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Year 2022, Volume: 12 Issue: 1, 435 - 445, 01.03.2022
https://doi.org/10.21597/jist.975119

Abstract

References

  • Canosa J, 1973. On a nonlinear diffusion equation describing population growth, IBM J Res Dev 17: 307–313.
  • Cattani C, Kudreyko A, 2008. Mutiscale Analysis of the Fisher Equation, ICCSA , Part I, Lecture Notes in Computer Science, Springer-Verlag, Berlin/Heidelberg, Vol. 5072: 1171–1180.
  • Dag I, Sahin A, Korkmaz A, 2010. Numerical investigation of the solution of Fisher’s equation via the B-spline Galerkin method. Numer Methods Partial Differ Equ 26(6): 1483–1503.
  • Dag I, Ersoy O, 2016. The exponential cubic B-spline algorithm for Fisher equation. Chaos Solitons Fractals 86: 101–106.
  • Dag I, 1994. Studies of B-spline finite elements, Ph.D. thesis, University College of North Wales, Bangor, Gwynedd.
  • Ersoy O, Dag I, 2015. The extended B-spline collocation method for numerical solutions of Fishers equation. AIP Conf Proc 1648: 370011.
  • Strang G. (1968) On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5: 506-517.
  • Gazdag J, Canosa J, 1974. Numerical solution of Fisher’s equation, J Appl Prob 11: 445–457.Geiser J, Bartecki K, 2008. Additive,multiplicative and iterative splitting methods for Maxwell equations, Algorithms andapplications, AIP Conf. Proc. vol. 1978 p. 470002.
  • Hundsdorfer W, Verwer J, 2003. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (First Edition), Springer-Verlag Berlin Heidelberg.
  • Kolmogoroff A, Petrovsky I, Piscounoff N, 1937. Study of the diffusion equation with growth of the quantity of matter and its application to biology problems, Bulletin de l’Université d’état à Moscou,Sére Internationale, Sec. A 1, 1–25.
  • Kapoor M, 2020. Solution of non-linear Fisher’s reaction-diffusion equation by using Hyperbolic B-spline based differential quadrature method Journal of Physics: Conference Series 1531 -012064 IOP Publishing doi:10.1088/1742-6596/1531/1/012064.
  • Madzvamuse A, 2006. Time stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J Comput Phys 214, 239–263.
  • Mittal R.C, Arora G. 2010. Efficient numerical solution of Fisher’s equation by using B-spline method Int. J. Comput. Math. 87 (13): 3039–51.
  • Mittal R.C, Jain R. (2012) Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions commun Nonlinear sci. Numer.Simulat 17: 4616-4625.
  • Mittal R.C, Jain R.K., (2013) Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B- spline collocation method Math. Sci. 7 (12): 1–10.
  • Qiu Y, Sloan D. M. (1998) Numerical solution of Fisher’s equation using amoving mesh method, J Comput Phys 146: 726–746.
  • Prenter P. M. (1975) Spline sandvariational methods, Wiley, New York.
  • Shukla H.S, Tamsir M. (2016) Extended modified cubic B-spline algorithm for nonlinear Fisher’s reaction-diffusion equation. Alexandria Engineering Journal 55(3): 2871-79.
  • Tamsir M, Srivastava V.K, Dhiman N. (2018) Chauhan, Numerical Computation of Nonlinear Fisher’s Reaction–Diffusion Equation with Exponential Modified Cubic B-Spline Differential Quadrature Method.Int. J. Appl. Comput. Math 4-6.
  • Zhao S, Wei G.W. (2003) Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation, SIAM J Sci Comput 25: 127–147.

Numerical Method for Approximate Solution of Fisher's Equation

Year 2022, Volume: 12 Issue: 1, 435 - 445, 01.03.2022
https://doi.org/10.21597/jist.975119

Abstract

In this paper, Fisher's reaction diffusion equation has been solved numerically by Strang splitting technique depending on collocation method with cubic B-spline. For our purpose, the initial and boundary value problem consisting of Fisher's equation is split into two sub-problems to be one linear and the other nonlinear such that each one contains the derivative in terms of time. Then, the whole problem is reduced to the algebraic equation system using finite element collocation method combined with the cubic B-spline for spatial discretization and the convenient classical finite difference approaches for time discretization. The effective and efficiency of the newly given method have been shown on the four examples. In addition, the newly obtained numerical results are shown in formats graphical profiles and tables to compare with studies available in the literature.

