1 Introduction

As a important model in describing a prototype of an excitable system, the FitzHugh–Nagumo model [1] has received great attention in recent years. The space fractional FitzHugh–Nagumo model is obtained by replacing standard Laplacian operator in FitzHugh–Nagumo model by Riesz derivatives. The purpose of introducing the spatial fractional FitzHugh–Nagumo model is mainly due to the fact that a spatial fractional derivative can capture the spatial connectivity of the extracellular domain more accurately.

Bu et al. [2] presented an ADI finite-element method to solve FitzHugh–Nagumo model. In [3], Liu et al. constructed a shifted Grünwald–Letnikov scheme to discretize the Riesz derivative of the fractional FitzHugh–Nagumo model. However, both the finite-element and finite-difference methods will produce large dense matrices due to the nonlocal term discretization when solving the linear systems. Yang et al. [4] and Cattani [5] proposed a fractional derivative of sinc function without singular kernel. In recent work [6,7,8,9], some general fractional calculus operators involving constant and variable order derivatives were used. Kumar et al. [10, 11] and Singh [12] studied the fractional exothermic reaction model with Mittag-Leffler law. The fractional Laplace decomposition technique was used to investigate the numerical solution of that model. More results on numerical solutions of fractional derivatives can be found in [13,14,15].

This paper uses the Legendre-spectral method to handle nonlocal terms. We propose an efficient numerical scheme to solve the spatial fractional FitzHugh–Nagumo model, the numerical scheme is performed by combining it with a second order method in time and Legendre-spectral method in space. Moveover, we prove that the obtained numerical scheme is unconditionally stable.

In Sect. 2, we will introduce the spatial fractional FitzHugh–Nagumo model. In Sect. 3, the numerical scheme and stability analysis are studied. In Sect. 4, numerical experiments are performed to demonstrate the effectiveness of the numerical methods. The conclusion of this article is given in last section.

2 FitzHugh–Nagumo model

We consider the following fractional FitzHugh–Nagumo model:

$$ \textstyle\begin{cases} u_{t}= \kappa \Delta ^{\alpha /2} u+u(1-u)(u-a)-v &\text{in } \varOmega _{d}\times [0, T], \\ v_{t}=\beta u-\gamma v+\eta &\text{in } \varOmega _{d} \times [0, T], \\ u(\cdot , 0)=u_{0},v(\cdot , 0)=v_{0} & \text{in } \varOmega _{d} \times [0, T], \\ u=0 &\text{on } \partial \varOmega _{d} \times [0, T], \end{cases} $$
(1)

where \(1.5\leq \alpha \leq 2\), \(\varOmega _{d}=(-1, 1)^{d}\), \(d=2,3\), and we define the following space fractional Laplace operator (see [16]):

$$ -\Delta ^{\alpha } := -\frac{1}{4}\sum ^{d}_{j=1}\bigl( D^{\alpha } _{x_{j}} -{{}_{x_{j}}} D^{\alpha }\bigr) \bigl({{}^{C}}D^{\alpha }_{x_{j}} - {{}^{C}_{x_{j}}} D^{\alpha }\bigr), $$
(2)

where \(D^{\alpha }_{x_{j}}\), \({{}_{x_{j}}} D_{1}^{\alpha }\), \({{}^{C}}D^{\alpha }_{x_{j}} \), \({{}^{C}_{x_{j}}} D_{1}^{\alpha }\) are left and right Riemann–Liouville and Captuo fractional derivatives, respectively. It is worth mentioning that our derivative is an extension of the Caputo and Riemann–Liouville derivative, so it contains a singular kernel around 0. But Yang–Srivastava–Machado fractional derivative does not contain a singular kernel.

We use \(A\lesssim B\) to mean that \(A\leq cB\), and \(A\simeq B\) to mean that \(A\lesssim B\) and \(B\lesssim A\).

Lemma 1

([17])

For \(0 < s <1\), \(s\neq \frac{1}{2}\), if \(w, v \in H^{s}_{0}(\varLambda )\), then

$$\begin{aligned}& \bigl({{}_{-1}} D^{s}_{x} {{}_{-1}} D^{s}_{x} u, v \bigr)_{\varLambda }= \bigl( {{}_{-1}} D ^{s}_{x} u, {{}_{x}} D^{s}_{1} v \bigr)_{\varLambda },\qquad \bigl({{}_{x}} D^{s}_{1} {{}_{x}} D^{s}_{1} u, v \bigr)_{\varLambda }= \bigl( {{}_{x}} D^{s} _{1} u, {_{-1}} D^{s}_{x} v \bigr)_{\varLambda }, \\& \bigl({{}_{x}} D^{s}_{1} {{}_{-1}} D^{s}_{x} u, v \bigr)_{\varLambda }= \bigl( {{}_{-1}} D ^{s}_{x} u, {{}_{-1}} D^{s}_{x} v \bigr)_{\varLambda },\qquad \bigl({{}_{-1}} D^{s}_{x} {{}_{x}} D^{s}_{1} u, v \bigr)_{\varLambda }= \bigl( {{}_{x}} D^{s} _{1} u, {{}_{x}} D^{s}_{1} v \bigr)_{\varLambda }. \end{aligned}$$

