# On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

Authors: Maryam Sarboland; Azim Aminataei
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2 and direct and indirect radial basis function networks (RBFNs) schemes. In
the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the
quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to
show the efficiency of the present methods, several experiments are considered. Our numerical solutions
are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore,
the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are
efficient, robust, and reliable for solving Burgers’ equation.


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## Body

## 1. Introduction

In this paper, we consider the one-dimensional nonlinear nonhomogeneous Burgers’ equation:(1)ut+uux=υuxx+F(x,t),x∈Ω=[a,b],t⩾t0,
with the initial condition,
(2)u(x,t0)=g(x),
and the boundary conditions,
(3)u(a,t)=g1(t),u(b,t)=g2(t),
where g(x), g1(t), and g2(t) are known functions, υ is the positive parameter that related to the Reynolds number R=1/υ, and F(x,t) is a known nonhomogeneous term.This equation was first derived from the hydrodynamics equations and used in surveying the laser generation of sound [1]. Later on, it was applied to other physical phenomena such as wind forcing the buildup of water waves, electrohydrodynamic field in plasma physics, and design of feedback control [2–4].WhenF(x,t)=0, (1) is the well-known Burgers’ equation:
(4)ut+uux=υuxx.
Burgers’ equation in homogeneous form was first introduced by Bateman [5] who found its steady solutions, descriptive of certain viscous flows. It was later presented by Burgers as one of class of equations describing mathematical models of turbulence [6]. In the context of gas dynamics, it was surveyed by Hopf [7] and Cole [8]. The homogeneous Burgers’ equation appears in various areas of applied mathematics and physics such as the phenomena of turbulence and supersonic flow, heat conduction, elasticity, and fusion [6–10].From an analytical point of view, the nonhomogeneous form is poorly studied, the complete analytical solution being closely dependent on the form of the nonhomogeneous term. For example, Karabutov et al. [1] obtained the analytical solution of the nonhomogeneous Burgers’ equation with F(x,t)=Asin(x), A>0, Ding et al. [11] studied the solution of (1) for the time-independent nonhomogeneous term F(x,t)=-kx, and Rao and Yadav [12] represented the solutions of the nonhomogeneous Burgers’ equation for the nonhomogeneous term F(x,t)=kx/(2βt+1)2 that k>0 and β>0 are constant. Recently, Moreau and vallée have obtained the analytical solution of the nonhomogeneous Burgers’ equation with an elastic forcing term F(x,t)=-kx+f(t), k∈ℝ [3].Up to now, various numerical methods are presented for the homogeneous Burgers’ equation such as finite difference, finite element, boundary element, and collocation methods. For a survey of these methods refer to [13–16] and references cited therein. Among the methods that are mentioned above, the spatial domain where the partial differential governing equations are defined is often discretized into meshes. In these methods, the creation of suitable meshes is very essential for getting accurate results. However, the procedure of mesh generation consumes a lot of time and labor for some problems, especially for discontinuous and high gradient problems, for which these methods will become complicate. The root of these difficulties is the use of mesh in the formulation step. To avoid the mesh generation, recently, a kind of so-called meshfree or meshless method has extended quickly. In these methods, the scattered nodes are only used instead of meshing the domain of the problem.For the last 20 years, the radial basis functions (RBFs) method was known as a powerful tool for the scattered data interpolation problem. The use of RBFs as a meshless process for the numerical solution of partial differential equations (PDEs) is based on the collocation scheme. The meshless methods based on RBFs were studied for approximating the solution of PDEs since initial development of Kansa’s work (1990) [17]. Kansa’s method was extended to solve various ordinary and partial differential equations [18–20]. In these works, the solution function is decomposed into RBFs and its derivatives are then arrived through differentiation that caused the reduction in convergence rate. In order to avoid this problem, Mai-Duy and Tran-Cong proposed an integrated MQ-RBFNs scheme for the approximation of function and its derivatives [21]. Numerical experiments and theoretical analysis demonstrate that for solving PDEs integrated RBF (IRBF) procedure is more accurate in comparison with direct RBF (DRBF) procedure. Also, IRBF scheme is more stable than DRBF for a range of PDEs [21, 22].In both DRBF and IRBF schemes, one must resolve a linear system of equations at each time step. In the past decade, the other meshless method was introduced by using a MQ quasi-interpolation without solving a linear system of equations. MQ quasi-interpolation is constructed directly from linear combination of MQ-RBF and the approximated function. In 1992, Beaston and Powell [23] presented three univariate MQ quasi-interpolations named as ℒ𝒜, ℒℬ, and ℒ𝒞. Wu and Schaback [24] proposed the MQ quasi-interpolation ℒ𝒟 and indicated that the scheme is shape preserving and convergent. Recently, Jiang et al. [25] have introduced a new multilevel univariate MQ quasi-interpolation approach with high approximation order compared with initial MQ quasi-interpolation scheme named as ℒ𝒲 and ℒ𝒲2. This approach is based on inverse multiquadric (IMQ) RBF interpolation and Wu and Schaback’s MQ quasi-interpolation operator ℒ𝒟. Chen and Wu applied initial MQ quasi-interpolation scheme for solving one-dimensional nonlinear homogeneous Burgers’ equation [26].In numerical solution of time dependent PDEs, such as Burgers’ and Sine-Gordon equations, by using MQ quasi-interpolation scheme, there is a limitation for discretization of the temporal derivative. One has to use low order finite difference approximation for discretization of time derivatives because one does not solve any system of equations at each time step; otherwise one must solve a system of equations [27]. Also, large number of nodes must be used for getting appropriate accuracy; see [26, 28, 29].In this paper, we present two numerical methods by using MQ quasi-interpolation for the numerical solution of the nonhomogeneous Burgers’ equation. In both of them, we use a two-order approximation for discretization of the time derivative. The main idea behind the discretization is to use more time derivatives in Taylor series expansion. This approach was demonstrated by Lax and Wendroff in finite difference [30] and used by Dağ et al. for the homogeneous Burgers’ equation [31]. By using this discretization, we have to solve a linear system of equations at each time step that the size of the system is equivalent to the number of the centers in the spatial domain. Also, because the IRBFN scheme requires fewer centers in comparison with the DRBFN scheme, we apply MQ quasi-interpolation scheme in the indirect form in order not to encounter with large scale matrix.The Jiang et al. MQ quasi-interpolation operatorℒ𝒲2 is summation of two series that the second series coefficients are combined with first series coefficients. By giving relation between two series coefficients based on function values, we can convert it to a compact form based on one series and use it in direct and indirect forms for the numerical solution of PDEs.The rest of present paper is organized as follows. A brief explanation of the MQ quasi-interpolation scheme is given in Section2. Our numerical methods are applied on the nonlinear Burgers’ equation in Section 3. In Section 4, the stability analysis of the methods is discussed. The results of several numerical experiments are reported in Section 5. Finally, some conclusions based on obtained results are drawn in Section 6.


