Special issue on mathematical models for collective dynamics

José Antonio Carrillo; Seung-Yeal Ha; Lorenzo Pareschi; Benedetto Piccoli; José Antonio Carrillo; Seung-Yeal Ha; Lorenzo Pareschi; Benedetto Piccoli

doi:10.3934/nhm.2020020

Networks and Heterogeneous Media

2020, Volume 15, Issue 3: i-i. doi: 10.3934/nhm.2020020

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Special issue on mathematical models for collective dynamics

1.
Mathematical Institute, University of Oxford, UK
2.
Seoul National University, Republic of Korea
3.
Ferrara University, Italy
4.
Rutgers University, Camden, USA

Published: 09 September 2020

Citation: José Antonio Carrillo, Seung-Yeal Ha, Lorenzo Pareschi, Benedetto Piccoli. 2020: Special issue on mathematical models for collective dynamics, Networks and Heterogeneous Media, 15(3): i-i. doi: 10.3934/nhm.2020020

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1. Introduction

Reaction-diffusion phenomena is ubiquitous occurring mostly in different fields of science and engineering in which the system components interact ^[1,2,3,4,5]. They are used to describe population growth, propagation of travelling waves, pattern formation, other intriguing phenomena arising in combustion theory, tumor invasion, and neural networks spanning across various temporal and spatial scales ^[6,7,8,9]. Mathematically, the reaction-diffusion model is a parabolic partial differential equation (PDE) involving terms which denote diffusion and local reaction kinetics as given below:

$\begin{equation} \partial_{t}\mathrm{Y}-D\nabla^{2}\mathrm{Y} = Q(\mathrm{Y}), \end{equation}$

(1.1)

where the first term on the left side represents the temporal part, the second term represents the diffusion component, and the term on the right side of equation denotes the local reaction kinetics. For the choice of $Q(\mathrm{Y}) = \mathrm{Y} (1-\mathrm{Y})$ , the resultant model is Fisher's equation that can be used to describe population evolution and how the wave propagates in certain medium ^[10]. If $Q(\mathrm{Y}) = \mathrm{Y} (1-\mathrm{Y}^{2})$ , the reaction-diffusion is called Newell-Whitehead-Segel (NWS) equation which is primarily used to describe convection phenomena in fluid thermodynamics ^[11]. Zeldovich-Frank-Kamenetskii equation (ZFK) is obtained for the choice of $Q(\mathrm{Y}) = \mathrm{Y} (1-\mathrm{Y})e^{-\beta (1-\mathrm{Y})}$ . In combustion theory, the ZFK equation is used to describe how flames propagate ^[12].

In theoretical neuroscience, the most well-studied reaction diffusion equation is the FitzHugh-Nagumo (FHN) model ^[13,14]. The FHN model is the simplification of the Hodgkin-Huxley model for the action potentials in squid giant axon ^[15,16,17]. Although FHN is not phenomenological (the involved parameters are not biophysical), it is a good candidate for studying how an action potential is generated and propagated. The main advantage of the FHN is that the solution space is two-dimensional and hence geometrical phase plane tools can be utilized to show how the trajectory evolves, which gives rise to the excitability and spike mechanism of spike generation. In the phase plane diagram of the FHN neuronal model, one of the nullclines has cubic nonlinearity. Over the last few decades, due to its model simplicity, FHN gained a lot of attention in the scientific community. Besides the generation and propagation of a train of spikes, FHN is also used to explained the dynamics of various physical systems in several fields of science, typically in the propagation of flame, Brownian motion process, autocatalytic chemical reaction, growth of logistic population, and neurophysiology ^[18,19].

Consider the following scaler reaction diffusion model with constant coefficients:

$\begin{equation} \partial_{t}\mathrm{Y}-\partial_{\xi\xi}\mathrm{Y} = \mathrm{Y}(1-\mathrm{Y})(\mathrm{Y}-\eta), \end{equation}$

(1.2)

where $\eta$ is a known real number and $\mathrm{Y}(\xi, t)$ is an unknown function to be determined. In this equation, the local reaction kinetics is denoted by the cubic function $\mathrm{Y}(1-\mathrm{Y})(\mathrm{Y}-\eta)$ . For $\eta = -1$ , Eq (1.2) reduces to the NWS equation.

In literature, various analytical and numerical strategies have been used to solve FHN-type equation. Shih et al. ^[20] used approximate conditional Lie point symmetry method for the numerical solution of the perturbed FHN-type equations. Mehta et al. ^[21] solved such nonlinear time dependent problems by using a novel block method coupled with compact finite difference schemes. In the perturbed model, the term $\epsilon \mathrm{Y}$ has been added to the cubic term and for $\epsilon = 0,$ the problem reduces to an unperturbed FHN-type model. Li and Guo implemented an integral approach for the exact solution of FHN-type equations ^[22]. Abbasbandy proposed a homotopy analysis method (HAM) to calculate a solitary-type solution of the nonlinear FHN-type model ^[23]. Kawahara and Tanaka derived the exact solution for the interaction of traveling fronts ^[24]. Gorder and Vajravelua obtained exact solutions for FHN-type equations and Nagumo telegraph equation using a variational method for fixed initial conditions in order to control the error ^[25]. Ali et al. proposed a Galerkin finite element method for the numerical solutions of FHN-type equation^[26]. Mehran and Sadegh used a non-conventional finite difference scheme for the numerical solutions of FHN-type equations ^[27]. Gorder implemented HAM for the approximate solutions of FHN-type reaction diffusion equations ^[28].

Reaction-diffusion of FHN-type equations discusses previously involved constant coefficients. However, in general, time-dependent coefficients and dispersion-reaction terms involved in such models are more realistic physically ^[29,30,31]. Due to variation in problem geometry and taking into account the factor of heterogeneity in the propagating medium, FHN-type models with time-dependent coefficients are quite sophisticated to study.

The main goal of this work is to demonstrate the numerical solutions of the generalized FitzHugh Nagumo (GFHN) equation with a linear dispersion term and time-dependent coefficients, which are given below:

$\begin{equation} \partial_{t}\mathrm{Y}+\alpha(t)\partial_{\xi}\mathrm{Y}-\vartheta(t)\partial_{\xi\xi}\mathrm{Y} -\delta(t)\mathrm{Y}(1-\mathrm{Y})(\mathrm{Y}-\eta) = 0,\; (\xi,t)\in[a,b]\times[0,t], \end{equation}$

(1.3)

where $\alpha(t)$ , $\vartheta(t)$ , and $\delta(t)$ are time-dependent coefficients. The associated initial condition is:

$\begin{equation} \mathrm{Y}(\xi,0) = \psi_{0}(\xi), \; \xi\in[a,b], \end{equation}$

(1.4)

and the boundary conditions are:

$\begin{equation} \mathrm{Y}(a,t) = \psi_{a}(t),\; \mathrm{Y}(b,t) = \psi_{b}(t), \,\, t\geq 0, \end{equation}$

(1.5)

where $\psi_{0}(\xi), \psi_{a}(t),$ and $\psi_{b}(t)$ are the known functions.

