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On the stability and convergence of an implicit logarithmic scheme for diffusion equations with nonlinear reaction

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Abstract

In this work, we investigate numerically a diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. The theoretical analysis of the numerical model is carried out also. In particular, we establish that the method is a consistent technique, we provide a priori bounds for the numerical solutions, and we prove the stability and the convergence of the scheme by applying a suitable Gronwall-type inequality. As one of the consequences of stability, we show not only that the solutions of the numerical model exist, but also that they are unique.

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References

  1. K. Al-Khaled, Numerical study of Fisher’s reaction–diffusion equation by the Sinc collocation method. J. Comput. Appl. Math. 137(2), 245–255 (2001)

    Google Scholar 

  2. K. Baba, M. Tabata, On a conservation upwind finite element scheme for convective diffusion equations. RAIRO-Analyse Numérique 15(1), 3–25 (1981)

    Google Scholar 

  3. C. Bo-Kui, M. Song-Qiang, W. Bing-Hong, Exact solutions of generalized Burgers–Fisher equation with variable coefficients. Commun. Theor. Phys. 53(3), 443 (2010)

    Google Scholar 

  4. G. Carey, Y. Shen, Least-squares finite element approximation of Fisher’s reaction–diffusion equation. Numer. Methods Partial Differ. Equ. 11(2), 175–186 (1995)

    Google Scholar 

  5. P. Das, S. Rana, H. Ramos, Homotopy perturbation method for solving caputo type fractional order Volterra–Fredholm integro-differential equations. Comput. Math. Methods 1, e1047 (2019)

    Google Scholar 

  6. X. Deng, Travelling wave solutions for the generalized Burgers–Huxley equation. Appl. Math. Comput. 204(2), 733–737 (2008)

    Google Scholar 

  7. S.A. El Morsy, M.S. El-Azab, Logarithmic finite difference method applied to KdVB equation. Am. Acad. Sch. Res. J. (AASRJ) 4(2), 41–48 (2012)

    Google Scholar 

  8. V. Ervin, J. Macías-Díaz, J. Ruiz-Ramírez, A positive and bounded finite element approximation of the generalized Burgers–Huxley equation. J. Math. Anal. Appl. 424(2), 1143–1160 (2015)

    Google Scholar 

  9. R.A. Fisher, The wave of advance of advantageous genes. Ann. Eug. 7(4), 355–369 (1937)

    Google Scholar 

  10. D. Furihata, Finite-difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134(1), 37–57 (2001)

    Google Scholar 

  11. D. Furihata, T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations (CRC Press, New York, 2010)

    Google Scholar 

  12. G. Hariharan, K. Kannan, K. Sharma, Haar wavelet method for solving Fisher’s equation. Appl. Math. Comput. 211(2), 284–292 (2009)

    Google Scholar 

  13. A.L. Hodgkin, A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952)

    CAS  PubMed  PubMed Central  Google Scholar 

  14. B. İnan, A logarithmic finite difference technique for numerical solution of the generalized Huxley equation, in Book of Abstracts, 7th International Eurasian Conference on Mathematical Sciences and Applications, Kiev, Ukraine (2018), pp. 100–101

  15. E.W. Jenkins, C. Paribello, N.E. Wilson, Discrete mass conservation for porous media saturated flow. Numer. Methods Partial Differ. Equ. 30(2), 625–640 (2014)

    Google Scholar 

  16. A.N. Kolmogorov, Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Moskow, Ser. Internat. Sec. A 1, 1–25 (1937)

    Google Scholar 

  17. J.A. López-Campos, A. Segade, E. Casarejos, J.R. Fernández, Behavior characterization of viscoelastic materials for the finite element method calculation applying Prony series. Comput. Math. Methods 1(1), e1014 (2019)

    Google Scholar 

  18. J. Macías-Díaz, A. Puri, On the propagation of binary signals in damped mechanical systems of oscillators. Physica D 228(2), 112–121 (2007)

    Google Scholar 

  19. J.E. Macías-Díaz, Numerical study of the transmission of energy in discrete arrays of sine-Gordon equations in two space dimensions. Phys. Rev. E 77(1), 016,602 (2008)

    Google Scholar 

  20. J.E. Macías-Díaz, Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping. Comput. Phys. Commun. 182(12), 2471–2478 (2011)

    Google Scholar 

  21. J.E. Macías-Díaz, A Mickens-type monotone discretization for bounded travelling-wave solutions of a Burgers–Fisher partial differential equation. J. Differ. Equ. Appl. 19(11), 1907–1920 (2013)

    Google Scholar 

  22. J.E. Macías-Díaz, Conciliating efficiency and dynamical consistency in the simulation of the effects of proliferation and motility of transforming growth factor \(\beta\) on cancer cells. Commun. Nonlinear Sci. Numer. Simul. 40, 173–188 (2016)

    Google Scholar 

  23. J.E. Macías-Díaz, A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives. J. Comput. Phys. 351, 40–58 (2017)

    Google Scholar 

  24. J.E. Macías-Díaz, Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines. Commun. Nonlinear Sci. Numer. Simul. 53, 31–43 (2017)

    Google Scholar 

  25. J.E. Macías-Díaz, Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains. Commun. Nonlinear Sci. Numer. Simul. 55, 248–264 (2018)

    Google Scholar 

  26. J.E. Macías-Díaz, On the numerical and structural properties of a logarithmic scheme for diffusion–reaction equations. Appl. Numer. Math. 140, 104–114 (2019)

    Google Scholar 

  27. J.E. Macías-Díaz, F.J. Avelar-González, R.S. Landry, On an efficient implementation and mass boundedness conditions for a discrete Dirichlet problem associated with a nonlinear system of singular partial differential equations. J. Differ. Equ. Appl. 21(11), 1021–1043 (2015)