## 2. The MQ Quasi-Interpolation Scheme

In this section, three univariate MQ quasi-interpolation schemes named asℒ𝒟, ℒ𝒲, and ℒ𝒲2 are described. Then, we describe our approach which converts operator ℒ𝒲2 to the compact form. More details can be seen in [23–25].For a given regionΩ=[a,b] and a finite set of distinct points,
(5)a=x0<x1<⋯<xN=b,h=max⁡1⩽i⩽N(xi-xi-1),
if we are supplied with a function f:[a,b]→ℝ, quasi-interpolation of f takes the form:
(6)ℒ(f)=∑i=0N‍f(xi)ϕi(x),
where each function ϕi(x) is a linear combination of the Hardy MQs basis function [32],
(7)ψi(x)=c2+(x-xi)2,
and low order polynomials and c∈ℝ+ is a shape parameter. This formula requires the derivative values of f at the end points that is not convenient for practical purposes [23]. Wu and Schaback [24] presented the univariate MQ quasi-interpolation operator ℒ𝒟 that is defined as
(8)ℒ𝒟f(x)=∑i=0N‍f(xi)ψ~i(x),
where
(9)ψ~0(x)=12+ψ1(x)-(x-x0)2(x1-x0),ψ~1(x)=ψ2(x)-ψ1(x)2(x2-x1)-ψ1(x)-(x-x0)2(x1-x0),ψ~i(x)=ψi+1(x)-ψi(x)2(xi+1-xi)-ψi(x)-ψi-1(x)2(xi-xi-1),111111111111111.111.2⩽i⩽N-2,ψ~N-1(x)=(xN-x)-ψN-1(x)2(xN-xN-1)-ψN-1(x)-ψN-2(x)2(xN-1-xN-2),ψ~N(x)=12+ψN-1(x)-(xN-x)2(xN-xN-1).
Suppose that {xki}i=1N- is a smaller set from the given points {xi}i=0N, where N- is a positive integer satisfying N-<N and 0=k0<k1<⋯<kN-+1=N. Using the IMQ-RBF, the second derivative of f(x) can be approximated by RBF interpolant Sf′′ as
(10)Sf′′=∑j=1N-‍αjφ-(|x-xkj|),
where
(11)φ-(r)=s2(s2+r2)3/2,
and s∈ℝ+ is a shape parameter.The coefficients{αj}j=1N- are uniquely determined by the interpolation condition
(12)Sf′′(xki)=∑j=1N-‍αjφ-(|xki-xkj|)=f′′(xki),1⩽i⩽N-.
Since (12) is solvable [33], so
(13)α=AX-1·fX′′,
where
(14)X={xk1,…,xkN-},α=[α1,…,αN-]T,AX=[φ-(|xki-xkj|)],fX′′=[f′′(xk1),…,f′′(xkN-)]T.
By using the f and the coefficient α defined in (13), a function e(x) is constructed in the form
(15)e(x)=f(x)-∑j=1N-‍αjs2+(x-xkj)2.
Now, the MQ quasi-interpolation operator ℒ𝒲 by using ℒ𝒟 defined by (8) and (9) on the data {(xi,e(xi))}i=1N with the shape parameter c is given by
(16)ℒ𝒲f(x)=∑j=1N-‍αjs2+(x-xkj)2+ℒ𝒟e(x).
The shape parameters c and s should not be the same constant in (16).In (12), the value of fxkj′′ can be replaced by
(17)fxkj′′=(+(xkj+1-xkj)f(xkj-1)])2[(xkj-xkj-1)f(xkj+1)-(xkj+1-xkj-1)f(xkj)+(xkj+1-xkj)f(xkj-1)])×((xkj-xkj-1)(xkj+1-xkj)(xkj+1-xkj-1))-1,
when the data’s {(xki,f(xki))}i=1N- are given. So, if fX′′ in (13) is replaced by
(18)FX′′=[fxk1′′,…,fxkN-′′]T,
then the quasi-interpolation operator defined by (15) and (16) is denoted by ℒ𝒲2 The linear reproducing property and the high convergence rate of ℒ𝒲2 were also studied in [25].The operatorℒ𝒲2 can be written in the compact form
(19)ℒ𝒲2f(x)=∑i=0N‍f(xi)ψ^i(x),
where the basis functions ψ^i(x) are obtained by substituting (13), (15), and (18) into (16). As such, let X={x0,x1,x2,x3,x4} and X′={x2}. So N=4, N-=1, k1=2, and
(20)α=(2[(xk1-xk0)f(xk2)-(xk2-xk0)f(xk1)+(xk2-xk1)f(xk0)])×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1,(21)e(x)=f(x)-αs2+(x-xk1)2.
Substituting (20) into (21) yields
(22)e(x)=f(x)-(×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1)(2[(xk1-xk0)f(xk2)11111111111-(xk2-xk0)f(xk1)11111111111+(xk2-xk1)f(xk0)])11111111×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1)×s2+(x-xk1)2.
Hence, the substitution of (20) and (22) into (16) leads to
(23)ℒ𝒲2f(x)=(×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1)(2[(xk1-xk0)f(xk2)1111111-(xk2-xk0)f(xk1)1111111+(xk2-xk1)f(xk0)])×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1)×s2+(x-xk1)2+∑i=04‍(×s2+(xi-xk1)2)f(xi)-(×(s(xk1-xk0)(xk2-xk1)(xk2-xk0))-1)(2[(xk1-xk0)f(xk2)11111111111111111-(xk2-xk0)f(xk1)11111111111111111+(xk2-xk1)f(xk0)])111111111111111×(s(xk1-xk0)(xk2-xk1)111111111111111111111×(xk2-xk0))-1)×s2+(xi-xk1)2)ψ~i(x),
whereas f(xk0)=f(x0), f(xk1)=f(x2), and f(xk2)=f(x4). Therewith, (23) can be rewritten as
(24)ℒ𝒲2f(x)=(×(s(x2-x0)(x4-x2)(x4-x0))-1)(2[(x2-x0)f(x4)-(x4-x0)f(x2)+(x4-x2)f(x0)])×(s(x2-x0)(x4-x2)(x4-x0))-1)×s2+(x-x2)2+∑i=04‍f(xi)ψ~i(x)-∑i=04(×(s(x2-x0)(x4-x2)(x4-x0))-1)(2[(x2-x0)f(x4)-(x4-x0)f(x2)+(x4-x2)f(x0)])×(s(x2-x0)(x4-x2)(x4-x0))-1)×s2+(xi-x2)2ψ~i(x).
Hence, the basis function ψ^i(x) are arrived as follows:
(25)ψ^0(x)=2s(x2-x0)(x4-x0)×[-∑i=04‍s2+(xi-x2)2ψ~i(x)]s2+(x-x2)2-∑i=04‍s2+(xi-x2)2ψ~i(x)]+ψ~0(x),ψ^2(x)=-2s(x2-x0)(x4-x2)×[-∑i=04‍s2+(xi-x2)2ψ~i(x)]s2+(x-x2)2-∑i=04‍s2+(xi-x2)2ψ~i(x)]+ψ~2(x),ψ^4(x)=2s(x4-x2)(x4-x0)×[-∑i=04‍s2+(xi-x2)2ψ~i(x)]s2+(x-x2)2-∑i=04‍s2+(xi-x2)2ψ~i(x)]+ψ~4(x),ψ^i(x)=ψ~i(x),i=1,3.By writing operatorℒ𝒲2 in the compact form (23), we can use it in two indirect and direct forms for the numerical solution of PDEs.