Lemma 2

([17])

Given \(s>0\), \(s\neq n-\frac{1}{2}\), \(n\in \mathbb{N}\), for \(w, v \in H ^{s}_{0}(\varLambda )\), we have

$$ \bigl({{}_{-1}} D^{s}_{x} {{}_{-1}} D^{s}_{x} u, v\bigr)_{\varLambda } \cong \cos ( \pi s) \bigl\Vert {{}_{-1}} D^{s}_{x} v \bigr\Vert ^{2}_{L^{2}(\varLambda )}\cong \cos ( \pi s) \bigl\Vert {{}_{x}} D^{s}_{1} v \bigr\Vert ^{2}_{L^{2}(\varLambda )} \cong \cos ( \pi s) \Vert v \Vert ^{2}_{s, \varLambda }. $$

From Lemmas 1 and 2, we obtain a bilinear form as follows:

$$\begin{aligned} a(u, w) :=& -\frac{1}{4} \sum^{d}_{j=1} \bigl[ \bigl({_{-1}}D^{\alpha }_{x_{j}} u, {{}_{x_{j}}} D_{1}^{\beta } w\bigr)+ \bigl({{}_{x_{j}}} D_{1}^{\alpha } u, {{}_{-1}}D^{\alpha }_{x_{j}} w\bigr) \\ &{}- \bigl({{}_{-1}}D^{\alpha }_{x_{j}} u, {{}_{-1}}D^{\alpha }_{x_{j}} w\bigr)- \bigl({{}_{x_{j}}} D_{1}^{\alpha } u, {{}_{x_{j}}} D_{1}^{\alpha } w \bigr) \bigr]. \end{aligned}$$

It is easy to check that \(\|u \|^{2}_{H^{\alpha }}\lesssim a(u, u)\).

Theorem 1

Suppose that \((u, v) \in H^{\alpha }_{0}(\varOmega )\), then we have the following estimates:

For \(\gamma \geq 1\),

$$ \Vert u \Vert ^{2}+ \frac{1}{\beta } \Vert v \Vert ^{2} \leq e^{-t} \biggl( \Vert u_{0} \Vert ^{2}+ \frac{1}{ \beta } \Vert v_{0} \Vert ^{2}\biggr) + \biggl(\frac{\eta ^{2}}{ \gamma \beta }+\biggl( \frac{a ^{2}+2}{2}\biggr)^{2} \biggr) \vert \varOmega \vert . $$
(3)

For \(1>\gamma > 0\),

$$ \Vert u \Vert ^{2}+ \frac{1}{\beta } \Vert v \Vert ^{2} \leq \Vert u_{0} \Vert ^{2}+ \frac{1}{ \beta } \Vert v_{0} \Vert ^{2} + \biggl(\frac{\eta ^{2}}{ \gamma \beta }+\biggl(\frac{a ^{2}+2}{2}\biggr)^{2} \biggr) \vert \varOmega \vert t. $$
(4)

Proof

Taking the inner product of the first equation of (2) with \(u(t)\) and of second equation with \(\frac{1}{ \beta } v(t)\), we have

$$ \frac{1}{2} \biggl( \frac{d}{dt} \Vert u \Vert ^{2}+\frac{1}{ \beta } \frac{d}{dt} \Vert v \Vert ^{2} +2\kappa \bigl\Vert D^{\frac{\alpha }{2}} u \bigr\Vert ^{2} \biggr)=\bigl(u(1-u) (u-a),u\bigr)+(-\gamma v+\eta , v). $$

