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A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations

  • P.G. Dlamini EMAIL logo and M. Khumalo
Published/Copyright: December 16, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.

Keywords: Compact finite differences; Quasilinearization; Nonlinear evolution equations
MSC 2010: 35K61; 65M06

1 Introduction

Many researchers are now using higher order compact finite difference schemes in place of the conventional second order finite difference to solve differential equations [1,2,3,4,5]. This is because significant improvements to the accuracy of numerical solutions have been obtained by using fourth or sixth order compact FD schemes. Another advantage is that the high accuracy is obtained on coarser grids which ensures greater computational efficiency. Lele [6] presented various compact finite difference schemes for applications such as evaluating high order derivatives, interpolation and filtering.

A number of researchers have used compact FD schemes to solve partial differential equations arising in various fields. In all the previous work the compact FD schemes were only applied on space variables and other discretization techniques are used for the time variables. Quite a number of numerical techniques have been paired with the compact FD schemes in order to discretize time. For example [7] used a two-step predictor-corrector algorithm called McCormack method to find the solution of Burgers equation. Li and Visbal [3] used the classical fourth-order four-stage Runge-Kutta method (RK4). A lot of of researchers have combined the compact FD schemes with the third order total variation diminishing Runge-Kutta (TVD-RK3) scheme (see [4, 8,9,10]). Dlamini et al [11] combined the compact FD schemes with the Crank-Nicolson technique in time to solve unsteady boundary layer flow problems. However these techniques limit the accuracy of the difference scheme to fourth order or less in time which means small mesh sizes have to be used in order to get desirable accuracy, and thus much more computational work is involved. The aim is to try the compact FD schemes (sixth order) on the time variable as well which will allow us to archive greater accuracy with a rough step size.

The purpose of this investigation is to explore the possibility of applying sixth order compact FD schemes in both space and time variables. We consider this on a few examples of nonlinear evolution PDEs. These are important equations arising in a number of fields of science and engineering. They are used to describe many complex nonlinear settings in applications such as vibration and wave propagation, fluid mechanics, plasma physics, quantum mechanics, nonlinear optics, solid state physics, chemical kinematics, physical chemistry, population dynamics, and many other areas of mathematical modelling. Understanding the behavior of these problems requires finding solutions of the differential equations and analyzing them. Most of the evolution equations are very complex to solve analytically and so we rely on approximate solutions. So the development of numerical solutions to solve such problems continues to be an active area of research.

Numerical experiments show that implementing the compact FD schemes on both space and time yield highly accurate results. This was experimented on the Fisher’s, Burgers-Fisher, Burgers-Huxley and Fitzhugh-Nagumo equations. Before using the compact FD schemes we first linearize the PDEs using a quasilinearization technique developed by Bellman and Kalaba. The method is therefore called the compact finite difference quasilinearization method (CFDQLM). The rest of the paper can be summarized as follows, in section 2 we give a description of how to use the sixth order compact finite differences to discretize in space and in time as well as giving a description of how the CFDQLM is used to solve evolution partial differential equations. In section 3 we outline the examples considered in this work. We then present and discuss results in section 4 and in the last section we conclude.

2 Compact finite difference schemes

In this section we introduce the compact finite difference schemes and illustrate how we use them to solve nonlinear evolution partial differential equations. We consider nonlinear PDEs of the form

∂u∂t=G(u,∂u∂x,∂2u∂x2) (1)

defined on the the region

t∈[0,T],x∈[a,b]

with boundary condition

u(a,t)=ua,u(b,t)=ub,

In the derivation of the compact FD schemes we consider the function u(x, t) that depends on space variable x and temporal variable t. We consider one-dimensional uniform meshes on the regions [a, b] and [0, T], with nodes xi (i = 1, 2, …, Nx) and tj (j = 1, 2, …, Nt) respectively, where

a=x1<x2<⋯<xNx=b (2)

and

0=t1<t2<⋯<tNx=T (3)

The distance between any two successive nodes is a constant Δ x = xi – xi–1 for the space variable and Δ t = tj–tj–1 for the temporal variable. We first show how we approximate the derivatives with respect to x using the compact FD schemes. Sixth order approximations of the first and second derivatives with respect to x, at interior nodes can be obtained using the following schemes (see [6] for details)

