MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation90158710.1155/2010/901587901587Research ArticleHigh Accuracy Combination Method for Solving the Systems of Nonlinear Volterra Integral and Integro-Differential Equations with Weakly Singular Kernels of the Second KindPanLu1HeXiaoming2Tao1MilovanovićGradimir V.1College of MathematicsSichuan UniversityChengduSichuan 610064Chinascu.edu.cn2Department of Scientific ComputingThe Florida State UniversityTallahasseeFL 32310USAfsu.edu20102452010201021102009010420102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.

1. Introduction

In this paper, we first consider the following system of differential equations: xi(t)=fi(t,x(t),z(t)),xi(0)=xi0,i=1,,m, where x(t)=(x1(t),,xm(t)), z(t)=(z11(t),,z1m(t),,zm1(t),,zmm(t)), and zij(t)=0t(t-s)αij(ln(t-s))βijkij(t,s,x(s))ds, 0>αij>-1, βij=0 or 1, i,j=1,,m as well as kij(t,s,x), i,j=1,,m, are continuous functions for 0stT and xRm.

The problem (1.1) can be transformed to the following system of nonlinear integral equations: find z(t) and x(t) satisfying zij(t)=0t(t-s)αij(ln(t-s))βijkij(t,s,x(s))ds,i,j=1,,m,xi(t)=xi0+0tfi(s,x(s),z(s))ds,i=1,,m, which is a special case of nonlinear Volterra integral system with weakly singular kernels of the second kind ui(t)=yi(t)+j=1m0t(t-s)αij(ln(t-s))βijkij(t,s,u(s))ds,i=1,,m, where -1<αij<0, and kij(t,s,u) is continuous function on 0stT and u=(u1,,um)Rm.

Nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind play important roles in the mathematical modeling of many physical and biological phenomena, particularly in such fields as heat transfer, nuclear reactor dynamics, and thermoelasticity, in which it is necessary to take into account the effect of past history. Plenty of work has been done to develop and analyze numerical methods for solving the Volterra integral and integro-differential equations with weakly singular kernels of the second kind; see  and reference therein. The recent progress in this research area has been achieved for the extrapolation method , the spline collocation method , and the Galerkin method [8, 9]. However, there are few works for solving the systems of these types of equations, which are more important than single equation for many applications. For example, the elastodynamic problems for piezoelectric and pyroelectric hollow cylinders under radial deformation can be successfully transformed into a system of two second kind Volterra integral equations.

The combination method as an accelerating convergence technique for solving integral equations was firstly presented in 1984 . Similar to the extrapolation method, the combination method combines several approximations to obtain an approximation of higher accuracy. One important advantage of the extrapolation method and the combination method is parallel computation since those original approximations can be computed independently. However, the extrapolation method uses a coarse grid and some finer grids. We must do much more work on the finer grids than the coarse one, which lowers the degree of parallelism. On the other hand, the loads of computing the approximations in the combination method are close to each other. Hence the combination method is an efficient parallel method to obtain an approximation of high accuracy with a high degree of parallelism. Recently this method has been used to solve the first kind Abel integral equations . In this paper, we will apply the high accuracy combination method for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind, for which there are few results, even for the general convergence to be proved in Section 4 of this paper.

In this paper, based on Lyness's  modified mid-point rectangular quadrature rule and modified trapezoidal quadrature rule, we will construct two quadrature methods to solve the systems. By means of the asymptotic expansions of the errors for both algorithms, we present a high accuracy combination algorithm. Then by using the generalization of discrete Gronwall inequality , the convergence rate, stability, and the asymptotic expansion of the error of the combination approximate solution are proved. Comparing to other algorithms, for example, extrapolation methods [6, 14], the combination algorithm has the advantage of less computation complexity because both of the modified mid-point rectangular quadrature rule and the modified trapezoidal quadrature rule have the same step size. Moreover an a posteriori error estimation is obtained, by which we can rectify the accuracy of our algorithm in processing.

2. Existence and Uniqueness of the Solution

Since the uniqueness, existence, and numerical methods of (1.2) can be decided by (1.1), we only discuss the problem (1.3) in the following.

Let kij(t,s,u) satisfy Lipschitz condition (A): |kij(t,s,u)-kij(t,s,v)|Lu-vu,vRm,0stT,1i,jm, then there is a unique solution in (1.3). In fact α=mini,jαij>-1, u(t)=(u1(t),,um(t))T, and u(t)=max1im|ui(t)|. We can choose η>0 such that α-η>-1. Thus we have N=max1i,jm,0<t<Ttαij-α+η|(lnt)βij|<. Let (C[0,T])m=C[0,T]××C[0,T] be a continuous function space which maps [0,T] into Rm and let F be a mapping from (C[0,T])m to itself. Then we have Fi(u)(t)=yi(t)+0tj=1m(t-s)αij(ln(t-s))ijβkij(t,s,u(s))ds,1im, where y=(y1,,ym)(C[0,T])m, and |Fi(u)(t)-Fi(v)(t)|=j=1m0t(t-s)α-η(t-s)αij-α+η|(ln(t-s))ijβ||kij(t,s,u(s))-kij(t,s,v(s))|dsj=1m0t(t-s)α-ηNL|u(s)-v(s)|ds=mNL0t(t-s)α-η|u(s)-v(s)|ds,1im, where f(t)=tα-ηL[0,T], which is due to α-η>-1.