Note that

$$\begin{aligned}& \begin{aligned} -u^{4}+(a+1)u^{3}-au^{2}&=- \frac{1}{2}u^{2} \bigl[ (u-a-1)^{2}+u^{2}- \bigl(a ^{2}+1\bigr) \bigr], \\ &=-\frac{1}{2}u^{2} \bigl[ (u-a-1)^{2} \bigr] -\frac{1}{2} \biggl[ \biggl(u^{2}-\frac{a ^{2}+2}{2} \biggr)^{2}+u^{2}-\biggl(\frac{a^{2}+2}{2} \biggr)^{2} \biggr], \end{aligned} \\& \begin{aligned} -\gamma v^{2}+\eta v&=-\frac{\gamma }{2} \biggl(v^{2}+ \frac{2\eta }{ \gamma }v+ \biggl(\frac{ \eta }{\gamma }\biggr)^{2}+v^{2}- \biggl(\frac{ \eta }{\gamma }\biggr)^{2} \biggr) \\ &=-\frac{\gamma }{2} \biggl(v+\frac{ \eta }{\gamma } \biggr)^{2}- \frac{\gamma }{2} v^{2} +\frac{\eta ^{2}}{2 \gamma }. \end{aligned} \end{aligned}$$

Then, we have

$$ \frac{d}{dt} \Vert u \Vert ^{2}+\frac{1}{ \beta } \frac{d}{dt} \Vert v \Vert ^{2} + \Vert u \Vert ^{2} +\frac{\gamma }{\beta } \Vert v \Vert ^{2} \leq \biggl(\frac{\eta ^{2}}{ \gamma \beta }+\biggl(\frac{a^{2}+2}{2}\biggr)^{2} \biggr) \vert \varOmega \vert . $$

This completes the proof. □

3 Numerical scheme and stability analysis

In this part, we will propose a C–N scheme in time, and we will also discuss the unconditional stability of our numerical scheme. First, we define the time step \(\delta t=T/M\), where M is a positive integer, \(t_{n}=n \delta t\), \(0 \leq n\leq M-1\). Consider the second-order time-discrete scheme as follows:

$$ \textstyle\begin{cases} \frac{u^{n+1}-u^{n}}{\delta t}=-\kappa (-\Delta )^{\alpha /2} u^{n+ \frac{1}{2}}+u^{n+\frac{1}{2}}(1-u^{*})(u^{*}-a)-v^{n+\frac{1}{2}}, \\ \frac{v^{n+1}-v^{n}}{\delta t}=\beta u^{n+\frac{1}{2}}-\gamma v^{n+ \frac{1}{2}}+\eta , \end{cases} $$
(5)

where \(u^{*}=\frac{3}{2}u^{n}-\frac{1}{2}u^{n-1}\), for \(n \geq 1\), and \(u^{*}=u^{0}\), for \(n=0\).

Theorem 2

The time discrete scheme (5) is unconditionally stable, and we have

$$ \bigl\Vert u^{k+1} \bigr\Vert ^{2}+ \frac{1}{\beta } \bigl\Vert v^{k+1} \bigr\Vert ^{2} \lesssim \bigl\Vert u^{0} \bigr\Vert ^{2}+ \bigl\Vert v^{0} \bigr\Vert ^{2}+1. $$
(6)

Proof

Taking the inner product of (5) with \(\delta t (u^{n+1}+u^{n})\) and \(\frac{1}{\beta } \delta t (v^{n+1}+v^{n})\), respectively, we get

$$\begin{aligned} & \bigl\Vert u^{n+1} \bigr\Vert ^{2}- \bigl\Vert u^{n} \bigr\Vert ^{2}+\frac{1}{\beta } \bigl( \bigl\Vert v^{n+1} \bigr\Vert ^{2}- \bigl\Vert v^{n} \bigr\Vert ^{2} \Vert \bigr)+2 \kappa \bigl\Vert D^{\frac{\alpha }{2}} u^{n+ \frac{1}{2}} \bigr\Vert ^{2} \\ &\quad =-\delta t \bigl( \bigl\Vert u^{n+\frac{1}{2}}\bigl(u^{*}-a-1 \bigr) \bigr\Vert ^{2}+ \bigl\Vert u^{n+ \frac{1}{2}} u^{*} \bigr\Vert ^{2}-\bigl(a^{2}+1\bigr) \bigl\Vert u^{n+\frac{1}{2}} \bigr\Vert ^{2} \bigr) \\ &\qquad {}-\delta t \biggl( \frac{\gamma }{\beta } \biggl\Vert v^{n+\frac{1}{2}}+ \frac{ \eta }{\gamma } \biggr\Vert ^{2}-\frac{\gamma }{\beta } \bigl\Vert v^{n+\frac{1}{2}} \bigr\Vert ^{2} +\frac{\eta ^{2}}{\beta \gamma } \vert \varOmega \vert \biggr). \end{aligned}$$