13ui−1,j′+ui,j′+13ui+1,j′=149ui+1,j−ui−1,j2Δx+19ui+2,j−yi−2,j4Δx, (4)

211ui−1,j″+ui,j″+211ui+1,j″=1211ui+1,j−2ui,j+ui−1,j(Δx)2+311ui+2,j−2ui,j+ui−2,j4(Δx)2, (5)

where ui,j = u(xi, tj) and the primes denote differentiation with respect to x. We apply the compact FD approximation for the first and second derivatives given by (4) and (5) respectively, at the interior nodes (i = 2, …, Nx − 1). Since we know boundary conditions at i = 1 and i = Nx, the compact FD schemes must be adjusted for the nodes near the boundary points. In order to maintain the order O(h6) accuracy at the boundary points as in the interior points and to maintain the same tridiagonal format, we use the following one sided scheme at i = 2,

u2,j′+13u3,j′=1Δx(−745u1,j−1712u2,j+8336u3,j−119u4,j+23u5,j−37180u6,j+136u7,j), (6)

and when i = Nx − 1 we use

13uNx−2,j′+uNx−1,j′=1Δx(745uNX,j+1712uNx−1,j−8336uNx−2,j+119uNx−3,j−23uNx−4,j+37180uNx−5,j+136uNx−6,j), (7)

Similarly, for the second derivatives, we use

u2,j″+211u3,j″=1(Δx)2(3145u1,j−19110u2,j−339110u3,j+1933396u4,j−4011u5,j+9655u6,j−479990u7,j+13220u8,j), (8)

at i = 2 and

211uNx−2,j″+uNx−1,j″=1(Δx)2(3145uNx,j+19110uNx−1,j+339110uNx−2,j+1933396uNx−3,j−4011uNx−4,j9655uNx−5,j+479990uNx−6,j+13220uNx−7,j), (9)

at i = Nx − 1. Using the above equations, the equations for approximating the first and second order derivatives can be expressed as

AxU⋅,j′=BxU⋅,j+Kx, (10)

AxxU⋅,j″=BxxU⋅,j+Kxx, (11)

where

Ax=1131311313113⋱⋱⋱13113131(NX−2)×(Nx−2),Kx=1Δx−745y1−y1360⋮⋮0−yN36745yN(NX−2)×1Bx=1Δx−17128336−11923−37180136−79079136−136−79079136⋱⋱⋱⋱⋱−136−79079136−136−79079−13637180−2311983361712(Nx−2)×(Nx−2)Axx=121121112112111211⋱⋱⋱21112112111(Nx−2)×(Nx−2),Kxx=1(Δx)23145y13y1440⋮⋮03yN443145yN(Nx−2)×1Bxx=1(Δx)2−19110−3391101933396−40119655−47999013220−1211−512212113443441211−51221211344⋱⋱⋱⋱⋱3441211−512212113443441211−51221211−13220−4799909655−40111933396−33911019110(Nx−2)×(Nx−2)U⋅,j′=[u2,j′,u3,j′,…,uNx−2,j′,uNx−1,j′]T,U⋅,j″=[u2,j″,u3,j″,…,uNx−2,j″,uNx−1,j″]T

From equations (10) and (11), U′ and U″ can be expressed as;

U⋅,j′=ExU⋅,j+Hx (12)

U⋅,j″=ExxU⋅,j+Hxx (13)

where

Ex=Ax−1Bx,Exx=Axx−1Bxx,Hx=Ax−1Kx,Hxx=Axx−1Kxx (14)

Next we show how we use the compact FD schemes to approximate the time derivatives. Approximation of the first derivative at interior points is given by

13u˙i,j−1+u˙i,j+13u˙i,j+1=149ui,j+1−ui,j−12Δt+19ui,j+2−yi,j−24Δt, (15)

where the dots denote differentiation with respect to time and Δ t is the distance between successive nodes. We adjust the schemes for i = 1, 2, Nt − 1 and Nt with the following one sided schemes respectively.

u˙i,1+13u˙i,2=1Δt(−451180ui,1+1003180ui,2−203ui,3+559ui,4−12536ui,5+6760ui,6−745ui,7), (16)

13u˙i,1+u˙i,2+13u˙i,3=1Δt(−3536ui,1+712ui,2−736ui,3+ui,4−712ui,5+736ui,6−136ui,7), (17)