Lemma 2.1.

Suppose that condition (A) is satisfied, then there is a unique solution to (1.3).

Proof.

Assume that u(t)=(u1(t),,um(t))T and v(t)=(v1(t),,vm(t))T are two different solutions to (1.3). Defining w(t)=u(t)-v(t), we get wi(t)=j=1m0t(t-s)αij(ln|t-s|)βij(kij(t,s,u(s))-kij(t,s,v(s)))ds,i=1,,m,w(t)mNL0ttα-ηw(s)ds. Therefore using Gronwall inequality, we get w(t)=0, which leads to the uniqueness.

In order to prove existence, we use a simple iterative process: let u(0)(t)=0, u(1)(t)=y(t) and ui(n)(t)=yi(t)+0tj=1m(t-s)αij(ln(t-s))βijkij(t,s)u(n-1)(s)ds,n=1,2,. Now we will prove that u(n)(t) is a convergent sequence. Let V(n)(t)=u(n)(t)-u(n-1)(t), then from (2.6) and (2.4) we get V(n+1)(t)M0t(t-s)α-ηV(n)(s)ds=Mtβ*V(n)=Mntβ**tβ*V(1), where tβ*V(n) denotes convolution, M=mNL, and β=α-η.

Taking Laplace transformation, we can easily deduce that V̂(n+1)(t)(MΓ(β+1)sβ+1)nV̂(1)(s). Taking inverse Laplace transformation, we get V(n+1)(t)(MΓ(β+1))nΓ(nβ+n)0tτn(β+1)V(0)(t-τ)dτ. Note that for n>m, we have u(n)(t)-u(m)(t)=(u(n)(t)-u(n-1)(t))+(u(n-1)(t)-u(n-2)(t))++(u(m+1)(t)-u(m)(t)) or u(n)(t)-u(m)(t)V(n)(t)++V(m+1)(t). But using (2.10), we easily prove that n=0V(n+1)(t)n=0(MΓ(β+1))nΓ(nβ+n)0tτn(β+1)V(0)(t-τ)dτ is a convergent series, which means that limn,mu(n)(t)-u(m)(t)=0oru(n)(t)u(t) and u(t) is the solution to (1.2).

3. The Numerical Methods

In this section two quadrature algorithms which are based on  will be given to solve the systems. Consider the weakly singular integral I(G)=ab(b-x)αg(x)dx,G(x)=(b-x)αg(x) and Navot's modified trapezoidal rule  QTh(G)=h2G(a)+hj=1N-1G(xj)-ζ(-α)h1+αg(b). There is an error estimate as follows: ETh(G)=QTh(G)-I(G)=-ζ(-α-1)h2+αg(b)+O(h2). We differentiate with respect to α in (3.3) and get ETh(G)=QTh(G)-I(G)=[ζ(-α-1)-ζ(-α-1)lnh]h2+αg(b)+O(h2). Then we have I(G)=ab(b-x)α(ln(b-x))βg(x)dx,G(x)=(ln(b-x))β(b-x)αg(x),β=0,1 such that QTh(G)=h2G(a)+hj=1N-1G(xj)-[-βζ(-α)+ζ(-α)(lnh)β]h1+αg(b). Thus we have ETh(G)=QTh(G)-I(G)=[βζ(-α-1)-ζ(-α-1)(lnh)β]h2+αg(b)+O(h2). Equations (3.5) and (3.6) are the integral functions with logarithm singularity and their modified trapezoidal rule, and (3.7) is an asymptotic expansion of the error.

From , we have the modified mid-point rectangular rule QMh(G)=hj=0N-1G(xj+1/2)-(2-α-1)ζ(-α)h1+αg(b). Hence we get EMh(G)=QMh(G)-I(G)=-(2-α-1)ζ(-α-1)h2+αg(b)+O(h2). From (3.4)–(3.7), we get EMh(G)=QMh(G)-I(G)=-[-β2-α-1ln2ζ(-α-1)-β(2-α-1-1)ζ(-α-1)+(2-α-1-1)ζ(-α-1)(lnh)β]h2+αg(b)+O(h2). More generally we have QMh(G)=hj=0N-1G(xj+1/2)-[-β2-αln2ζ(-α)-β(2-α-1)ζ(-α)+(2-α-1)ζ(-α)(lnh)β]h1+αg(b). Therefore from (3.7) and (3.10), if β=0, we get (2-α-1-1)ETh(G)-EMh(G)=O(h2) or (1-2-α-12-2-α-1QTh(G)+12-2-α-1QMh(G))-I(G)=O(h2). Note that the combination has a high accuracy order O(h2), which is higher than O(h2+α) in (3.7) and (3.10).

