A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses

Amit Prakash; Hardish Kaur

doi:10.1515/nleng-2018-0057

Article Open Access

A reliable numerical algorithm for a fractional model of Fitzhugh-Nagumo equation arising in the transmission of nerve impulses

Amit Prakash and Hardish Kaur

Published/Copyright: July 12, 2019

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 8 Issue 1

Abstract

The key objective of this paper is to study the fractional model of Fitzhugh-Nagumo equation (FNE) with a reliable computationally effective numerical scheme, which is compilation of homotopy perturbation method with Laplace transform approach. Homotopy polynomials are employed to simplify the nonlinear terms. The convergence and error analysis of the proposed technique are presented. Numerical outcomes are shown graphically to prove the efficiency of proposed scheme.

Keywords: Fractional Fitzhugh-Nagumo equation; Homotopy polynomials; Homotopy perturbation transform technique (HPTT)

1 Introduction

Fractional order derivatives are playing a significant role in the modelling of several phenomena in diffusion procedures, reaction processes, relaxation vibrations, fluid mechanics and numerous fields of science and engineering. Fractional partial differential equations are an important class of differential equations as the fractional calculus operators are nonlocal operators and are suitable to describe the nonlocal effects characterizing most of the real-world phenomena. Many vigorous methods have been constructed for solving fractional differential equations such as homotopy perturbation method [1], q-homotopy analysis transform method [2], new iterative Sumudu transform method [3], reduced differential transform method [4], finite element method [5], operational matrix method [6], Adomian decomposition method [7], variational iteration method [8, 9, 10, 11, 12] and many more. In [13], W. P. Bu and A. G. Xiao have developed a novel test basis function for the Petrov- Galerkin finite element method to solve one dimensional fractional differential equation with Dirichlet boundary condition. And W. P. Bu et al. [14] have studied the distributed- order time fractional diffusion equation with finite element method and they developed a higher order finite difference scheme of the Caputo fractional derivative to improve the time convergence rate of discussed techniques. Z. Hammouch and T. Mekkaoui discussed a very powerful technique Adomian decomposition method for finding the convergent series solution of fractional KPP-like equations. M. Stynes et al. presented a new analysis of finite difference scheme in [15] to deal with problems having weak singularity of the solution and a reaction diffusion problem with Caputo fractional derivative is considered. Also, convergence and error estimation results are discussed.

In the present research work, a combined scheme of Laplace transform and homotopy perturbation method is proposed to investigate the fractional model of FNE. Homotopy perturbation method was firstly introduced by J. H. He [16] as a powerful tool to approach various kinds of nonlinear problems and the Laplace transform has proved to be an effective mechanism for solving several linear and nonlinear differential equations. The homotopy perturbation method combined with Laplace transform produces a more effective and simple technique for handling many nonlinear problems in the realm of science, in comparison with other mathematical techniques. One of the most remarkable advantage of using the proposed technique is that usually just few perturbation terms are sufficient to yield a reasonably accurate solution. A nonlinear fractional-order Fitzhugh-Nagumo equation is expressed as

Dtβw−wxx=ww−ρ1−w,0<β≤1,(1)

where ρ is arbitrary constant. For β = 1, Eq. (1) reduces to classical nonlinear Fitzhugh-Nagumo equation

wt−wxx=ww−ρ1−w,0<β≤1,(2)

and if we take ρ = −1, then Fitzhugh-Nagumo equation model is converted into the famous Newell-Whitehead model. FNE is a famous reaction-diffusion system which was firstly introduced by Hodgkin and Huxley and is used to model the transmission of nerve impulses. FNE has also been used in biology, circuit theory and in the area of population genetics. Many researchers have investigated FNE by different techniques [17, 18, 19, 20, 21, 22, 23, 24] as variational iteration method, Adomian decomposition method [17], homotopy analysis method [25] and many more. Fractional model of Fitzhugh-Nagumo equation has been studied by fractional reduced differential transform method and q- homotopy analysis transform method [26], Haar wavelet method [27] etc.

The remaining part of this paper is organized as follows: In Section 2, some basic definitions of fractional derivatives and Laplace transform are discussed, section 3 gives the basic plan of HPTT. In section 4, convergence analysis and error estimation results are discussed. In section 5, numerical solutions of two fractional models of FNE obtained by HPTT are discussed to show the efficiency of proposed technique. Finally, section 6 presents the conclusion of the current research work.