13u˙i,Nt−1+u˙i,Nt=1Δt(3536ui,Nt−712ui,Nt−1+736ui,Nt−2−ui,Nt−3+712ui,Nt−4−736ui,Nt−5+136ui,Nt−6), (18)

13u˙i,Nt−2+u˙i,Nt−1+13u˙i,Nt=1Δt(451180ui,Nt−1003180ui,Nt−1+203ui,Nt−2−559ui,Nt−3−12536ui,Nt−4+6760ui,Nt−5−745ui,Nt−6), (19)

The equation for approximating the first time derivative is obtained by combining equations (15)-(19) and is given by

AtU˙i,.=BtUi,. (20)

where At = Ax and

Bt=1Δt−4511801003180−203559−125366760−745−3536712−7361−712736−136−136−79079136⋱⋱⋱⋱⋱−136−79079136136−736712−1736−7123536745−676012536−559203−1003180451180Nt×Nt

From equation (20), U̇ can be expressed as

U˙i,⋅=EtUi,⋅=∑k=1Ntej,ku(xi,tk),i=2,3,…,Nx−1,j=1,2,…,Nt (21)

where Et=At−1Bt and ejk are the elements of Et and Ui,⋅ = [ui,1, ui,2, …, ui,Nt].

To solve the PDE (1) we start by linearizing it by using the quasilinearization method which was proposed by Bellman and Kalaba [12]. It is convenient to split the function G in (1) into its linear and nonlinear components and rewrite the governing equation in the form,

L[u,u′,u″]+N[u,u′,u″]−u˙=0, (22)

where the dot and primes denote the time and space derivatives, respectively. L is a linear operator and N is a non-linear operator. If we assume that the difference ur+1 − ur and all its space derivatives is small, then we can approximate the non-linear operator N using the linear terms of the Taylor series and hence

N[u,u′,u″]≈N[ur,ur′,ur″]+∑k=02∂N∂u(k)(ur+1(k)−ur(k)) (23)

where r and r + 1 denote previous and current iterations respectively.

Equation (23) can be expressed as

N[u,u′,u″]≈N[ur,ur′,ur″]+∑k=02ϕk,r[ur,ur′,ur″]ur+1(k)−∑k=02ϕk,r[ur,ur′,ur″]ur(k) (24)

where

ϕk,r[ur,ur′,ur″]=∂N∂u(k)[ur,ur′,ur″]. (25)

Substituting equation (24) into equation (22), we get

L[ur+1,ur+1′,ur+1″]+∑k=02ϕk,rur+1(k)−u˙r+1=Rr[ur,ur′,ur″] (26)

where

Rr[ur,ur′,ur″]=∑k=02ϕk,rur(k)−N[ur2,ur′,ur″].

Substituting (21) into (26) we get

L[Ur+1,j,Ur+1,j′,Ur+1,j″]+∑k=02Φk,rUr+1,j(k)−2∑k=0NtejkUr+1,k=Rr[Ur,j,Ur,j′,Ur,j″] (27)

for j = 1, 2, 3, …, Nt, where

Φk,r=ϕk,r(x2,tj)ϕk,r(x1,tj)⋱ϕk,r(xNx−1,tj). (28)

and

Ur+1,j=[ur+1,j(x2),ur+1,j(x3),…,ur+1,j(xNx−1)] (29)

Since the initial condition is known, then we express equation (27) as

L[Ur+1,j,Ur+1,j′,Ur+1,j″]+∑k=02Φk,rUr+1,j(k)−∑k=1Ntej,kUr+1,k=Rj (30)

where

Rj=Rr[Ur,j,Ur,j′,Ur,j″]+ei,1U1−L[Hx,Hxx]−Φ1,rHx−Φ2,rHxx,j=2,3,…,Nt. (31)

Equation (30) can be expressed as the following (Nt − 1)(Nx − 1)×(Nt − 1)(Nx − 1) matrix system

X2,2X2,3⋯X2,NtX3,2X3,3⋯X3,Nt⋮⋮⋱⋮XNt,2XNt,3⋯XNt,NtU2U3⋮UNt=R2R3⋮RNt, (32)

where

Xi,i=L[I,Ex,Exx]+Φ0,r+Φ1,rEx+Φ2,rExx−ei,iI (33)

Xi,j=−ei,jI,when i≠j,i,j=2,3,…,Nt (34)

and I is the identity matrix of size (Nx − 1) × (Nx − 1). Solving equation (28) gives u(xi, tj).