If α=0, β=1 in (3.5), then QTh(G)=h2G(a)+hj=1N-1G(xj)+12ln(h2π)hg(b),QMh(G)=hj=0N-1G(xj+1/2)-12ln2hg(b), where ζ(0)=-0.5, ζ(0)=-ln(2π)/2, and (3.13) becomes (13QTh(G)+23QMh(G))-I(G)=O(h2). Using (3.14) and (3.15), we can construct two algorithms for solving Volterra integral system of equations. Since the kernels of the systems are weakly singular, the following Navot's quadrature rule is used. Setting t=tl, we get ui(tl)=yi(tl)+j=1m0tl(tl-s)αij(ln|tl-s|)βijkij(tl,s,u(s))ds,i=1,2,,m. Now we recall the following lemma from .

Lemma 3.1.

Let g(t)C2r[a,b], G(x)=(b-t)α(ln|b-t|)βg(t), h=(b-a)/N, tk=a+kh(k=0,,N), then modified trapezoidal rule TN(G)=h2G(t0)+hj=1N-1-[-βζ(-α)+ζ(-α)(lnh)β]g(b)h1+α has an asymptotic error expansion EN(G)=j=1r-1B2j(2j)!G(2j-1)(a)h2j+j=12r-1(-1)j[-βζ(-α-j)+ζ(-α-j)(lnh)β]g(j)hj+α+1j!+O(h2r), where α>-1, β=0,1, ζ(t) and ζ(t) are Riemann-Zeta function and its derivative function, and B2j are Bernoulli numbers.

From the above, we obtain the following discrete system of equations for modified trapezoidal quadrature method: ui0=yi(t0),uil=yi(tl)+j=1m[hk=0l-1wlk(tl-tk)αij×(ln|tl-tk|)βijkij(tl,tk,Uk)+wll,ijh1+αijkij(tl,tl,Ul)],i=1,2,,m,l=1,2,,N, where h=1/N, tk=kh, Uj=(u1j,u2j,,umj), j=1,2,,m, and wlk={12ifk=0,1if1kl-1,βijζ(-αij)+ζ(-αij)(lnh)βij-βijζ(-αij)ifk=l,l=1,2,,N.

On the other hand, we also obtain the discrete system for modified mid-point rectangular quadrature method ui0=yi(t0),uil=yi(tl)+j=1m[hk=0l-1wlk(tl-tk+1/2)αij×((ln|tl-tk+1/2|)βijkij(tl,tk,Uk+Uk+1/2)2)+wll,ijh1+αijkij(tl,tl,Ul)k=0l-1],i=1,2,,m,l=1,2,,N, where wlk={1if0kl-1,-[-βij2-αln2ζ(-α)-βij(2-α-1)ζ(-α)ifk=l,l=1,2,,N,+(2-α-1)ζ(-α)(lnh)βij]βij=0,or1.

Because the discrete system is nonlinear diagonal system of equations, we introduce the following iterative algorithms.

Algorithm 3.2 (modified trapezoidal quadrature method).

We have the following steps.Step 1.

Take ε>0 sufficiently small and let u0(tj)=(u1j0,,umj0)=(y1(tj),,ym(tj)),j=1,2,,N and n:=0.

Step 2.

Compute uiln+1(lN,i=1,2,,m) in parallel by the following simple iteration: uiln+1=yi(tl)+j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijkij(tl,tk,Uk)+wllh1+αijkij(tl,tl,Uln)], where Uk=(u1k,,umk), Uln=(u1ln,,umln), and wlk is defined by (3.20).

Step 3.

If max1im|uiln+1-uiln|ε, then let uiln+1=ũil and stop the iteration, otherwise set n:=n+1, go to Step 2.

Algorithm 3.3 (modified mid-point rectangular quadrature method).

We have the following steps.Step 1.

Take ε>0 sufficiently small and let u̅0(tj)=(u̅1j0,,u̅mj0)=(y1(tj),,ym(tj)),j=1,2,,N and n:=0.

Step 2.

Compute u̅iln+1(lN,i=1,2,,m) in parallel by the following simple iteration: u̅iln+1=yi(tl)+j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βij×kij(tl,tk,(U̅kn+U̅k+1/2n)2)+wll,ijh1+αijkij(tl,tl,U̅ln)], where U̅k=(u1k,,umk), U̅ln=(u̅1ln,,u̅mln), and wlk is defined by (3.22).

Step 3.

If max1im|u̅iln+1-u̅iln|ε, then let u̅iln+1=ũil and stop the iteration, otherwise set n:=n+1, go to Step 2.

