Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow

Khalil Ur Rehman; Nosheen Fatima; Wasfi Shatanawi; Nabeela Kousar; Khalil Ur Rehman; Nosheen Fatima; Wasfi Shatanawi; Nabeela Kousar

doi:10.3934/math.2025027

AIMS Mathematics

2025, Volume 10, Issue 1: 598-633. doi: 10.3934/math.2025027

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Research article Special Issues

Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow

1.
Department of Mathematics and Sciences, College of Humanities and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia
2.
Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000, Pakistan
3.
Department of Mathematics, Faculty of Science, The Hashemite University, P.O. Box 330127, Zarqa, 13133, Jordan

Received: 10 September 2024 Revised: 23 December 2024 Accepted: 02 January 2025 Published: 10 January 2025
MSC : 35A25, 65M06, 76D05

The mathematical formulation of fluid flow problems often results in coupled nonlinear partial differential equations (PDEs); hence, their solutions remain a challenging task for researchers. The present study offers a solution for the flow differential equations describing a bio-inspired flow field of non-Newtonian fluid with gyrotactic microorganisms. A methanol-based nanofluid with ferrous ferric oxide, copper, and silver nanoparticles was considered in a stretching permeable cylinder. The chemical reaction, activation energy, viscous dissipation, and convective boundary conditions were considered. The Casson fluid, a non-Newtonian fluid model, was used as flowing over a cylinder. The fundamental PDEs were established using boundary layer theory in a cylindrical coordinate system for concentration, mass, momentum, and microorganisms' field. These PDEs were then transformed into nonlinear ODEs by applying transforming variables. ODEs were then numerically solved in MATLAB software using the built-in solver bvp4c algorithm. We established an artificial neural network (ANN) model, incorporating Tan-Sig and Purelin transfer functions, to enhance the accuracy of predicting skin friction coefficient (SFC) values along the surface. The networks were trained using the Levenberg–Marquardt method. Quantitative results show that the ferrous ferric oxide nanofluid is superior in increasing Nusselt number, Sherwood number, velocity, and microorganism density number; silver nanofluid is superior in increasing skin friction coefficient, temperature, and concentration. Interestingly, heat transfer rate decreases with the magnetic and curvature parameters and Eckert number, whereas the skin friction coefficient increases with the magnetic parameter and Darcy–Forchheimer number. The present results are validated with the previous existing studies.

Keywords:

Citation: Khalil Ur Rehman, Nosheen Fatima, Wasfi Shatanawi, Nabeela Kousar. Mathematical solutions for coupled nonlinear equations based on bioconvection in MHD Casson nanofluid flow[J]. AIMS Mathematics, 2025, 10(1): 598-633. doi: 10.3934/math.2025027

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Abstract

1. Introduction

Lots of physical phenomena can be expressed by non-linear partial differential equations (PDE), including, inter alia, dissipative and dispersive PDE. In this paper, we consider the Kuramoto-Sivashinsky (KS) equation

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = \varphi(s, t)\; 0 \le s\le 1, \; 0 \le t \le T, \; \gamma > 0, \end{eqnarray}$

(1.1)

$\begin{eqnarray} &&\phi(0, t) = 0, \; \phi(1, t) = 0, \; \phi_{ss}(0, t) = 0, \phi_{ss}(1, t) = 0, \; 0 < t < T, \end{eqnarray}$

(1.2)

$\begin{eqnarray} &&\phi(s, 0) = \varphi(s), \; 0\le s \le 1, \end{eqnarray}$

(1.3)

where $\gamma \in R$ is the constant.

The KS equation plays an important role in physics such as in diffusion, convection and so on. Lots of attention has been paid by researchers in recent years. An $H_1$ -Galerkin mixed finite element method for the KS equation was proposed in ^[1], lattice Boltzmann models for the Kuramoto-Sivashinsky equation were studied in ^[2], Backward difference formulae (BDF) methods for the KS equation were investigate in ^[3]. Stability regions and results for the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations were given in ^[4,5], respectively. In ^[6], an improvised quintic B-spline extrapolated collocation technique was used to solve the KS equation, and the stability of the technique was analyzed using the von Neumann scheme, which was found to be unconditionally stable. In ^[7], a septic Hermite collocation method (SHCM) was proposed to simulate the KS equation, and the nonlinear terms of the KS equation were linearized using the quasi-linearization process. In ^[8], a semidiscrete approach was presented to solve the variable-order (VO) time fractional 2D KS equation, and the differentiation operational matrices and the collocation technique were used to get a linear system of algebraic equations. In ^[9] the discrete Legendre polynomials (LPs) and the collocation scheme for nonlinear space-time fractional KdV-Burgers-Kuramoto equation were presented.

