# Numerical Solution of a Class of Nonlinear Partial Differential Equations by Using Barycentric Interpolation Collocation Method

Authors: Hongchun Wu; Yulan Wang; Wei Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Partial differential equations (PDEs) are widely used in mechanics, control processes, ecological and economic systems, chemical cycling systems, and epidemiology. Although there are some numerical methods for solving PDEs, simple and efficient methods have always been the direction that scholars strive to pursue. Based on this problem, we give the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs. Four numerical experiments are carried out and compared with other methods; the accuracy of the numerical solution obtained by the present method is obviously improved.


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## Body

## 1. Introduction

Consider the following partial differential equation:(1)utt+γuxx+gu=fx,t,x∈a,b,t∈0,T,the initial conditions are expressed as(2)ux,0=ϕx,utx,0=ψx,x∈a,b,and the boundary conditions are expressed as(3)ua,t=p0t,ub,t=p1t,t≥t0,or(4)uxa,t=p0t,uxb,t=p1t,t≥t0,where u=u(x,t) represents an unknown function at position x and time t. γ is a known constant. If g(u) is nonlinear, the equation is a nonlinear PDEs. If g(u) is linear, then the equation is linear PDEs.PDEs are paid more attention, and there are many studies on the numerical solution of PDEs. For PDEs, a quintic polynomial spline technique is used for the numerical solutions of fourth-order PDEs with Caputo time fractional derivative on a finite domain [1]. The adaptive grid Haar wavelet collocation method is used for the numerical solution of parabolic PDEs [2]. The numerical stability of PDEs with piecewise constant arguments is considered [3] along with a quantum algorithm for preparing states that encode solutions of nonhomogeneous linear PDEs [4]. A time discretization scheme approximates the time fractional PDEs [5]. We also consider the application of a domain-type method of fundamental solutions together with a Picard iteration scheme for solving nonlinear elliptic PDEs [6] and a deep learning-based approach that can handle general high-dimensional parabolic PDEs [7]. Couple double Laplace transform with iterative method solves nonlinear Klein-Gordon equation subject to initial and boundary conditions [8]. The variational iteration method (VIM) and homotopy perturbation method (HPM) are used for various kinds of nonlinear problems ([9–12]). Numerical solutions to linear and nonlinear PDEs use the peridynamic differential operator [13]. Numerical collocation approach solves PDEs [14]. A modified method using wavelet intergrid operators known as new wavelet based full-approximation scheme similar to multigrid full-approximation scheme is developed for the numerical solution of nonlinear elliptic PDEs [15]. B-spline collocation method ([16–23]) had solved various kinds of nonlinear PDEs. Meshless Barycentric interpolation collocation method ([24–30]) is a high precision numerical method. This paper gives the meshless barycentric interpolation collocation method (MBICM) for solving a class of PDEs.This paper is organized as follows. In Section2, the meshless barycentric interpolation collocation method (MBICM) is introduced in detail. Several numerical examples are included to demonstrate the practicability of the method and some relevant figures, tables, and analyes are provided in Section 3. Some concluding remarks are given in Section 4.


## 2. Meshless Barycentric Interpolation Collocation Method

The MBICM uses the barycentric interpolation as the approximate function and uses the differential matrix of the barycentric interpolation to discrete the differential operator. It does not need to divide the calculation grid, and the calculation program is easy to write. It is an efficient and high precision meshless numerical method for calculating linear differential equations. For nonlinear PDEs, we construct a linearized iterative scheme to transform the equations into linear equations.First, we introduce the MBICM for linear PDEs.


