### A New Analysis of the Time-Fractional and Space-Time Fractional-Order Nagumo Equation

#### Abstract

In this paper, we present an algorithm by using the *Adomian Decomposition Method* (ADM) in order to solve the time-fractional Nagumo equation and the space-time fractional-order Nagumo equation. In the space-time fractional case, we expand the \(tanh(\cdot)\) initial condition in the basis functions \(e^{−n\zeta}\). The fractional-order derivative could then be easily calculated. An important point in our investigation is that many earlier authors avoided this initial condition as there was no direct method to calculate its fractional derivative. We have studied the convergence analysis and applied it to the time-fractional Nagumo equation and the space-time fractional-order Nagumo equation. We compare the ADM solution with the exact solution and find a very good agreement. We also graphically illustrate the behavior of the ADM solutions.

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K. Abbaoui and Y. Cherruault, Convergence of Adomian’s method applied to differential equations, Computers and Mathematics with Applications 28(5) (1994), 103 – 109, DOI: 10.1016/0898-1221(94)00144-8.

S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method, Applied Mathematical Modelling 32 (2008), 2706 – 2714, DOI: 10.1016/j.apm.2007.09.019.

M.A. Abdelkawy, M.A. Zaky, A.H. Bhrawy and D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Romanian Reports in Physics 67 (2015), 773 – 791.

G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dordrecht, Boston and London (1999).

G. Adomian, A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135 (1988), 501 – 544, DOI: 10.1016/0022-247X(88)90170-9.

M. Alabdullatif, H.A. Abdusalam and E.S. Fahmy, Adomian decomposition method for nonlinear reaction diffusion system of Lotka-Volterra type, International Mathematical Forum 2(2) (2007), 87 – 96, DOI: 10.12988/imf.2007.07006.

J. Biazar, E. Babolian and R. Islam, Solution of the system of ordinary differential equations by Adomian decomposition method, Applied Mathematics and Computation 147 (2004), 713 – 719, DOI: 10.1016/S0096-3003(02)00806-8.

J. Biazar, E. Babolian and R. Islam, Solution of the system of Volterra integral equations of the first kind by Adomian decomposition method, Applied Mathematics and Computation 139 (2003), 249 – 258.

M. Caputo, Linear models of dissipation whose Q is almost frequency independent - II, Geophysical Journal of the Royal Astronomical Society 13 (1967), 529 – 539.

Y. Cherruault and G. Adomian, Decomposition methods: A new proof of convergence, Mathematical and Computer Modelling 18(12) (1993), 103 – 106, DOI: 10.1016/0895-7177(93)90233-O.

Y. Cherruault, Convergence of Adomian’s method, Mathematical and Computer Modelling 14 (1990), 83 – 86, DOI: 10.1016/0895-7177(90)90152-D.

P.A. Clarkson and M.C. Nucci, The nonclassical method is more general than the direct method for symmetry reductions: An example of the Fitzhugh-Nagumo equation, Physics Letters A 164 (1992), 49 – 56, DOI: 10.1016/0375-9601(92)90904-Z.

V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 316 (2006), 753 – 763, DOI: 10.1016/j.jmaa.2005.05.009.

I.L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Applied Mathematics and Computation 21 (2008), 372 – 376, DOI: 10.1016/j.aml.2007.05.008.

R. Fitzhugh, Impulse and physiological states in models of nerve membrane, Biophysical Journal 1 (1961), 445 – 466.

H.-F. Fu and H. Wang, A preconditioned fast finite difference method for space-time fractional partial differential equations, Fractional Calculus and Applied Analysis 20 (2017), 88 – 116.

M. Giona and H.E. Roman, Fractional diffusion equation for transport phenomena in random media, Physica A: Statistical Mechanics and Its Applications 185 (1992), 87 – 97, DOI: 10.1016/0378-4371(92)90441-R.

Y.-C. Guo and H.-Y. Li, New exact solutions to the Fitzhugh-Nagumo equation, Applied Mathematics and Computation 180 (2006), 524 – 528.

R. Hilfer (Editor), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore — New Jersey — London — Hong Kong (2000).

S. Gh. Hosseini, E. Babolian and S. Abbasbandy, A new algorithm for solving Van der Pol equation based on piecewise spectral Adomian decomposition method, International Journal of Industrial Mathematics 8 (2016), 177 – 184.

M. Javidi and A. Golbabai, A spectral domain decomposition approach for the generalized Burgers- Fisher equation, Chaos Solitons and Fractals 39 (2009), 385 – 392.

D.S. Jones, M. Plank and B.D. Sleeman, Differential Equations and Mathematical Biology, Second edition, Chapman and Hall (CRC Press), New York (2009).

T. Kawahara and M. Tanaka, Interaction of travelling fronts: An exact solution of a nonlinear diffusion equation, Physics Letters A 97 (1983), 311 – 314, DOI: 10.1016/0375-9601(83)90648-5.

