Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines

Aslam Khan; Abdul Ghafoor; Emel Khan; Kamal Shah; Thabet Abdeljawad; Aslam Khan; Abdul Ghafoor; Emel Khan; Kamal Shah; Thabet Abdeljawad

doi:10.3934/nhm.2024028

Networks and Heterogeneous Media

2024, Volume 19, Issue 2: 634-654. doi: 10.3934/nhm.2024028

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Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines

1.
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Mathematics, University of Malakand, Chakdara, Dir (L), KP, Pakistan
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefaco Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 06 February 2024 Revised: 24 May 2024 Accepted: 11 June 2024 Published: 24 June 2024

This paper aims to conduct the numerical solutions of the scalar reaction diffusion model with cubic non-linearity having constant and time-dependent coefficients. The proposed method is hybrid in nature in which Haar wavelets are used to approximate the spatial derivatives and the Runge-Kutta (RK) routines are used to solve the resultant system of ordinary differential equations. We illustrate the applicability and efficiency of the proposed method by computing

$L_2$ ,

$L_{\infty}$ , and

$L_{rms}$ error estimates for various test models. The numerical accuracy and stability of the Haar wavelet-based method of lines for solving the scaler reaction-diffusion model provides further insight into the use of this scheme for model equations across various disciplines.

Keywords:

Haar wavelet and its integrals,
Runge-Kutta method,
generalized FitzHuge-Naguma reaction diffusion models

Citation: Aslam Khan, Abdul Ghafoor, Emel Khan, Kamal Shah, Thabet Abdeljawad. Solving scalar reaction diffusion equations with cubic non-linearity having time-dependent coefficients by the wavelet method of lines[J]. Networks and Heterogeneous Media, 2024, 19(2): 634-654. doi: 10.3934/nhm.2024028

