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On the Implicit Exponential Finite Difference Method for the Generalized Burgers-Fisher Equation

Year 2021, Volume: 2 Issue: 2, 83 - 100, 31.07.2021

Abstract

In this paper, we construct implicit exponential finite difference method to solve the Fisher’s, the Burgers-Fisher and the generalized Burgers-Fisher equations with specified initial and boundary conditions. We obtain plots of absolute errors vs x at some values of time t . We compare some of our numerical results with those obtained by other authors using methods such as Adomian Decomposition method, Exp-function method hybridized with Heuristic Computation and Optimal Homotopy Asymptotic
method by computing absolute errors at some values of space x and time t .

References

  • [1] Wang T.J., Generalized Laguerre spectral method for Fisher’s equation on a semi-infinite interval, International Journal of Computer Mathematics, 92, 1039-1052, 2015.
  • [2] Wang X.Y., Zhu Z.S., Lu Y.K., Solitary wave solutions of the generalized Burgers-Huxley equation, Journal of Physics A: Mathematical and General, 23, 271-274, 1990.
  • [3] Parekh N., Puri S., A new numerical scheme for the Fisher equation, Journal of Physics A: Mathematical and General, 23, 1085-1091, 1990.
  • [4] Tang S., Weber R.O., Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method, Journal of the Australian Mathematical Society Series B, 33, 27-38, 1991.
  • [5] Mavoungou T., Cherruault Y., Numerical study of Fisher’s equation by Adomian’s Method, Mathematical and Computer Modelling, 19, 89-95, 1994.
  • [6] Mickens R.E., A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffussion, Computers and Mathematics with Applications, 45, 429-436, 2003.
  • [7] Zhao S., Wei G.W., Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation, SIAM Journal on Scientific Computing, 25, 127-147, 2003.
  • [8] Ismail H.N.A., Rabboh A.A.A., A restricitve Padé approximation for the solution of the generalized Fisher and Burgers-Fisher equations, Applied Mathematics and Computation, 154, 203-210, 2004.
  • [9] Tan Y., Xu H., Liao S.J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos, Solitons and Fractals, 31, 462-472, 2007.
  • [10] Şahin A., Dağ İ., Saka B., A B-spline algorithm for the numerical solution of Fisher’s equation, Kybernetes, 37, 326-342, 2008.
  • [11] Babolian E., Saeidian J., Analytic approxiamte solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations, Communications in Nonlinear Science and Numerical Simulation, 14, 1984-1992, 2009.
  • [12] Mittal R.C., Arora G., Efficient numerical solution of Fisher’s equation by using B-spline method, Interanational Journal of Computer Mathematics, 87, 3039-3051, 2010.
  • [13] Zarebnia M., Jalili S., Application of spectral collocation method to a class of nonlinear PDEs, Communications in Numerical Analysis, 1-14, 2013.
  • [14] Mittal R.C., Jain R.K., Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B-spline collocation method, Mathematical Sciences, 7(12), 10 pages, 2013.
  • [15] Aghamohamadi M., Rashidinia J., Ezzati R., Tension spline method for solution non-linear Fisher equation, Applied Mathematics and Computation, 249, 399-407, 2014.
  • [16] Dag I., Ersoy O., The exponential cubic B-spline algorithm for Fisher equation, Chaos, Soliton and Fractals, 86, 101-106, 2016.