4. Convergence and Error Estimation

In this section we will analyze the convergence of Algorithm 3.2 proposed in Section 3. The proof of convergence rate of Algorithm 3.3 is similar. When t=tl, the system can be expressed as ui(tl)=yi(tl)+j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijkij(tl,tk,Uk)+wllh1+αijkij(tl,tl,Ul)]+j=1m[Eij,l,t((tl-t)αij(ln|tl-t|)βijkij(tl,t,u(t)))]. By Lemma 3.1, the remainder has the following estimate: j=1m[Eij,l,t((tl-t)αij(ln|tl-t|)βijkij(tl,t,u(t)))]=O(h2+α(lnh)β), where α=min1i,jmαij>-1 and β=max1i,jmβij.

Letting eil=ui(tl)-uil and {uil} be the solution of the discrete equations, we derive ei0=0,eil=j=1mk=0l-1hwlk(tl-tk)αij(ln|tl-tk|)βij×(kij(tl,tk,u1(tk),,um(tk))-kij(tl,tk,u1k,,umk))+j=1mwllh1+αij(kij(tl,tl,u1(tl),,um(tl))-kij(tl,tl,u1l,,uml))+j=1m[Eij,l,t((tl-t)αij(ln|tl-t|)βijkij(tl,t,u1(t),,um(t)))],i=1,2,,m,l=1,2,,N.

Lemma 4.1.

Suppose that the sequence {ei}i=1N satisfies e0=0,|ei|j=1i-1Bij|ej|+A,1iN, where Bij=2Lh(ti-tj)α(ln|ti-tj|)β, -1<α0, β=0,1, and h is sufficiently small such that Lhwii1/2. Then |ei|HA, where H=k=0Rk/(k!)s, R=2L(b-a)sΓ(s)e1/12s(e/s)hs, and s=1+α.

Theorem 4.2.

Assume that h is sufficiently small, then the system of the nonlinear discrete equation (3.19) has a unique solution and the simple iteration (3.24) is geometrically convergent.

Proof.

Firstly, if Ul=(u1l,u2l,,uml) and Vl=(v1l,v2l,,vml)(l=1,2,,N) are solutions to (3.19), then {Zl=Ul-Vl} satisfies the inequality |uil-vil|j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βij|kij(tl,tk,Uk)-kij(tl,tk,Vk)|]+j=1m[wllh1+αij|kij(tl,tl,Ul)-kij(tl,tk,Vl)|]j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijLUk-Vk+wllh1+αijLUl-Vl],i=1,2,,m,l=1,2,,N, where we use ui0=vi0=y(ti). Note that Zk=max1jm|zjk|, then we easily deduce that Z0=0,ZlM1hk=0l-1(tl-tk)α(ln|tl-tk|)βZk,l=1,2,,N, where M1=mmax0k<lN|wlk(tl-tk)αij-α(1+c)(ln|tl-tk|)βLij|.

If h is sufficiently small such that M1h1+α1/2, we have Z0=0;Zl2M1hk=0l-1(tl-tk)α(ln|tl-tk|)βZk,l=1,2,,N. Then by Lemma 4.1, we get Zl=0, l=1,2,,N. Hence the uniqueness is shown.

Secondly, from the iteration (3.24) we have |uiln+1-uiln|=j=1m[wllh1+αij|kij(tl,tl,Uln)-kij(tl,tl,Uln-1)|]j=1m[    wllh1+αijLUln-Uln-1],i=1,2,,m;l=1,2,,N. Then Uln+1-UlnMh1+αUln-Uln-112Uln-Uln-1,l=1,2,,N, where M=mwllL. We assume that h is small enough such that Mh1+α1/2. Thus we prove that the simple iteration (3.24) is geometrically convergent, and its limit is the unique solution of (3.19).

Theorem 4.3.

There is a positive constant C independent of h such that Ul-ŨlCεh1+α, where Ũl=(ũ1l,ũ2l,,ũml)=(u1ln+1,u2ln+1,,umln+1) is defined in Algorithm 3.2.

Proof.

Letting Vl=Ul-Ũl, we get ui0-ũi0=0,uil-ũil=j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βij(kij(tl,tk,Uk)-kij(tl,tk,Ũk))]+j=1mwllh1+αij(kij(tl,tl,Ul)-kij(tl,tl,Uln))=j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βij(kij(tl,tk,Uk)-kij(tl,tk,Ũk))]+j=1mwllh1+αij(kij(tl,tl,Ul)-kij(tl,tl,Uln+1))+j=1mwllh1+αij(kij(tl,tl,Uln+1)-kij(tl,tl,Uln)),i=1,2,,m;l=1,2,,N. By Algorithm 3.2 and Lipschitz condition, we have VlM1hk=0l-1(tl-tk)α(ln|tl-tk|)βVk+2M1εh1+α. Using Lemma 3.1, we get Ul-ŨlCεh1+α.

Theorem 4.4.

There is a positive constant C independent of h such that the errors eil=ui(tl)-uil(i=1,2,,m,l=1,2,,N) have the following error bound: max0lN;1im|eil|Ch2+α|lnh|β.

Proof.