References

  • Canosa J, 1973. On a nonlinear diffusion equation describing population growth, IBM J Res Dev 17: 307–313.
  • Cattani C, Kudreyko A, 2008. Mutiscale Analysis of the Fisher Equation, ICCSA , Part I, Lecture Notes in Computer Science, Springer-Verlag, Berlin/Heidelberg, Vol. 5072: 1171–1180.
  • Dag I, Sahin A, Korkmaz A, 2010. Numerical investigation of the solution of Fisher’s equation via the B-spline Galerkin method. Numer Methods Partial Differ Equ 26(6): 1483–1503.
  • Dag I, Ersoy O, 2016. The exponential cubic B-spline algorithm for Fisher equation. Chaos Solitons Fractals 86: 101–106.
  • Dag I, 1994. Studies of B-spline finite elements, Ph.D. thesis, University College of North Wales, Bangor, Gwynedd.
  • Ersoy O, Dag I, 2015. The extended B-spline collocation method for numerical solutions of Fishers equation. AIP Conf Proc 1648: 370011.
  • Strang G. (1968) On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5: 506-517.
  • Gazdag J, Canosa J, 1974. Numerical solution of Fisher’s equation, J Appl Prob 11: 445–457.Geiser J, Bartecki K, 2008. Additive,multiplicative and iterative splitting methods for Maxwell equations, Algorithms andapplications, AIP Conf. Proc. vol. 1978 p. 470002.
  • Hundsdorfer W, Verwer J, 2003. Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (First Edition), Springer-Verlag Berlin Heidelberg.
  • Kolmogoroff A, Petrovsky I, Piscounoff N, 1937. Study of the diffusion equation with growth of the quantity of matter and its application to biology problems, Bulletin de l’Université d’état à Moscou,Sére Internationale, Sec. A 1, 1–25.
  • Kapoor M, 2020. Solution of non-linear Fisher’s reaction-diffusion equation by using Hyperbolic B-spline based differential quadrature method Journal of Physics: Conference Series 1531 -012064 IOP Publishing doi:10.1088/1742-6596/1531/1/012064.
  • Madzvamuse A, 2006. Time stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J Comput Phys 214, 239–263.
  • Mittal R.C, Arora G. 2010. Efficient numerical solution of Fisher’s equation by using B-spline method Int. J. Comput. Math. 87 (13): 3039–51.
  • Mittal R.C, Jain R. (2012) Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions commun Nonlinear sci. Numer.Simulat 17: 4616-4625.
  • Mittal R.C, Jain R.K., (2013) Numerical solutions of nonlinear Fisher’s reaction-diffusion equation with modified cubic B- spline collocation method Math. Sci. 7 (12): 1–10.
  • Qiu Y, Sloan D. M. (1998) Numerical solution of Fisher’s equation using amoving mesh method, J Comput Phys 146: 726–746.
  • Prenter P. M. (1975) Spline sandvariational methods, Wiley, New York.
  • Shukla H.S, Tamsir M. (2016) Extended modified cubic B-spline algorithm for nonlinear Fisher’s reaction-diffusion equation. Alexandria Engineering Journal 55(3): 2871-79.
  • Tamsir M, Srivastava V.K, Dhiman N. (2018) Chauhan, Numerical Computation of Nonlinear Fisher’s Reaction–Diffusion Equation with Exponential Modified Cubic B-Spline Differential Quadrature Method.Int. J. Appl. Comput. Math 4-6.
  • Zhao S, Wei G.W. (2003) Comparison of the discrete singular convolution and three other numerical schemes for solving Fisher’s equation, SIAM J Sci Comput 25: 127–147.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Matematik / Mathematics
Authors

Melike Karta 0000-0003-3412-4370

Publication Date March 1, 2022
Submission Date July 27, 2021
Acceptance Date October 15, 2021
Published in Issue Year 2022 Volume: 12 Issue: 1

Cite

APA Karta, M. (2022). Numerical Method for Approximate Solution of Fisher’s Equation. Journal of the Institute of Science and Technology, 12(1), 435-445. https://doi.org/10.21597/jist.975119
AMA Karta M. Numerical Method for Approximate Solution of Fisher’s Equation. J. Inst. Sci. and Tech. March 2022;12(1):435-445. doi:10.21597/jist.975119
Chicago Karta, Melike. “Numerical Method for Approximate Solution of Fisher’s Equation”. Journal of the Institute of Science and Technology 12, no. 1 (March 2022): 435-45. https://doi.org/10.21597/jist.975119.
EndNote Karta M (March 1, 2022) Numerical Method for Approximate Solution of Fisher’s Equation. Journal of the Institute of Science and Technology 12 1 435–445.
IEEE M. Karta, “Numerical Method for Approximate Solution of Fisher’s Equation”, J. Inst. Sci. and Tech., vol. 12, no. 1, pp. 435–445, 2022, doi: 10.21597/jist.975119.
ISNAD Karta, Melike. “Numerical Method for Approximate Solution of Fisher’s Equation”. Journal of the Institute of Science and Technology 12/1 (March 2022), 435-445. https://doi.org/10.21597/jist.975119.
JAMA Karta M. Numerical Method for Approximate Solution of Fisher’s Equation. J. Inst. Sci. and Tech. 2022;12:435–445.
MLA Karta, Melike. “Numerical Method for Approximate Solution of Fisher’s Equation”. Journal of the Institute of Science and Technology, vol. 12, no. 1, 2022, pp. 435-4, doi:10.21597/jist.975119.
Vancouver Karta M. Numerical Method for Approximate Solution of Fisher’s Equation. J. Inst. Sci. and Tech. 2022;12(1):435-4.

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