  • [17] Chandraker V., Awasthi A., Jayaraj S., A numerical treatment of Fisher Equation, Procedia Engineering, 127, 1256-1262, 2015.
  • [18] Bhattacharya M.C., An explicit conditionally stable finite difference equation for heat conduction problems, International Journal for Numerical Methods in Engineering, 21, 239-265, 1985.
  • [19] Bhattacharya M.C., Finite difference solutions of partial differential equations, International Journal for Numerical Methods in Biomedical Engineering, 6(3), 173-184, 1990.
  • [20] Handschuh R.F., Keith T.G., Applications of an exponential finite-difference technique, Numerical Heat Transfer, 22, 363-378, 1992.
  • [21] Bahadır A.R., Exponential finite-difference method applied to Korteweg-de Vries equation for small times, Applied Mathematics and Computation, 160, 675-682, 2005.
  • [22] İnan B., Bahadır A.R., Numerical solution of the one-dimensional Burgers equation: Implicit and fully implicit exponential finite difference methods, Pramana Journal of Physics, 81, 547-556, 2013.
  • [23] İnan B., Bahadır A.R., A numerical solution of the Burgers’ equation using a Crank-Nicolson exponential finite difference method, Journal of Mathematical and Computational Science, 4, 849-860, 2014.
  • [24] İnan B., Bahadır A.R., An explicit exponential finite difference method for the Burger’s equation, European International Journal of Science and Technology, 2, 61-72, 2013.
  • [25] İnan B., Bahadır A.R., Two different exponential finite difference methods for numerical solutions of the linearized Burgers’ equation, International Journal of Modern Mathematical Sciences, 13, 449-461, 2015.
  • [26] Ismail H.N.A., Raslan K., Rabboh A.A.A., Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Applied Mathematics and Computation, 159, 291-301, 2004.
  • [27] Nawaz R., Ullah H., Islam S. and Idrees M., Application of optimal homotopy asymptotic method to Burger equations, Journal of Applied Mathematics, http://dx.doi.org/10.1155/2013/387478, Article ID 387478, 8 pages, 2013.
  • [28] Mittal R.C., Triphathi A., Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic B-splines, International Journal of Computer Mathematics, 92, 1053-1077, 2015.
  • [29] Malik S.A., Qureshi I.M., Amir M., Malik A.N., Haq I., Numerical solution to generalized Burgers-Fisher equation using exp-function method hybridized with heuristic compuation, Plos One, Doi: 10.1371/journal.pone.0121728
  • [30] Agbavon K.M., Appadu A.R., Khumalo M., On the numerical solution of Fisher’s equation with coefficient of diffusion term much smaller than coefficient of reaction term, Advances in Difference Equations, 146, 33 pages, 2019.
  • [31] Jebreen H.B., On the numerical solution of Fisher’s equation by an e cient algorithm based on multiwavelets, AIMS Mathematics, 6(3), 2369-2384, 2020.
  • [32] Singh A., Dahiya S., Singh S.P., A fourth order spline collocation method for nonlinear Burgers-Fisher equation, Mathematical Sciences, 14, 75-85, 2020.
  • [33] Hussain M., Haq S., Numerical solutions of strongly nonlinear generalized Burgers-Fisher equation via meshfree spectral technique, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2020.1846729, 2020.
Year 2021, Volume: 2 Issue: 2, 83 - 100, 31.07.2021