3 Numerical experiments

We apply the proposed algorithm to well-known nonlinear PDEs of the form (1) with exact solutions. In order to determine the level of accuracy of the CFDQLM approximate solution, at a particular time level, in comparison with the exact solution we report maximum error which is defined by

EN=maxr{|u(xr,t)−u~(xr,t)|,:0≤r≤N}, (35)

where ũ(xr, t) is the approximate solution and u(xr, t) is the exact solution at the time level t.

Example 3.1

(Fisher’s equation). We consider the Fisher’s equation which represents a reactive-diffusive system and is encountered in chemical kinetics andpopulation dynamics applications. The equation is given by

∂u∂t=∂2U∂x2+αu(1−u), (36)

subject to the initial condition

u(x,0)=1(1+eα/6x)2, (37)

and exact solution [13]

u(x,t)=1(1+eα/6x−5αt/6)2, (38)

where α is a constant. For this example, the appropriate linear operator L and nonlinear operator N are chosen as

L(u)=u″+αu,N(u)=−αu2. (39)

We linearize the nonlinear operator N by expanding using the Taylor series expansion. We assume that the difference ur+1−ur and it’s derivatives is very small.

N≈N[ur,ur′,ur″]+∑k=02ϕk,rur+1(k)−∑k=02ϕk,rur(k) (40)

=αur2−2αur(ur+1−ur) (41)

The coefficients are obtained by taking only the linear terms of the Taylor expansion and they are given as

ϕ0,r=∂N∂u[ur,ur′,ur″]=−2αur (42)

ϕ1,r=∂N∂u′[ur,ur′,ur″]=0 (43)

ϕ2,r=∂N∂u′[ur,ur′,ur″]=0 (44)

and

Rr=∑k=02ϕk,rur(k)−N[uγ2,ur′,ur″]=−αur2 (45)

which is consistent with the quasilinearization discussed in the previous section.

Therefore, the linearized equation can be expressed as

ur+1″+ϕ0,rUr+1+αur+1−u˙=Rr (46)

Applying the compact FD schemes both in x and t, and initial condition, weget

ExxUr+1,j+Φ0,rUr+1,j+αUr+1,j−∑k=2Ntej,kUr+1,k=Rj (47)

Equation (47) can be expressed as

X2,2X2,3⋯X2,NtX3,2X3,3⋯X3,Nt⋮⋮⋱⋮XNt,2XNt,3⋯XNt,NtU2U3⋮UNt=R2R3⋮RNt, (48)

where

Xii=Exx+Φ0,r(i)+(α−ei,i)I (49)

Xi,j=−ei,jI,wheni≠j, (50)

Rj=Rr+ei,1U1−Hxxj,j=2,3,...,Nt (51)

The approximate solution is obtained by solving (48). The rest of the examples are solved in a similar manner.

Example 3.2

(Burgers-Fisher equation). We consider the generalized Burgers-Fisher equation [14]

∂u∂t+αuδ∂u∂x=∂2U∂x2+βu(1−uδ), (52)

with initial condition

u(x,0)={12+12tanh⁡(−αδ2(δ+1)x)}1δ, (53)

and exact solution

u(x,t)={12+12tanh⁡(−αδ2(δ+1)[x−(αδ+1+β(δ+1)α)t])}1δ, (54)

where α, β and δ are parameters. In this work the parameters are chosen to be α = β = δ = 1.

Example 3.3

(Burgers-Huxley equation). We consider the Burgers-Huxley equation

∂u∂t+αuδuX=∂2u∂x2+βu(1−uδ)(uδ−γ), (55)

where α, β≥0 are constant parameters, δ is a positive integer (set to be δ = 1 in this study) and γ∈(0,1) . The exact solution subject to the initial condition

u(x,0)=12−12tanh⁡[βr−αx], (56)

is

u(x,t)=12−12tanh⁡[βr−α(x−ct)], (57)

where

r=α2+8βandc=(α−r)(2γ−1)+2α4 (58)

The general solution (57) is reported in [15, 16].