2 Preliminaries

In this section, some basic definitions of fractional-order derivatives and integrals [28, 29] are discussed.

Definition 2.1

Areal function g(μ), μ > 0, in the space C_ζ, ζ ∈ ℝ if there exist a real number p > ζ such that g(μ) = μ^p g₁(μ) where g₁(μ) ∈ C[0, ∞) and in the space Cζl if g^l ∈ C_ζ, l ∈ N.

Definition 2.2

The Riemann-Liouville fractional integral of order β ≥ 0, of a function g(μ) ∈ C_ζ, ζ ≥ −1 is presented as:

Iβgμ=1Γβ∫0μfημ−η1−βdη=1Γ(β+1)∫0μf(ζ)(dζ)β,

I0gμ=gμ,

where Γ is the well-known Gamma function.

Definition 2.3

The Caputo fractional derivative of g(μ), g ∈ C−1m, m ∈ N, m > 0, is defined as

Dβgμ=Im−βDmgμ

=1Γ(m−β)∫0μμ−ηm−β−1g(m)ηdη,

where m − 1 < β ≤ m.

The operator D^β has following basic properties

D^β I^β g(μ) = g(μ),
IβDβgμ=gμ−∑k=0m−1gk0+μΓk+1,m>0.

Definition 2.4

The Laplace transform of the Caputo fractional derivative Dμβg(μ) is defined as

LDμβg(μ)=sβLgμ−∑k=0l−1sβ−k−1g(k)0,l−1<β≤l.

3 Proposed Homotopy perturbation transform technique (HPTT)

Consider the nonlinear differential equation of arbitrary order

Dtβwx,t+Rwx,t+Nwx,t=fx,t,l−1<β≤l,(3)

with the condition

wmx,0=fmx,m=0,1,2,…,l−1,(4)

where Dtβw (x, t) represents arbitrary derivative of w(x, t) in Caputo sense, R stands for the linear differential operator, N represents the nonlinear differential operator and f(x, t) represents the source term.

Firstly, exerting Laplace transform operator on Eq. (3) and after simplification it yields

Lwx,t−∑k=0l−1s−k−1wkx,0+1sβLRwx,t+LNwx,t−Lf(x,t)=0.(5)

Taking inverse Laplace transform on both sides of Eq. (5), it gives

wx,t=L−1∑k=0l−1s−k−1wkx,0+Lf(x,t)−L−11sβLRwx,t+LNwx,t.(6)

HPM defines a solution by an infinite series of components given by

wx,t=∑i=0∞piwi(x,t),(7)

and the nonlinear term can be given as

Nwx,t=∑i=0∞piHi(w),(8)

where H_i(w) represents the homotopy polynomial and given in the following form:

Hiw=1i!∂i∂piN∑j=0∞pjwjp=0,i=0,1,2,3,…(9)

Using Eq. (7) and Eq. (8) in Eq. (6), we get

∑i=0∞piwi(x,t)=L−1∑k=0l−1s−k−1wkx,0+Lf(x,t)−pL−11sβLR∑i=0∞piwi(x,t)+∑i=0∞piHi(w).(10)

Equating on both sides the coefficients of like powers of p, it gives the following iterates

p0:w0x,t=L−1∑k=0l−1s−k−1wkx,0+Lf(x,t),p1:w1x,t=−L−11sβLRw0(x,t)+H0(w),p2:w2x,t=−L−11sβLRw1(x,t)+H1(w),p3:w3x,t=−L−11sβLRw2(x,t)+H2(w),

in the same pattern, we can calculate the remaining iterates. Therefore, the series solution is given by

wx,t=limp→1limN→∞⁡∑i=0Npiwix,t,=limN→∞⁡∑i=0Nwix,t.(11)

4 Convergence and Error Analysis

Convergence and absolute error of the proposed HPTT can be analysed by the following theorems:

Theorem 4.1

Homotopy perturbation transform technique used to obtain the solution of Eq. (3) is equivalent to determining the following sequence

si=w1+w2+w3+⋯+wi,

s0=w0,

by iterative scheme

si+1=−L−11sβLR∑i=0nwi(x,t)+∑i=0nHi(w),(12)

where

N∑i=0∞piwi(x,t)=∑i=0∞piHi(w).