In order to avoid the Runge's phenomenon, barycentric interpolation ^[10,11,12] was developed. In recent years, linear rational interpolation (LRI) was proposed by Floater ^[13,14,15], and error of linear rational interpolation was also proved. The barycentric interpolation collocation method (BICM) has been developed by Wang et al.^{[22,23,24,25]}, and the algorithm of BICM has been used for linear/non-linear problems ^[21]. Volterra integro-differential equation (VIDE)^[16,20], heat equation (HE) ^[17], biharmonic equation (BE) ^[18], the Kolmogorov-Petrovskii-Piskunov (KPP) equation ^[19], fractional differential equations ^[20], fractional reaction-diffusion equation ^[28], semi-infinite domain problems ^[27] and biharmonic equation ^[26], plane elastic problems ^[29] have been studied by the linear barycentric interpolation collocation method (LBICM), and their convergence rates also have been proved.

In order to solve the KS equation efficiently, the LBRIM is presented. Because the nonlinear part of the KS equation cannot be solved directly, three kinds of linearization methods, including direct linearization, partial linearization and Newton linearization, are presented. Then, the nonlinear part of the KS equation is translated into the linear part, three kinds of iterative schemes are presented, and matrix equation of the linearization schemes are constructed. The convergence rate of the LBRCM for the KS equation is also given. At last, two numerical examples are presented to validate the theoretical analysis.

2. Linearization for KS equation

In the following, the KS equation is changed into the linear equation by the linearization scheme, including direct linearization, partial linearization and Newton linearization.

2.1. Direct linearization

For the Kuramoto-Sivashinskyr equation with the initial value of nonlinear term $\phi\frac{\partial \phi}{\partial s}$ is changed to $\phi_{0}\frac{\partial \phi_{0}}{\partial s}$ ,

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi_{0}\frac{\partial \phi_{0}}{\partial s} = \varphi(s, t), \end{eqnarray}$

(2.1)

and then we get the linear scheme as

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2} = -\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}+\varphi(s, t), a \le s\le b, 0 \le t \le T. \end{eqnarray}$

(2.2)

2.2. Partial linearization

By the partial linearization, nonlinear term $\phi\frac{\partial \phi}{\partial s}$ is changed to $\phi_{0}\frac{\partial \phi}{\partial s}$ ,

(2.3)

and then we have

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2}+\phi_{n-1}\frac{\partial \phi_{n}}{\partial s} = \varphi(s, t), a \le s\le b, 0 \le t \le T. \end{eqnarray}$

(2.4)

2.3. Newton linearization

For the initial value $\phi\frac{\partial \phi}{\partial s} = \phi_{0}\frac{\partial \phi_{0}}{\partial s}+(\frac{\partial \phi_{0}}{\partial s}+\phi_{0}\frac{\partial \phi_{0}}{\partial s})(\phi-\phi_{0})$ , we have

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi_{0}}{\partial s}+ \phi_{0}\frac{\partial \phi_{0}}{\partial s}\phi = \varphi(s, t)+\phi_{0}\frac{\partial \phi_{0}}{\partial s}\phi_{0}, \end{eqnarray}$

(2.5)

and then we have

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2}+\phi_{n}\frac{\partial \phi_{n-1}}{\partial s}+\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}\phi_{n} = \varphi(s, t)+\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}\phi_{n-1}, \end{eqnarray}$

(2.6)

where $n = 1, 2, \cdots.$

3. Differentiation matrices of KS equation

Interval $[a, b]$ is divided into $a = s_{0} < s_{1} < s_{2} < \cdots < s_{m-1} < s_{m} = b$ , for uniform partition with $h_{s} = \frac{b-a}{m}$ and nonuniform partition to be the second kind of Chebychev point. Time $[0, T]$ is divided into $0 = t_{0} < t_{1} < t_{2} < \cdots < t_{n-1} < t_{n} = T$ and $h_{t} = \frac{T}{n}$ for uniform partition. Then, we take $\phi_{nm}(s, t)$ to approximate $\phi(s, t)$ as