### 2.1. The Partial Differential Matrix of Barycentric Interpolation

For the nonlinear (1), we give an initial value u0(x,t) and construct a linear iterative scheme(5)untt+γunxx+gun-1=fx,t,n=0,1,2,…,Next, we give meshless barycentric interpolation collocation method of (5).In the interval[a,b] takes M different nodes, a≤x1<x2<⋯<xM≤b, respectively, in the interval [0,T] takes N different nodes, 0≤t1<t2<⋯<tN≤T, respectively. These nodes can generate two dimensional nodes on the rectangular area Ω=[a,b]×[0,T], as follows: {(xi,tj),i=1,2,…,M; j=1,2,…,N}.The barycenter interpolation form of functionu(x,t) at the nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N} can be expressed as(6)ux,t=∑i=1M∑j=1Nξixηjtuij,i=1,2,…,M,j=1,2,…,N,where ξi(x)=∏k=1,k≠iM(x-xk)/∏k=1,k≠iM(xi-xk),i=1,2,…,M, ηj(t)=∏k=1,k≠jN(t-tk)/∏k=1,k≠jN(tj-tk),j=1,2,…,N. Moreover, ξi(x) is the barycentric interpolation basis function of M nodes on interval [a,b], ηj(t) is the barycentric interpolation basis function of N nodes on interval [0,T]. uij=u(xi,tj),i=1,2,…,M;j=1,2,…,N is the function value of u(x,t) at the nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N}.According to formula (6), the partial derivative of l+k order of function u(x,t) can be written as(7)∂l+ku∂xl∂tk=∑i=1M∑j=1Nξilxηjktuij,l,k=1,2,…,and the function values of the formula (7) at the nodes (xp,tq) are(8)ul,kxp,tq≔∂l+kuxp,tq∂xl∂tk=∑i=1M∑j=1Nξilxpηjktquij,p=1,2,…,M;q=1,2,…,N.The nodes on the interval[a,b] and [0,T] constitute two groups of column vectors respectively. They are defined as(9)x0=x1,x2,…,xMT,t0=t1,t2,…,tNT,then the matrixes X, T above the region Ω can be generated by the above column vectors x0,t0, and the form of matrixes X and T is as follows:(10)XT=x0T,x0T,…,x0T,T=t0,t0,…,t0.The matrixesX, T are respectively written as column vectors x, t of N×M dimension:(11)x=X1,X2,…,XM×NT,t=T1,T2,…,TM×NT.The relations between the componentsxi,yj of the formula (8) and the components XK,YK of formula (10) are as follows:(12)Xk=Xi-1N+j=xi,Tk=Ti-1N+j=tj,i=1,2,…,M;j=1,2,…,N;k=1,2,…,M×N.The function values of formula (6) and formula (7) at the node form column vectors u,u(l,k), and they are as follows:(13)u=u1,u2,…,uM×NT,ul,k=u1l,k,u2l,k,…,uM×Nl,kT,up=uXp,Tp,upl,k=ul,kXp,Tp,p=1,2,…,M×N.Therefore, formula (8) can be expressed in the following matrix form:(14)ul,k=Dl,ku,in formula (12), D(l,k)=C(l)⊗D(k) is the Kronecker product of matrixes C(l) and D(k), and it is also called l+k order partial differential matrix at nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N}. C(l) and D(k) are l,k order differential matrixes formed by barycenter interpolation at node interval [a,b] and interval [0,T], respectively.Denote that(15)C0=IM,D0=IN,where IM and IN are M order unit matrix and N order unit matrix, respectively.By using notation (6), (7), and (14), the calculation formula of BICM for model (1) can be written in following matrix form:(16)D0,2+γD2,0+diaggun-1u=fx,t.Here,diag is a symbol of diagonal matrix composed of vectors. If the linear iterative scheme (5) is convergent, then un(x,t)→u(x,t)(n→∞). In the calculation process, we set a calculation accuracy ε=1e-15 or ε=1e-10, if |un(x,t)-un-1(x,t)|⩽ε, then the iteration stops, and at this point, approximate solution of nonlinear differential equations is obtained.


### 2.2. The Discrete Formula and Application Method of Initial Boundary Conditions

The discretization of initial boundary conditions requires the use of the barycenter interpolation formula (6). By acting on the initial boundary conditions, the discrete formulae of initial boundary conditions are given:(17)∑i=1M∑j=1Nξixpηjt0uij=ϕxp,p=1,2,…,M;(18)∑i=1M∑j=1Nξixpηj′t0uij=φxp,p=1,2,…,M;(19)∑i=1M∑j=1Nξiaηjtquij=p0tq,q=1,2,…,N;(20)∑i=1M∑j=1Nξibηjtquij=p1tq,q=1,2,…,N.The boundary conditions are replaced by displacement method. The following is a description of the specific displacement process:The firstN+1 equation, the first 2N+1 equation, …, and the first (M-2)N+1 equation of the system of equations are replaced by M equations of the initial condition (17); the first 2N equation, the first 3N equation, …, and the first (M-1)N of the system of equations are replaced by M equations of the initial condition (18); the first equation to the first N equation of the system of equations are separated by N equations of the boundary condition (19); the first (M-1)N+1 equation to the first M×N equation of the system of equations are replaced by N equations of the boundary condition (20).