## 3. The Numerical Methods

In this section, the numerical schemes are presented for solving the nonlinear Burgers’ equation (1) by using the MQ quasi-interpolation ℒ𝒲2 In our approach, the MQ quasi-interpolation approximates the solution function and the spatial derivatives of the differential equation and Taylor’s series expansion is employed to approximate the temporal derivative similar to the work that Dağ et al. did in [31]. The MQ quasi-interpolation method is applied in direct and indirect forms.We discretize the problem using the following Taylor’s series expansion with step sizeΔt:
(26)utn=un+1-unΔt-Δt2uttn+O(Δt2),
where utn=ut(x,tn) and tn=t0+nΔt.Differentiating (1) with respect to time, uttn can be written as follows:
(27)uttn=(-unuxn+υuxxn+Fn)t=-un(utn)x-uxnutn+υ(utn)xx+Ftn,
where Ftn=Ft(x,tn). Using forward difference formula for the time derivative utn in (27), uttn can be rewritten as
(28)Δtuttn=-un(uxn+1-uxn)-uxn(un+1-un)+υ(uxxn+1-uxxn)+ΔtFtn.
Substituting (28) into (26) and using the expression achieved in (1), the following time discretized form of nonlinear Burgers’ equation is yielded:
(29)un+1+Δt2unuxn+1+Δt2uxnun+1-υΔt2uxxn+1=un+υΔt2uxxn+Δt(Fn+Δt2Ftn).


### 3.1. The Direct MQ Quasi-Interpolation Scheme

In this scheme, the unknown functionun is approximated by using MQ quasi-interpolation scheme, and its spatial derivatives uxn and uxxn are calculated by differentiating such closed form of quasi approximation as follows:
(30)un(x)=∑i=0N‍uinψ^i(x),(31)uxn(x)=∑i=0N‍uin∂ψ^i∂x(x)=∑i=0N‍uinψˇi(x),(32)uxxn(x)=∑i=0N‍uin∂2ψ^i∂x2(x)=∑i=0N‍uinψ-i(x),
where ∂ψ^i/∂x=ψˇi and ∂2ψ^i/∂x2=ψ-i.Now, replacing (30)–(32) into (29) and applying collocation method yield
(33)∑k=0Nukn+1[(1+Δt2∑j=0N‍ujnψˇij)ψ^ik+Δt2(∑j=0N‍ujnψ^ij)ψˇik-υΔt2ψ-ik]=∑j=0N‍ujn(ψ^ij+υΔt2ψ-ij)+Δt(Fin+Δt2Ftn(xi)),1⩽i⩽N-1,
where Fin=F(xi,tn), Ftn(xi)=Ft(xi,tn), ψ^ij=ψ^j(xi), ψˇij=ψˇj(xi), and ψ-ij=ψ-j(xi), whereas, according to (3), we have
(34)u0n=u(x0,tn)=u(a,tn)=g1(tn),(35)uNn=u(xN,tn)=u(b,tn)=g2(tn).
Substituting (34) and (35) into (33), wherein (33) generates a system of N-1 linear equations in N-1 unknown parameters uin+1.Equation (33) can be written in the matrix form
(36)[A1+Δt2uxn*A1+Δt2un*D1-υΔt2D¨1]un+1=[A1+υΔt2D¨1-Δt2∑j=12‍gjn+1(Ψ^j*D1+Ψˇj*A1)]un-∑k=01‍∑j=12‍gjn+k[Ψ^j-υΔt2Ψ-j]-Δt∑j=12‍gjn+1gjnΨ^j*Ψˇj-Δt2[g1n+1g2n+g2n+1g1n][Ψˇ2*Ψ^1+Ψˇ1*Ψ^2]+ΔtF1n+Δt22F´1n,
where symbol * stands for component by component multiplication,
(37)A1ij=[ψ^ij]i,j=1N-1,D1ij=[ψˇij]i,j=1N-1,D¨1ij=[ψ-ij]i,j=1N-1,Ψ^1=[ψ^0(x1),ψ^0(x2),…,ψ^0(xN-1)]T,Ψ^2=[ψ^N(x1),ψ^N(x2),…,ψ^N(xN-1)]T,Ψˇ1=[ψˇ0(x1),ψˇ0(x2),…,ψˇ0(xN-1)]T,Ψ˘1=[ψˇN(x1),ψˇN(x2),…,ψˇN(xN-1)]T,Ψ-1=[ψ-0(x1),ψ-0(x2),…,ψ-0(xN-1)]T,Ψ-2=[ψ-N(x1),ψ-N(x2),…,ψ-N(xN-1)]T,F1n=[Fn(x1),Fn(x2),…,Fn(xN-1)]T,F´1n=[Ftn(x1),Ftn(x2),…,Ftn(xN-1)]T.
Subsequently, (36) can be written as
(38)un+1=M1-1N1un+M1-1Ψ,
where
(39)M1=A1+Δt2uxn*A1+Δt2un*D1-υΔt2D¨1,N1=A1+υΔt2D¨1-Δt2∑j=12‍gjn+1(Ψ^j*D1+Ψˇj*A1),Ψ=-∑k=01‍∑j=12‍gjn+k[Ψ^j-υΔt2Ψ-j]+Δt(-∑j=12‍gjn+1gjnΨ^j*Ψˇj-12[g1n+1g2n+g2n+1g1n][Ψˇ2*Ψ^1+Ψˇ1*Ψ^2]+F1n+Δt2F´1n(∑j=12‍gjn+1gjnΨ^j*Ψˇj).
In order to make reduction in error, the obtained ui from (38) is substituted in the right hand side of (30) that can be written as follows:
(40)un=A1un+g1nΨ^1+g2nΨ^2,
and the obtained value is considered as ui. Therefore, from (38) and (40), it yields that
(41)un+1=A1M1-1N1A1-1un+A1M1-1Ψ-A1M1-1N1A1-1(g1nΨ^1+g2nΨ^2)+g1n+1Ψ^1+g2n+1Ψ^2.
Hence, the unknown parameters ui are specified from (41) instead of (38).