$\begin{eqnarray} \phi_{nm}(s, t) = \sum\limits_{i = 0}^{m} \sum\limits_{j = 0}^{n} r_{i}(s) r_{j}(t) \phi_{ij} \end{eqnarray}$

(3.1)

where $\phi_{ij} = \phi(s_{i}, t_{j})$ ,

$\begin{eqnarray} r_{i}(s) = \frac{ \frac{w_{i}}{s-s_{i}} }{ \sum\limits_{j = 0}^{m} \frac{w_{j}}{s-s_{j}}}, \quad r_{j}(t) = \frac{ \frac{w_{j}}{t-t_{j}} }{ \sum\limits_{i = 0}^{n} \frac{w_{i}}{t-t_{i}}} \end{eqnarray}$

(3.2)

is the barycentric interpolation basis ^[26], and

$\begin{eqnarray} w_{i} = \sum\limits_{k \in J_{i}}(-1)^{k} \prod\limits_{j = k, j \neq i}^{k+d_{s}} \frac{1}{s_{i}-s_{j}}, \quad w_{j} = \sum\limits_{k \in J_{j}}(-1)^{k} \prod\limits_{i = k, k \neq j}^{k+d_{t}} \frac{1}{t_{j}-t_{i}} \end{eqnarray}$

(3.3)

where $J_{i} = \{k \in I, i-d_{s} \le k \le i\}, I = \{0, 1, \cdots, m-d_{s} \}$ . See ^[26]. We get the barycentric rational interpolation.

For the case

$\begin{eqnarray} w_{i} = \frac{1}{\prod\nolimits_{i \neq k}\left(s_{i}-s_{k}\right)}, w_{j} = \frac{1}{\prod\nolimits_{j \neq k}\left(t_{j}-t_{k}\right)}, \end{eqnarray}$

(3.4)

we get the barycentric Lagrange interpolation.

So,

$\begin{eqnarray} r_{j}^{\prime }\left(s_{i}\right) = \frac{w_{j} / w_{i}}{s_{i}-s_{j}}, \quad j \neq i, \quad r_{i}^{\prime }\left(s_{i}\right) = -\sum\limits_{j \neq i} r_{j}^{\prime }\left(s_{i}\right), \end{eqnarray}$

(3.5)

$\begin{eqnarray} r_{j}^{(k)}\left(s_{i}\right) = k \left(r_{i}^{(k-1)}\left(s_{i}\right) r_{i}^{\prime}\left(s_{j}\right) - \frac{r_{i}^{(k-1)}\left(s_{j}\right) }{s_{i}-s_{j}}\right), \quad j \neq i, \end{eqnarray}$

(3.6)

$\begin{eqnarray} r_{i}^{(k)}\left(s_{i}\right) = -\sum\limits_{j \neq i} r_{j}^{(k)}\left(s_{i}\right). \end{eqnarray}$

(3.7)

Then, we have

$\begin{eqnarray} \mathrm{{D}}^{(0, 1)}_{ij}& = & r_{i}^{\prime }\left(t_{j}\right), \end{eqnarray}$

(3.8)

$\begin{eqnarray} \mathrm{{D}}^{(1, 0)}_{ij}& = & r_{i}^{\prime }\left(s_{j}\right), \end{eqnarray}$

(3.9)

$\begin{eqnarray} \mathrm{{D}}^{(k, 0)}_{ij}& = & r_{i}^{(k) }\left(s_{j}\right), k = 2, 3, \cdots. \end{eqnarray}$

(3.10)

3.1. Matrix equation of direct linearization

Combining (3.1) and (2.2), we have

$\begin{eqnarray} \left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} = {\Psi}-diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}, \end{eqnarray}$

(3.11)

and then we have

$\begin{eqnarray} \mathcal{L} \mathcal{\phi}_{n} = {\Psi}_{n-1} \end{eqnarray}$

(3.12)

where

$\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n},$

${\Psi}_{n-1} = {\Psi}-diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}$

and $\otimes$ is the Kronecher product ^[17].