## 2.1. The Partial Differential Matrix of Barycentric Interpolation

For the nonlinear (1), we give an initial value u0(x,t) and construct a linear iterative scheme(5)untt+γunxx+gun-1=fx,t,n=0,1,2,…,Next, we give meshless barycentric interpolation collocation method of (5).In the interval[a,b] takes M different nodes, a≤x1<x2<⋯<xM≤b, respectively, in the interval [0,T] takes N different nodes, 0≤t1<t2<⋯<tN≤T, respectively. These nodes can generate two dimensional nodes on the rectangular area Ω=[a,b]×[0,T], as follows: {(xi,tj),i=1,2,…,M; j=1,2,…,N}.The barycenter interpolation form of functionu(x,t) at the nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N} can be expressed as(6)ux,t=∑i=1M∑j=1Nξixηjtuij,i=1,2,…,M,j=1,2,…,N,where ξi(x)=∏k=1,k≠iM(x-xk)/∏k=1,k≠iM(xi-xk),i=1,2,…,M, ηj(t)=∏k=1,k≠jN(t-tk)/∏k=1,k≠jN(tj-tk),j=1,2,…,N. Moreover, ξi(x) is the barycentric interpolation basis function of M nodes on interval [a,b], ηj(t) is the barycentric interpolation basis function of N nodes on interval [0,T]. uij=u(xi,tj),i=1,2,…,M;j=1,2,…,N is the function value of u(x,t) at the nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N}.According to formula (6), the partial derivative of l+k order of function u(x,t) can be written as(7)∂l+ku∂xl∂tk=∑i=1M∑j=1Nξilxηjktuij,l,k=1,2,…,and the function values of the formula (7) at the nodes (xp,tq) are(8)ul,kxp,tq≔∂l+kuxp,tq∂xl∂tk=∑i=1M∑j=1Nξilxpηjktquij,p=1,2,…,M;q=1,2,…,N.The nodes on the interval[a,b] and [0,T] constitute two groups of column vectors respectively. They are defined as(9)x0=x1,x2,…,xMT,t0=t1,t2,…,tNT,then the matrixes X, T above the region Ω can be generated by the above column vectors x0,t0, and the form of matrixes X and T is as follows:(10)XT=x0T,x0T,…,x0T,T=t0,t0,…,t0.The matrixesX, T are respectively written as column vectors x, t of N×M dimension:(11)x=X1,X2,…,XM×NT,t=T1,T2,…,TM×NT.The relations between the componentsxi,yj of the formula (8) and the components XK,YK of formula (10) are as follows:(12)Xk=Xi-1N+j=xi,Tk=Ti-1N+j=tj,i=1,2,…,M;j=1,2,…,N;k=1,2,…,M×N.The function values of formula (6) and formula (7) at the node form column vectors u,u(l,k), and they are as follows:(13)u=u1,u2,…,uM×NT,ul,k=u1l,k,u2l,k,…,uM×Nl,kT,up=uXp,Tp,upl,k=ul,kXp,Tp,p=1,2,…,M×N.Therefore, formula (8) can be expressed in the following matrix form:(14)ul,k=Dl,ku,in formula (12), D(l,k)=C(l)⊗D(k) is the Kronecker product of matrixes C(l) and D(k), and it is also called l+k order partial differential matrix at nodes {(xi,tj),i=1,2,…,M; j=1,2,…,N}. C(l) and D(k) are l,k order differential matrixes formed by barycenter interpolation at node interval [a,b] and interval [0,T], respectively.Denote that(15)C0=IM,D0=IN,where IM and IN are M order unit matrix and N order unit matrix, respectively.By using notation (6), (7), and (14), the calculation formula of BICM for model (1) can be written in following matrix form:(16)D0,2+γD2,0+diaggun-1u=fx,t.Here,diag is a symbol of diagonal matrix composed of vectors. If the linear iterative scheme (5) is convergent, then un(x,t)→u(x,t)(n→∞). In the calculation process, we set a calculation accuracy ε=1e-15 or ε=1e-10, if |un(x,t)-un-1(x,t)|⩽ε, then the iteration stops, and at this point, approximate solution of nonlinear differential equations is obtained.