Proof

For i = 0, from Eq. (12), we have

s1=−L−11sβLRw0+H0(w),

Then as s₁ = w₁, so

w1=−L−11sβLRw0+H0(w),

for i = 1, we have

s2=−L−11sβLRw0+w1+H0w+H1(w)=−L−11sβLRw0+H0w−L−11sβLRw1+H1(w),

as s₂ = w₁ + w₂, we have

w1+w2=w1−L−11sβLRw1+H1(w),

thus

w2=−L−11sβLRw1+H1(w),

proceeding in this way, suppose by induction that wk = −L−11sβLRwk−1+Hk−1(w), where k = 1, 2, …, i, so

si+1=−L−11sβLR∑i=0nwi(x,t)+∑i=0nHi(w)=−L−11sβLRw0+H0w−L−11sβLRw1+H1w−…−L−11sβLRwi−1+Hi−1(w)−L−11sβLRwi+Hi(w)=w1+w2+…+wi−L−11sβLRwi+Hi(w).

Thus, by using s_i+1 = w₁ + w₂ + … + w_i + w_i+1, we have

wi+1=−L−11sβLRwi+Hi(w).

Which is same as the result obtained in HPTT and hence the theorem is proved. □

Theorem 4.2

[1] Let w_i(x, t) and w(x, t)be defined in Banach space (C[0, 1], ∥.∥), then the series ∑i=0∞w_i(x, t) converges to the solution w(x, t) of Eq. (3) if ∃ 0 < y < 1, such that ∥w_i+1(x, t)∥ ≤ y ∥w_i(x, t)∥, ∀ i ∈ N.

Proof

Let {s_i} beasequence of partial sums of the series (11), then

si+1−si=wi+1≤ywi≤y2wi−1≤⋯≤yi+1w0.

For any i, j ∈ N, i ≥ j,

si−sj=(si−si−1)+(si−1−si−2)+⋯+(sj+1−sj)≤(si−si−1)+(si−1−si−2)+⋯+(sj+1−sj)≤yiw0+yi−1w0+⋯+yj+1w0≤yj+11+y+y2+⋯+yi+…w0≤1−yi−j1−yyj+1w0.(13)

Since 0 < y < 1, we have 1 − y^i−j < 1, then

si−sj≤yj+11−yw0.

So ∥s_i − s_j∥ → 0 as i, j → ∞ as w₀ is bounded. Thus {s_i} is a Cauchy sequence in Banach space and hence convergent. Therefore ∃ w(x, t) ∈ B such that ∑i=0∞w_i(x, t) = w(x, t). □

Theorem 4.3

[1] If there exists 0 < y < 1 in such a way that ∥w_i+1(x, t)∥ ≤ y ∥w_i(x, t)∥, ∀ i ∈ N, then the maximum absolute truncation error of the series solution (11) is determined as

wx,t−∑i=0jwix,t≤yj+1(1−y)w0x,t.

Proof

By using Eq. (13) and from Theorem 4.2, as i → ∞, s_i → w(x, t), we get

wx,t−∑i=0jwix,t≤yj+1(1−y)w0x,t.

5 Numerical Examples

In this section, we apply the proposed technique on some test examples.

Example 5.1

Consider the fractional model of Fitzhugh-Nagumo equation

Dtβw−wxx=ww−ρ1−w,0<β≤1,t>0,x∈R,(14)

with corresponding initial conditions as

wx,0=12+12tanh2x4.(15)

The solution in closed form for fractional FNE (14) with condition (15) for β = 1 is given as

wx,t=12+12tanh2x+(1−2ρ)t4.(16)

Firstly, operating Laplace transform on Eq. (14) and simplifying, it gives

Lw−wx,0s+1sβL−wxx−w2+w3+wρ−w2ρ=0.(17)

Taking inverse Laplace transform on Eq. (17), we have

wx,t=wx,0−L−11sβL−wxx−w2+w3+wρ−w2ρ.(18)

Applying HPM, we have

∑i=0∞piwix,t=wx,0+pL−11sβL∑i=0∞piwi(x,t)xx+∑i=0∞piHiw+pL−11sβL−ρ∑i=0∞piwix,t,(19)

where

∑i=0∞piHiw=w21+ρ−w3.