3.2. Matrix equation of partial linearization

Combining (3.1) and (2.4), we have

$\begin{eqnarray} \left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} = {\Psi}, \end{eqnarray}$

(3.13)

$n = 1, 2, \cdots$ , and then we have

$\begin{eqnarray} \mathcal{L} \mathcal{\phi} = {\Psi} \end{eqnarray}$

(3.14)

where $\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}$ .

3.3. Matrix equation of Newton linearization

Combining (3.1) and (2.6), we have

$\begin{eqnarray} &&\left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} \\&& = {\Psi} + [diag(\mathcal{\phi}_{n})-diag(\mathcal{\phi}_{n-1})]\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}, \end{eqnarray}$

(3.15)

and then we get

$\begin{eqnarray} \mathcal{L} \mathcal{\phi} = {\Psi}_{n-1} \end{eqnarray}$

(3.16)

where

$\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n},$

and

${\Psi}_{n-1} = {\Psi} + [diag(\mathcal{\phi}_{n})-diag(\mathcal{\phi}_{n-1})]\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}.$

4. Convergence rate of KS equation

In this part, an error estimate of the KS equation is given with $r_n(s) = \sum\limits_{i = 0}^{n} r_{i}(s) \phi_{i}$ to replace $\phi(s)$ , where $r_{i}(s)$ is defined as (3.2), and $\phi_{i} = \phi(s_{i})$ . We also define

$\begin{eqnarray} e(s): = \phi(s)-r_n(s) = (s-s_{i}) \cdots(s-s_{i+d})\phi\left[s_{i}, s_{i+1}, \ldots, s_{i+d}, s\right]. \end{eqnarray}$

(4.1)

Then, we have the following.

Lemma 1. For $e(s)$ defined by (4.1) and $\phi(s)\in C^{d+2}[a, b]$ , there is

$\begin{equation} \left|e^{(k)}(s)\right| \leq C h^{d-k+1}, k = 0, 1, \cdots. \end{equation}$

(4.2)

For KS equation, rational interpolation function of $\phi(s, t)$ is defined as $r_{mn}(s, t)$

$\begin{equation} r_{mn}(s, t) = \frac{ \sum\limits_{i = 0}^{m+d_{s}} \sum\limits_{j = 0}^{n+d_{t}} \frac{w_{i, j}}{\left(s-s_{i}\right)\left(t-t_{j}\right)} {\phi}_{i, j}}{ \sum\limits_{i = 0}^{m+d_{s}} \sum\limits_{j = 0}^{n+d_{t}} \frac{w_{i, j}}{\left(s-s_{i}\right)\left(t-t_{j}\right)}} \end{equation}$

(4.3)

where

$\begin{equation} w_{i, j} = (-1)^{i-d_{s}+j-d_{t}} \sum\limits_{k_{1} \in J_{i}}\prod\limits_{h_{1} = k_{1}, h_{1} \neq j}^{k_{1}+d_{s}}\frac{1}{\left|s_{i}-s_{h_{1}}\right|} \sum\limits_{k_{2} \in J_{i}} \prod\limits_{h_{2} = k_{2}, h_{2} \neq j}^{k_{2}+d_{t}} \frac{1}{\left|t_{j}-t_{h_{2}}\right|}. \end{equation}$

(4.4)

We define $e(s, t)$ to be the error of $\phi(s, t)$ as

$\begin{eqnarray} e(s, t) :& = & \phi(s, t)-r_{mn}(s, t) \\ & = & \left(s-s_{i}\right) \cdots\left(s-s_{i+d_{s}}\right)\phi\left[s_{i}, s_{i+1}, \ldots, s_{i+d_1}, s, t\right] \\ &+&\left(t-t_{j}\right) \cdots\left(t-t_{j+d_{t}}\right)\phi\left[s, t_{j}, t_{j+1}, \ldots, t_{j+d_2}, t\right]. \end{eqnarray}$

(4.5)

With similar analysis of Lemma 1, we have the following

Theorem 1. For $e(s, t)$ defined as (4.5) and $\phi(s, t) \in C^{d_{s}+2}[a, b]\times C^{d_{t}+2}[0, T]$ , we have