## 2.2. The Discrete Formula and Application Method of Initial Boundary Conditions

The discretization of initial boundary conditions requires the use of the barycenter interpolation formula (6). By acting on the initial boundary conditions, the discrete formulae of initial boundary conditions are given:(17)∑i=1M∑j=1Nξixpηjt0uij=ϕxp,p=1,2,…,M;(18)∑i=1M∑j=1Nξixpηj′t0uij=φxp,p=1,2,…,M;(19)∑i=1M∑j=1Nξiaηjtquij=p0tq,q=1,2,…,N;(20)∑i=1M∑j=1Nξibηjtquij=p1tq,q=1,2,…,N.The boundary conditions are replaced by displacement method. The following is a description of the specific displacement process:The firstN+1 equation, the first 2N+1 equation, …, and the first (M-2)N+1 equation of the system of equations are replaced by M equations of the initial condition (17); the first 2N equation, the first 3N equation, …, and the first (M-1)N of the system of equations are replaced by M equations of the initial condition (18); the first equation to the first N equation of the system of equations are separated by N equations of the boundary condition (19); the first (M-1)N+1 equation to the first M×N equation of the system of equations are replaced by N equations of the boundary condition (20).


## 3. Numerical Experiment

Now, numerical examples of nonlinear Klein-Gordon equations, other types of equations, and systems of equations are solved by using MBICM. In the following numerical examples, we compare the errors of different methods to illustrate the accuracy of the barycentric interpolation collocation method. The errors are expressed by the error normL2, L∞, and L2, L∞ error norms and absolute error are expressed as follows:L 2 Error Norm (21) L 2 u = ∑ i = 1 M u x i , t - u n x i , t 2 △ x 1 / 2 ,L ∞ Error Norm (22) L ∞ u = max 1 ≤ i ≤ M ⁡ u x i , t - u n x i , t ,Absolute Error (23) A b u = u x i , t - u n x i , t .where u(xi,t) and un(xi,t) are exact and numerical solution, respectively.Example 1 (see [31]).
Consider the nonlinear Klein-Gordon equation (1) with γ=-1 and g(u)=u2(24)utt-uxx+u2=-xcos⁡t+x2cos2⁡t.where x∈(-1,1), t>0. Equation (24) satisfies the initial conditions:(25)ux,0=x,utx,0=0,x∈-1,1,and with Dirichlet boundary conditions(26)u-1,t=-cos⁡t,u1,t=cos⁡t,t≥0.The exact solution for this problem is(27)ux,t=xcos⁡t.In Table1, we enumerate the error norms in different time t=1,t=3,t=5,t=7 and t=10 using different methods. By comparison, it is found that the present method is more accurate than other methods.Table 1
Comparison of error norms at different timet for Example 1.
t O(k2+k2h2+h2)method[31] O(k2+k2h2+h4)method[31] Present method L ∞ u L 2 u L ∞ u L 2 u L ∞ u L 2 ( u ) 1 1.03E-09 7.01E-09 7.68E-10 4.91E-09 7.9192E-13 1.5457E-12 3 1.00E-09 6.59E-09 7.52E-10 4.69E-09 1.0825E-13 2.3010E-13 5 2.56E-10 1.29E-09 1.76E-10 9.46E-10 3.9573E-13 7.6876E-13 7 1.13E-09 7.47E-09 7.63E-10 5.11E-09 9.6356E-13 2.1556E-12 10 9.46E-10 5.84E-09 6.55E-10 3.98E-09 2.4178E-10 5.8307E-10In Figure1, the exact solution of u obtained by the present method at t=5 with the number of nodes M=20,N=20. Figure 3, the numerical and exact solutions of u obtained by present method at different time t=1,t=3,t=5 and t=7 with the number of nodes M=20,N=20. From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high.Figure 1
The exact solution ofu for Example 1.In Figure2, the absolute error of u obtained by present method at t=5 with the number of nodes M=20,N=20. In Figure 4, the absolute errors of u are obtained by the present method at different times t=1,t=3,t=5, and t=7 with the number of nodes M=20,N=20. We can get the absolute error of u changing with x at t=3 as the most stable.Figure 2
The absolute error ofu for Example 1 (t=5,M=20,N=20).Figure 3
Comparison of numerical and exact solutions ofu for Example 1 (M=20,N=20).Figure 4
The absolute errors ofu at different time t for Example 1 (M=20,N=20).Example 2 (see [31]).