Solving above equations, we have the following successive components of solution

w0x,t=12+12tanh2x4,

w1x,t=(1−2ρ)8sech22x4tβΓ(β+1),

w2x,t=−(1−2ρ)216t2βΓ(2β+1)sech22x4tanh2x4,

w3x,t=(1−2ρ)364t3βΓ(3β+1)sech42x4−3+2cosh22x4.

In the same pattern, we can calculate the remaining iterates of series solution. Thus, the series solution obtained by HPTT is given as

wx,t=w0x,t+w1x,t+w2x,t+w3x,t+….

Fig. 1 shows the graphical behaviour of 3}^rd order HPTT solution and exact solution for fixed ρ = −1 and for different values of order of time-fractional derivative β. Fig. 2 shows the plots of HPTT solution corresponding to diverse values of β and demonstrates the result that as values of β decreases w increases. It can be observed from Table 1 and Table 2 that exact solution for given problem 5.1 is in close agreement with 3^rd order HPTT solution. Here only 3^rd order HPTT solution is used to calculate the numerical solution and HPTT can yield more accurate solution with less absolute error by computing higher order approximation.

Fig. 1

(a) Exact solution when ρ = -1, for Ex. 5.1; (b) Numerical simulation of w(x, t) at β = 0.25 when ρ = −1, for Ex. 5.1; (c) Numerical simulation of w(x, t) at β = 0.50 when ρ = −1, for Ex. 5.1; (d) Numerical simulation of w(x, t) at β = 1 when ρ = −1, for Ex. 5.1

Fig. 2

Plots of w(x, t) w.r.t x for varying β, for Ex. 5.1

Table 1

Absolute errors for differences between the exact solution and 3^rd order HPTT solution for diverse values x and t when ρ = −1, β = 1, for Ex. 5.1

X	t	Exact solution	HPTT solution	\|u_exa.(x, t) − u₃(x, t)\|
	0.1	0.5549027314	0.5550423076	1.4 × 10⁻⁴
0.1	0.2	0.5915797906	0.5926897865	1.1 × 10⁻³
	0.3	0.6272646756	0.6309809686	3.7 × 10⁻³
	0.4	0.6616151086	0.6703366818	8.7 × 10⁻³

	0.01	0.5390453402	0.5390454786	1.4 × 10⁻⁷
0. 2	0.02	0.5427702206	0.5427713276	0
	0.03	0.5464903249	0.5464940587	0
	0.04	0.5502052444	0.55020140897	0

Table 2

Absolute errors for differences between the exact solution and 3^rd order HPTT solution for diverse values x and t when ρ = 0.45, β = 0.5, for Ex. 5.1

X	t	Exact solution	HPTT solution	\|u_exa.(x, t) − u₃(x, t)\|
	0.1	0.5189186334	0.5221224713	3.2 × 10⁻³
0.1	0.2	0.5201667232	0.5239648028	3.7 × 10⁻³
	0.3	0.5214145613	0.5253776035	3.96 × 10⁻³
	0.4	0.5226621322	0.5265680048	3.9 × 10⁻³

	0.01	0.5354209060	0.5366995256	1.3 × 10⁻³
0.2	0.02	0.5355452764	0.5372803916	1.7 × 10⁻³
	0.03	0.5356696425	0.5377259923	2.0 × 10⁻³
	0.04	0.5357940041	0.5381015730	2.3 × 10⁻³

Example 5.2

Consider the fractional model of Fitzhugh-Nagumo equation

Dtβw−wxx=ww−ρ1−w,0<β≤1,t>0,x∈R,(20)

with the initial condition

wx,0=11+e−x2.(21)

The solution in closed form for fractional FNE (20) with condition (21) for β = 1 is given by

wx,t=11+ex2+yt,wherey=12−2ρ.(22)

Firstly, operating Laplace transform on Eq. (20) and simplifying, it gives

Lw−wx,0s+1sβL−wxx−w2+w3+wρ−w2ρ=0.(23)

Taking inverse Laplace transform on Eq. (23), we get

wx,t=wx,0−L−11sβL−wxx−w2+w3+wρ−w2ρ.(24)

Applying HPM, we have

∑i=0∞piwix,t=wx,0+pL−11sβL∑i=0∞piwi(x,t)xx+∑i=0∞piHiw+pL−11sβL−ρ∑i=0∞piwix,t.(25)

where

∑i=0∞piHiw=w21+ρ−w3.