$\begin{equation} \left|e^{(k_{1}, k_{2})}\left(s, t\right)\right| \leq C (h_{s}^{d_{s}-k_{1}+1}+h_{t}^{d_{t}-k_{2}+1}), k_{1}, k_{2} = 0, 1, \cdots. \end{equation}$

(4.6)

We take the direct linearization of the KS equation as an example prove the convergence rate. Let $\phi(s_{m}, t_{n})$ be the approximate function of $\phi(s, t)$ and $L$ be a bounded operator. There holds

$\begin{eqnarray} L \phi(s_{m}, t_{n}) = \varphi(s_{m}, t_{n}), \end{eqnarray}$

(4.7)

and

$\begin{eqnarray} \lim\limits_{m, n\rightarrow \infty} \phi(s_{m}, t_{n}) = \phi(s, t). \end{eqnarray}$

(4.8)

Then, we get the following

Theorem 2. For $\phi(s_{m}, t_{n}):L\phi(s_{m}, t_{n}) = \varphi(s, t)$ and $L$ defined as (4.7), there

$|\phi(s, t)-\phi(s_{m}, t_{n})|\le C(h^{d_{s}-3}+\tau^{d_{t}}).$

Proof. As

$\begin{eqnarray} \begin{array}{ll} &L \phi(s, t)-L \phi(s_{m}, t_{n}) \\[0.5cm]& = \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}- \phi_{0}\frac{\partial \phi_{0}}{\partial s}-\varphi(s, t) \\[0.5cm]&- \left[\frac{\partial \phi(s_{m}, t_{n})}{\partial t}+\gamma \frac{\partial^4 \phi(s_{m}, t_{n})}{\partial s^4}+\frac{\partial^2 \phi(s_{m}, t_{n})}{\partial s^2}+ \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}(s_{m}, t_{n})}{\partial s}-\varphi(s, t)\right] \\[0.5cm]& = \frac{\partial \phi}{\partial t}-\frac{\partial \phi}{\partial t}(s_{m}, t_{n}) +\gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\[0.5cm]&+ \frac{\partial^2 \phi}{\partial s^2}-\frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n})+\left[\phi_{0}\frac{\partial \phi_{0}}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}}{\partial s}(s_{m}, t_{n}) \right] \\[0.5cm]&: = E_{1}(s, t)+E_{2}(s, t)+E_{3}(s, t)+E_{4}(s, t). \end{array} \end{eqnarray}$

(4.9)

Here,

$E_{1}(s, t) = \frac{\partial \phi}{\partial t}-\frac{\partial \phi}{\partial t}(s_{m}, t_{n}),$

$E_{2}(s, t) = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right],$

$E_{3}(s, t) = \frac{\partial^2 \phi}{\partial s^2}-\frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n}),$

$E_{4}(s, t) = \phi_{0}\frac{\partial \phi_{0}}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}}{\partial s}(s_{m}, t_{n}) .$

With $E_{2}(s, t)$ , we have

$\begin{equation} \begin{array} {ll} E_{2}(s, t)& = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\& = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t)+ \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t)- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\& = \frac{ \sum\limits_{i = 0}^{m-d_{s}} (-1)^{i}\frac{\partial^4 \phi}{\partial s^4}[s_{i}, s_{i+1}, \ldots, s_{i+d_1}, s_{m}, t]} { \sum\limits_{i = 0}^{m-d_{s}} \lambda_{i}(s)} \\ \nonumber&+ \frac{ \sum\limits_{j = 0}^{n-d_{t}}(-1)^{j}\frac{\partial^4 \phi}{\partial s^4}[ t_{j}, t_{j+1}, \ldots, t_{j+d_2}, s_{m}, t_{n}]} { \sum\limits_{j = 0}^{n-d_{t}}\lambda_{j}(t)} \\& = \frac{\partial^4 e}{\partial s^4}(s_{m}, t)+\frac{\partial^4 e}{\partial s^4}(s_{m}, t_{n}). \end{array} \end{equation}$

For $E_{2}(s, t)$ we get

$\begin{equation} |E_{2}(s, t)|\le \left|\frac{\partial^4 e}{\partial s^4}(s_{m}, x)+\frac{\partial^4 e}{\partial s^4}(s_{m}, t_{n})\right|\le C (h^{d_{s}-3}+\tau^{d_{t}+1}). \end{equation}$