Consider the nonlinear Klein-Gordon equation (1) with γ=-1 and g(u)=π2/4u+u2,(28)utt-uxx+π24u+u2=x2sin2⁡π2t.where x∈(-1,1), t>0. Equation (28) satisfies the initial conditions:(29)ux,0=0,utx,0=π2x,x∈-1,1,and with Dirichlet boundary conditions(30)u-1,t=-sin⁡π2t,u1,t=sin⁡π2t,t≥0.The exact solution for this problem is(31)ux,t=xsin⁡π2t.In Table2, we list the error norms in different times t=1,t=2,t=3,t=4, and t=5 using different methods. By comparison, we found that the present method is much more accurate than other methods.Table 2
Comparison of error norms at different timet for Example 2.
t O(k2+k2h2+h2)method[31] Ok2+k2h2+h4method[31] Present method L ∞ ( u ) L 2 ( u ) L ∞ ( u ) L 2 ( u ) L ∞ ( u ) L 2 ( u ) 1 3.97E-06 2.71E-05 3.97E-06 2.71E-05 2.9723E-12 6.5955E-12 2 1.51E-06 8.97E-06 1.51E-06 8.97E-06 3.0783E-13 6.1985E-13 3 2.14E-06 1.49E-05 2.14E-06 1.49E-05 5.5911E-13 1.4550E-12 4 1.86E-06 1.05E-05 1.86E-06 1.05E-05 8.4829E-13 1.8202E-12 5 5.08E-06 3.36E-05 5.08E-06 3.36E-05 9.6175E-12 2.3494E-11In Figure5, the exact solution of u is obtained by the present method at t=5 with the number of nodes M=20,N=20. In Figure 7, numerical and exact solutions of u are obtained by the present method at different times t=1,t=2,t=3,t=4, and t=5 with the number of nodes M=20,N=20. From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high. Moreover, we find the monotonicity of function u at different time.Figure 5
The exact solution ofu for Example 2 (t=5,M=20,N=20).In Figure6, the absolute error of u is obtained by the present method at t=5 with the number of nodes M=20,N=20. In Figure 8, the absolute errors of u are obtained by the present method at different times t=1,t=2,t=3,t=4, and t=5 with the number of nodes M=20,N=20. We can get the absolute errors of u changing with x at t=2,t=3 and t=4 are the most stable. But the absolute error of u changing with x fluctuates greatly at t=5.Figure 6
The absolute error ofu for Example 2 (t=5,M=20,N=20).Figure 7
Comparison of numerical and exact solutions ofu for Example 2 (M=20,N=20).Figure 8
The absolute errors ofu at different time t for Example 2 (M=20,N=20).Example 3 (see [32]).
We consider nonlinear Hunter-Saxton equation:(32)uxt=-uuxx-12ux2.where 0≤x≤1, 0≤t≤T. Equation (32) satisfies the initial conditions:(33)ux,0=2x,utx,0=-2x,x∈0,1,and the boundary conditions(34)u0,t=0,ux0,t=21+t.The exact solution of this problem is as follows:(35)ux,t=2x1+t.In Tables3 and 4, the absolute errors of u are obtained by the present method for different x with the number of nodes M=15,N=15 at t=1 and t=0.1, respectively. It can be found that the absolute error decreases with the decrease of time t in the same number of nodes and x.Table 3
Comparing the resultst=1, M=N=15 for Example 3.
x Numerical solution Exact solution Absolute error 0 0.0000 0.0000 0.0002E-10 0.0432 0.0432 0.0432 0.0035E-10 0.1654 0.1654 0.1654 0.0166E-10 0.3455 0.3455 0.3455 0.0836E-10 0.5523 0.5523 0.5523 0.0829E-10 0.7500 0.7500 0.7500 0.0797E-10 0.9045 0.9045 0.9045 0.0971E-10 0.9891 0.9891 0.9891 0.1047E-10 1.0000 1.0000 1.0000 0.1053E-10Table 4
Comparing the results att=0.1, M=N=15 for Example 3.
x Numerical solution Exact solution Absolute error 0 0.0000 0 0.0017E-12 0.0432 0.0786 0.0786 0.0709E-12 0.1654 0.3008 0.3008 0.3931E-12 0.3455 0.6282 0.6282 0.2666E-12 0.5523 1.0041 1.0041 0.1452E-12 0.7500 1.3636 1.3636 0.1499E-12 0.9045 1.6446 1.6446 0.0708E-12 0.9891 1.7983 1.7983 0.0315E-12 1.0000 1.8182 1.8182 0.0298E-12In Figure9, the numerical and exact solutions of u are obtained by the present method at different times t=1 and t=0.1 with the number of nodes M=15,N=15. From this we can see that the numerical solution and the exact solution have high reunification, so the accuracy of the present method is high. Moreover, the smaller the value of t is, the faster the function u changes with x. In Figure 10, the absolute error of u is obtained by the present method with the number of nodes M=15,N=15.Figure 9