### 3.2. The Indirect MQ Quasi-Interpolation Scheme

In indirect scheme, the highest order derivatives (second order in this paper) of the solution function are first approximated by (19), and their lower order derivatives and the solution function are then obtained by symbolic integration. Therefore, uxxn can be approximated by MQ quasi-interpolation ℒ𝒲2 on data {xj}j=1N-1 as follows:
(42)uxxn(x)=∑j=1N-1‍uxxn(xj)ψ^j(x).
Now, integrating (42) yields
(43)uxn(x)=∑j=1N-1‍uxxn(xj)∫‍ψ^j(x)dx+C1,(44)un(x)=∑j=1N-1‍uxxn(xj)∫‍∫‍ψ^j(x)dxdx+C1x+C2.
Equations (42)–(44) can be rewritten in the compact form as follows:
(45)un(x)=∑j=0N‍wjnφj(x),uxn(x)=∑j=0N‍wjnφ˙j(x),uxxn(x)=∑j=0N‍wjnφ¨j(x),
where
(46)φj(x)=∫‍∫‍ψ^j(x)dxdx,1⩽j⩽N-1,φ0(x)=x,φN(x)=1,φ˙j(x)=∫‍ψ^j(x)dx,1⩽j⩽N-1,φ˙0(x)=1,φ˙N(x)=0,φ¨j(x)=ψ^j(x),1⩽j⩽N-1,φ¨0(x)=0,φ¨N(x)=0,wjn=uxxn(xj),1⩽j⩽N-1,w0n=C1,wNn=C2.
Similar to direct scheme, replacing (45) into (29) and (3) and applying collocation method lead to
(47)∑k=0N‍wkn+1[(1+Δt2∑j=0N‍wjnφ˙ij)φik1111111111+Δt2(∑j=0N‍wjnφij)φ˙ik-υΔt2φ¨ik]=∑j=0N‍wjn(φij+υΔt2tφ¨ij)+Δt(Fin+Δt2Ftn(xi)),1⩽i⩽N-1,∑k=0N‍wkn+1φk(x0)=g1n+1,∑k=0N‍wkn+1φk(xN)=g2n+1,
where φij=φj(xi), φ˙ij=φ˙j(xi), φ¨ij=φ¨j(xi), and gin+1=gi(tn+1), i=1,2.Equations (47) generate a system of N+1 linear equations in N+1 unknown parameters win+1.Similar to the direct quasi-interpolation scheme, (47) can be written in matrix form
(48)[Ad+Ab+Δt2uxn*Ad+Δt2un*D2-υΔt2D¨2]wn+1=[Ad+υΔt2D¨2]wn+Gn+1+ΔtF2n+Δt22F´2n,
where, in this case,
(49)Ad(i+1)(j+1)=φij,Ab(i+1)(j+1)=0,D2(i+1)(j+1)=φ˙ij,D¨2(i+1)(j+1)=φ¨ij,
for i=1,…,N-1; j=0,1,…,N and
(50)Ad(i+1)(j+1)=0,Ab(i+1)(j+1)=φij,D2(i+1)(j+1)=0,D¨2(i+1)(j+1)=0,
for i=0,N; j=0,1,…,N and
(51)Gn+1=[g1n+1,0,…,0,g2n+1]T,F2n=[0,Fn(x1),…,Fn(xN-1),0]T,F´2n=[0,Ftn(x1),…,Ftn(xN-1),0]T.
Subsequently, (48) can be written as
(52)wn+1=M2-1N2wn+M2-1Gn+1+M2-1F¨n,
where
(53)M2=A2+Δt2uxn*Ad+Δt2un*D2-υΔt2D¨2,N2=Ad+υΔt2D¨2,F¨n=ΔtF2n+Δt22F´2n,
and A2=Ad+Ab. From (42), it yields that
(54)un=A2wn.
Hence, the combination of (52) and (54) is given as
(55)un+1=A2M2-1N2A2-1un+A2M2-1Gn+1+A2M2-1F¨n.


## 3.1. The Direct MQ Quasi-Interpolation Scheme

In this scheme, the unknown functionun is approximated by using MQ quasi-interpolation scheme, and its spatial derivatives uxn and uxxn are calculated by differentiating such closed form of quasi approximation as follows:
(30)un(x)=∑i=0N‍uinψ^i(x),(31)uxn(x)=∑i=0N‍uin∂ψ^i∂x(x)=∑i=0N‍uinψˇi(x),(32)uxxn(x)=∑i=0N‍uin∂2ψ^i∂x2(x)=∑i=0N‍uinψ-i(x),
where ∂ψ^i/∂x=ψˇi and ∂2ψ^i/∂x2=ψ-i.Now, replacing (30)–(32) into (29) and applying collocation method yield
(33)∑k=0Nukn+1[(1+Δt2∑j=0N‍ujnψˇij)ψ^ik+Δt2(∑j=0N‍ujnψ^ij)ψˇik-υΔt2ψ-ik]=∑j=0N‍ujn(ψ^ij+υΔt2ψ-ij)+Δt(Fin+Δt2Ftn(xi)),1⩽i⩽N-1,
where Fin=F(xi,tn), Ftn(xi)=Ft(xi,tn), ψ^ij=ψ^j(xi), ψˇij=ψˇj(xi), and ψ-ij=ψ-j(xi), whereas, according to (3), we have
(34)u0n=u(x0,tn)=u(a,tn)=g1(tn),(35)uNn=u(xN,tn)=u(b,tn)=g2(tn).
Substituting (34) and (35) into (33), wherein (33) generates a system of N-1 linear equations in N-1 unknown parameters uin+1.Equation (33) can be written in the matrix form
(36)[A1+Δt2uxn*A1+Δt2un*D1-υΔt2D¨1]un+1=[A1+υΔt2D¨1-Δt2∑j=12‍gjn+1(Ψ^j*D1+Ψˇj*A1)]un-∑k=01‍∑j=12‍gjn+k[Ψ^j-υΔt2Ψ-j]-Δt∑j=12‍gjn+1gjnΨ^j*Ψˇj-Δt2[g1n+1g2n+g2n+1g1n][Ψˇ2*Ψ^1+Ψˇ1*Ψ^2]+ΔtF1n+Δt22F´1n,
where symbol * stands for component by component multiplication,
(37)A1ij=[ψ^ij]i,j=1N-1,D1ij=[ψˇij]i,j=1N-1,D¨1ij=[ψ-ij]i,j=1N-1,Ψ^1=[ψ^0(x1),ψ^0(x2),…,ψ^0(xN-1)]T,Ψ^2=[ψ^N(x1),ψ^N(x2),…,ψ^N(xN-1)]T,Ψˇ1=[ψˇ0(x1),ψˇ0(x2),…,ψˇ0(xN-1)]T,Ψ˘1=[ψˇN(x1),ψˇN(x2),…,ψˇN(xN-1)]T,Ψ-1=[ψ-0(x1),ψ-0(x2),…,ψ-0(xN-1)]T,Ψ-2=[ψ-N(x1),ψ-N(x2),…,ψ-N(xN-1)]T,F1n=[Fn(x1),Fn(x2),…,Fn(xN-1)]T,F´1n=[Ftn(x1),Ftn(x2),…,Ftn(xN-1)]T.
Subsequently, (36) can be written as
(38)un+1=M1-1N1un+M1-1Ψ,
where
(39)M1=A1+Δt2uxn*A1+Δt2un*D1-υΔt2D¨1,N1=A1+υΔt2D¨1-Δt2∑j=12‍gjn+1(Ψ^j*D1+Ψˇj*A1),Ψ=-∑k=01‍∑j=12‍gjn+k[Ψ^j-υΔt2Ψ-j]+Δt(-∑j=12‍gjn+1gjnΨ^j*Ψˇj-12[g1n+1g2n+g2n+1g1n][Ψˇ2*Ψ^1+Ψˇ1*Ψ^2]+F1n+Δt2F´1n(∑j=12‍gjn+1gjnΨ^j*Ψˇj).
In order to make reduction in error, the obtained ui from (38) is substituted in the right hand side of (30) that can be written as follows:
(40)un=A1un+g1nΨ^1+g2nΨ^2,
and the obtained value is considered as ui. Therefore, from (38) and (40), it yields that
(41)un+1=A1M1-1N1A1-1un+A1M1-1Ψ-A1M1-1N1A1-1(g1nΨ^1+g2nΨ^2)+g1n+1Ψ^1+g2n+1Ψ^2.
Hence, the unknown parameters ui are specified from (41) instead of (38).