    Google Scholar 

  28. J.E. Macías-Díaz, A. González, A convergent and dynamically consistent finite-difference method to approximate the positive and bounded solutions of the classical Burgers–Fisher equation. J. Comput. Appl. Math. 318, 604–615 (2017)

    Google Scholar 

  29. J.E. Macías-Díaz, A.S. Hendy, R.H. De Staelen, A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations. Appl. Math. Comput. 325, 1–14 (2018)

    Google Scholar 

  30. J.E. Macías-Díaz, A.S. Hendy, R.H. De Staelen, A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives. Comput. Phys. Commun. 224, 98–107 (2018)

    Google Scholar 

  31. J.E. Macías-Díaz, B. İnan et al., Structural and numerical analysis of an implicit logarithmic scheme for diffusion equations with nonlinear reaction. Int. J. Mod. Phys. C (IJMPC) 30(09), 1–14 (2019)

    Google Scholar 

  32. J.E. Macías-Díaz, I.E. Medina-Ramírez, A. Puri, Numerical treatment of the spherically symmetric solutions of a generalized Fisher–Kolmogorov–Petrovsky–Piscounov equation. J. Comput. Appl. Math. 231(2), 851–868 (2009)

    Google Scholar 

  33. J.E. Macías-Díaz, A. Puri, A boundedness-preserving finite-difference scheme for a damped nonlinear wave equation. Appl. Numer. Math. 60(9), 934–948 (2010)

    Google Scholar 

  34. J.E. Macías-Díaz, A. Puri, An explicit positivity-preserving finite-difference scheme for the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. Appl. Math. Comput. 218(9), 5829–5837 (2012)

    Google Scholar 

  35. F. Martínez, I. Martínez-Vidal, S. Paredes, Conformable Euler’s Theorem on homogeneous functions. Comput. Math. Methods 1, e1048 (2019)

    Google Scholar 

  36. T. Matsuo, D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys. 171(2), 425–447 (2001)

    Google Scholar 

  37. R.E. Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 11, 645–653 (2005)

    Google Scholar 

  38. R.E. Mickens, Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 11(7), 645–653 (2005)

    Google Scholar 

  39. R. Mittal, R. Jiwari, Numerical study of Fisher’s equation by using differential quadrature method. Int. J. Inf. Syst. Sci. 5(1), 143–160 (2009)

    Google Scholar 

  40. R.C. Mittal, R.K. Jain, Numerical solutions of nonlinear Fisher’s reaction–diffusion equation with modified cubic B-spline collocation method. Math. Sci. 7(1), 12 (2013)

    Google Scholar 

  41. M.D. Morales-Hernández, I.E. Medina-Ramírez, F.J. Avelar-González, J.E. Macías-Díaz, An efficient recursive algorithm in the computational simulation of the bounded growth of biological films. Int. J. Comput. Methods 9(04), 1250,050 (2012)

    Google Scholar 

  42. D. Prakasha, P. Veeresha, H.M. Baskonus, Two novel computational techniques for fractional Gardner and Cahn–Hilliard equations. Comput. Math. Methods 1(2), e1021 (2019)

    Google Scholar 

  43. H. Ramos, Development of a new Runge–Kutta method and its economical implementation. Comput. Math. Methods 1(2), e1016 (2019)

    Google Scholar 

  44. M. Ran, C. Zhang, A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator. Int. J. Comput. Math. 93(7), 1103–1118 (2016)

    Google Scholar 

  45. J. Ruiz-Ramírez, J.E. Macías-Díaz, A finite-difference scheme to approximate non-negative and bounded solutions of a FitzHugh–Nagumo equation. Int. J. Comput. Math. 88(15), 3186–3201 (2011)

    Google Scholar 

  46. R. Santos, J.P. Martins, M. Felgueiras, S. Ferreira, Revisiting random polygonal lines iteratively generated on the plane. Comput. Math. Methods 1(4), e1036 (2019)

    Google Scholar 

  47. W. Tang, J. Zhang, Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)

    Google Scholar 

  48. T. Wang, B. Guo, Analysis of some finite difference schemes for two-dimensional Ginzburg–Landau equation. Numer. Methods Partial Differ. Equ. 27(5), 1340–1363 (2011)

    Google Scholar 

  49. X. Wang, Z. Zhu, Y. Lu, Solitary wave solutions of the generalised Burgers–Huxley equation. J. Phys. A Math. Gen. 23(3), 271 (1990)

    Google Scholar 

  50. B. Wongsaijai, T. Mouktonglang, N. Sukantamala, K. Poochinapan, Compact structure-preserving approach to solitary wave in shallow water modeled by the Rosenau-RLW equation. Appl. Math. Comput. 340, 84–100 (2019)

    Google Scholar 

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Acknowledgements

The corresponding author wishes to thank the anonymous reviewers for their comments and criticisms. All of their comments were taken into account in the revised version of the paper, resulting in a substantial improvement with respect to the original submission. The first author wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through the Grant A1-S-45928.

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Author notes

  1. Ahmed S. Hendy

    Present address: Department of Computational Mathematics and Computer Science, Institute of Natural sciences and Mathematics, Ural Federal University, ul. Mira. 19, Yekaterinburg, Russia, 620002

Authors and Affiliations

  1. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, 20131, Aguascalientes, AGS, Mexico

    Jorge E. Macías-Díaz

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    Ahmed S. Hendy

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  1. Jorge E. Macías-Díaz
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Macías-Díaz, J.E., Hendy, A.S. On the stability and convergence of an implicit logarithmic scheme for diffusion equations with nonlinear reaction. J Math Chem 58, 735–749 (2020). https://doi.org/10.1007/s10910-020-01103-8

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  • DOI: https://doi.org/10.1007/s10910-020-01103-8

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