Example 3.4

(Fitzhugh-Nagumo equation). The Fitzhugh-Nagumo equation is given by

∂u∂t=∂2U∂x2+u(u−α)(1−u) (59)

with initial condition

u(x,0)=12[1−coth⁡(−x22)]. (60)

This equation has the exact solution [17]

u(x,t)=12[1−coth⁡(−x22+2α−14t)], (61)

where α is a parameter.

4 Results and discussion

In this section we present the results of the implementation of the CFDQLM on the nonlinear evolution equations mentioned above. In Tables 1 - 4 we show the maximum errors between the exact solutions and the CFDQLM results for the Fisher’s, Fishers-Burger, Burgers-Huxely and Fitzhugh-Nagumo equations respectively. The results were obtained with grid points NX = 30 for the space variable x. As shown in the tables, for the time variable t, the number of grid points Nt varied from 10 to 50. It can be seen that after Nt = 40 there is a small change in the error. This means that Nt = 40 is sufficient to give sufficient accuracy. As shown by the results, the method gives highly accurate results of at least 10−12 which is achieved with remarkably few grid points in both x and t. The same observation is seen in Figure 5 which shows the errors for each of the equations considered. It can be seen that the average error is around 10−12 for the values of t considered. Another remarkable observation is that the method gives accurate results even for t > 1. A lot of methods loose significant accuracy when t > 1.

Table 1

Maximum errors EN for Fisher’s equation when α = 1 using Nx = 30

t\Nt 10 20 30 40 50
0.2 3.396e-007 1.086e-009 3.592e-011 2.725e-012 2.734e-013
0.4 2.605e-008 1.951e-010 7.812e-012 7.934e-013 1.291e-013
0.6 8.689e-009 7.175e-011 3.761e-012 7.182e-013 1.475e-013
0.8 3.480e-009 6.084e-011 4.795e-012 8.085e-013 2.018e-013
1.0 3.342e-009 5.979e-011 2.542e-012 7.814e-013 1.212e-013
1.2 1.376e-009 5.649e-011 4.247e-012 6.779e-013 1.575e-013
1.4 2.766e-009 5.487e-011 1.562e-012 5.637e-013 2.598e-014
1.6 3.614e-009 6.219e-011 3.941e-012 5.197e-013 9.348e-014
1.8 6.478e-009 8.894e-011 7.390e-012 6.311e-013 2.415e-013
2.0 2.625e-008 5.748e-010 3.641e-011 4.902e-012 1.015e-012

Table 2

Maximum errors EN for Burgers-Fisher equation when α = γ = δ = 1 using Nx = 30

t\Nt 10 20 30 40 50
0.2 8.512e-007 1.357e-008 9.097e-010 1.249e-010 2.632e-011
0.4 7.122e-008 2.127e-009 1.448e-010 2.060e-011 4.494e-012
0.6 1.655e-008 4.160e-010 2.949e-011 4.601e-012 1.080e-012
0.8 1.678e-008 6.381e-011 3.567e-012 5.065e-013 1.065e-013
1.0 3.027e-008 7.375e-011 1.006e-011 1.684e-012 4.735e-013
1.2 2.395e-008 9.651e-011 1.218e-011 2.303e-012 6.368e-013
1.4 5.896e-008 4.515e-011 1.330e-011 2.010e-012 5.536e-013
1.6 6.942e-008 6.562e-011 4.527e-012 1.202e-012 3.662e-013
1.8 1.183e-007 2.337e-010 9.879e-012 5.163e-013 1.923e-013
2.0 4.251e-007 1.529e-009 5.956e-011 6.207e-012 1.095e-012

Table 3

Maximum errors EN for Burgers-Huxley equation when α = β = 1 and γ = 0.5 using Nx = 30

t\Nt 10 20 30 40 50
0.2 3.247e-009 1.403e-011 6.778e-013 1.793e-014 1.066e-013
0.4 2.054e-010 2.187e-012 7.661e-014 6.317e-014 8.055e-014
0.6 1.191e-010 4.759e-013 3.342e-014 7.988e-014 6.156e-014
0.8 2.602e-011 1.812e-013 6.134e-014 5.929e-014 5.534e-014
1.0 1.151e-010 6.362e-014 3.586e-014 3.625e-014 4.829e-014
1.2 9.804e-011 9.576e-014 3.125e-014 4.452e-014 4.186e-014
1.4 2.114e-010 3.592e-013 5.662e-014 2.798e-014 5.851e-014
1.6 2.506e-010 9.730e-013 2.920e-014 2.542e-014 3.930e-014
1.8 4.486e-010 1.959e-012 1.222e-013 4.019e-014 2.298e-014
2.0 1.721e-009 1.364e-011 8.060e-013 1.065e-013 3.186e-014