Abstract

References

  • [1] Wang T.J., Generalized Laguerre spectral method for Fisher’s equation on a semi-infinite interval, International Journal of Computer Mathematics, 92, 1039-1052, 2015.
  • [2] Wang X.Y., Zhu Z.S., Lu Y.K., Solitary wave solutions of the generalized Burgers-Huxley equation, Journal of Physics A: Mathematical and General, 23, 271-274, 1990.
  • [3] Parekh N., Puri S., A new numerical scheme for the Fisher equation, Journal of Physics A: Mathematical and General, 23, 1085-1091, 1990.
  • [4] Tang S., Weber R.O., Numerical study of Fisher’s equation by a Petrov-Galerkin finite element method, Journal of the Australian Mathematical Society Series B, 33, 27-38, 1991.
  • [5] Mavoungou T., Cherruault Y., Numerical study of Fisher’s equation by Adomian’s Method, Mathematical and Computer Modelling, 19, 89-95, 1994.
  • [6] Mickens R.E., A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffussion, Computers and Mathematics with Applications, 45, 429-436, 2003.
  • [7] Zhao S., Wei G.W., Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation, SIAM Journal on Scientific Computing, 25, 127-147, 2003.
  • [8] Ismail H.N.A., Rabboh A.A.A., A restricitve Padé approximation for the solution of the generalized Fisher and Burgers-Fisher equations, Applied Mathematics and Computation, 154, 203-210, 2004.
  • [9] Tan Y., Xu H., Liao S.J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos, Solitons and Fractals, 31, 462-472, 2007.
  • [10] Şahin A., Dağ İ., Saka B., A B-spline algorithm for the numerical solution of Fisher’s equation, Kybernetes, 37, 326-342, 2008.
  • [11] Babolian E., Saeidian J., Analytic approxiamte solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations, Communications in Nonlinear Science and Numerical Simulation, 14, 1984-1992, 2009.
  • [12] Mittal R.C., Arora G., Efficient numerical solution of Fisher’s equation by using B-spline method, Interanational Journal of Computer Mathematics, 87, 3039-3051, 2010.
  • [13] Zarebnia M., Jalili S., Application of spectral collocation method to a class of nonlinear PDEs, Communications in Numerical Analysis, 1-14, 2013.
  • [14] Mittal R.C., Jain R.K., Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B-spline collocation method, Mathematical Sciences, 7(12), 10 pages, 2013.
  • [15] Aghamohamadi M., Rashidinia J., Ezzati R., Tension spline method for solution non-linear Fisher equation, Applied Mathematics and Computation, 249, 399-407, 2014.
  • [16] Dag I., Ersoy O., The exponential cubic B-spline algorithm for Fisher equation, Chaos, Soliton and Fractals, 86, 101-106, 2016.
  • [17] Chandraker V., Awasthi A., Jayaraj S., A numerical treatment of Fisher Equation, Procedia Engineering, 127, 1256-1262, 2015.
  • [18] Bhattacharya M.C., An explicit conditionally stable finite difference equation for heat conduction problems, International Journal for Numerical Methods in Engineering, 21, 239-265, 1985.
  • [19] Bhattacharya M.C., Finite difference solutions of partial differential equations, International Journal for Numerical Methods in Biomedical Engineering, 6(3), 173-184, 1990.
  • [20] Handschuh R.F., Keith T.G., Applications of an exponential finite-difference technique, Numerical Heat Transfer, 22, 363-378, 1992.
  • [21] Bahadır A.R., Exponential finite-difference method applied to Korteweg-de Vries equation for small times, Applied Mathematics and Computation, 160, 675-682, 2005.
  • [22] İnan B., Bahadır A.R., Numerical solution of the one-dimensional Burgers equation: Implicit and fully implicit exponential finite difference methods, Pramana Journal of Physics, 81, 547-556, 2013.
  • [23] İnan B., Bahadır A.R., A numerical solution of the Burgers’ equation using a Crank-Nicolson exponential finite difference method, Journal of Mathematical and Computational Science, 4, 849-860, 2014.
  • [24] İnan B., Bahadır A.R., An explicit exponential finite difference method for the Burger’s equation, European International Journal of Science and Technology, 2, 61-72, 2013.
  • [25] İnan B., Bahadır A.R., Two different exponential finite difference methods for numerical solutions of the linearized Burgers’ equation, International Journal of Modern Mathematical Sciences, 13, 449-461, 2015.
  • [26] Ismail H.N.A., Raslan K., Rabboh A.A.A., Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Applied Mathematics and Computation, 159, 291-301, 2004.
  • [27] Nawaz R., Ullah H., Islam S. and Idrees M., Application of optimal homotopy asymptotic method to Burger equations, Journal of Applied Mathematics, http://dx.doi.org/10.1155/2013/387478, Article ID 387478, 8 pages, 2013.
  • [28] Mittal R.C., Triphathi A., Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic B-splines, International Journal of Computer Mathematics, 92, 1053-1077, 2015.
  • [29] Malik S.A., Qureshi I.M., Amir M., Malik A.N., Haq I., Numerical solution to generalized Burgers-Fisher equation using exp-function method hybridized with heuristic compuation, Plos One, Doi: 10.1371/journal.pone.0121728
  • [30] Agbavon K.M., Appadu A.R., Khumalo M., On the numerical solution of Fisher’s equation with coefficient of diffusion term much smaller than coefficient of reaction term, Advances in Difference Equations, 146, 33 pages, 2019.
  • [31] Jebreen H.B., On the numerical solution of Fisher’s equation by an e cient algorithm based on multiwavelets, AIMS Mathematics, 6(3), 2369-2384, 2020.
  • [32] Singh A., Dahiya S., Singh S.P., A fourth order spline collocation method for nonlinear Burgers-Fisher equation, Mathematical Sciences, 14, 75-85, 2020.
  • [33] Hussain M., Haq S., Numerical solutions of strongly nonlinear generalized Burgers-Fisher equation via meshfree spectral technique, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2020.1846729, 2020.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Bilge Inan 0000-0002-6339-5172

Appanah Rao Appadu This is me 0000-0001-9783-9790

Publication Date July 31, 2021
Published in Issue Year 2021 Volume: 2 Issue: 2

Cite

19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.