## 3.2. The Indirect MQ Quasi-Interpolation Scheme

In indirect scheme, the highest order derivatives (second order in this paper) of the solution function are first approximated by (19), and their lower order derivatives and the solution function are then obtained by symbolic integration. Therefore, uxxn can be approximated by MQ quasi-interpolation ℒ𝒲2 on data {xj}j=1N-1 as follows:
(42)uxxn(x)=∑j=1N-1‍uxxn(xj)ψ^j(x).
Now, integrating (42) yields
(43)uxn(x)=∑j=1N-1‍uxxn(xj)∫‍ψ^j(x)dx+C1,(44)un(x)=∑j=1N-1‍uxxn(xj)∫‍∫‍ψ^j(x)dxdx+C1x+C2.
Equations (42)–(44) can be rewritten in the compact form as follows:
(45)un(x)=∑j=0N‍wjnφj(x),uxn(x)=∑j=0N‍wjnφ˙j(x),uxxn(x)=∑j=0N‍wjnφ¨j(x),
where
(46)φj(x)=∫‍∫‍ψ^j(x)dxdx,1⩽j⩽N-1,φ0(x)=x,φN(x)=1,φ˙j(x)=∫‍ψ^j(x)dx,1⩽j⩽N-1,φ˙0(x)=1,φ˙N(x)=0,φ¨j(x)=ψ^j(x),1⩽j⩽N-1,φ¨0(x)=0,φ¨N(x)=0,wjn=uxxn(xj),1⩽j⩽N-1,w0n=C1,wNn=C2.
Similar to direct scheme, replacing (45) into (29) and (3) and applying collocation method lead to
(47)∑k=0N‍wkn+1[(1+Δt2∑j=0N‍wjnφ˙ij)φik1111111111+Δt2(∑j=0N‍wjnφij)φ˙ik-υΔt2φ¨ik]=∑j=0N‍wjn(φij+υΔt2tφ¨ij)+Δt(Fin+Δt2Ftn(xi)),1⩽i⩽N-1,∑k=0N‍wkn+1φk(x0)=g1n+1,∑k=0N‍wkn+1φk(xN)=g2n+1,
where φij=φj(xi), φ˙ij=φ˙j(xi), φ¨ij=φ¨j(xi), and gin+1=gi(tn+1), i=1,2.Equations (47) generate a system of N+1 linear equations in N+1 unknown parameters win+1.Similar to the direct quasi-interpolation scheme, (47) can be written in matrix form
(48)[Ad+Ab+Δt2uxn*Ad+Δt2un*D2-υΔt2D¨2]wn+1=[Ad+υΔt2D¨2]wn+Gn+1+ΔtF2n+Δt22F´2n,
where, in this case,
(49)Ad(i+1)(j+1)=φij,Ab(i+1)(j+1)=0,D2(i+1)(j+1)=φ˙ij,D¨2(i+1)(j+1)=φ¨ij,
for i=1,…,N-1; j=0,1,…,N and
(50)Ad(i+1)(j+1)=0,Ab(i+1)(j+1)=φij,D2(i+1)(j+1)=0,D¨2(i+1)(j+1)=0,
for i=0,N; j=0,1,…,N and
(51)Gn+1=[g1n+1,0,…,0,g2n+1]T,F2n=[0,Fn(x1),…,Fn(xN-1),0]T,F´2n=[0,Ftn(x1),…,Ftn(xN-1),0]T.
Subsequently, (48) can be written as
(52)wn+1=M2-1N2wn+M2-1Gn+1+M2-1F¨n,
where
(53)M2=A2+Δt2uxn*Ad+Δt2un*D2-υΔt2D¨2,N2=Ad+υΔt2D¨2,F¨n=ΔtF2n+Δt22F´2n,
and A2=Ad+Ab. From (42), it yields that
(54)un=A2wn.
Hence, the combination of (52) and (54) is given as
(55)un+1=A2M2-1N2A2-1un+A2M2-1Gn+1+A2M2-1F¨n.


## 4. The Stability Analysis

In this section, the stability analysis from direct and indirect quasi-interpolation schemes is presented by using spectral radius of the amplification matrix similar to the work that Siraj-ul-Islam et al. did in [34]. Let u be the exact and u* the numerical solution of (1); then the error vector εn+1=un+1-u*n+1 in the direct and indirect quasi-interpolation schemes can be written as
(56)εn+1=un+1-u*n+1=A1M1-1N1A1-1εn=E1εn,εn+1=un+1-u*n+1=A2M2-1N2A2-1εn=E2εn,
where E1=A1M1-1N1A1-1 and E2=A2M2-1N2A2-1. For the stability of the numerical schemes, we must have εn→0 as n→∞; that is, ρ(E1)⩽1, ρ(E2)⩽1, which is the necessary and sufficient condition for the numerical schemes to be stable, where ρ(E1) and ρ(E2) denote the spectral radius of the amplification matrices E1 and E2, respectively. Equations (56) can be written as
(57)M1A1-1εn+1=N1A1-1εn,M2A2-1εn+1=N2A2-1εn.
Equations (57) can be written into the following forms by using the values of M1, N1, M2, and N2 defined in (39) and (53):
(58)[I+(Δt2)R1]εn+1=[I+(Δt2)R2]εn,[I+(Δt2)S1]εn+1=[K+(Δt2)S2]εn,
where
(59)R1=[uxn*A1+un*D1-υD¨1]A1-1,R2=[υD¨1-∑j=12‍gjn+1(Ψ^j*D1+Ψˇj*A1)]A1-1,S1=A2-1[uxn*Ad+un*D2-υD¨2],K=A2-1Ad,S2=A2-1[υD¨2].
The condition of stability will be satisfied if maximum eigenvalue of the matrix E1=[I+(Δt/2)R1]-1[I+(Δt/2)R2] and maximum eigenvalue of the matrix E2=[I+(Δt/2)S1]-1[K+(Δt/2)S2] are less than unity (in direct and indirect MQ quasi-interpolation schemes, resp.); that is,
(60)|1+(Δt/2)ηR21+(Δt/2)ηR1|⩽1,|ηK+(Δt/2)ηS21+(Δt/2)ηS1|⩽1,
where ηR1, ηR2, ηS1, ηS2, and ηK denote the eigenvalues of the matrices R1, R2, S1, S2, and K, respectively. It is clear from (60) that the stability of the methods depends on the time step Δt and eigenvalues of the matrices ηR1, ηR2, ηS1, ηS2, and ηK. The condition numbers and magnitude of the eigenvalues of the matrices R1, R2, S1, S2, and K depend on the shape parameter and the number of collocation points. Hence, the condition number and the spectral radius of the matrices E1 and E2 are dependent on the shape parameter and the number of collocation points. Since it is not possible to find explicit relationship among the spectral radius of the matrices and the shape parameter, this dependency is approximated numerically by keeping the number of collocation points fixed.