In recent literature, various authors computed the numerical solutions of GFHN equation with time-dependent coefficients. For instance, tanh and the Jacobi Gauss Lobatto collocation technique ^[18,29], the polynomial differential quadrature method ^[32], and the mixed-types discontinuous Galerkin method ^[33,34] were used.

Recently, wavelet numerical procedures attracted different researchers in various scientific communities because of their good properties and easy implementation. Some interesting aspects of wavelet methods include, compact support, localization in time and space, and multi-resistant analysis. Among different families of wavelet, Haar wavelet (HW) is simple and popular. The importance of the HW has been explored in various articles: Lepik proposed HW-based methods for different problems ^[35], Jiwari ^[36] utilized the HW method for the solution of Burger's equation, Kumar and Pandit ^[37] solved coupled Burger's equations using the HW method, Shiralashetti et al. ^[38] used the HW scheme for the numerical solutions of the initial valued problems. For further analysis of the HW, interested readers are referred to ^[39,40,41]. The remaining parts of the paper are presented in the following pattern:

● The underlying motivation and preliminaries are given in Sections 2 and 3, respectively.

● The proposed methodology is presented in Section 4.

● Numerical results and stability are reported in Sections 5 and 6.

● Finally, conclusions are drawn in Section 7.

2. Main motive

The analytical solution of the time-dependent coefficients partial differential equations is quite complicated to compute. Therefore, numerical treatment is an alternative and an efficient way to cope with this issue. According to our analysis, the method of lines using the HW is not proposed for FHN-type models. In this work, we propose an HW-based method of lines for the solutions of FHN-type equations. The theoretical stability and its numerical verification will also be a part of this work.

3. Haar wavelet and its integrals

Here, we address some basic results. Consider an arbitrary interval $[a, b)$ and divide it into $2L$ equal subintervals of length $\partial \xi = \frac{b-a}{2L}$ , where $L = 2^\mathcal{J}$ represents the maximum level of resolution. Define the dilation and translational parameters as $j = 0, 1, ..., \mathcal{J}$ , $\mathcal{K} = 0, 1, ..., l-1$ , where $l = 2^{j}$ , respectively. Using the dilation and translational parameters, define the wavelet number $\iota = l+\mathcal{K}+1$ . Now the first and $\iota$ -th HW are given below ^[35]:

$\begin{eqnarray} \mathrm{h}_{1}(\xi) = \left\{ \begin{array}{ll} 1, & { \xi\in[a,b) },\\ 0, & \mbox{elsewhere} .\end{array} \right. \end{eqnarray}$

(3.1)

$\begin{eqnarray} \mathrm{h}_{\iota}(\xi) = \left\{ \begin{array}{lll} 1, & { \xi\in[\zeta_{1}(\iota), \zeta_{2}(\iota)) },\\ -1, & { \xi\in[\zeta_{2}(\iota), \zeta_{3}(\iota)) },\\ 0, & \mbox{elsewhere,}\end{array} \right. \end{eqnarray}$

(3.2)

where $\zeta_{1}(\iota) = a+2\mathcal{K}\omega\partial \xi$ , $\zeta_{1}(\iota) = a+(2\mathcal{K}+1)\omega\delta \xi$ , $\zeta_{3}(\iota) = a+2(\mathcal{K}+1)\omega\partial \xi,$ and $\omega = \frac{\mathrm{L}}{l}$ .

In our analysis, we will use the following integrals:

$\begin{equation} \beta_{\iota,\alpha}(\xi) = \displaystyle {\int}_{a}^{\xi}\displaystyle {\int}_{a}^{\xi},...,\displaystyle {\int}_{a}^{\xi}\mathrm{h}_{\iota}(z) dz^{\alpha} = \frac{1}{(\alpha-1)!}\displaystyle {\int}_{a}^{\xi}(\xi-z)^{\alpha-1}\mathrm{h}_{\iota}(z) dz, \end{equation}$

(3.3)

where $\alpha = 1, 2, 3, ..., n$ and $\iota = 1, 2, 3, ..., 2L$ . Analytically, the evaluation of these integrals for Eqs (3.1) and (3.2) are given below:

$\begin{equation} \beta_{1,\alpha}(\xi) = \frac{(\xi-a)^\alpha}{\alpha!}, \end{equation}$

(3.4)

$\begin{eqnarray} \begin{gathered} \beta_{\iota,\alpha}(\xi) = \left\{ \begin{array}{llll} 0, & { \xi < \zeta_{1}(\iota) },\\ \frac{1}{\alpha!}[\xi-\zeta_{1}(\iota)]^{\alpha}, & { \xi\in[\zeta_{1}(\iota),\zeta_{2}(\iota)] },\\ \frac{1}{\alpha!}[(\xi-\zeta_{1}(\iota))^{\alpha}-2(\xi-\zeta_{2}(\iota))^{\alpha}], & { \xi\in[\zeta_{2}(\iota),\zeta_{3}(\iota)] },\\ \frac{1}{\alpha!}[(\xi-\zeta_{1}(\iota))^{\alpha}-2(\xi-\zeta_{2}(\iota))^{\alpha}+(\xi-\zeta_{3}(\iota))^{\alpha}],& { \xi > \zeta_{3}(\iota) }.\end{array} \right. \ \end{gathered} \end{eqnarray}$

(3.5)

4. Proposed scheme

In this section, the proposed method is described in detail. Here, an integral approach is utilized, therefore, the greatest order spatial derivative in Eq (1.3) can be approximated via the HW series as:

$\begin{equation} \partial_{\xi\xi}\mathrm{Y}(\xi,t) = \sum\limits_{\iota = 1}^{2L}\lambda_\iota(t) \mathrm{h}_\iota(\xi), \end{equation}$

(4.1)

where $\lambda_\iota (t)$ denotes the unknown HW coefficients and $\mathrm{h}_\iota(\xi)$ are HW basis. From Eq (4.1), we deduce the following equations via twice integration:

$\begin{equation} \partial_{\xi}\mathrm{Y}(\xi,t) = \sum\limits_{\iota = 1}^{2L}\lambda_\iota(t) \beta_{1,\iota}(\xi)+\partial_\xi\mathrm{Y}(\xi,t)|_{\xi = a}, \end{equation}$

(4.2)

$\begin{equation} \mathrm{Y}(\xi,t) = \sum\limits_{\iota = 1}^{2L}\lambda_\iota \beta_{2,\iota}(\xi)+(\xi-a)\partial_\xi\mathrm{Y}(\xi,t)|_{\xi = a}+\mathrm{Y}(\xi,t)|_{\xi = a}. \end{equation}$

(4.3)

The evaluation of Eq (4.3) at $\xi = b$ gives:

$\begin{equation} \partial _{\xi}\mathrm{Y}(a,t) = \frac{\psi_{b}(t)-\psi_{a}(t)}{b-a}-\frac{1}{b-a}\sum\limits_{\iota = 1}^{2L}\lambda_\iota(t) \beta_{2,\iota}(b). \end{equation}$

(4.4)

Plugging $\partial _{\xi}\mathrm{Y}(a, t)$ in Eqs (4.2) and (4.3), we get:

$\begin{equation} \partial_{\xi}\mathrm{Y}(\xi,t) = \sum\limits_{\iota = 1}^{2L}\lambda_\iota(t)\big[\beta_{1,\iota}(\xi)-\frac{1}{(b-a)}\beta_{2,\iota}(b)\big]+\frac{1}{(b-a)}\big(\psi_{b}(t) -\psi_{a}(t) \big), \end{equation}$

(4.5)

$\begin{equation} \mathrm{Y}(\xi,t) = \sum\limits_{\iota = 1}^{2L}\lambda_\iota(t)\big[\beta_{2,\iota}(\xi)-\frac{(\xi-a)}{(b-a)}\beta_{2,\iota}(b)\big]+\frac{(\xi-a)}{(b-a)}\psi_{b}(t)+\frac{(b-\xi)}{(b-a)}\psi_{a}(t). \end{equation}$

(4.6)

Now, the corresponding matrix form of Eqs (4.1)–(4.5), and (4.6) by using $\xi \rightarrow \xi_{m}$ are:

$\begin{equation} \partial_{\xi\xi}\mathrm{Y}(\xi,t) = H(\xi_{m})\lambda(t), \end{equation}$

(4.7)

$\begin{equation} \partial_{\xi}\mathrm{Y}(\xi,t) = \mathrm{M}_{1}(\xi_{m})\lambda(t)+\mathrm{M}_{2}(t), \end{equation}$

(4.8)

$\begin{equation} \mathrm{Y}(\xi,t) = \mathrm{N}_{1}(\xi_{m})\lambda(t)+\mathrm{N}_{2}(t), \end{equation}$

(4.9)

where

$\mathrm{M}_{1}(\xi_{m}) = \sum\limits_{\iota = 1}^{2L}\big[\beta_{1,\iota}(\xi_{m})-\frac{1}{(b-a)}\beta_{2,\iota}(b)\big],\; \mathrm{M}_{2}(t) = \frac{1}{(b-a)}\big(\psi_{b}(t) - \psi_{a}(t) \big),$

$\mathrm{N}_{1}(\xi_{m}) = \big[\beta_{2,\iota}(\xi_{m})-\frac{(\xi_{m}-a)}{(b-a)}\beta_{2,\iota}(b)\big],\; \mathrm{N}_{2}(t) = \frac{(\xi_{m}-a)}{(b-a)}\psi_{b}(t)+\frac{(b-\xi_{m})}{(b-a)}\psi_{a}(t).$

From Eqs (4.7)–(4.9) and (1.3), the following equation can be obtained:

$\begin{equation} \begin{split} &\frac{d\mathrm{Y}_{m}(t)}{dt} = -\alpha(t)\Big(\mathrm{M}_{1}(\xi_{m})\lambda(t)-\mathrm{M}_{2}(t)\Big) +\vartheta(t)H(\xi_{m}) \big)+\delta(t)\Big[(\mathrm{N}_{1}(\xi_{m})\lambda(t) +\mathrm{N}_{2}(t))*\\ &\Big(1-(\mathrm{N}_{1}(\xi_{m})\lambda(t)+\mathrm{N}_{2}(t))\Big)*\Big((\mathrm{N}_{1}(\xi_{m})\lambda(t)+\mathrm{N}_{2}(t))-\eta \Big)\Big],\; \xi\in[a,b], \; t > 0, \end{split} \end{equation}$

(4.10)

where "*" represents an element wise product. In a more compact form, Eq (4.10) can be written as:

$\begin{equation} \frac{d \mathrm{Y}}{dt} = \mathbf{F}(t,\mathrm{Y}),\; \mathrm{Y}(0) = \psi_{0}, \end{equation}$

(4.11)

where $\mathbf{F}(t, \mathrm{Y}) = \vartheta(t)\big(\mathrm{h}\lambda(t) \big)- \alpha(t)\Big(\mathrm{M}_{1}\lambda(t)+\mathrm{M}_{2}(t)\Big)$ - $\delta(t) \Big[(\mathrm{N}_{1}\lambda(t)+\mathrm{N}_{2}(t))*\Big(1-(\mathrm{N}_{1}\lambda(t) +\mathrm{N}_{2}(t))\Big)*\Big((\mathrm{N}_{1}\lambda(t)+\mathrm{N}_{2}(t))-\eta \Big)\Big].$

Here, Eq (4.11) represents the system of first-order ordinary differential equations, which can be solved via the RK-4 scheme discussed in a later section.

4.1. Temporal discretization

To obtain the solutions of Eq (4.11), we use the following $\mbox{RK-4}$ scheme:

$\begin{gathered} \mathcal{K}_1 = \mathbf{F}(t^m ,\mathrm{Y}^m), \\ \mathcal{K}_2 = \mathbf{F}(t^m + \frac{\partial t}{2} , \mathrm{Y}^m+\frac{\mathcal{K}_1}{2}), \\ \mathcal{K}_3 = \mathbf{F}(t^m + \frac{\partial t}{2},\mathrm{Y}^m+\frac{\mathcal{K}_2}{2}), \\ \mathcal{K}_4 = \mathbf{F}(t^m + {\partial t},\mathrm{Y}+{\mathcal{K}_3}).\\ \mathrm{Y}^{m+1} = \mathrm{Y}^{m}+\frac{\partial t}{6}\big(\mathcal{K}_1 +2(\mathcal{K}_2+\mathcal{K}_3)+\mathcal{K}_4\big), \,\,\, m\geq 0, \end{gathered}$

(4.12)

where $\partial t$ is the time step size.

5. Simulations and comparison

In this section, numerical solutions of GFHN with constant and time-dependent coefficient are listed. The efficiency of the proposed scheme is checked by applying different error measures, namely: $L_2$ , $L_\infty$ , and $L_{rms}$ as described below:

$\begin{equation} L_2 = \|\mathrm{Y}^E-\mathrm{Y}^N\|_{2}\cong\sqrt{\delta \xi \sum\limits_{\iota = 1}^{2L} {\mid{\mathrm{Y}_{\iota}^E-\mathrm{Y}_{\iota}^N}\mid}^2}, \end{equation}$

(5.1)

$\begin{equation} L_\infty = \|\mathrm{Y}^E-\mathrm{Y}^N\|_{\infty}\cong max_{\infty} {\mid{\mathrm{Y}_{\iota}^E-\mathrm{Y}_{\iota}^N}\mid}, \end{equation}$

(5.2)

$\begin{equation} L_{rms} = \sqrt{\sum\limits_{\iota = 1}^{2L}\big(\frac{\mathrm{Y}^E-\mathrm{Y}^N}{\sqrt{2L}} \big)^2}. \end{equation}$

(5.3)

5.1. Problem 1

Here, we consider Eq (1.2) with constant coefficients and various choices of the exact solutions:

$\begin{equation*} \label{e1.1} \begin{split} &\mbox{case}\; (i) :\; \mathrm{Y}(\xi,t) = \frac{1}{2}+\frac{1}{2}\tanh\Big[\frac{1}{2\sqrt{2}}\big( \xi-\frac{2\eta-1}{\sqrt{2}}t\big) \Big],\\ &\mbox{case}\; (ii) :\; \mathrm{Y}(\xi,t) = \frac{1}{2}(1+\alpha)+\frac{1}{2}(1-\alpha)\tanh\Big[\frac{(1-\alpha)}{2\sqrt{2}} \xi+\frac{(1-\alpha^2)}{2\sqrt{2}}t \Big],\\ &\mbox{case}\; (iii) :\; \mathrm{Y}(\xi,t) = \frac{1}{1+\exp\big[\frac{-\xi+(\sqrt{2}(\frac{1}{2}-\eta))t}{\sqrt{2}} \big]},\\ &\mbox{case}\; (iv) :\; \mathrm{Y}(\xi,t) = \Big[\frac{1}{2}\Big]\Big[1-\coth\big(\frac{\xi}{2\sqrt{2}}+\frac{2\eta-1}{4}t+\frac{\pi}{4}\big)\Big].\\ \end{split} \end{equation*}$

The associated initial and boundary conditions for all cases are used from the exact solutions. Simulations are done in different spatial domains to test the technique, and a comparison is made with the previously published work. The spatial domains used for case $(i)$ are $[0, 1]$ and $[-10, 10]$ , for case $(ii)$ is $[-22, 22]$ , for case $(iii)$ are $[0, 1]$ and $[-10, 10]$ , while for case $(iv)$ is the interval $[-10, 10]$ . The obtained results of case $(i)$ in $[-10, 10]$ with different values of $\eta$ are given in –, while the outcomes in the interval $[0, 1]$ with various values of $\eta$ are given in Table 4.

Table 1. Outcomes of problem 1 for case

$(i)$ with

$\eta = 0.75$ .

	Present ( $\partial t=0.0001$ )			Present ( $\partial t=0.001$ )			Present ( $\partial t=0.1$ )
$t$	$L_\infty$	$L_{rms}$	CPU time (Seconds)	$L_\infty$	$L_{rms}$	CPU time (Seconds)	$L_\infty$	$L_{rms}$	CPU time (Seconds)
0.2	$6.387\times10^{-6}$	$1.844\times10^{-7}$	6.240240	$1.203\times10^{-5}$	$1.782\times10^{-6}$	2.195968	$8.339\times10^{-4}$	$1.835\times10^{-4}$	1.624090
0.5	$1.414\times10^{-5}$	$4.959\times10^{-7}$	8.821217	$2.809\times10^{-5}$	$4.515\times10^{-6}$	2.954488	$2.027\times10^{-3}$	$4.626\times10^{-4}$	1.805044
1.0	$2.419\times10^{-5}$	$1.124\times10^{-6}$	13.053880	$5.211\times10^{-5}$	$9.234\times10^{-6}$	3.111479	$3.917\times10^{-3}$	$9.362\times10^{-4}$	1.839039
1.5	$3.230\times10^{-5}$	$1.899\times10^{-6}$	18.199838	$7.429\times10^{-5}$	$1.415\times10^{-5}$	5.149131	$5.822\times10^{-3}$	$1.418\times10^{-3}$	1.942381
2.0	$3.926\times10^{-5}$	$2.816\times10^{-6}$	23.565382	$9.548\times10^{-5}$	$1.924\times10^{-5}$	4.562245	$7.718\times10^{-3}$	$1.906\times10^{-3}$	1.919129
5.0	$7.276\times10^{-5}$	$1.0461\times10^{-5}$	54.294368	$2.174\times10^{-4}$	$5.217\times10^{-5}$	8.126805	$1.840\times10^{-2}$	$4.896\times10^{-3}$	2.256312
	^[42] ( $\partial t=0.0001$ )			^[32] ( $\partial t=0.001$ )			^[43] ( $\partial t=0.1$ )
$t$	$L_\infty$	$L_{rms}$		$L_\infty$	$L_{rms}$		$L_\infty$	$L_{rms}$
0.2	$1.889\times10^{-5}$	$2.196\times10^{-7}$		$4.741\times10^{-5}$	$1.588\times10^{-5}$		$1.887\times10^{-5}$	$7.455\times10^{-6}$
0.5	$4.155\times10^{-5}$	$1.569\times10^{-6}$		$1.231\times10^{-4}$	$3.843\times10^{-5}$		$4.151\times10^{-5}$	$1.641\times10^{-5}$
1.0	$6.989\times10^{-5}$	$7.144\times10^{-6}$		$2.626\times10^{-4}$	$8.187\times10^{-5}$		$6.973\times10^{-5}$	$2.743\times10^{-5}$
1.5	$9.168\times10^{-5}$	$1.726\times10^{-5}$		$4.209\times10^{-4}$	$1.338\times10^{-4}$		$9.118\times10^{-5}$	$3.534\times10^{-5}$
2.0	$1.096\times10^{-4}$	$3.185\times10^{-5}$		$5.999\times10^{-3}$	$1.943\times10^{-4}$		$1.085\times10^{-4}$	$4.128\times10^{-5}$
5.0	$1.896\times10^{-4}$	$1.880\times10^{-4}$		$2.305\times10^{-3}$	$7.863\times10^{-4}$		$1.800\times10^{-4}$	$6.134\times10^{-5}$

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Table 2. Outcomes of problem 1 for case

$(i)$ with

$\eta = -1$ .