Solving above equations, we get the following successive components of solution

w0x,t=11+e−x2,w1x,t=1−2ρe−2x+(3−4ρ)e−x221+e−x23tβΓ(β+1),

w2x,t=t2βΓ(2β+1)(2−ρ)1−2ρe−2x21+e−x23−1−2ρe−3x21+e−x24+31−2ρe−22x1+e−x25+3−4ρ1−2ρ4e−x21+e−x23+e−2x1+e−x24−93−4ρ4+2+2ρ1−2ρ2+33−4ρe−3x21+e−x25+1+ρ3−4ρe−x21+e−x24−31−2ρe−2x21+e−x25−−3(3−4ρ)e−x221+e−x25,

w3x,t=t3βΓ(3β+1)−15ρ(2−ρ)(1−2ρ)8−9ρ(1−2ρ)2+ρ1−2ρe−3x21+e−x24+3ρ(1−2ρ)(2−ρ)2+21ρ1−2ρe−22x1+e−x25+ρe−2x1+e−x24−93−4ρ(1−2ρ)8−2+2ρ1−2ρ2+9(3−4ρ)4+6(3−4ρ)(1−2ρ)4−3(3−4ρ)ρe−3x21+e−x25+…+….

In the same pattern, we can calculate the remaining iterates of series solution. Thus, the series solution obtained by HPTT is given as

wx,t=w0x,t+w1x,t+w2x,t+w3x,t+….

Fig. 3 shows the graphical behaviour of 3^rd order HPTT approximate solution for fixed ρ = −1 and for different values of order of time-fractional derivative β. It can be observed from Table 3 and Table 4 that exact solution for given problem 5.2 is in close agreement with 3^rd order HPTT solution. Fig. 4 shows the plots of HPTT solution corresponding to diverse values of β and demonstrates the result that as values of β decreases w increases and also with increasing values of x, w(x, t) decreases. Here only 3^rd order approximation is used to calculate the numerical solution and HPTT can yield more accurate solution with less absolute error by computing higher order approximations.

Fig. 3

(a) Numerical simulation of w(x, t) at β = 1 when ρ = −1, for Ex. 5.2; (b) Numerical simulation of w(x, t) at β = 0.75 when ρ = −1, for Ex. 5.2; (c) Numerical simulation of w(x, t) at β = 0.5 when ρ = −1, for Ex. 5.2; (d) Numerical simulation of w(x, t) at β = 0.25 when ρ = −1, for Ex. 5.2

Fig. 4

Plots of w(x, t) w.r.t x for varying β, for Ex. 5.2

Table 3

Absolute errors for differences between the exact solution and 3^rd order HPTT solution for diverse values x and t when ρ = −1, β = 1, for Ex. 5.2

X	t	Exact solution	HPTT solution	\|u_exa.(x, t) − u₃(x, t)\|
0.001	0.001	0.4992928937	0.5008021225	1.5 × 10⁻³
0.002	0.002	0.4985857903	0.5016049345	3.0 × 10⁻³
0.003	0.003	0.4978786925	0.5024084333	4.5 × 10⁻³
0.004	0.004	0.4971716029	0.5032126161	6.0 × 10⁻³
0.005	0.005	0.4964645249	0.5040174801	7.5 × 10⁻³
0.006	0.006	0.4957574611	0.5048230224	9.1 × 10⁻³
0.007	0.007	0.4950504142	0.5056292406	1.0 × 10⁻²
0.008	0.008	0.4943433871	0.5064361311	1.2 × 10⁻²
0.009	0.009	0.4936363826	0.5072436920	1.3 × 10⁻²
0.01	0.01	0.4929294036	0.5080519202	1.5 × 10⁻²

Table 4

Absolute errors for differences between the exact solution and 3^rd order HPTT solution for diverse values x and t when ρ = 0.45, β = 0.5, for Ex. 5.2