Dropping some positive terms and summing up over \(n=1, 2, \ldots , k \), we get

$$ \bigl\Vert u^{k+1} \bigr\Vert ^{2}+ \frac{1}{\beta } \bigl\Vert v^{k+1} \bigr\Vert ^{2}\leq \delta t \bigl(a ^{2}+1\bigr) \sum ^{k}_{n=1} \bigl\Vert u^{n+\frac{1}{2}} \bigr\Vert ^{2}+ \biggl( \bigl\Vert u ^{1} \bigr\Vert + \frac{1}{\beta } \bigl\Vert v^{1} \bigr\Vert ^{2}+\frac{\eta ^{2} T}{\beta \gamma } \vert \varOmega \vert \biggr). $$

Using discrete Gronwall lemma, one has

$$ \bigl\Vert u^{k+1} \bigr\Vert ^{2}+ \frac{1}{\beta } \bigl\Vert v^{k+1} \bigr\Vert ^{2} \lesssim \bigl\Vert u^{1} \bigr\Vert + \frac{1}{\beta } \bigl\Vert v^{1} \bigr\Vert ^{2}+1. $$

Linking the first step, we can prove that

$$ \bigl\Vert u^{1} \bigr\Vert ^{2}+ \frac{1}{\beta } \bigl\Vert v^{1} \bigr\Vert ^{2} \lesssim \bigl\Vert u^{0} \bigr\Vert ^{2}+ \bigl\Vert v^{0} \bigr\Vert ^{1} +1. $$

This ends the proof. □

4 Numerical implementation and numerical results

We consider the Legendre-spectral method to discretize in the spatial direction. We obtain the full discrete scheme of problem (5) as follows:

$$ \textstyle\begin{cases} (\frac{u^{n+1}_{N}-u^{n}_{N}}{\delta t}, \phi _{N})=-\kappa a(u^{n+ \frac{1}{2}}_{N}, \phi _{N}) +(u^{n+\frac{1}{2}}_{N}(1-u^{*}_{N})(u ^{*}_{N}-a)-v^{n+\frac{1}{2}}_{N},\phi _{N}), \quad \phi _{N}\in \mathbb{P}_{N}, \\ (\frac{v^{n+1}_{N}-v^{n}_{n}}{\delta t}, \varphi _{N})=\beta (u^{n+ \frac{1}{2}}_{N}, \varphi _{N})-\gamma (v^{n+\frac{1}{2}}_{N},\varphi _{N})+(\eta , \varphi _{N}), \quad \varphi _{N}\in \mathbb{P}_{N}. \end{cases} $$
(7)

Define

$$ u^{n}_{N}=\sum^{N-1}_{i,j=1}u^{n}_{i,j}L_{N,i}(x)L_{N,j}(y), v ^{n}_{N}=\sum^{N-1}_{i,j=1}v^{n}_{i, j}L_{N,i}(x)L_{N,j}(y), $$

where \(L_{N,i}(x)\) are Lagrangian polynomials, \(u^{n}_{i,j}\) denote the unknown coefficients.

In order to verify the asymptotic behavior of the solutions, the effect of space-fractional derivative α will also be investigated. The numerical method (7) is computed in the square \([0, 2.5] \times [0, 2.5]\) with \(\kappa =10^{-4}\), \(\mu =0.1\), \(\beta =5\times 10^{-3}\), \(\gamma =10^{-2}\), and \(\eta =0\). Tables 1–4 display the temporal convergence orders and the errors in the \(L^{2}\) and \(H^{\alpha /2}\) norms for \(\alpha =1.5\) and 1.7. It is confirmed that our numerical scheme can achieve second-order accuracy in time, and the numerical solutions are in good agreement with the exact solution. In addition, using scheme (7), we simulate the dynamic behavior for different α, where the results are summarized in Fig. 1. As can be seen from this figure, due to the long-tail mechanism of the fractional Laplace operator, the wavelength becomes larger when the fractional diffusion coefficient α becomes larger. This shows that a fractional equation with diffusion mechanisms is a powerful tool to describe dynamic state.

Figure 1
figure 1

The dynamic behavior for different α

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Table 1 The \(L^{2}\) and \(H^{\alpha /2}\) numerical errors at \(\alpha =1.5\) for various temporal resolutions u
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Table 2 The \(L^{2}\) and \(H^{\alpha /2}\) numerical errors at \(\alpha =1.5\) for various temporal resolutions v
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Table 3 The \(L^{2}\) and \(H^{\alpha /2}\) numerical errors at \(\alpha =1.7\) for various temporal resolutions u
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Table 4 The \(L^{2}\) and \(H^{\alpha /2}\) numerical errors at \(\alpha =1.7\) for various temporal resolutions v
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5 Conclusions

An efficient linearized numerical scheme is constructed to solve the space fractional FitzHugh–Nagumo equation. The numerical scheme is proved to be stable. Numerical examples show that the proposed scheme is effective. Moreover, the fractional diffusion coefficient α has a significant effect on the dynamic behavior.