		$t=0.1$			$t=0.5$			$t=1.0$
	$\xi$	Exact	Numerical	Error	Exact	Numerical	Error	Exact	Numerical	Error
Present
	-10	$1.013383\times10^{-3}$	$1.013382\times10^{-3}$	$8.713\times10^{-10}$	$1.844968\times10^{-3}$	1.844963 $\times10^{-3}$	$4.687\times10^{-9}$	$3.897765\times10^{-3}$	$3.897749\times10^{-3}$	$1.573\times10^{-8}$
	-7	$8.209472\times10^{-3}$	$8.209430\times10^{-3}$	$4.275\times10^{-8}$	$1.485835\times10^{-2}$	1.485800 $\times10^{-2}$	$3.444\times10^{-7}$	$3.094160\times10^{-2}$	$3.094052\times10^{-2}$	$1.078\times10^{-6}$
	-3	$1.216556\times10^{-1}$	$1.216557\times10^{-1}$	$1.920\times10^{-7}$	$2.015162\times10^{-1}$	2.015188 $\times10^{-1}$	$2.553\times10^{-6}$	$3.482263\times10^{-1}$	$3.482340\times10^{-1}$	$7.646\times10^{-6}$
	0	$5.305565\times10^{-1}$	$5.305566\times10^{-1}$	$1.619\times10^{-7}$	$6.731305\times10^{-1}$	6.731300 $\times10^{-1}$	$5.918\times10^{-7}$	$8.134186\times10^{-1}$	$8.134174\times10^{-1}$	$1.247\times10^{-6}$
	3	$9.069410\times10^{-1}$	$9.069410\times10^{-1}$	$2.120\times10^{-7}$	$9.466898\times10^{-1}$	9.466899 $\times10^{-1}$	$1.552\times10^{-7}$	$9.740892\times10^{-1}$	$9.740895\times10^{-1}$	$3.233\times10^{-7}$
	7	$9.939053\times10^{-1}$	$9.939053\times10^{-1}$	$3.713\times10^{-8}$	$9.966459\times10^{-1}$	9.966460 $\times10^{-1}$	$1.017\times10^{-7}$	$9.984128\times10^{-1}$	$9.984129\times10^{-1}$	$9.444\times10^{-8}$
	10	$9.992490\times10^{-1}$	$9.992490\times10^{-1}$	$7.370\times10^{-10}$	$9.995877\times10^{-1}$	9.995877 $\times10^{-1}$	$1.214\times10^{-9}$	$9.998052\times10^{-1}$	$9.998052\times10^{-1}$	$9.589\times10^{-10}$
^[33]
	-10	$0.123172E\times10^{-2}$	$0.123147\times10^{-2}$	$2.974\times10^{-6}$	$0.273123\times10^{-2}$	0.273299 $\times10^{-2}$	$1.086\times10^{-6}$	$0.639709\times10^{-2}$	$0.639771\times10^{-2}$	$4.544\times10^{-6}$
	-7	$0.799954\times10^{-2}$	$0.800058\times10^{-2}$	$1.005\times10^{-6}$	$0.149620\times10^{-1}$	0.149662 $\times10^{-1}$	$1.086\times10^{-6}$	$0.639709\times10^{-2}$	$0.639771\times10^{-2}$	$4.544\times10^{-6}$
	-3	$0.122814\times10^{+0}$	$0.122808\times10^{+0}$	$1.352\times10^{-6}$	$0.198331\times10^{+0}$	0.198340 $\times10^{+0}$	$2.984\times10^{-6}$	$0.351539\times10^{+0}$	$0.351533\times10^{+0}$	$9.543\times10^{-3}$
	0	$0.537579\times10^{+0}$	$0.537585\times10^{+0}$	$1.550\times10^{-6}$	$0.680017\times10^{+0}$	0.680007 $\times10^{+0}$	$8.287\times10^{-6}$	$0.819036\times10^{+0}$	$0.819048\times10^{+0}$	$4.715\times10^{-6}$
	3	$0.909059\times10^{+0}$	$0.909004\times10^{+0}$	$2.442\times10^{-6}$	$0.9487161\times10^{+0}$	0.948717 $\times10^{+0}$	$9.415\times10^{-6}$	$0.975932\times10^{+0}$	$0.975967\times10^{+0}$	$3.343\times10^{-6}$
	7	$0.994343\times10^{+0}$	$0.994331\times10^{+0}$	$2.683\times10^{-6}$	$0.997526\times10^{+0}$	0.997569 $\times10^{+0}$	$9.821\times10^{-7}$	$0.999603\times10^{+0}$	$0.999694\times10^{+0}$	$1.900\times10^{-6}$
	10	$0.999376\times10^{+0}$	$0.999372\times10^{+0}$	$9.138\times10^{-7}$	$0.9999983\times10^{+0}$	0.100002 $\times10^{+1}$	$7.363\times10^{-7}$	$0.999998\times10^{+0}$	$0.100001\times10^{+1}$	$1.930\times10^{-7}$

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Table 3. Solutions of problem 1 for case

$(i)$ with

$\eta = -2$ .

	$t=0.1$			$t=0.5$			$t=1.0$
$\xi$	Exact	Numerical	Error	Exact	Numerical	Error	Exact	Numerical	Error
-10	$1.119842\times10^{-3}$	$1.119841\times10^{-3}$	$9.172\times10^{-10}$	$3.038202\times10^{-3}$	3.038195 $\times10^{-3}$	$7.248\times10^{-9}$	$1.052473\times10^{-2}$	$1.052469\times10^{-2}$	$3.677\times10^{-8}$
-7	$9.065044\times10^{-3}$	$9.064999\times10^{-3}$	$4.448\times10^{-8}$	$2.426341\times10^{-2}$	2.426292 $\times10^{-2}$	$4.832\times10^{-7}$	$7.986202\times10^{-2}$	$7.986054\times10^{-2}$	$1.479\times10^{-6}$
-3	$1.327517\times10^{-1}$	$1.327520\times10^{-1}$	$2.791\times10^{-7}$	$2.938320\times10^{-1}$	2.938363 $\times10^{-1}$	$4.239\times10^{-6}$	$5.922212\times10^{-1}$	$9.922280\times10^{-1}$	$6.850\times10^{-6}$
0	$5.553666\times10^{-1}$	$5.553666\times10^{-1}$	$2.417\times10^{-7}$	$7.724818\times10^{-1}$	7.724813 $\times10^{-1}$	$4.934\times10^{-7}$	$9.221826\times10^{-1}$	$9.221826\times10^{-1}$	$3.423\times10^{-8}$
3	$9.150444\times10^{-1}$	$9.150445\times10^{-1}$	$4.741\times10^{-8}$	$9.669729\times10^{-1}$	9.669732 $\times10^{-1}$	$2.824\times10^{-7}$	$9.903092\times10^{-1}$	$9.903095\times10^{-1}$	$2.634\times10^{-7}$
7	$9.944820\times10^{-1}$	$9.944821\times10^{-1}$	$3.521\times10^{-8}$	$9.979629\times10^{-1}$	9.979630 $\times10^{-1}$	$6.559\times10^{-8}$	$9.994155\times10^{-1}$	$9.994155\times10^{-1}$	$3.733\times10^{-8}$
10	$9.993204\times10^{-1}$	$9.993204\times10^{-1}$	$6.943\times10^{-10}$	$9.997499\times10^{-1}$	9.997477 $\times10^{-1}$	$7.674\times10^{-10}$	$9.999283\times10^{-1}$	$9.999283\times10^{-1}$	$3.684\times10^{-10}$

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Table 4.

$L_{\infty}$ norm of problem 1 for case

$(i)$ with various values of

$\eta$ .

	$\eta=0.25$		${\eta=0.5}$		$\eta=0.75$
$t$	Present	^[44]	Present	^[44]	Present	^[44]
0.01	$4\times10^{-9}$	$5\times10^{-8}$	$7\times10^{-11}$	$5\times10^{-8}$	$4\times10^{-10}$	$5\times10^{-8}$
0.1	$3\times10^{-8}$	$3\times10^{-7}$	$4\times10^{-9}$	$3\times10^{-7}$	$3\times10^{-8}$	$3\times10^{-7}$
1.0	$1\times10^{-7}$	$1\times10^{-6}$	$6\times10^{-9}$	$5\times10^{-7}$	$1\times10^{-7}$	$1\times10^{-6}$
$L_{2}$
	$\eta=0.25$		${\eta=0.5}$		$\eta=0.75$
$t$	Present	^[44]	Present	^[44]	Present	^[44]
0.01	$5\times10^{-8}$	$5\times10^{-8}$	$8\times10^{-10}$	$5\times10^{-8}$	$5\times10^{-9}$	$5\times10^{-8}$
0.1	$4\times10^{-7}$	$3\times10^{-7}$	$4\times10^{-8}$	$3\times10^{-7}$	$4\times10^{-7}$	$3\times10^{-7}$
1.0	$2\times10^{-6}$	$1\times10^{-6}$	$7\times10^{-8}$	$5\times10^{-7}$	$2\times10^{-7}$	$9\times10^{-7}$

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In these tables, the obtained solutions are compared with the existing results in the literature ^{[32,42,43,44]}. Through comparison, it is obvious that the present technique shows better performance than the cited work ^[32,42,44]. The present method is based on RK-4 which gives good results using a small step size.