X	t	Exact solution	HPTT solution	\|u_exa.(x, t) − u₃(x, t)\|
0.001	0.001	0.50026800	0.5030386021	2.8 × 10⁻²
0.002	0.002	0.50027473	0.5044177330	4.1 × 10⁻²
0.003	0.003	0.50017995	0.5054409553	5.3 × 10⁻²
0.004	0.004	0.50010722	0.5063580427	6.2 × 10⁻²
0.005	0.005	0.5002438	0.5071827865	6.9 × 10⁻²
0.006	0.006	0.49993390	0.5079419255	8.0 × 10⁻²
0.007	0.007	0.49998373	0.5086513397	8.7 × 10⁻²
0.008	0.008	0.49992356	0.5093213703	9.4 × 10⁻²
0.009	0.009	0.49973585	0.5099592005	1.0 × 10⁻²
0.01	0.01	0.49963020	0.5105700571	1.1 × 10⁻²

6 Conclusion

In this paper, we have presented an effective numerical algorithm HPTT, which is amalgamation of HPM, Laplace transform and homotopy polynomials, to study the fractional model of FNE. The prominence of the proposed numerical approach lies in its potential of compiling two powerful computational approaches for investigating nonlinear fractional differential equations. Numerical simulation results are demonstrated graphically and in tables. The obtained results prove that the proposed numerical approach is highly understandable and logical to investigate several nonlinear fractional-order mathematical models arising in numerous real-world problems.

amitmath0185@gmail.com

Acknowledgement

The authors are extremely thankful to the reviewers for carefully reading the paper and useful comments and suggestions which have helped to improve the paper.

References

[1] M. G. Sakar, F. Uludag, F. Erdogan, Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method, Applied Mathematical Model., 40(2016), 6639-6649.10.1016/j.apm.2016.02.005Search in Google Scholar

[2] A. Prakash, H. Kaur, Numerical solution for fractional model of Fokker- Planck equation by using q-HATM, Chaos, Solitons and Fractals, 105 (2017), 99-110.10.1016/j.chaos.2017.10.003Search in Google Scholar

[3] A. Prakash, M. Kumar, D. Baleanu, A new iterative technique for a fractional model of nonlinear Zakharov- Kuznetsov equations via Sumudu transform, Applied Mathematics and Computation, 334 (2018), 30-40.10.1016/j.amc.2018.03.097Search in Google Scholar

[4] P. K. Gupta, Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method, Comput. Math. Appl., 58(2011), 2829-842.10.1016/j.camwa.2011.03.057Search in Google Scholar

[5] M. Jingtang, J. Liu, Z. Zhou, Convergence analysis of moving finite element methods for space fractional differential equations, Journal of Computational and Applied Mathematics, 255(2014), 661-670.10.1016/j.cam.2013.06.021Search in Google Scholar

[6] M. Yi, J. Huang, wavelet operational matrix method for solving fractional equations with variable coefficients, Applied Mathematics and Computation, 230 (2014), 383-39.10.1016/j.amc.2013.06.102Search in Google Scholar

[7] Z. Hammouch. T. Mekkaoui, Approximate analytical and numerical solutions to fractional KPP-like equations, Gen. Math Notes, 14(2) (2013), 1-9.10.14419/ijpr.v1i2.849Search in Google Scholar

[8] Z. Hammouch. T. Mekkaoui, A Laplace-Variational Iteration method for solving the homogeneous Smoluchowski Coagulation equation, Applied Mathematical Sciences, 6(18) (2012), 879-886.Search in Google Scholar

[9] A. Prakash and M. Kumar, He’s Variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation, Journal of Applied Analysis and Computation, 6(3) (2016), 738-748.10.11948/2016048Search in Google Scholar

[10] A. Prakash and M. Kumar, Numerical method for time-fractional Gas dynamic equations, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., (2018), org/10.1007/s40010-018.Search in Google Scholar

[11] A. Prakash, M. Goyal and S. Gupta, Fractional variational iteration method for solving time- fractional Newell-Whitehead-Segel equation, Nonlinear Engineering, (2018), org/10.1515/nleng-2018-0001.Search in Google Scholar