(4.10)

Then, we have

$\begin{equation} |E_{1}(s, t)|\le \left|\frac{\partial e}{\partial t}(s_{m}, t)+\frac{\partial e}{\partial t}(s_{m}, t_{n})\right|\le C (h^{d_{s}+1}+\tau^{d_{t}}). \end{equation}$

(4.11)

Similarly, for $E_{3}(s, t)$ we have

$\begin{equation} \begin{array}{ll} E_{3}(s, t) = \frac{\partial^2 \phi}{\partial s^2}(s, t)- \frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n}) = \frac{\partial^2 e}{\partial s^2}(s, t_{n})+ \frac{\partial^2 e}{\partial s^2}(s_{m}, t_{n}), \end{array} \end{equation}$

(4.12)

and

$\begin{equation} |E_{3}(s, t)|\le \left| \frac{\partial^2 e}{\partial s^2}(s, t_{n})+ \frac{\partial^2 e}{\partial s^2}(s_{m}, t_{n}) \right|\le C (h^{d_{s}-1}+\tau^{d_{t}+1}). \end{equation}$

(4.13)

For $E_{4}(s, t)$ we get

$\begin{equation} \begin{array}{ll} |E_{4}(s, t)|& = \left|\phi_{0}\frac{\partial \phi}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi}{\partial s}(s_{m}, t_{n})\right| \\ &\le \left|\frac{\partial e}{\partial t}(s_{m}, t)+\frac{\partial e}{\partial t}(s_{m}, t_{n})\right|\le C (h^{d_{s}+1}+\tau^{d_{t}}). \end{array} \end{equation}$

(4.14)

Combining (4.9) and (4.11)–(4.14) together, the proof of Theorem 2 is completed. □

5. Numerical examples

All the examples are carried on a computer with Intel(R) Core(TM) i5-8265U CPU @ 1.60 GHz 1.80 GHz operating system, 16 G radon access running memory and a 512 G solid state disk memory. All simulation experiments were realized by the software Matlab (Version: R2016a). In this part, two examples are presented to test the theorem.

Example 1. Consider the KS equation

$\frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = \varphi(s, t)$

with the condition is

$\phi(0, t) = 0, \phi(1, t) = 0,$

and

$\phi(s, 0) = \sin(2\pi s).$

$\phi_{ss}(0, t) = 0, \phi_{ss}(1, t) = 0,$

and

$\varphi(s, t) = e^{-t}\sin(2\pi s)[2\pi e^{-t}\cos(2\pi s)-1+16\pi^4-4\pi^2].$

The solution of the KS equation is

$\phi(s, t) = e^{-t}\sin(2\pi s).$

In Figures 1–3, errors of unform partition with direct linearization, partial linearization, Newton linearization for the KS equation are presented. In Figures 4–6, errors of non-uniform partition with direct linearization, partial linearization, Newton linearization for the KS equation are presented.

Figure 1. Errors of nonuniform partition by direct linearization with

$m = n = 19$ .

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Figure 2. Errors of nonuniform partition by partial linearization with

$m = n = 19$ .

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Figure 3. Errors of nonuniform partition by Newton linearization with

$m = n = 19$ .

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Figure 4. Errors of uniform partition by direct linearization with

$m = n = 19$ .

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Figure 5. Errors of uniform partition by partial linearization with

$m = n = 19$ .

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Figure 6. Errors of uniform partition by Newton linearization with

$m = n = 19$ .

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In Tables 1 and 2, errors of LBCM and LBRCM for the KS equation with boundary condition dealt with by the method of substitution and method of addition are given. From Table 1, we know that the accuracy of LBCM is higher than LBRCM, and from Table 2 the accuracy of the method of additional is higher than the method of substitution.

Table 1. Errors of LBCM for KS equation with

$m = n = 17$ .