N. Khodabakhshi, S.M. Vaezpour and D. Baleanu, Numerical solutions of the initial value problem for fractional differential equations by modification of the Adomian decomposition method, Fractional Calculus and Applied Analysis 20 (2014), 382 – 400.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam — London — New York (2006).

J.W. Kirchner, X.-H. Feng and C. Neal, Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403 (2000), 524 – 527.

D. Lesnic, A nonlinear reaction-diffusion process using the Adomian decomposition method, International Communications in Heat and Mass Transfer 34(2) (2007), 129 – 135.

C.-P. Li and W.-H. Deng, Remarks on fractional derivatives, Applied Mathematics and Computation 187 (2007), 777 – 784, DOI: 10.1016/j.amc.2006.08.163.

S.-J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation 147 (2004), 499 – 513, https://www.researchgate.net/journal/0096-3003_Applied_Mathematics_and_Computation.

R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers Incorporated, Danbury (Connecticut) — São Paulo — New Delhi (2006).

T. Mavoungou and Y. Cherruault, Convergence of Adomian’s method and applications to non-linear partial differential equation, Kybernetes 21(6) (1992), 13 – 25.

K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons, New York — Chichester — Brisbane — Toronto — Singapore (1993).

J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE 50 (1962), 2061 – 2070.

Z. Odibat and S. Momani, Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling 32 (2008), 28 – 39, DOI: 10.1016/j.apm.2006.10.025.

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego — London — Toronto (1999).

M. Popolizio and R. Garrappa, On the use of matrix functions for fractional partial differential equations, Mathematics and Computers in Simulation 81 (2011), 1045 – 1056.

K.M. Saad and A.A. Al-Shomrani, An application of homotopy analysis transform method for Riccati differential equation of fractional order, Journal of Fractional Calculus and Applications 7 (2016), 61 – 72.

K.M. Saad and E.H.F. Al-Sharif, Analytical study for time and time-space fractional Burgers’ equation, Advances in Difference Equations 2017 (2017), Article ID 300, 1 – 15.

K.M. Saad, An approximate analytical solutions of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications 5 (1) (2014), 58 – 70, http://fcag-egypt.com/Journals/JFCA/.

K.M. Saad, Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system, European Physical Journal Plus 133 (2018), Article ID 94, DOI: 10.1140/epjp/i2018-11947-6.

K.M. Saad, E.H.F. Al-Sharif, S.M. Mohamed and X.-J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, European Physical Journal Plus 132 (2017), Article ID 23, DOI: 10.1140/epjp/i2017-11303-6.

K.M. Saad, H.M. Srivastava and D. Kumar, A reliable analytical algorithm for cubic isothermal auto-catalytic chemical system, Alexandria Engineering Journal (submitted).

X.-C. Shi, L.-L. Huang and Y. Zeng, Fast Adomian decomposition method for the Cauchy problem of the time-fractional reaction diffusion equation, Advances in Mechanical Engineering 8(2) (2016), 1 – 5.

M. Shih, E. Momoniat and F.M. Mahomed, Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh-Nagumo equation, Journal of Mathematical Physics 46 (2005), Article ID 023503, 1 – 11.

H.M. Srivastava and K.M. Saad, Some new models of the time-fractional gas dynamics equation, Advanced Mathematical Models and Applications 3(1) (2018), 5 – 17.

Y. Takeuchi, Y. Yoshimoto and R. Suda, Second order accuracy finite difference methods for space fractional partial differential equations, Journal of Computational and Applied Mathematics 320 (2017), 101 – 119, DOI: 10.1016/j.cam.2017.01.013.

M. Tataria, M. Dehghana and M. Razzaghi, Application of the Adomian decomposition method for the Fokker-Planck equation, Mathematical and Computer Modelling 45 (2007), 639 – 650, DOI: 10.1016/j.mcm.2006.07.010.

A.-M. Wazwaz, R. Rach and J.-S. Duan, A study on the systems of the Volterra integral forms of the Lane-Emden equations by the Adomian decomposition method, Mathematical Methods in the Applied Sciences 37 (2014), 10 – 19, DOI: 10.1002/mma.2776 .

Ç. Yöcel, K. Yildiray and K. Aydin, The solution of the Bagley-Torvik equation with the generalized Taylor collocation method, Journal of the Franklin Institute 347 (2010), 452 – 466.

X.-J. Yang, F. Gao and H.M. Srivastava, Exact travelling wave equations for the local fractional two-dimensional Burgers-type equations, Computers and Mathematics with Applications 73 (2017), 203 – 210.

X.-J. Yang, J. Hristov, H.M. Srivastava and B. Ahmad, Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation, Abstract and Applied Analysis 2014 (2014), Article ID 278672, 1 – 10.

DOI: http://dx.doi.org/10.26713%2Fjims.v10i4.961

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