Comparison of numerical and exact solutions ofu for Example 3(M=15,N=15).Figure 10
The absolute error ofu for Example 3 (t=1,M=15,N=15).In Figure11, the logarithmic absolute errors of u are obtained by the present method at different times t=1 and t=0.1 with the number of nodes M=15,N=15. In Figure 12, the absolute errors of u are obtained by the present method at different times t=1 and t=0.1 with the number of nodes M=15,N=15. Both figures show that the errors fluctuate and then tend to be stable with the increase of x at t=1; the errors fluctuate slightly and remain stable with the increase of x at t=0.1.Figure 11
The logarithmic absolute errors ofu for Example 3 (M=15,N=15).Figure 12
The absolute errors ofu at different time t for Example 3 (M=15,N=15).Example 4 (see [32]).
For nonlinear Hunter-Saxton equation (32) with the following initial conditions (36)ux,0=2+3x2/3+2x+2,utx,0=-4x+12+3x-1/3+1,x∈0,1.The exact solution is as follows: (37)ux,t=2+3ν2/3+2ν+21+t2where ν=x(1+t).In Table5, there are the error norms of u with different values nodes M,N at t=1,t=0.1 and t=0.01. We can see that the accuracy of u is improved with the increase of the number of nodes; when the number of nodes increases to a certain number, the accuracy tends to be stable; it shows that the convergence of the present method is better. On the other hand, u has the highest accuracy at t=0.1.Table 5
Comparison of error norms at different timet for Example 4.
t M = N = 15 M = N = 19 M = N = 23 L ∞ ( u ) L 2 ( u ) L ∞ ( u ) L 2 ( u ) L ∞ ( u ) L 2 ( u ) 0.01 2.1370E-10 5.2854E-10 3.3950E-11 8.8235E-11 2.5417E-11 8.0727E-11 0.1 9.2474E-10 2.2981E-09 2.5642E-12 5.9925E-12 3.3769E-12 9.2082E-12 1 9.8737E-09 2.5369E-08 8.4526E-11 2.4854E-10 2.5882E-12 6.6491E-12In Figure13, the numerical and exact solutions of u are obtained by the present method at different times t=1,t=0.1, and t=0.01 with the number of nodes M=19,N=19. From this we can see that the numerical solution and the exact solution have high reunification. Moreover, the bigger the t, the smaller the u value. In Figure 14, the absolute error of u is obtained by the present method with the number of nodes M=23,N=23. The accuracy of the present method can be very high.Figure 13
Comparison of numerical and exact solutions ofu for Example 4 (M=19,N=19).Figure 14
The absolute error ofu for Example 4 (t=1,M=23,N=23).In Figure15, the logarithmic absolute errors of u are obtained by the present method at different times t=1,t=0.1, and t=0.01 with the number of nodes M=19,N=19. In Figure 16, the absolute errors of u are obtained by the present method at different times t=1,t=0.1, and t=0.01 with the number of nodes M=19,N=19. We can get that the absolute error of function u has obvious fluctuation at t=1, and the absolute error of u is always relatively stable at t=0.1.Figure 15
The logarithmic absolute errors ofu for Example 4 (M=19,N=19).Figure 16
The absolute errors ofu at different time t for Example 4 (M=19,N=19).


## 4. Conclusions and Remarks

In this paper, we first transform a nonlinear partial differential equation into a linear differential equation by using the direct linearization iterative method and then solve the linear partial differential equation by using the meshless barycentric interpolation collocation method. The error estimation and convergence analysis of the method are proved. Through four examples and their charts, it is found that this method has higher computation accuracy than the existing methods. Finally, this method is applied to nonlinear partial differential equations and the calculated accuracy is higher.All computations are performed by the MatlabR2007b software packages.



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