Taking El=max1im|eil|, we get E0=0,ElM1hk=0l-1(tl-tk)α(ln|tl-tk|)βEk+M1h1+αEl+O(h2+α),l=1,2,,N. If h is sufficiently small such that M1h1+α1/2, we have El2M1hk=0l-1(tl-tk)α(ln|tl-tk|)βEk+O(h2+α),l=1,2,,N. Using inequality (4.5), we can get max0lN;1im|eil|Ch2+α|lnh|β.

Corollary 4.5.

Assume that ε=O(h) in Algorithm 3.2, one can obtain the estimate max1im;1lN|ui(tl)-ũil|=O(h2+α(lnh)β).

5. Asymptotic Expansion, Combination, and a Posteriori Error Estimate

In the following we only derive the asymptotic expansions of the errors and the a posteriori error estimation of Algorithm 3.2. For Algorithm 3.3 we just simply present the corresponding result.

From Lemma 3.1 and (4.1) by using Taylor's expansion, we have ui(tl)=yi(tl)+j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijkij(tl,tk,Uk)+wllh1+αijkij(tl,tl,Ul)]+h2+α(lnh)βj=1mTij0(tl)+h2+αj=1mTij1(tl)+O(h2+α1|lnh|β1),i=1,,m, where α1=min1i,jm{αij:αij>α}, β1=max1i,jm{βij:αij=α1}, and Tij0(t)={-ζ(-α-1)ddtkij(tl,t,u(t))ifαij=α,0ifαij>α,Tij1(t)={βζ(-α-1)ddtkij(tl,t,u(t))ifαij=α,0ifαij>α.

For eil=ui(tl)-uil, we have ei0=ui(t0)-ui0=0,eil=ui(tl)-uil=j=1mk=0l-1hwlk(tl-tk)αij(ln|tl-tk|)βij(kij(tl,tk,u(tk))-kij(tl,tk,Uk))+j=1mwllh1+αij(kij(tl,tl,u(tl))-kij(tl,tl,Ul))+h2+α(lnh)βTi0(tl)+h2+αTi1(tl)+O(h2+α1|lnh|β1),i=1,2,,m,l=1,2,,N. Here j=1mTijk=Tik(tl), k=0,1.

Using Theorem 4.4 and Taylor's expansion, we get kij(tl,tk,u1(tk),,um(tk))-kij(tl,tk,u1k,,umk)=kij,u1(tl,tk,u1(tk),,um(tk))(u1(tk)-u1k)++kij,um(tl,tk,u1(tk),,um(tk))(um(tk)-umk)+o(h2)=kij,u1(tl,tk,u1(tk),,um(tk))e1k++kij,um(tl,tk,u1(tk),,um(tk))emk+o(h2), where we assume that kij(s,t,u) is derivable for ul, l=1,,m and let kij,ul(s,t,u)=ulkij(s,t,u),l=1,,m. Then ei0=ui(t0)-ui0=0,eil=ui(tl)-uil=j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijr=1mkij,ur(tl,tk,u(tk))erk]+j=1mwllh1+αijr=1mkij,ur(tl,tl,u(tl))erl+h2+α(lnh)βTi0(tl)+h2+αTi1(tl)+O(h2+α1|lnh|β1),i=1,2,,m,l=1,2,,N. Obviously if β=0, then Ti1(tl)=0.

Now we construct the following auxiliary system of linear Volterra integral equations: find {Φjr(s),j=1,2,,m,r=0,1} satisfying Φir(t)=Tir(t)+j=1m0t(t-s)αij(ln|t-s|)βij[p=1mkij,up(t,s,u(s))Φ1r(s)],i=1,2,,m,r=0,1, and their discrete system of equations: find {Φilr,i=1,2,,m,l=1,,N} satisfying Φi0r=0,Φilr=Tir(tl)+j=1m[hk=0l-1wlk(tl-tk)αij(ln|tl-tk|)βijp=1mkij,up(tl,tk,u(tk))Φpkr]+j=1mwllh1+αijp=1mkij,up(tl,tl,u(tk))Φpkr,i=1,2,,m,l=1,2,,N;r=0,1. From Theorem 4.4, we have max1im,1lN|Φilr-Φir(tl)|=O(h2+α1|lnh|β1),r=0,1. Substituting (5.8) and (5.7) into (5.6), we get Ei0=0,Eil=j=1m[hk=0l-1wlk(tl-tk)αijkij,ul(tl,tk,uj(tk))Ejk]+j=1mwllh1+αijkij,ul(tl,tl,uj(tl))Ejl+O(h2+α1(lnh)β1),i=1,2,,m,l=1,2,,N. Note that Eil=eil-h2+α(lnh)βTil0-h2+αTil1. Using Lemma 4.1, we get ui(tl)-uil-h2+α(lnh)βΦil0(tl)-h2+αΦil1(tl)=O(h2+α1(lnh)β1),i=1,2,,m,l=1,2,,N. From (5.9) we obtain ui(tl)-uil-h2+α(lnh)βΦi0(tl)-h2+αΦi1(tl)=O(h2+α1|lnh|β1),i=1,2,,m,l=1,2,,N, where Φi1(t)=0 if β=0. Similarly for Algorithm 3.3 we have ui(tl)-u̅il-h2+α(lnh)βΦ̅il0-h2+αΦ̅il1=O(h2+α1|lnh|β1) or ui(tl)-u̅il-h2+α(lnh)βΦ̅i0(tl)-h2+αΦ̅i1(tl)=O(h2+α1|lnh|β1),i=1,2,,m,l=1,2,,N, where Φ̅i1(t)=0 if β=0 and Φ̅i1(tl)=-[2-α-1ln2ζ(-α-1)ζ'(-α-1)+(2-α-1-1)]Φi1(tl),Φ̅i0(tl)=(2-α-1-1)Φi0(tl). From the above discussion, we have proved the following theorem for the combination method.