	Method of substitution		Method of additional
Linearization	Uniform partition	Nonuniform partition	Uniform partition	Nonuniform partition
direct	1.3278e-07	5.6616e-10	1.7050e-08	4.6293e-10
partial	5.5563e-07	2.6381e-09	1.1492e-07	5.0974e-10
Newton	6.6705e-07	4.8875e-10	8.8609e-08	2.5867e-11

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Table 2. Errors of LBRCM for KS equation with

$m = n = 17, d_{s} = d_{t} = 12$ .

	Method of substitution		Method of additional
Linearization	Uniform partition	Nonuniform partition	Uniform partition	Nonuniform partition
direct	4.4575e-06	3.2280e-08	4.1010e-08	2.2749e-09
partial	4.4573e-06	3.2245e-08	5.4191e-07	1.5951e-07
Newton	4.4560e-06	3.2215e-08	1.2972e-06	3.5137e-07

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In , we choose the Newton linearization to solve the KS equation, and the error of LBRCM for uniform and nonuniform partitions are presented with $t = 0.3, 0.9, 2, 4, 8, 16$ .

Table 3. Errors of Newton linearization for

$t$ .

	Uniform partition		Nonuniform partition
$t$	$(8, 8) d_{s}=d_{t}=7$	$(16, 16) d_{s}=d_{t}=15$	$(8, 8) d_{s}=d_{t}=7$	$(16, 16) d_{s}=d_{t}=15$
0.3	1.5449e-01	1.3163e-06	6.2692e-02	2.4769e-08
0.9	1.4211e-01	1.1737e-06	6.1721e-02	2.3846e-08
2	1.2162e-01	1.0785e-06	5.8680e-02	2.3685e-08
4	9.1544e-02	9.4383e-07	5.3241e-02	2.3353e-08
8	5.1798e-02	7.2283e-07	4.3721e-02	2.2440e-08
16	1.6540e-02	4.1712e-07	2.9435e-02	1.9220e-08

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The errors of LBRCM of uniform and Chebyshev partitions are presented with $(m, n, d_{s}, d_{t}) = (8, 8, 7, 7), (16, 16, 15, 15)$ . From the table, comparing $(m, n) = (8, 8)$ with $(m, n) = (16, 16)$ , the accuracy was higher when the number became bigger.

In the following table, we take Newton linearization to present numerical results. From and , with errors of Newton linearization for uniform partition $d_{t} = 6;t = 1$ are given and convergence rate is $O(h^{d_{s}})$ . From , with space variable $s, d_{s} = 6$ , and there is superconvergence rate $O(h^{d_{s}-1})$ at $t = 1$ .

Table 4. Errors of Newton linearization for uniform partition

$d_{t} = 6$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	4.1317e-01		3.2652e-03		3.3180e-01
16, 16	1.8608e-01	1.1508	3.1257e-02	-	3.3919e-02	3.2902
32, 32	9.5437e-02	0.9633	1.0198e-02	1.6159	3.3873e-03	3.3239
64, 64	4.7221e-02	1.0151	2.6490e-03	1.9448	3.5472e-04	3.2554

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Table 5. Errors of Newton linearization for uniform partition

$d_{s} = 6$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	1.3997e-01		1.4004e-01		1.4008e-01
16, 16	5.4923e-03	4.6716	5.4957e-03	4.6714	5.4973e-03	4.6714
32, 32	1.2850e-04	5.4176	1.2883e-04	5.4148	1.2891e-04	5.4143
64, 64	2.9976e-06	5.4218	3.0728e-06	5.3898	3.0798e-06	5.3874

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For and , the errors of Chebyshev partition for Newton linearization with $s$ and $t$ are presented. For $d_{t} = 6$ , the convergence rate is $O(h^{d_{s}})$ in Table 6, while in Table 7, there are also superconvergence phenomena.

Table 6. Errors of Newton linearization for Chebyshev partition

$d_{t} = 6$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	5.4754e-01		2.9399e-02		8.5922e-02
16, 16	1.0318e-01	2.4078	4.6815e-03	2.6507	1.2658e-03	6.0849
32, 32	9.6912e-02	0.0904	8.0675e-04	2.5368	1.9577e-05	6.0148
64, 64	4.8014e-01	-	1.7672e-03	-	2.2716e-05	-

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Table 7. Errors of Newton linearization for Chebyshev partition