[12] A. Prakash, M. Kumar, Numerical solution of two-dimensional time-fractional order biological population model, Open Physics, 14 (2016), 177-186.10.1515/phys-2016-0021Search in Google Scholar

[13] W. P. Bu, A. G. Xiao, An h-p version of the continuous Petrov-Galerkin finite element method for Riemann- Liouville fractional differential equation with novel test basis functions, Numer. Algorithms, 2018 org/10.1007/s11075-018-0559-2.Search in Google Scholar

[14] W. P. Bu, A. G. Xiao, W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 72(2017), 422-441.10.1007/s10915-017-0360-8Search in Google Scholar

[15] M. Stynes, E. O’Riordan, J. L. Gracia, Error analysis of a finite difference method on graded mashes or a time fractional diffusion equation, Siam Journal on Numerical Analysis, 55(2) (2017), 1057-1079.10.1137/16M1082329Search in Google Scholar

[16] J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178 (1999), 257-262.10.1016/S0045-7825(99)00018-3Search in Google Scholar

[17] A. A. Soliman, Numerical simulation of the Fitzhugh- Nagumo equations, Abstract and Applied Analysis, Volume 2012, Article ID 762516.10.1155/2012/762516Search in Google Scholar

[18] M. C. Nucci, P.A. Clarkson, the nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation, Phys. Lett. A, 164 (1992), 49-56.10.1016/0375-9601(92)90904-ZSearch in Google Scholar

[19] H. Li, Y. Guo, New exact solutions to the Fitzhugh-Nagumo equation, Appl. Math. Comput., 180 (2006), 524-528.10.1016/j.amc.2005.12.035Search in Google Scholar

[20] M. E. Schonbek, Boundary value problems for the Fitzhugh-Nagumo equations, J. Differ. Equ., 30 (1978), 119-147.10.1016/0022-0396(78)90027-XSearch in Google Scholar

[21] E. Yanagida, Stability of travelling front solutions of the Fitzhugh-Nagumo equations, Math Comput. Model., 12 (1989), 289-301.10.1016/0895-7177(89)90106-4Search in Google Scholar

[22] D. E. Jackson, error estimates for the semidiscrete Galerkin approximations of the Fitzhug -Nagumo equations, Appl. Math. Comput., 50 (1992), 93-114.10.1016/0096-3003(92)90013-QSearch in Google Scholar

[23] W. Gao, J. Wang, Existence of wavefronts and impulses to Fitzhugh-Nagumo equations, Nonlinear Anal., 57 (2004), 5-6.10.1016/j.na.2004.03.009Search in Google Scholar

[24] D. Olmos, B. D. Shizgal, Pseudospectral method of solution of the Fitzhugh-Nagumo equation, Math. Comput. Simul., 79 (2009), 2258-2278.10.1016/j.matcom.2009.01.001Search in Google Scholar

[25] S. Abbasbandy, Soliton solutions for the Fitzhugh-Nagumo equation with homotopy analysis method Appl. Math. Model., 32 (2008), 2706-2714.10.1016/j.apm.2007.09.019Search in Google Scholar

[26] D. Kumar, J. Singh, D. Baleanu, A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses, Nonlinear Dyn., (2017), 10.1007/s11071-017-3870-x.Search in Google Scholar

[27] G. Hariharan, R. Rajaraman, two reliable wavelet methods to Fitzhugh-Nagumo (FN) and fractional FN equations, J. Math. Chem., 51(2013), 2432-2454.10.1007/978-981-32-9960-3_9Search in Google Scholar

[28] I. Podlubny, Fractional differential equations, Academic Press, San Diego (1999).Search in Google Scholar

[29] A. A. Kilbas, H. M. Srivastva, J. J. Trujillo, Theory and applications of fractional differential equations North-Holland Mathematical Studies (Elsevier Publications), 204 (2006).10.1016/S0304-0208(06)80001-0Search in Google Scholar

Received: 2018-03-22

Revised: 2018-08-22

Accepted: 2018-11-06

Published Online: 2019-07-12

Published in Print: 2019-01-28

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/nleng-2018-0057

Keywords for this article

Fractional Fitzhugh-Nagumo equation; Homotopy polynomials; Homotopy perturbation transform technique (HPTT)

Creative Commons

BY 4.0