Theorem 5.1.

(1) If β=0, then uilc=(1-2-1-α)uil+u̅il2-2-1-α=ui(tl)+O(h2+α1),i=1,2,,m,l=1,2,,N. Furthermore, one has the following a posteriori error estimate: min{uil,u̅il}uilcmax{uil,u̅il}or|ui(tl)-uil+u̅il2||uil-u̅il2|. That is, one can estimate the average errors |ui(tl)-(uil+u̅il)/2| by |(uil-u̅il)/2|.

(2) If β=1, then uilc=(1-2-1-α)uil+u̅il2-2-1-α=ui(tl)+O(h2+α),i=1,2,,m,l=1,2,,N. Hence one can obtain a high order of accuracy, which is better than O(h2+α(lnh)). One also easily deduces that the modified mid-point rectangular quadrature method is better than modified trapezoidal quadrature method when 1-2-1-α0.

6. Numerical Examples

In this section, we present two numerical examples to illustrate the features of the combination method discussed in this paper. Let ET denote the error of modified trapezoidal quadrature method, EM the error of modified mid-point rectangular quadrature method, and EC the error of combination method.

Example 6.1.

Consider the following system of integral equations with algebraic singularity: x(s)=95s5/3+s-0s1s-t3(x(t)+y(t))dt,0s1,y(s)=π2s-1615s5/2+s+0s1s-t(12x2(t)+12y2(t)-y(t))dt,0s1 with the exact solution x(s)=s, y(s)=s.

The errors of the approximation solutions obtained by Algorithms 3.2 and 3.3 and their combination are presented in Tables 1, 2, 3, and 4. Numerical results show that the combination method has higher order convergence rate than the two original algorithms. Tables 14 also show that the error ratios of Algorithms 3.2 and 3.3 are close to the theoretic value, which is 22+(-1/2)=2.828 and the error ratio of the combination method is better than the 22+(-1/2).

The errors and the a posterior error estimate of x(s) in Example 6.1 at mesh point (h=1/10, α=-1/2 in combination coefficient).

sETEMEC|x(si)-(Txi+Mxi)/2||(Txi-Mxi)/2|
0.1 −3.871E−03 9.519E−04 −1.406E−04 1.459E−032.411E−03
0.2 −4.559E−03 1.313E−03 −1.764E−05 1.623E−032.936E−03
0.3 −4.664E−03 1.381E−03 1.129E−051.642E−033.022E−03
0.4 −4.758E−03 1.428E−03 2.692E−051.665E−033.093E−03
0.5 −4.908E−03 1.480E−03 3.323E−051.714E−033.194E−03
0.6 −5.130E−03 1.546E−03 3.369E−051.792E−033.338E−03
0.7 −5.434E−03 1.630E−03 2.996E−051.902E−033.532E−03
0.8 −5.824E−03 1.735E−03 2.281E−052.044E−033.779E−03
0.9 −6.310E−03 1.864E−03 1.249E−052.223E−034.087E−03
1.0 −6.904E−03 2.021E−03 −1.165E−062.442E−034.462E−03

Maximum error 6.904E−03 2.021E−03 1.406E−04 2.442E−03 4.462E−03

The errors and the a posterior error estimate of x(s) in Example 6.1 at mesh point (h=1/20, α=-1/2 in combination coefficient).

sETEMEC|x(si)-(Txi+Mxi)/2||(Txi-Mxi)/2|
0.1 −1.526E−03 4.376E−04 −7.358E−06 5.444E−04 9.820E−04
0.2 −1.560E−03 4.709E−04 1.082E−05 5.446E−04 1.016E−03
0.3 −1.557E−03 4.786E−04 1.750E−05 5.391E−04 1.018E−03
0.4 −1.576E−03 4.871E−04 1.963E−05 5.446E−04 1.032E−03
0.5 −1.625E−03 5.013E−04 1.957E−05 5.620E−04 1.063E−03
0.6 −1.705E−03 5.228E−04 1.821E−05 5.909E−04 1.114E−03
0.7 −1.815E−03 5.522E−04 1.596E−05 6.314E−04 1.184E−03
0.8 −1.958E−03 5.901E−04 1.297E−05 6.837E−04 1.274E−03
0.9 −2.135E−03 6.373E−04 9.312E−06 7.487E−04 1.386E−03
1.0 −2.350E−03 6.948E−04 4.944E−06 8.278E−04 1.523E−03