$d_{s} = 6$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	6.1344e-02		6.1386e-02		6.1415e-02
16, 16	8.1492e-05	9.5561	8.1163e-05	9.5629	8.0977e-05	9.5669
32, 32	1.4204e-07	9.1642	1.4183e-07	9.1606	1.5487e-07	9.0303
64, 64	6.3190e-06	-	3.8960e-06	-	1.4861e-06	-

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Example 2. Consider the KS equation

$\frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = 0,$

with the analytic solution

$\phi(s, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(s-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(s-ct+s_{0})\right],$

and boundary condition

$\phi(-10, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(-10-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(-10-ct+s_{0})\right],$

$\phi(10, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(10-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(10-ct+s_{0})\right],$

and initial condition

$\phi(s, 0) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(s+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(s+s_{0})\right],$

with $c = 2, x_{0} = 10$ .

In –, errors of direct linearization, partial linearization, Newton linearization with $m = n = 19$ KS equation are presented, respectively.

Figure 7. Errors of direct linearization with

$m = n = 19$ .

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Figure 8. Errors of partial linearization with

$m = n = 19$ .

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Figure 9. Errors of Newton linearization with

$m = n = 19$ .

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In the following table, direct linearization is chosen to present numerical results. From and , errors of direct linearization for uniform partition $d_{t} = 7$ with different $d_{s}$ are given and the convergence rate is $O(h^{d_{s}}-1)$ . From , with space variable $s, d_{s} = 7$ , and there are also superconvergence phenomena.

Table 8. Errors of direct linearization for uniform partition for

$d_{t} = 7$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	1.3587e+00		8.9361e-01		6.3703e-01
16, 16	2.1617e-01	2.6520	2.7467e-01	1.7019	2.5682e-01	1.3106
32, 32	6.7743e-02	1.6740	6.8822e-02	1.9967	4.7078e-02	2.4476
64, 64	2.5175e-02	1.4281	1.3216e-02	2.3806	4.3739e-03	3.4281

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Table 9. Errors of direct linearization for uniform partition for

$d_{s} = 7$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	3.6253e-01		3.6380e-01		3.6446e-01
16, 16	1.8147e-01	0.9984	1.8124e-01	1.0052	1.8121e-01	1.0081
32, 32	6.4076e-02	1.5019	6.4158e-02	1.4982	6.4141e-02	1.4983
64, 64	8.9037e-04	6.1692	8.9840e-04	6.1581	8.9863e-04	6.1574

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For and , the errors of Chebyshev partition for direct linearization with $s$ and $t$ are presented. For $d_{t} = 7$ , the convergence rate is $O(h^{d_{s}})$ in Table 10, while in Table 11, there are also superconvergence phenomena.

Table 10. Errors of direct linearization for Chebyshev partition for

$d_{t} = 7$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	6.5990e-01		4.0742e-01		3.6175e-01
16, 16	1.1154e-01	2.5646	1.7539e-01	1.2160	2.1752e-01	0.7338
32, 32	4.3052e-02	1.3735	8.6654e-03	4.3391	1.2511e-03	7.4418
64, 64	3.9204e-02	0.1351	2.3776e-03	1.8658	3.5682e-04	1.8099

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Table 11. Errors of direct linearization for Chebyshev partition for

$d_{s} = 7$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	4.3760e-01		4.3745e-01		4.3739e-01
16, 16	1.1801e-01	1.8908	1.1801e-01	1.8902	1.1801e-01	1.8900
32, 32	9.9842e-04	6.8850	9.9854e-04	6.8849	9.9801e-04	6.8857
64, 64	2.5749e-06	8.5990	2.5052e-06	8.6388	4.8401e-06	7.6879

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6. Conclusions

In this paper, LBRCM is used to solve the (1+1) dimensional SK equation. Three kinds of linearization methods are taken to translate the nonlinear part into a linear part. Matrix equations of the discrete SK equation are obtained from corresponding linearization schemes. The convergence rate of LBRCM is also presented. In the future work, LBRCM can be developed for the (2+1) dimensional SK equation and other partial differential equations classes, including Kolmogorov-Petrovskii-Piskunov (KPP) equation and, fractional reaction-diffusion equation and so on.

Acknowledgments

The work of Jin Li was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003).

The authors also gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation.

Conflict of interest

The author declares no conflict of interest.

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