In , the computed values of the constants $\mathcal{K}_{1}, \mathcal{K}_{2}, \mathcal{K}_{3}, \mbox{and}\; \mathcal{K}_{4},$ for $\eta$ = 0.75, $\partial$ t = 0.001, and $L$ = 2 in the spatial domain $[-10, 10]$ at distinct time levels are reported.

Table 5. The coefficients values

$\mathcal{K}_{1}, \mathcal{K}_{2}, \mathcal{K}_{3}, \mbox{and}\; \mathcal{K}_{4}$ of problem 1 for case

$(i)$ with

$\eta = 0.75$ ,

$\partial t = 0.001$ , and

$L = 2$ .

$t$	$\mathcal{K}_1$	$\mathcal{K}_2$	$\mathcal{K}_3$	$\mathcal{K}_4$	Exact solution	Approximate Solution
0.1	$-0.000550127435708$	$-0.000550134504350$	-0.000550134504441	$-0.000550141573170$	$0.002000842010538$	0.001998228498937
	$-0.003944143032133$	$-0.003944200185466$	-0.003944200186294	$-0.003944257340322$	$0.011607626920875$	0.011488174877494
	$-0.013201078042744$	$-0.013201383972046$	-0.013201383979136	$-0.013201689914171$	$0.064365301508760$	0.064570374111189
	$-0.051365677465119$	$-0.051368265656579$	-0.051368265786992	$-0.051370854097156$	$0.287228366819168$	0.287197745055223
	$-0.051813540724938$	$-0.051816722723024$	-0.051816722918439	$-0.051819905116762$	$0.702427327811607$	0.702470418192179
	$-0.017916749446516$	$-0.017917560559869$	-0.017917560596589	$-0.017918371748347$	$0.932557185568827$	0.932334691176706
	$-0.001827678052741$	$-0.001827749308296$	-0.001827749311074	$-0.001827820569427$	$0.987804495301507$	0.987929599908556
	$-0.000469739827336$	$-0.000469757565601$	-0.000469757566270	$-0.000469775305206$	$0.997896788384635$	0.997899530906305
0.5	$-0.000577059786102$	$-0.000577067177572$	-0.000577067177666	$-0.000577074569228$	$0.001810781501093$	0.001767463721627
	$-0.003051640090534$	$-0.003051683586188$	-0.003051683586808	$-0.003051727083001$	$0.010514629756167$	0.010105627339338
	$-0.013080724367551$	$-0.013081016714371$	-0.013081016720905	$-0.013081309072915$	$0.058599062346241$	0.059295267046785
	$-0.048329726367191$	$-0.048332073109897$	-0.048332073223848	$-0.048334420069440$	$0.267198413052228$	0.267262778290452
	$-0.053763565954023$	$-0.053766904699846$	-0.053766904907184	$-0.053770243864621$	$0.681111275379947$	0.681347627307882
	$-0.019155289006543$	$-0.019156170889244$	-0.019156170929845	$-0.019157052855031$	$0.925989123684090$	0.924926841938356
	$-0.002566491484946$	$-0.002566591819423$	-0.002566591823345	$-0.002566692161784$	$0.986539147932092$	0.987046273168589
	$-0.000386965519072$	$-0.000386980139663$	-0.000386980140215	$-0.000386994761360$	$0.997676105725876$	0.997732428060447
1.0	$-0.000492580802813$	$-0.000492587089040$	-0.000492587089121	-0.000492593375426	0.001598349150613	0.001497994290761
	$-0.002423286399790$	$-0.002423320374624$	-0.002423320375100	-0.002423354350360	0.009290606750817	0.008752036768966
	$-0.012362148938809$	$-0.012362412752656$	-0.012362412758286	$-0.012362676576548$	0.052072036738247	0.052916126005743
	$-0.044718979445185$	$-0.044721048624543$	-0.044721048720285	$-0.044723117985214$	0.243445131323980	0.244007643848355
	$-0.055982628284796$	$-0.055986143692980$	-0.055986143913729	$-0.055989659546008$	0.653369506436006	0.653900188317205
	$-0.020946593575349$	$-0.020947579113019$	-0.020947579159389	$-0.020948564745616$	0.916953070453016	0.914910265106938
	$-0.003363560269563$	$-0.003363692368992$	-0.003363692374180	$-0.003363824478866$	0.984774145473912	0.985560005959086
	$-0.000411345006904$	$-0.000411360558612$	-0.000411360559200	$-0.000411376111496$	0.997367497354861	0.997536946415740

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In , numerical and exact solutions are plotted for various values of $\eta$ for case $(i)$ while in , the absolute error at distinct points are plotted with various time levels. Similar simulations are obtained for case ( $ii$ ) and its absolute error at distinct points are compared with the existing results ^[33,45,46] in Table 6, which show the superiority of the method.

Figure 1. Exact and numerical measures of problem 1 for case

$(i)$ . The arrow shows the direction of the travelling wave.

	$t=0.001$				$t=0.002$				$t=0.003$
$\xi$	^[45]	^[46]	^[33]	Present	^[45]	^[46]	^[33]	Present	^[45]	^[46]	^[33]	Present
-22	$1.680\times10^{-7}$	$1.367\times10^{-7}$	$1.435\times10^{-7}$	$2.139\times10^{-10}$	$3.362\times10^{-7}$	2.888 $\times10^{-7}$	$2.955\times10^{-7}$	$3.158\times10^{-10}$	$5.045\times10^{-7}$	$4.561\times10^{-7}$	$4.097\times10^{-7}$	$3.955\times10^{-10}$
-6	$5.184\times10^{-6}$	$4.611\times10^{-6}$	$4.844\times10^{-7}$	$4.987\times10^{-6}$	$1.037\times10^{-5}$	9.740 $\times10^{-6}$	$9.990\times10^{-6}$	$9.980\times10^{-6}$	$1.556\times10^{-5}$	$1.538\times10^{-5}$	$1.567\times10^{-5}$	$1.497\times10^{-5}$
2	$2.956\times10^{-5}$	$2.669\times10^{-5}$	$2.751\times10^{-5}$	$2.924\times10^{-5}$	$5.915\times10^{-5}$	5.633 $\times10^{-5}$	$5.811\times10^{-5}$	$5.847\times10^{-5}$	$8.876\times10^{-5}$	$8.875\times10^{-5}$	$8.620\times10^{-5}$	$8.768\times10^{-5}$
10	$1.073\times10^{-6}$	$9.152\times10^{-7}$	$1.006\times10^{-7}$	$5.511\times10^{-7}$	$2.146\times10^{-6}$	1.931 $\times10^{-6}$	$2.011\times10^{-6}$	$1.101\times10^{-6}$	$3.218\times10^{-6}$	$3.047\times10^{-6}$	$4.992\times10^{-6}$	$1.651\times10^{-6}$
18	$5.458\times10^{-7}$	$4.272\times10^{-7}$	$5.130\times10^{-7}$	$6.036\times10^{-9}$	$1.091\times10^{-6}$	9.016 $\times10^{-7}$	$9.215\times10^{-7}$	$1.206\times10^{-8}$	$1.636\times10^{-6}$	$1.423\times10^{-6}$	$1.610\times10^{-6}$	$1.808\times10^{-8}$