Table 4

Maximum errors EN for Fitzhugh-Nagumo when α = 1 using Nx = 30

t\Nt 10 20 30 40 50
0.2 3.872e-009 2.016e-011 1.180e-012 1.693e-013 3.142e-014
0.4 2.486e-010 3.313e-012 1.953e-013 3.686e-014 2.287e-014
0.6 1.230e-010 8.290e-013 6.362e-014 2.059e-014 2.010e-014
0.8 6.523e-012 4.283e-013 5.429e-014 2.759e-014 3.003e-014
1.0 9.432e-011 3.036e-013 7.488e-014 2.803e-014 2.587e-014
1.2 6.756e-011 1.702e-013 6.334e-014 2.942e-014 3.675e-014
1.4 1.609e-010 4.746e-014 6.606e-014 3.830e-014 2.831e-014
1.6 1.924e-010 4.236e-013 3.775e-014 2.681e-014 4.013e-014
1.8 3.477e-010 1.222e-012 9.048e-014 4.147e-014 4.041e-014
2.0 1.355e-009 9.256e-012 5.085e-013 6.395e-014 2.986e-014

A graphical comparison between the CFDQLM results and the exact solutions is shown in Figures 1-4 for the Fisher’s, Burgers-Fisher, Burgers-Huxely, Fitzhugh-Nagumo equations respectively. A good agreement is observed between the two sets of results.

Fig. 1 Solution of the Fisher’s equation
Fig. 1

Solution of the Fisher’s equation

Fig. 2 Solution of Burgers-Fisher equation
Fig. 2

Solution of Burgers-Fisher equation

Fig. 3 Solution of the Burgers-Huxley equation
Fig. 3

Solution of the Burgers-Huxley equation

Fig. 4 Solution of the Fitzhugh-Nagumo equation
Fig. 4

Solution of the Fitzhugh-Nagumo equation

Fig. 5 Error norms at different time levels for the Fishers, Burgers-Fisher, Burgers-Huxley and the Fitzhugh-Nagumo equations
Fig. 5

Error norms at different time levels for the Fishers, Burgers-Fisher, Burgers-Huxley and the Fitzhugh-Nagumo equations

Figure 6 shows the convergence plots of the CFDQLM for the the four equations. It can be seen that the method reaches full convergence after 4 iterations. The convergence criteria is defined as the infinity norm of the difference between the solution at current iteration and previous iteration, ie,

||Ur+1−Ur||∞,

where r and r+1 denote previous and current iterations. This shows that the method quickly converges to the true solution.

Fig. 6 Convergence graphs for the Fishers, Burgers-Fisher, Burgers-Huxley and the Fitzhugh-Nagumo equations
Fig. 6

Convergence graphs for the Fishers, Burgers-Fisher, Burgers-Huxley and the Fitzhugh-Nagumo equations

5 Conclusion

In this paper, we have extended the use of compact finite difference schemes to both space and time variable when solving parabolic differential equations. This idea was tested on nonlinear evolution equations, namely Fisher’s, Burgers-Fisher, Burgers-Huxley and Fitzhugh-Nagumo equations. The quasilinearization technique was used to first linearize the equations. The results obtained show that the method is highly accurate. A noteworthy result is that the high accuracy is obtained using few grid points on both space and time variables. This makes the method more computationally efficient.

For future work, we plan to extend the application of the method to:

  1. systems of PDEs

  2. PDEs with higher spatial domains

  3. hyperbolic PDEs

References

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Received: 2016-4-20
Accepted: 2017-6-13
Published Online: 2017-12-16

© 2017 Dlamini and Khumalo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2017-0118
Keywords for this article
Compact finite differences; Quasilinearization; Nonlinear evolution equations
Creative Commons
BY-NC-ND 4.0

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  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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