJ, USA; 1955.
Canosa J. Diusion in nonlinear multiplicative media, Journal of Math- ematical Physics, 1969;10(10):1862-1868.
Malet W. Solitary wave solutions of nonlinear wave equations, Amer- ican Journal of Physics. 1992;60(7):6650-654.
Yokus A, Yavuz M. Novel comparison of numerical and analytical meth- ods for frectional Burger-Fisher equation, Discrete and Continous Dy- namical Systems series. 2021;14:2591-2606.
Li J. Geometric properties and exact travelling wave solutions for the generalized Burger-Fisher equation and the sharma-Tasso-olver equa- tion. Journal of Nonlinear Modeling and analysis. 2019;1:1-10.
Kapoor M, Joshi V. Uniform Algberaic Hyperbolic tension B-spline DQM for the solution of Fisher's Reaction-Diusion equation, Third International Conference in Applied Research in Engeneering and sci- ence Technology, Paris France; 2020.
Oyewumi and Oderinu; AJPAM, 4(3): 437-450, 2022; Article no.AJPAM.1091
Saad KM, Atangana A, Baleanu D. New fractional derivatives with non- singular kernel applied to the Burgers equation. Chaos. 2018;28:063109.
Bastani M, Salkuyeh DK. A highly accurate method to solve Fisher's equation, Pramana, 2012;78(3):335-346.
Gunzburger MD, Hou LS, Zhu W. Fully discrete nite element ap- proximations of the forced Fisher equation, Journal of Mathematical Analysis and Applications. 2016;313(2):419-440.
Mittal RC, Jain RK. Numerical solutions of nonlinear Fisher's reaction- diusion equation with modied cubic B-spline collocation method, Mathematical Sciences. 2013;7(12).
Dag I, Sahin A, Korkmaz A. Numerical investigation of the solution of Fisher's equation via B-spline garlekin method, Numerical Methods for Partial Dierential Equations. 2010;26(6):1483-1503.
Tan Y, Xu H, Liao SJ. Explicit series solution of travelling waves with a front of Fisher equation. Chaos, Solitons and Fractals. 2007;31(2):462- 472.
Roessler J, Hussner H. Numerical solution of the 1+ 2 dimensional Fisher's equation by nite elements and the Galerkin method, Mathe- matical and Computer Modelling. 1997;25(3):57-67.
Gurarslan G. Numerical modelling of linear and nonlinear diusion equations by compact nite dierence method. Applied Mathematics and Computation. 2010;216(8):2472-2478.
Chen XY. Numerical methods for the Burger-Fisher equation[M.S. the- sis], University of Aeronautics and Astronautics, Nanjing, China; 2007.
Mickens RE, Gumel AB. Construction and analysis of a non-standard - nite dierence scheme for the Burgers-Fisher equation, Journal of Sound and Vibration. 2002;257(4):791-797.
Mickens RE. Relation between the time and space step size in nonstand nite-dierence schemes for the Fisher equation, Numerical Methods for Partial Dierential equations, 1997;15(1):51-55.
Mickens RE. A best nite-dierence scheme for the Fisher equation, Numerical Methods for Partial Dierential equations. 1994;10(10):581- 585.
Parek N, Puri S. A new numerical scheme for the Fisher equation. Journal of Physics A, 1990;23(21):L1085-L1091.
Oyewumi and Oderinu; AJPAM, 4(3): 437-450, 2022; Article no.AJPAM.1091
Rizwan-Uddin R. Comparison of the nodal integral method and non standard nite-dierence scheme for Fisher equation, SIAM Journal on Scientic Computing. 2001;22(6):1926-1942.
Chernma R. Exact and numerical solutions of tie generalized sher equation, Reports on Mathematical Physics. 2001;47:393-411.
Gazdag J, canosa J. Numerical solutions of Fisher's equation, Journal of Applied Probability, 1974;11:445-457.
Zhao T, Li C, Zang Z, Wu Y. Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation, Applied mathe- matical Modelling. 2012;36(3):1046-1056.
Sahin A, Dag I, Saka B. A B-spline algorithm for the numerical solution of Fisher's equation. Kybernetes. 2008;37(2):326-342.
Keskin Y, Oturanc G. Reduced Dierential Transform Method for Par- tial Dierential Equations. International Journal of Nonlinear Sciences and Numerical Simulation. 2009;10 (6):741-749.
Chandraker V, Awasthi A, Jayaraj S. Numerical Treatment of Burger- Fisher equation, Global Colloquium in Recent Advancement and Eec- tual Researches in Engeneering, Science and Technology. 2016;25:1217- 1225.