	$\partial t=10^{-4}$		$\partial t=10^{-5}$
$t$	Present	^[44]	Present	^[44]
0.01	$4\times10^{-8}$	$2\times10^{-7}$	$6\times10^{-9}$	$2\times10^{-7}$
0.1	$3\times10^{-7}$	$9\times10^{-7}$	$4\times10^{-8}$	$9\times10^{-7}$
10.0	$9\times10^{-8}$	$3\times10^{-7}$	$7\times10^{-9}$	$3\times10^{-7}$
$L_{2}$
	$\partial t=10^{-4}$		$\partial t=10^{-5}$
$t$	Present	^[44]		Present	^[44]
0.01	$2\times10^{-7}$	$1\times10^{-7}$	$3\times10^{-8}$	$1\times10^{-7}$
0.1	$1\times10^{-6}$	$6\times10^{-7}$	$2\times10^{-7}$	$6\times10^{-7}$
10.0	$3\times10^{-7}$	$2\times10^{-7}$	$4\times10^{-8}$	$2\times10^{-7}$

	$\eta=-1$			$\eta=0.75$			$\eta=4$
$t$	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$
0.1	$9.677\times10^{-7}$	$4.969\times10^{-6}$	$5.559\times10^{-8}$	$7.961\times10^{-7}$	4.699 $\times10^{-6}$	$9.270\times10^{-9}$	$3.662\times10^{-6}$	$1.361\times10^{-5}$	$1.464\times10^{-7}$
0.2	$1.890\times10^{-6}$	$9.619\times10^{-6}$	$1.166\times10^{-7}$	$1.514\times10^{-76}$	8.965 $\times10^{-6}$	$1.950\times10^{-8}$	$7.200\times10^{-6}$	$2.596\times10^{-5}$	$3.349\times10^{-7}$
0.3	$2.773\times10^{-6}$	$1.399\times10^{-5}$	$1.834\times10^{-7}$	$2.168\times10^{-6}$	1.286 $\times10^{-5}$	$3.081\times10^{-8}$	$1.042\times10^{-6}$	$3.687\times10^{-5}$	$5.497\times10^{-7}$
0.4	$3.617\times10^{-6}$	$1.814\times10^{-5}$	$2.555\times10^{-7}$	$2.771\times10^{-6}$	1.643 $\times10^{-5}$	$4.330\times10^{-8}$	$1.320\times10^{-5}$	$4.624\times10^{-5}$	$7.665\times10^{-7}$
0.5	$4.424\times10^{-6}$	$2.208\times10^{-5}$	$3.325\times10^{-7}$	$3.326\times10^{-6}$	1.973 $\times10^{-5}$	$5.702\times10^{-8}$	$1.544\times10^{-5}$	$5.399\times10^{-5}$	$9.573\times10^{-7}$
1.0	$7.881\times10^{-6}$	$3.949\times10^{-5}$	$7.631\times10^{-7}$	$5.641\times10^{-6}$	3.310 $\times10^{-5}$	$1.460\times10^{-7}$	$1.650\times10^{-5}$	$6.607\times10^{-5}$	$9.066\times10^{-7}$
1.5	$1.031\times10^{-5}$	$5.316\times10^{-5}$	$1.186\times10^{-7}$	$7.452\times10^{-6}$	4.300 $\times10^{-5}$	$2.707\times10^{-7}$	$8.097\times10^{-6}$	$2.564\times10^{-5}$	$1.411\times10^{-7}$
2.0	$1.168\times10^{-5}$	$6.187\times10^{-5}$	$1.525\times10^{-6}$	$8.979\times10^{-6}$	5.088 $\times10^{-5}$	$4.305\times10^{-7}$	$5.428\times10^{-6}$	$1.746\times10^{-5}$	$3.026\times10^{-7}$