## 5. The Numerical Experiments

Five test experiments are studied to investigate the robustness and the accuracy of the proposed methods. The solution function of Burgers’ equation is approximated by direct MQ quasi-interpolation (DMQQI) and indirect MQ quasi-interpolation (IMQQI) schemes and the results are compared with analytical solutions and the results in [18, 26, 31, 35]. The L∞ and L2 error norms which are defined by
(61)L∞=∥u*n-un∥∞=max⁡0⩽j⩽N|u*n(xj)-un(xj)|,L2=∥u*n-un∥2=h∑j=0N‍(u*n(xj)-un(xj))2
are used to measure the accuracy. Also, the stability analysis of the methods is considered for first experiment. In all experiments, the shape parameter s is considered twice the shape parameter c and N is chosen twice N¯. Also, the centers and the collocation points have been chosen as the same and equidistant.The computations associated with the experiments discussed above were performed in Maple 16 on a PC with a CPU of 2.4 GHZ.Experiment 1.
In this experiment, we consider nonlinear Burgers’ equation (1) with F(x,t)=0 and the initial and the boundary conditions:
(62)u(x,0)=sin(πx),0⩽x⩽1,u(0,t)=u(1,t)=0,t⩾0.
The exact series solution of this experiment was given by Cole [8]:
(63)u(x,t)=2πυ∑k=1∞kaksin(kπx)exp⁡(-k2π2υt)a0+∑k=1∞akcos⁡(kπx)exp⁡(-k2π2υt),
where
(64)a0=∫01‍exp⁡(-1-cos⁡⁡(πx)2πυ)dx,ak=2∫01‍cos⁡(kπx)exp⁡(-1-cos⁡⁡(πx)2πυ)dx,1111111111111111111.(k=1,2,3,…).
Numerical results are presented for υ=0.1 and υ=0.01 with Δt=0.001 and compared with the exact solutions and the results of the MQ quasi-interpolation scheme (MQQI; see [26]) and adaptive MQ scheme (AMQ; see [18]) for the cases of υ=0.1 and υ=0.01 in Tables 1 and 2, respectively. Also, the numerical solutions are compared with the results obtained by MQQI scheme, AMQ scheme, and Galerkin scheme [35] for υ=0.0001 in Table 3. In the cases υ=0.1 and υ=0.01, the shape parameter c is 0.815h. In the case υ=0.0001, the parameter c is 2.78×10-1 and 1.389×10-4 for N=36 and N=72, respectively. The space-time graph of the estimated solution for υ=0.1 and υ=0.01 is presented in Figures 1 and 2.Table 1
Comparison of results with the exact solution and the results in [26] of υ=0.1 with Δt=0.001 at t=1 for different values of x of Experiment 1.
x
Exact
MQQI [26]
IMQQI
DMQQI
Error
Error
N
=
100
N
=
10
N
=
20
N
=
10
N
=
20
IMQQI
DMQQI
0.1
0.0663
0.0712
0.0663
0.0663
0.0662
0.0662
8.59
E
-
07
4.41
E
-
05
0.2
0.1312
0.1343
0.1313
0.1312
0.1310
0.1311
2.18
E
-
06
7.77
E
-
05
0.3
0.1928
0.1934
0.1929
0.1928
0.1925
0.1926
4.49
E
-
06
1.21
E
-
04
0.4
0.2480
0.2454
0.2483
0.2480
0.2476
0.2478
8.59
E
-
06
1.88
E
-
04
0.5
0.2919
0.2852
0.2924
0.2919
0.2913
0.2916
1.55
E
-
05
2.85
E
-
04
0.6
0.3161
0.3047
0.3169
0.3161
0.3150
0.3155
2.26
E
-
05
4.15
E
-
04
0.7
0.3081
0.2929
0.3095
0.3081
0.3064
0.3072
3.39
E
-
05
5.53
E
-
04
0.8
0.2537
0.2378
0.2558
0.2538
0.2515
0.2526
4.96
E
-
05
5.98
E
-
04
0.9
0.1461
0.1354
0.1481
0.1461
0.1441
0.1455
4.47
E
-
05
4.16
E
-
04Table 2
Comparison of results with the exact solution and the results in [26] of υ=0.01 with Δt=0.001 at t=1 for different values of x of Experiment 1.
x
Exact
MQQI [26]
IMQQI
DMQQI
Error
Error
N
=
100
N
=
20
N
=
30
N
=
30
N
=
60
IMQQI
DMQQI
0.1
0.0754
0.0787
0.0754
0.0754
0.0727
0.0751
4.17
E
-
07
2.84
E
-
04
0.2
0.1506
0.1520
0.1506
0.1506
0.1496
0.1505
1.43
E
-
07
1.12
E
-
04
0.3
0.2257
0.2255
0.2256
0.2257
0.2253
0.2256
3.20
E
-
08
3.16
E
-
05
0.4
0.3003
0.2990
0.3003
0.3003
0.3002
0.3003
4.40
E
-
10
8.83
E
-
06
0.5
0.3744
0.3723
0.3744
0.3744
0.3744
0.3744
6.09
E
-
08
4.67
E
-
06
0.6
0.4448
0.4478
0.4478
0.4478
0.4478
0.4478
6.97
E
-
08
4.55
E
-
06
0.7
0.5203
0.5164
0.5205
0.5203
0.5203
0.5203
5.19
E
-
07
4.99
E
-
06
0.8
0.5915
0.5862
0.5910
0.5915
0.5915
0.5915
5.05
E
-
06
4.54
E
-
06
0.9
0.6600
0.6296
0.6539
0.6604
0.6598
0.6600
7.78
E
-
05
1.10
E
-
06Table 3
Comparison of results with the results of [18, 26, 35] for υ=0.0001 and Δt=0.001 at t=1 for different values of x of Experiment 1.
x
Galerkinmethod [35]
AMQ [18]
MQQI [26]
IMQQI
DMQQI
N
=
10
N
=
72
N
=
36
N
=
72
N
=
72
0.056
0.0422
0.0424
0.0424
0.0562
0.0421
0.0421
0.167
0.1263
0.1263
0.1262
0.1260
0.1263
0.1263
0.278
0.2103
0.2103
0.2096
0.2103
0.2103
0.2103
0.389
0.2939
0.2939
0.2928
0.2939
0.2939
0.2938
0.500
0.3769
0.3769
0.3754
0.3769
0.3769
0.3766
0.611
0.4592
0.4592
0.4572
0.4592
0.4592
0.4565
0.722
0.5404
0.5404
0.5381
0.5404
0.5404
0.5231
0.833
0.6203
0.6201
0.6174
0.6203
0.6203
0.5296
0.944
0.6983
0.6957
0.6948
0.6967
0.6982
0.3815The space-time graph of the estimated solution of Burgers’ equation by using IMQQI forx∈[0,1], t∈[0,1], υ=0.1 (a), and υ=0.01 (b) of Experiment 1.
(a)
(b)The space-time graph of the estimated solution of Burgers’ equation by using DMQQI forx∈[0,1], t∈[0,1], υ=0.1 (a), and υ=0.01 (b) of Experiment 1.
(a)
(b)Numerical comparison in these cases shows that the obtained results, particularly in IMQQI scheme, are in good agreement with the exact solutions and the results of the other schemes.Relation between the spectral radius of the matricesE1 and E2 and the different values of the shape parameter c is shown in Table 4 by keeping the number of collocation points fixed. It is clear from Table 4 that if the values of shape parameter c are greater than the critical value c=0.1 (c=0.01), then the solution obtained from the IMQQI (DMQQI) method breaks down and hence the IMQQI and DMQQI methods become unstable. Therefore, the interval stability of IMQQI and DMQQI schemes is (0,0.1) and (0.004,0.01), respectively.Table 4