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Abstract

$L_2$ ,

$L_{\infty}$ , and

References

[1]	N. F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, London: Academic Press, 1986.
[2]	R. S. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2004. https://doi.org/10.1002/0470871296
[3]	J. Smoller, Shock Waves and Reaction–-Diffusion Equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4612-0873-0
[4]	J. D. Murray, Animal coat patterns and other practical applications of reaction diffusion mechanisms, Mathematical Biology II: Spatial Models and Biomedical Applications, New York: Springer, 3 (2003), 141–191. https://doi.org/10.1007/b98869
[5]	M. G. Neubert, H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613–1628. https://doi.org/10.1890/0012-9658(2000)081[1613:DADCAS]2.0.CO;2 doi: 10.1890/0012-9658(2000)081[1613:DADCAS]2.0.CO;2
[6]	J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev., 17 (1973), 307–313. https://doi.org/10.1147/rd.174.0307 doi: 10.1147/rd.174.0307
[7]	R. Luther, Propagation of chemical reactions in space, J. Chem. Educ., 64 (1987), 740. https://doi.org/10.1021/ed064p740 doi: 10.1021/ed064p740
[8]	M. A. J. Chaplain, Reaction-diffusion prepatterning and its potential role in tumour invasion, J. Biol. Syst., 3 (1995), 929–936. https://doi.org/10.1142/S0218339095000824 doi: 10.1142/S0218339095000824
[9]	J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189–192. https://doi.org/10.1126/science.261.5118.189 doi: 10.1126/science.261.5118.189
[10]	M. M. Tang, P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980), 69–77.
[11]	E. Bodenschatz, W. Pesch, G. Ahlers, Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluid Mech., 32 (2000), 709–778. https://doi.org/10.1146/annurev.fluid.32.1.709 doi: 10.1146/annurev.fluid.32.1.709
[12]	Y. B. Zeldovich, D. A. Frank-Kamenetskii, The theory of thermal propagation of flames, Zh. Fiz. Khim, 12 (1938), 100–105.
[13]	R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445–466. https://doi.org/10.1016/S0006-3495(61)86902-6 doi: 10.1016/S0006-3495(61)86902-6
[14]	J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061–2070. https://doi.org/10.1109/JRPROC.1962.288235 doi: 10.1109/JRPROC.1962.288235
[15]	A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol., 117 (1952), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764 doi: 10.1113/jphysiol.1952.sp004764
[16]	M. Argentina, P. Coullet, V. Krinsky, Head-on collisions of waves in an excitable Fitzhugh-Nagumo system: A transition from wave annihilation to classical wave behavior, J. Theor. Biol., 205 (2000), 47–52. https://doi.org/10.1006/jtbi.2000.2044 doi: 10.1006/jtbi.2000.2044
[17]	M. P. Zorzano, L. Vázquez, Emergence of synchronous oscillations in neural networks excited by noise, Physica D, 179 (2003), 105–114. https://doi.org/10.1016/S0167-2789(03)00007-1 doi: 10.1016/S0167-2789(03)00007-1
[18]	A. H. Bhrawy, A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Comput., 222 (2013), 255–264. https://doi.org/10.1016/j.amc.2013.07.056 doi: 10.1016/j.amc.2013.07.056
[19]	D. Cebrían-Lacasa, P. Parra-Rivas, D. Ruiz-Reynés, L. Gelens, Six decades of the Fitzhugh-Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines, preprint, arXiv: 2404.11403
[20]	M. Shih, E. Momoniat, F. M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, J. Math. Phys., 46 (2005), 023503. https://doi.org/10.1063/1.1839276 doi: 10.1063/1.1839276
[21]	A. Mehta, G. Singh, H. Ramos, Numerical solution of time dependent nonlinear partial differential equations using a novel block method coupled with compact finite difference schemes, Comput. Appl. Math., 42 (2023), 201. https://doi.org/10.1007/s40314-023-02345-3 doi: 10.1007/s40314-023-02345-3
[22]	H. Li, Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Appl. Math. Comput., 180 (2006), 524–528. https://doi.org/10.1016/j.amc.2005.12.035 doi: 10.1016/j.amc.2005.12.035
[23]	S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32 (2008), 2706–2714. https://doi.org/10.1016/j.apm.2007.09.019 doi: 10.1016/j.apm.2007.09.019
[24]	T. Kawahara, M. Tanaka, Interactions of traveling fronts: An exact solution of a nonlinear diffusion equation, Phys. Lett. A, 97 (1983), 311–314. https://doi.org/10.1016/0375-9601(83)90648-5 doi: 10.1016/0375-9601(83)90648-5
[25]	R. A. Van Gorder, K. Vajravelu, A variational formulation of the Nagumo reaction–diffusion equation and the Nagumo telegraph equation, Nonlinear Anal. Real World Appl., 11 (2010), 2957–2962. https://doi.org/10.1016/j.nonrwa.2009.10.016 doi: 10.1016/j.nonrwa.2009.10.016
[26]	H. Ali, M. Kamrujjaman, M. S. Islam, Numerical computation of Fitzhugh-Nagumo equation: A novel galerkin finite element approach, Int. J. Math. Res., 9 (2020), 20–27. https://doi.org/10.18488/journal.24.2020.91.20.27 doi: 10.18488/journal.24.2020.91.20.27
[27]	M. Namjoo, S. Zibaei, Numerical solutions of Fitzhugh-Nagumo equation by exact finite-difference and NSFD schemes, Comput. Appl. Math., 37 (2018), 1395–1411. https://doi.org/10.1007/s40314-016-0406-9 doi: 10.1007/s40314-016-0406-9