		$\eta=-1$			$\eta=1$			$\eta=0.75$
$t$	L	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$	$L_{\infty}$	$L_{2}$	$L_{rms}$
0.01	1	$2.145\times10^{-4}$	$2.173\times10^{-4}$	$6.515\times10^{-5}$	$2.022\times10^{-4}$	2.049 $\times10^{-4}$	$6.193\times10^{-5}$	$2.307\times10^{-4}$	$2.064\times10^{-4}$	$6.232\times10^{-5}$
	3	$3.599\times10^{-5}$	$5.573\times10^{-5}$	$9.275\times10^{-6}$	$3.319\times10^{-5}$	5.171 $\times10^{-5}$	$8.652\times10^{-6}$	$3.352\times10^{-5}$	$5.219\times10^{-5}$	$8.727\times10^{-6}$
	5	$2.625\times10^{-6}$	$8.187\times10^{-6}$	$6.643\times10^{-7}$	$2.234\times10^{-6}$	7.016 $\times10^{-6}$	$5.751\times10^{-7}$	$2.281\times10^{-6}$	$7.155\times10^{-6}$	$5.857\times10^{-7}$
	7	$3.131\times10^{-7}$	$1.946\times10^{-6}$	$7.687\times10^{-8}$	$9.539\times10^{-8}$	5.991 $\times10^{-7}$	$2.520\times10^{-8}$	$1.209\times10^{-7}$	$7.573\times10^{-7}$	$3.128\times10^{-8}$
0.1	1	$2.132\times10^{-3}$	$2.288\times10^{-3}$	$8.423\times10^{-4}$	$1.142\times10^{-3}$	1.270 $\times10^{-3}$	$4.984\times10^{-4}$	$1.228\times10^{-3}$	$1.359\times10^{-3}$	$5.304\times10^{-4}$
	3	$2.519\times10^{-4}$	$5.146\times10^{-4}$	$9.770\times10^{-5}$	$1.177\times10^{-4}$	2.562 $\times10^{-4}$	$5.065\times10^{-5}$	$1.284\times10^{-4}$	$2.774\times10^{-4}$	$5.461\times10^{-5}$
	5	$1.753\times10^{-5}$	$7.158\times10^{-5}$	$6.734\times10^{-6}$	$7.536\times10^{-6}$	3.263 $\times10^{-5}$	$3.211\times10^{-6}$	$8.298\times10^{-6}$	$3.571\times10^{-5}$	$3.497\times10^{-6}$
	7	$2.065\times10^{-6}$	$1.679\times10^{-5}$	$7.777\times10^{-7}$	$3.194\times10^{-7}$	2.818 $\times10^{-6}$	$1.422\times10^{-7}$	$4.636\times10^{-7}$	$3.783\times10^{-6}$	$1.870\times10^{-7}$
0.2	1	$5.512\times10^{-3}$	$6.009\times10^{-3}$	$2.238\times10^{-3}$	$1.147\times10^{-3}$	1.763 $\times10^{-3}$	$7.469\times10^{-4}$	$1.693\times10^{-3}$	$2.009\times10^{-3}$	$8.449\times10^{-4}$
	3	$6.614\times10^{-4}$	$1.396\times10^{-3}$	$2.723\times10^{-4}$	$1.335\times10^{-4}$	3.243 $\times10^{-4}$	$6.846\times10^{-5}$	$1.565\times10^{-4}$	$3.765\times10^{-4}$	$7.901\times10^{-5}$
	5	$4.613\times10^{-5}$	$1.948\times10^{-4}$	$1.887\times10^{-5}$	$8.413\times10^{-6}$	4.091 $\times10^{-5}$	$4.311\times10^{-6}$	$9.990\times10^{-6}$	$4.802\times10^{-5}$	$5.027\times10^{-6}$
	7	$5.276\times10^{-6}$	$4.482\times10^{-5}$	$2.163\times10^{-6}$	$3.610\times10^{-7}$	3.597 $\times10^{-6}$	$1.931\times10^{-7}$	$5.271\times10^{-7}$	$5.111\times10^{-6}$	$2.695\times10^{-7}$
0.3	1	$1.295\times10^{-2}$	$1.394\times10^{-2}$	$4.660\times10^{-3}$	$1.537\times10^{-3}$	1.926 $\times10^{-3}$	$8.404\times10^{-4}$	$1.886\times10^{-3}$	$2.331\times10^{-3}$	$1.008\times10^{-3}$
	3	$1.796\times10^{-3}$	$3.661\times10^{-3}$	$6.980\times10^{-4}$	$1.295\times10^{-4}$	3.375 $\times10^{-4}$	$7.358\times10^{-5}$	$1.636\times10^{-4}$	$4.182\times10^{-4}$	$9.059\times10^{-5}$
	5	$1.306\times10^{-4}$	$5.218\times10^{-4}$	$4.930\times10^{-5}$	$8.177\times10^{-6}$	4.242 $\times10^{-5}$	$4.621\times10^{-6}$	$1.040\times10^{-5}$	$5.313\times10^{-5}$	$5.748\times10^{-6}$
	7	$1.400\times10^{-5}$	$1.144\times10^{-4}$	$5.458\times10^{-6}$	$3.564\times10^{-7}$	3.791 $\times10^{-6}$	$2.094\times10^{-7}$	$5.505\times10^{-7}$	$5.677\times10^{-6}$	$3.088\times10^{-7}$
0.4	1	$3.446\times10^{-2}$	$3.905\times10^{-2}$	$7.273\times10^{-3}$	$1.487\times10^{-3}$	1.917 $\times10^{-3}$	$8.488\times10^{-4}$	$1.941\times10^{-3}$	$2.461\times10^{-3}$	$1.080\times10^{-3}$
	3	$7.093\times10^{-3}$	$1.220\times10^{-2}$	$2.140\times10^{-3}$	$1.211\times10^{-4}$	3.264 $\times10^{-4}$	$7.239\times10^{-5}$	$1.616\times10^{-4}$	$4.305\times10^{-4}$	$9.485\times10^{-5}$
	5	$5.414\times10^{-4}$	$1.867\times10^{-3}$	$1.605\times10^{-4}$	$7.588\times10^{-6}$	4.095 $\times10^{-5}$	$4.540\times10^{-6}$	$1.029\times10^{-5}$	$5.458\times10^{-5}$	$6.009\times10^{-6}$
	7	$5.169\times10^{-5}$	$3.698\times10^{-4}$	$1.635\times10^{-5}$	$3.368\times10^{-7}$	3.721 $\times10^{-6}$	$2.085\times10^{-7}$	$5.465\times10^{-7}$	$5.851\times10^{-6}$	$3.236\times10^{-7}$
0.5	1	$1.921\times10^{-1}$	$2.391\times10^{-1}$	$5.616\times10^{-1}$	$1.386\times10^{-3}$	1.821 $\times10^{-3}$	$8.130\times10^{-4}$	$1.921\times10^{-3}$	$2.477\times10^{-3}$	$1.095\times10^{-3}$
	3	$6.222\times10^{-2}$	$7.995\times10^{-2}$	$1.088\times10^{-2}$	$1.103\times10^{-4}$	3.042 $\times10^{-4}$	$6.809\times10^{-5}$	$1.570\times10^{-4}$	$4.261\times10^{-4}$	$9.479\times10^{-5}$
	5	$7.028\times10^{-3}$	$1.727\times10^{-2}$	$1.144\times10^{-3}$	$6.897\times10^{-6}$	3.813 $\times10^{-5}$	$4.268\times10^{-6}$	$9.935\times10^{-6}$	$5.395\times10^{-5}$	$5.999\times10^{-6}$
	7	$5.741\times10^{-4}$	$2.903\times10^{-3}$	$9.948\times10^{-5}$	$3.121\times10^{-7}$	3.525 $\times10^{-6}$	$1.990\times10^{-7}$	$5.295\times10^{-7}$	$5.804\times10^{-6}$	$3.139\times10^{-7}$

	$L_{\infty}$			$L_{2}$			$L_{rms}$
$t$	Present	^[32]	^[33]	Present	^[32]	^[33]	Present	^[32]	^[33]
0.2	$5.325\times10^{-7}$	$1.235\times10^{-5}$	$1.122\times10^{-5}$	$3.626\times10^{-6}$	1.312 $\times10^{-6}$	$7.154\times10^{-6}$	$1.124\times10^{-8}$	$4.567\times10^{-5}$	....
0.5	$1.203\times10^{-6}$	$5.198\times10^{-4}$	$7.584\times10^{-5}$	$8.234\times10^{-6}$	6.099 $\times10^{-6}$	$7.489\times10^{-6}$	$6.344\times10^{-8}$	$5.642\times10^{-5}$	....
1.0	$2.158\times10^{-6}$	$6.328\times10^{-4}$	$9.127\times10^{-5}$	$1.521\times10^{-5}$	2.121 $\times10^{-5}$	$8.689\times10^{-5}$	$2.164\times10^{-7}$	$8.167\times10^{-5}$	....
1.5	$2.996\times10^{-6}$	$8.538\times10^{-4}$	$9.242\times10^{-5}$	$2.213\times10^{-5}$	$3.234\times10^{-5}$	$7.027\times10^{-5}$	$4.160\times10^{-7}$	$2.368\times10^{-4}$	....

Networks and Heterogeneous Media

Special issue on mathematical models for collective dynamics

Related Papers:

1. Introduction

2. Main motive

3. Haar wavelet and its integrals

4. Proposed scheme

4.1. Temporal discretization

5. Simulations and comparison

5.1. Problem 1

5.2. Problem 2

6. Stability analysis

7. Concluding remarks

Author contributions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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通讯作者: 陈斌, bchen63@163.com

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