Maximum error 2.350E−03 6.948E−04 1.963E−05 8.278E−04 1.523E−03

Maximum error ratios 2.937 2.908 7.162 2.950 2.931

The errors and the a posterior error estimate of y(s) in Example 6.1 at mesh point (h=1/10, α=-1/2 in combination coefficient).

sETEMEC|y(si)-(Tyi+Myi)/2||(Tyi-Myi)/2|
0.1 7.256E−03 −2.096E−03 2.280E−05 2.580E−03 4.676E−03
0.2 4.454E−03 −1.246E−03 4.548E−05 1.604E−03 2.850E−03
0.3 4.207E−03 −1.184E−03 3.705E−05 1.511E−03 2.696E−03
0.4 4.472E−03 −1.254E−03 4.324E−05 1.609E−03 2.863E−03
0.5 4.976E−03 −1.388E−03 5.327E−05 1.794E−03 3.182E−03
0.6 5.646E−03 −1.568E−03 6.621E−05 2.039E−03 3.607E−03
0.7 6.465E−03 −1.787E−03 8.219E−05 2.339E−03 4.126E−03
0.8 7.436E−03 −2.046E−03 1.017E−04 2.695E−03 4.741E−03
0.9 8.577E−03 −2.350E−03 1.257E−04 3.114E−03 5.463E−03
1.0 9.917E−03 −2.704E−03 1.555E−04 3.607E−03 6.310E−03

Maximum error 9.917E−03 2.704E−03 1.555E−04 3.607E−03 6.310E−03

The errors and the a posterior error estimate of y(s) in Example 6.1 at mesh point (h=1/20, α=-1/2 in combination coefficient).

sETEMEC|y(si)-(Tyi+Myi)/2||(Tyi-Myi)/2|
0.1 1.748E−03 −4.874E−04 1.914E−05 6.305E−04 1.118E−03
0.2 1.298E−03 −3.659E−04 1.110E−05 4.662E−04 8.321E−04
0.3 1.311E−03 −3.680E−04 1.234E−05 4.715E−04 8.396E−04
0.4 1.450E−03 −4.058E−04 1.467E−05 5.222E−04 9.280E−04
0.5 1.657E−03 −4.622E−04 1.778E−05 5.972E−04 1.059E−03
0.6 1.915E−03 −5.328E−04 2.164E−05 6.909E−04 1.224E−03
0.7 2.221E−03 −6.166E−04 2.628E−05 8.023E−04 1.419E−03
0.8 2.579E−03 −7.143E−04 3.179E−05 9.324E−04 1.647E−03
0.9 2.995E−03 −8.277E−04 3.834E−05 1.084E−03 1.912E−03
1.0 3.481E−03 −9.597E−04 4.618E−05 1.260E−03 2.220E−03

Maximum error 3.481E−03 9.597E−04 4.618E−05 1.260E−03 2.220E−03

Maximum error ratios 2.817 2.800 3.137 2.824 2.814
Example 6.2.

Consider the following system of integro-differential equations with algebraic singularity: x(s)=528x(s)7/5-12584s12/5+1+0s1(s-t)35(x2(t)-110x(t))dt,0s1,x(0)=0 with the exact solution x(s)=s. Let u(s)=x(s),v(s)=528x7/5(s)-12584s12/5+1+0s1(s-t)35(x2(t)-110x(t))dt. Then the original system of integro-differential equations can be transformed to the following system of Volterra integral equations: u(s)=0sv(t)dt,u(0)=0,v(s)=528u7/5(s)-12584s12/5+1+0s1(s-t)35(u2(t)-110u(t))dt,v(0)=1. The two equations contain algebraic singularity with the coefficient α=-3/5. The exact solution is u(s)=s,v(s)=1.

The errors of the numerical solutions obtained by Algorithms 3.2 and 3.3 and their combination are presented in Tables 5 and 6. Numerical results still show that the combination method has obviously higher convergence rate than the two algorithms, as well as better maximum error ratio, which is greater than 22+(-3/5)=2.639.

The errors and the a posterior error estimate of x(s), that is, u(s) in Example 6.2 at mesh point (h=1/10, α=-3/5 in combination coefficient).

sETEMEC|u(si)-(Tui+Mui)/2||(Tui-Mui)/2|
0.1 5.977E−05 −1.630E−05 −1.470E−06 2.174E−05 3.804E−05
0.2 2.767E−04 −7.095E−05 −3.177E−06 1.029E−04 1.738E−04
0.3 7.027E−04 −1.738E−04 −2.926E−06 2.645E−04 4.382E−04
0.4 1.374E−03 −3.342E−04 −1.214E−06 5.199E−04 8.541E−04
0.5 2.348E−03 −5.659E−04 2.229E−06 8.912E−04 1.457E−03
0.6 3.717E−03 −8.898E−04 8.219E−06 1.414E−03 2.303E−03
0.7 5.621E−03 −1.338E−03 1.830E−05 2.141E−03 3.480E−03
0.8 8.278E−03 −1.961E−03 3.525E−05 3.159E−03 5.119E−03
0.9 1.202E−02 −2.832E−03 6.397E−05 4.595E−03 7.427E−03
1.0 1.737E−02 −4.066E−03 1.134E−04 6.653E−03 1.072E−02