The spectral radius andL∞ and L2 error norms versus shape parameter c when  Δt=0.001, N=100, and υ=0.01 at t=1 of Experiment 1.
IMQQI
DMQQI
c
ρ
(
E
1
)
L
∞
L
2
c
ρ
(
E
2
)
L
∞
L
2
1
.
00
E
-
20
0.99925
2.4628
E
-
04
5.1531
E
-
04
4.00
E
-
03
0.99925
5.0580
E
-
02
8.8134
E
-
03
1.00
E
-
10
0.99925
2.4628
E
-
03
5.1531
E
-
04
6.00
E
-
03
0.99925
9.7461
E
-
03
1.6022
E
-
03
1.00
E
-
05
0.99925
2.4642
E
-
03
5.1413
E
-
04
8.00
E
-
03
0.99925
4.7521
E
-
03
7.5521
E
-
04
1.00
E
-
02
0.99925
2.1783
E
-
03
2.8715
E
-
04
1.00
E
-
02
0.99925
5.1918
E
-
03
6.7218
E
-
04
1.00
E
-
01
0.99924
2.8855
E
-
04
3.3028
E
-
05
3.00
E
-
02
1.00049
8.3973
E
-
01
1.2667
E
-
01
1.20
E
-
01
5.10176
1.3242
E
+
04
1.7896
E
+
04
5.00
E
-
02
1.05591
9.3609
E
-
01
1.7319
E
-
01It can be seen from Table4 that the schemes are very sensitive to the values of the shape parameter c and the interval stability of methods is a small interval.Experiment 2.
In this experiment, we consider the shock propagation solution of the homogeneous Burgers’ equation [31] as a numerical experiment. This solution is given by
(65)u(x,t)=xt(1+t/t*exp⁡(x2/4υt)),t⩾1,t*=exp⁡(18υ),0⩽x⩽1.2.
The initial condition of the problem is obtained from (65) at time t=1 and the boundary conditions in (3) can be obtained from the exact solution. Propagation of the shock is studied with Δt=0.01 for υ=0.005 and υ=0.001. The shape parameter c is denoted by  6.0×10-2, 2.4×10-2, and 1.2×10-2 for N=20, N=50, and N=100, respectively. The L2 and L∞ error norms are calculated in 240 and 2400 points for υ=0.005 and υ=0.001, respectively, and compared with the results of [31] in Table 5 at different times. The space-time graph of the estimated solution for υ=0.005 is showed in Figure 3.Table 5
The comparison ofL2 and L∞ errors between the numerical results by using our schemes and the results of [31] with Δt=0.01, υ=0.005, and υ=0.001 of Experiment 2.
υ
=
0.005
L
∞
L
2
t
=
1.7
t
=
2.4
t
=
3.1
t
=
1.7
t
=
2.4
t
=
3.1
IMQQI;N=20
9.88
E
-
03
3.74
E
-
03
9.70
E
-
04
2.18
E
-
03
7.85
E
-
04
2.48
E
-
04
IMQQI;N=50
7.63
E
-
05
2.88
E
-
05
1.45
E
-
05
1.79
E
-
05
8.35
E
-
06
4.86
E
-
06
DMQQl;N=50
1.82
E
-
04
1.58
E
-
04
9.88
E
-
05
4.12
E
-
05
3.24
E
-
05
2.09
E
-
05
TCM [31]; N=240
6.48
E
-
05
4.32
E
-
05
3.13
E
-
05
1.69
E
-
05
1.21
E
-
05
9.20
E
-
06
TGM [31]; N=240
1.78
E
-
03
1.28
E
-
03
1.00
E
-
03
3.23
E
-
04
2.99
E
-
04
2.75
E
-
04
υ
=
0.001
t
=
1.7
t
=
3.0
t
=
3.5
t
=
1.7
t
=
3.0
t
=
3.5
IMQQI;N=100
7.57
E
-
03
2.85
E
-
03
1.70
E
-
03
1.07
E
-
03
3.20
E
-
04
2.21
E
-
04
DMQQI;N=100
1.71
E
-
02
4.03
E
-
03
5.29
E
-
03
2.26
E
-
03
5.37
E
-
04
6.43
E
-
04
TCM [31]; N=2400
1.54
E
-
03
3.58
E
-
04
2.41
E
-
04
1.89
E
-
04
5.12
E
-
05
3.59
E
-
05
TGM [31]; N=2400
1.56
E
-
03
3.67
E
-
04
2.49
E
-
04
1.93
E
-
04
5.54
E
-
05
4.07
E
-
05The space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) forx∈[0,1.2] and t∈[1,3.5] of Experiment 2.
(a)
(b)Experiment 3.
In this experiment, we study the fusion phenomenon of the two solitary waves of the homogeneous Burgers’ equation. The fusion phenomenon happens when two or more solitons will fusion to one soliton at a specific time. In [9], Wang et al. studied the following Burgers’ equation:
(66)ut+2uux-uxx=0.
They obtained the two-solitary-wave solution
(67)u(x,t)=-k1ek1(x+k1t)+k2ek2(x+k2t)1+ek1(x+k1t)+ek2(x+k2t),
wherek1 and k2 are constant. Let
(68)t⟶t,x⟶x2.
Hence, (66) converts to Burgers’ equation form (1) wherein F(x,t)=0 and υ=0.25. In this case, two-solitary-wave fusion happens at a specific time t=0. Because we can show the fusion phenomenon, we consider an interval [-5,5] for t and x. For this purpose, we introduce a new time variable τ=t+5 and approximate the solution u(x,τ) of (1) by using our schemes for τ∈[0,10]. Then, we obtain u(x,t) for t∈[-5,5]. The initial condition can be obtained from the exact solution at τ=0. The boundary conditions can be also taken from the exact solution.
TheL2 and L∞ errors of IMQQI scheme are calculated in 1000 points for k1=1, k2=-1, Δt=0.001, and c=h and listed in Table 6. From our numerical experiment whose results are not given here, we see that the accuracy of the DMQQI scheme is very bad in this experiment but we can clearly see from Table 6 that the results of the IMQQI scheme are in good agreement with the exact solutions.
The space-time graph of the estimated solution by using IMQQI is presented in Figure4.Table 6
TheL2 and L∞ errors of the IMQQI scheme with Δt=0.001 at different times of Experiment 3.
L
∞
L
2
t
=
-
4
t
=
0
t
=
5
t
=
-
4
t
=
0
t
=
5
IMQQI;N=20
3.907
E
-
04
3.242
E
-
03
1.592
E
-
02
5.660
E
-
04
3.181
E
-
03
1.728
E
-
02
IMQQI;N=40
7.678
E
-
06
2.661
E
-
05
3.379
E
-
04
1.174
E
-
05
1.670
E
-
05
2.119
E
-
04Figure 4
The space-time graph of the estimated solution of Burgers’ equation by using IMQQI forx∈[-5,5] and t∈[-5,5] of Experiment 3.Experiment 4.