[28]	R. A. V. Gorder, Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function h (x, t) in the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1233–1240. https://doi.org/10.1016/j.cnsns.2011.07.036 doi: 10.1016/j.cnsns.2011.07.036
[29]	H. Triki, A. M. Wazwaz, On soliton solutions for the Fitzhugh-Nagumo equation with time-dependent coefficients, Appl. Math. Model., 37 (2013), 3821–3828. https://doi.org/10.1016/j.apm.2012.07.031 doi: 10.1016/j.apm.2012.07.031
[30]	K. U. Rehman, W. Shatanawi, M. Y. Malik, Group theoretic thermal analysis (GTTA) of Powell-Eyring fluid flow with identical free stream (FS) and heated stretched porous (HSP) boundaries: AI Decisions, Case Stud. Therm. Eng., 55 (2024), 104101. https://doi.org/10.1016/j.csite.2024.104101 doi: 10.1016/j.csite.2024.104101
[31]	N. Jamal, M. Sarwar, N. Mlaiki, A. Aloqaily, Solution of linear correlated fuzzy differential equations in the linear correlated fuzzy spaces, AIMS Math., 9 (2024), 2695–2721. https://doi.org/10.3934/math.2024134 doi: 10.3934/math.2024134
[32]	R. Jiwari, R. Gupta, V. Kumar, Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Ain Shams Eng. J., 5 (2014), 1343–1350. https://doi.org/10.1016/j.asej.2014.06.005 doi: 10.1016/j.asej.2014.06.005
[33]	S. Singh, Mixed-type discontinuous galerkin approach for solving the generalized Fitzhugh-Nagumo reaction-diffusion model, Int. J. Appl. Comput. Math., 7 (2021), 207. https://doi.org/10.1007/s40819-021-01153-9 doi: 10.1007/s40819-021-01153-9
[34]	S. S. Khayyam, M. Sarwar, A. Khan, N. Mlaiki, F. M. Azmi, Solving integral equations via fixed point results involving rational-type inequalities, Axioms, 12 (2023), 685. https://doi.org/10.3390/axioms12070685 doi: 10.3390/axioms12070685
[35]	Ü. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68 (2005), 127–143. https://doi.org/10.1016/j.matcom.2004.10.005 doi: 10.1016/j.matcom.2004.10.005
[36]	R. Jiwari, A Haar wavelet quasilinearization approach for numerical simulation of Burgers' equation, Comput. Phys. Commun., 183 (2012), 2413–2423. https://doi.org/10.1016/j.cpc.2012.06.009 doi: 10.1016/j.cpc.2012.06.009
[37]	M. Kumar, S. Pandit, A composite numerical scheme for the numerical simulation of coupled Burgers' equation, Comput. Phys. Commun., 185 (2014), 809–817. https://doi.org/10.1016/j.cpc.2013.11.012 doi: 10.1016/j.cpc.2013.11.012
[38]	S. C. Shiralashetti, A. B. Deshi, P.B. M. Desai, Haar wavelet collocation method for the numerical solution of singular initial value problems, Ain Shams Eng. J., 7 (2016), 663–670. https://doi.org/10.1016/j.asej.2015.06.006 doi: 10.1016/j.asej.2015.06.006
[39]	A. Ghafoor, Numerical solutions of time dependent partial differential equations via Haar wavelets, Nat. Appl. Sci. Int. J., 1 (2020), 39–52. https://doi.org/10.47264/idea.nasij/1.1.4 doi: 10.47264/idea.nasij/1.1.4
[40]	A. Ghafoor, N. Khan, M. Hussain, R. Ullah, A hybrid collocation method for the computational study of multi-term time fractional partial differential equations, Comput. Math. Appl., 128 (2022), 130–144. https://doi.org/10.1016/j.camwa.2022.10.005 doi: 10.1016/j.camwa.2022.10.005
[41]	S. Haq, A. Ghafoor, M. Hussain, S. Arifeen, Numerical solutions of two dimensional sobolev and generalized Benjamin-Bona-Mahony-Burgers equations via Haar wavelets, Comput. Math. Appl., 77 (2019), 565–575. https://doi.org/10.1016/j.camwa.2018.09.058 doi: 10.1016/j.camwa.2018.09.058
[42]	I. Ahmad, M. Ahsan, Z. Din, A. Masood, P. Kumam, An efficient local formulation for time-dependent PDEs, Mathematics, 7 (2019), 216. https://doi.org/10.3390/math7030216 doi: 10.3390/math7030216
[43]	H. Ramos, A. Kaur, V. Kanwar, Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations, Comput. Appl. Math., 41 (2022), 34. https://doi.org/10.1007/s40314-021-01729-7 doi: 10.1007/s40314-021-01729-7
[44]	B. Inan, K. K. Ali, A. Saha, T. Ak, Analytical and numerical solutions of the Fitzhugh-Nagumo equation and their multistability behavior, Numer. Methods Partial Differ. Equations, 37 (2021), 7–23. https://doi.org/10.1002/num.22516 doi: 10.1002/num.22516
[45]	G. Arora, V. Joshi, A computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes, Ain Shams Eng. J., 9 (2018), 1141–1150. https://doi.org/10.1016/j.asej.2016.06.013 doi: 10.1016/j.asej.2016.06.013
[46]	N. Dhiman, M. Tamsir, A collocation technique based on modified form of trigonometric cubic B-spline basis functions for Fisher's reaction-diffusion equation, Multidiscip. Model. Mater. Struct., 14 (2018), 923–939. https://doi.org/10.1108/MMMS-12-2017-0150 doi: 10.1108/MMMS-12-2017-0150
[47]	J. Butcher, Runge-kutta methods, Scholarpedia, 2 (2007), 3147. https://doi.org/10.4249/scholarpedia.3147
[48]	Ö. Oruç, Numerical simulation of two-dimensional and three-dimensional generalized Klein-Gordon-Zakharov equations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation, Numer. Methods Partial Differ. Equations, 38 (2022), 1068–1089. https://doi.org/10.1002/num.22806 doi: 10.1002/num.22806

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