Benattia ME, Belghaba K. Application of Aboodh Transform for solv- ing First order constant coecients complex Equation, General Letters in Mathematics. 2019;6(1):28-34.
Mohand M, Mahgoub A, Alshikh A. On The Relationship Between Aboodh Transform and New Integral Transform "ZZ Transform" Math- ematical Theory and Modelling. 2013;6(9).
Mahgob MA, Hassan Sedeeg AK. The Solution of Porous Medium Equation by Aboodh Homotopy Perturbation Method, American Jour- nal of Applied Mathematics. 2016;5(4):217-221.
Aboodh KS, Farah RA, Almardy IA, Osman AK. Solving delay dif- ferential equations by Aboodh transformation method. International Journal of Applied Mathematics and Statistical Sciences. 2018;7(1):21- 30.
Hassan Sedeeg AK, Mahgoub MA. Aboodh Transform Homotopy Per- turbation Method For Solving System Of Nonlinear Partial Dierential Equations. Mathematical Theory and Modeling. 2016;6(8). 448 Oyewumi and Oderinu; AJPAM, 4(3): 437-450, 2022; Article no.AJPAM.1091
Aboodh KS, Idris A, Nurudeen RI. On the Aboodh Transform Connec- tions with Some Famous Integral Transforms. International Journal of Engineering and Information Systems, 2017;1(9):143-151.
Keskin Y, Caglar I, Koc AB. Numerical solution of sine-gordon equation by reduced dierential transform method, In Proceedings of the World Congress on Engineering. 2011; 6{8.
Ibis B. Application of reduced dierential transformation method for solving Fourth-Order Parabolic Partial Dierential Equations, Journal of Mathematics and Computer Science. 2014;12:124-131.
K. Sulaiman, K.S. Aboodh, The New integral Transform Aboodh Transform, Global journal of Pure and Applied Mathematics. 2013;9 (1):35-43.
Mohand M, Mahgob A. Homotopy Perturbation Method and Aboodh Transform for Solving Sine {Gordon and Klein { Gordon Equations. International Journal of Engineering Sciences and Research Technology. 2016;10.
Brastos AG, Khaliq AQM. An exponential time dierencing method of lines for Burgers-Fisher and coupled Burgers equations, J.Comput. Appl. Math. 2019;182-197.
Chandraker V, Awasthi A, Jayaraj S. Numerical treatment of Burger- Fisher equation. Procedia Technology. 2016;25:1217{1225.
Hamzah DA, Tuwankotta JM, Soeharyadi Y. On the numerical solu- tion of Fisher's equation by iterative operator-splitting method. Neural, Parallel, and Scientic Computations. 2017;25:395-401
Sulaiman TA, Yavuz M, Bulut H, Baskonus HM. Investigation of the fractional coupled viscous Burgers equation involving Mittag-Leer kernel, Phys. A. 2019;527:121126.
Wang X, Lu Y. Exact solutions of the extended Burgers{Fisher equa- tion. Chinese Phys. Lett. 1990;7:145{147.
Wazwaz AM. The tanh method for generalized forms of nonlinear heat conduction and Burger{ Fisher equations, Appl. Math. Comput. 2005;321-338.
Zhou Y, Lou Q, Zhang W. Bounded traveling waves of the gen- eralized Burger Fisher equation. Int. J. of Bifurcation and Chaos. 2013;23(3):1350054, 1{11. 449 Oyewumi and Oderinu; AJPAM, 4(3): 437-450, 2022; Article no.AJPAM.1091
Xu Z, Xian D. Application of Exp-function method to generalized Burgers Fisher equation, Acta Math. Appl. Sinica, English Series. 2010;26(4):669{676.
Wazwaz AM, Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers - Huxley equations, Applied Mathematics and Computation. 2005;169:639{656.
Wang S, Tang X, Lou S. Soliton ssion and fusion: Burgers equa- tion and Sharma- Tasso - Olver equation. Chaos Solitons and Fractals. 2014;21:231{239.
Yokus A, Kaya D. Numerical and exact solutions for time fractional burgers equation. J. Nonlinear Sci. Appl. 2017;10:3419{3428.
Ismail HNA, Rabboh AA. A restrictive pad e approximation for the solution of the generalized Fisher and Burger-Fisher equations, Appl. Math. Comput. 2004;154:203{210.
Kaya D, El-Sayed S. A numerical simulation and explicit solutions of the generalized Burgers - Fisher equation, Appl. Math. Comput. 2004;152:403{413.