In this experiment, we consider Burgers’ equation (1) with the nonhomogeneous term F(x,t)=kx/(2βt+1)2 that k>0 and β⩾0 are constant. So, Burgers’ equation (1) has the following form:
(69)ut+uux=υuxx+kx(2βt+1)2.
In 2010, Rao and Yadav [12] obtained the solution of Burgers’ equation (69) with the initial condition u(x,0)=u0(x)∈L2(ℝ,ex2/2). They showed that the solution of (69) is given by
(70)u(x,t)=A0x2βt+1,A0=β+β2+k,
for the initial condition u(x,0)=kx, k>β.
In this paper, we simulate this solution fork=5 and β=2 in x∈[-1,1]. The boundary conditions in (3) can be obtained from the exact solution (70). The L2 and L∞ error norms are calculated in 1000 points for arbitrary υ, Δt=0.01, and c=h and listed in Table 7. Also, the space-time graph of the estimated solution by using IMQQI and DMQQI schemes is presented in Figure 5.
Table7 shows that the accuracy of the DMQQI scheme is low even if the number of collocation points increases, whereas the IMQQI scheme provides the good results with a small number of points.Table 7
The comparison ofL2 and L∞ errors between the numerical results of our schemes with Δt=0.01 of Experiment 4.
L
∞
L
2
t
=
1
t
=
5
t
=
10
t
=
1
t
=
5
t
=
10
IMQQI;N=10
1.171
E
-
06
2.816
E
-
09
1.876
E
-
10
8.394
E
-
08
2.020
E
-
09
1.345
E
-
10
DMQQI;N=20
1.533
E
-
01
4.146
E
-
02
2.159
E
-
02
7.441
E
-
02
2.102
E
-
02
1.101
E
-
02
DMQQI;N=40
1.234
E
-
01
3.446
E
-
02
2.034
E
-
02
6.612
E
-
02
2.102
E
-
02
1.037
E
-
02The space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) forx∈[-1,1] and t∈[0,10] of Experiment 4.
(a)
(b)Experiment 5.
We finally closed our analysis by considering the following Burgers’ equation with the nonhomogeneous term:(71)ut+uux=υuxx+Asin(x),A>0,
with the initial condition
(72)u(x,0)=0,x∈[0,π],
which was discussed in [1], as a nonlinear model for describing hypersound generation in prescribed light field. The solution of the differential equation (71) can be written as a series of Mathieu’s functions [1]:
(73)u(x,t)=2υ∂∂xln⁡[∑k=0∞‍a2kexp⁡(-υλ2k(q)4t)ce2k(π-x2,q)],
where
(74)a2k=[∫02π‍ce0(x/2,q)dx][∫02π‍ce2k2(x/2,q)dx]
and q=A/υ2. The notations used here correspond to those from the book by Strutt [36]. The boundary conditions are
(75)u(0,t)=u(π,t)=0,t⩾0.
The numerical results compared with the exact solutions for υ=1 with q=20 and υ=0.1 with q=100 in Tables 8 and 9, respectively. The numerical calculations are performed with Δt=0.001 and c=h. The space-time graph of the estimated solutions is also presented for υ=0.1 in Figure 6.Table 8
Comparison of numerical results with the exact solutions forυ=1 with q=20 and Δt=0.001 at t=3 of Experiment 5.
x
Exact
IMQQI
DMQQI
Error
Error
N
=
10
N
=
20
N
=
20
N
=
30
IMQQI
DMQQI
0.5
2.1481
2.1500
2.1481
2.1343
2.1474
1.025
E
-
05
6.383
E
-
04
1.0
4.1562
4.1563
4.1562
4.1512
4.1558
6.026
E
-
06
4.374
E
-
04
1.5
5.8928
5.9011
5.8928
5.8842
5.8924
1.619
E
-
05
3.516
E
-
04
2.0
7.2404
7.2593
7.2404
7.2264
7.2400
3.306
E
-
05
3.083
E
-
04
2.5
8.0302
8.0156
8.0358
8.0056
8.0298
2.109
E
-
04
3.599
E
-
04
3.0
4.5140
4.8006
4.5143
4.4773
4.4965
2.169
E
-
04
1.750
E
-
02Table 9
Comparison of numerical results with the exact solutions forυ=0.1 with q=100 and Δt=0.001 at t=3 of Experiment 5.
x
Exact
IMQQI
DMQQI
Error
Error
N
=
20
N
=
30
N
=
30
N
=
60
IMQQI
DMQQI
0.5
0.4824
0.4853
0.4851
0.4822
0.4851
2.679
E
-
03
2.696
E
-
03
1.0
0.9331
0.9392
0.9392
0.9383
0.9391
6.056
E
-
03
6.006
E
-
03
1.5
1.3221
1.3318
1.3300
1.3317
1.3320
7.894
E
-
03
9.878
E
-
03
2.0
1.6222
1.6382
1.6271
1.6370
1.6371
4.911
E
-
03
1.490
E
-
02
2.5
1.8102
1.8281
1.8122
1.8322
1.8322
1.833
E
-
03
2.195
E
-
02
3.0
1.6155
1.6657
1.6207
1.5897
1.6476
5.433
E
-
03
3.238
E
-
02The space-time graph of the estimated solution of Burgers’ equation by using IMQQI (a) and DMQQI (b) forυ=0.1, x∈[0,π] and t∈[0,3] of Experiment 5.
(a)
(b)


## 6. Conclusion

In this paper, two numerical schemes based on high accuracy MQ quasi-interpolation scheme and RBFs approximation schemes (IRBF and DRBF approximation schemes) have been presented for solving the nonlinear nonhomogeneous Burgers’ equation. The accuracy of the methods can be increased by selecting the appropriate shape parameter. The choice of the shape parameter is still a pendent question.The numerical results which are given in the previous section indicate that the performance of the methods specially IMQQI is in excellent agreement with the exact solutions. Tables1–9 show that the IMQQI scheme is more accurate than DMQQI scheme as expected, and the interval stability of the IMQQI method is greater than the DMQQI method. Also, the IMQQI scheme required less nodes in comparison with the DMQQI scheme. We can even get good results with less number of points specially in IMQQI method but the results are bad at the ends of interval that we can improve it by using the knot method [37]. Hence, we will not encounter with large scale matrix. Besides, we use equidistant points in our numerical experiments but our schemes can be used for the scattered points.



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*Source: 598432-2014-04-15.xml*