Maximum error 1.737E−02 4.066E−03 1.134E−04 6.653E−03 1.072E−02

The errors and the a posterior error estimate of x(s), that is, u(s) in Example 6.2 at mesh point (h=1/20, α=-3/5 in combination coefficient).

sETEMEC|u(si)-(Tui+Mui)/2||(Tui-Mui)/2|
0.1 1.576E−05 −5.207E−06 −1.119E−06 5.279E−06 1.049E−05
0.2 9.596E−05 −2.543E−05 −1.763E−06 3.527E−05 6.069E−05
0.3 2.555E−04 −6.436E−05 −2.019E−06 9.554E−05 1.599E−04
0.4 5.067E−04 −1.252E−04 −2.052E−06 1.907E−04 3.159E−04
0.5 8.701E−04 −2.130E−04 −1.870E−06 3.286E−04 5.416E−04
0.6 1.379E−03 −3.356E−04 −1.398E−06 5.216E−04 8.571E−04
0.7 2.083E−03 −5.049E−04 −4.622E−07 7.890E−04 1.294E−03
0.8 3.060E−03 −7.394E−04 1.267E−06 1.160E−03 1.900E−03
0.9 4.428E−03 −1.067E−03 4.395E−06 1.680E−03 2.747E−03
1.0 6.366E−03 −1.529E−03 1.003E−05 2.419E−03 3.948E−03

Maximum error 6.366E−03 1.529E−03 1.003E−05 2.419E−03 3.948E−03

Maximum error ratio 2.729 2.659 11.3072.751 2.715
7. Conclusions

In this paper we use the combination method to solve the systems of integral and integro-differential equations with weakly singular kernels of the second kind, which are important to many applications but have few results. High accuracy and high parallelism are two features of this method. Our numerical results also confirm the theoretical conclusions.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (10671136).

BrunnerH.The numerical solution of weakly singular Volterra integral equations by collocation on graded meshesMathematics of Computation198545172417437MR80493310.2307/2008134ZBL0584.65093BrunnerH.Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernelsIMA Journal of Numerical Analysis198662221239MR96766410.1093/imanum/6.2.221ZBL0634.65142AtkinsonK. E.The Numerical Solution of Integral Equations of the Second Kind19974Cambridge, UKCambridge University Pressxvi+552Cambridge Monographs on Applied and Computational MathematicsMR1464941HackbuschW.Integralgleichungen1989Stuttgart, GermanyB. G. Teubner374Teubner Studienbücher MathematikMR1010893ChenZ.XuY.ZhaoJ.The discrete Petrov-Galerkin method for weakly singular integral equationsJournal of Integral Equations and Applications1999111135MR168517810.1216/jiea/1181074260ZBL0974.65122T.YongH.Extrapolation method for solving weakly singular nonlinear Volterra integral equations of the second kindJournal of Mathematical Analysis and Applications20063241225237MR226246710.1016/j.jmaa.2005.12.013ZBL1115.65129PedasA.TammeE.Spline collocation method for integro-differential equations with weakly singular kernelsJournal of Computational and Applied Mathematics20061971253269MR225606610.1016/j.cam.2005.07.035ZBL1104.65129PedasA.TammeE.Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernelsJournal of Computational and Applied Mathematics20082131111126MR238271010.1016/j.cam.2006.12.024ZBL1156.65107BrunnerH.LinY.ZhangS.Higher accuracy methods for second-kind Volterra integral equations based on asymptotic expansions of iterated Galerkin methodsJournal of Integral Equations and Applications1998104375396MR166966710.1216/jiea/1181074245ZBL0944.65140QunL.T.The combination of approximate solutions for accelerating the convergenceRAIRO Analyse Numérique1984182153160MR743882ZBL0544.65037LiuY.T.High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equationsApplied Mathematics and Computation2006178244145110.1016/j.amc.2005.07.022MR2248503ZBL1104.65127LynessJ. N.NinhamB. W.Numerical quadrature and asymptotic expansionsMathematics of Computation196721162178MR022548810.2307/2004157ZBL0178.18402T.HuangY.A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kindJournal of Mathematical Analysis and Applications200328215662MR200032910.1016/S0022-247X(02)00369-4ZBL1030.65140LiuY.T.Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equationsJournal of Computational and Applied Mathematics20072011300313MR229355610.1016/j.cam.2006.02.021ZBL1113.65123NavotJ.A further extension of Euler-Maclaurin summation formulaJournal of Mathematical Physics196241155184T.HuangY.A high accuracy combinatorial algorithm and a posteriori error estimation for solving second-kind Abel integral equationsJournal of Systems Science and Mathematical Sciences2004241110117MR2034994