Mohammad ZAREBNIA Reza PARVAZ

Department of Mathematics,University of Mohaghegh Ardabili,56199-11367 Ardabil,Iran

E-mail:zarebnia@uma.ac.ir;rparvaz@uma.ac.ir

Amir SABOOR BAGHERZADEH

Department of Applied Mathematics,Faculty of Mathematics,Ferdowsi University of Mashhad,Mashhad,Iran

E-mail:saboorbagherzadeh.a@gmail.com

Abstract In this paper,we study an efficient asymptotically correction of a-posteriori error estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method.The deviation of the error for Volterra integrodifferential equations by using the defect correction principle is defined.Also,it is shown that for m degree piecewise polynomial collocation method,our method provides O(hm+1)as the order of the deviation of the error.The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.

Key words Volterra integro-differential;defect correction principle;piecewise polynomial;collocation; finite difference;error analysis

1 Introduction

In this work,we consider Volterra integro-differential(VID)equations as follows

We say F is semilinear if we can write F as follows

Also in this paper,we say z[y](t)is linear if we can write z[y](t)as

where Λ(t,s)is sufficiently smooth in J:={(t,s)|0≤ s≤ t≤ T}.We shall assume that F and K are uniformly continuous in W and S,respectively,where

It is well-known that under the following conditions,VID problems(1.1)–(1.2)have a unique solution y∈C1(I)[1],

where C1,C2and C3are nonnegative and finite constants.

For(1.1)–(1.2),the following conditions can be assumed,

? F is semilinear and z[·]is linear.

? F and z[·]are nonlinear.

? F is nonlinear and z[·]is linear.

? F is semilinear and z[·]is nonlinear.

In this paper,we study the deviation of the error for all of the above conditions.When we use m degree piecewise polynomial collocation method for VID problems,we prove that the order of the deviation of the error is O(hm+1).The piecewise polynomial collocation method for VID problem was studied in[2].Also other methods for the integro-differential equations were studied in refs.[3–6].

The general structure of defect correction was introduced in[7],and the Brakhage’s defect correction for integral equations was studied in[8].The deviation of the error estimation based on piecewise polynomial collocation method was studied in refs.[9,10]for linear and nonlinear second order boundary value problem.

The layout of this paper is organized as follows.In Section 2,piecewise polynomial collocation method, finite differences scheme and exact difference scheme are described.In Section 3,we perform analysis of the deviation of the error for linear and nonlinear cases.In Section 4,we present the results of numerical experiments that demonstrate our findings.A summary is given at the end of the paper in Section 5.

2 Description of the Method

In this section,we describe some details about piecewise polynomial collocation method,finite differences scheme and exact difference scheme.

2.1 Piecewise Polynomial Collocation Method

We give a brief introduction to the use of piecewise polynomial collocation method for solution of the VID equation(1.1)–(1.2).

Step 1 Let

we define the set?nas

also we define hi:= τi+1? τi,h′:=minihiand h:=maxihi.Let

In each subinterval[τi,τi+1],we define collocation points as

Step 2 In each subinterval[τi,τi+1](i=0,···,n ? 1),we define a polynomial as

We define a continuous collocation solution as

Step 3 The unknown coefficients ci,k(k=0,···,m,i=0,···,n ? 1)in(2.5),will be determined by using the following conditions

Definition 2.1 We define(Lagrange polynomials)

Remark 2.2 In Step 3,since always we can not determine exact value for z[p](ti,j),therefore we use the following method to determine z[p](ti,j),

where

Lemma 2.3 For sufficiently smooth f,the following estimate holds

Proof When z[·](t)is nonlinear we can write

where

Therefore by using the interpolation error theorem(see[11,Section 2.1]),we have

Similarly for linear z[·](t)we can prove(2.10). ?

For above collocation method,the following theorem holds.

Theorem 2.4 Assume that the VID problem(1.1)–(1.2)has a unique and sufficiently smooth solution y(t).Also assume that p(t)is a piecewise polynomial collocation solution of degree≤m.Then for sufficiently small h,the collocation solution p(t)is well-defined and the following uniform estimate at least hold

Proof The proof see[2].?

Remark 2.5 By using numerical experiment,we can see that for equidistant collocation gird points with odd m the following uniform estimate hold

2.2 Finite Difference Scheme

In this section,we define

Considering(2.4),we can write a general one-step finite difference scheme as

Definition 2.6 For any function u,we define

also we define

By using Taylor expansions,the following lemma is obtained easily.

Lemma 2.7 If the function f has a continuous first derivative in[xj,xj+1],then there exists a numer ξj∈ [xj,xj+1],such that

Lemma 2.8 For sufficiently smooth f the following estimate holds

where χ[·]i,jis defined by(2.20).

Proof When z[·](t)is nonlinear by using Lemma 2.7,we get

also we can write

and we can say that j≤m.Then we have

Similarly for linear z[·](t),we can prove this lemma. ?

For above finite difference scheme by using Taylor expansion and Lemma 2.8,we have the following estimate

2.3 Deviation of the Error Estimation for VID Equations

In this subsection,by using the defect correction principle,we find the deviation of the error estimation for(1.1)–(1.2).For y′(t)=f(t),0 ≤ t ≤ T,where f(t)is permitted to have jump discontinuities in the points belonging to?n,by using Taylor expansion we obtain

In fact,we find “exact finite difference scheme” for y′(t)=f(t),which is satisfied by the exact solution.Moreover a solution of problem(1.1)–(1.2)satisfies the exact finite difference scheme

We know that the following values in the collocation points are zero,

We define defect at ti,jas follows

In order to compute integral in(2.33),we use quadrature formula.Then we find

where

With standard arguments,for sufficiently smooth f,we can show that the following error holds

In the special case where m is odd and the nodes ρiare symmetrically,we have

Now let π ={πi,j;(i,j)∈ A}be defined as the solution of the following finite difference scheme

We define D:={Di,j;(i,j)∈A?{(n,0)}}.For small value D,we can say

where η is defined in(2.18)–(2.19).We define ε and e as

We remember that an estimate for the error e is given in Theorem 2.4.The deviation of the error is defined as follows

In the next section,we will prove that the order of the deviation of the error estimate for VID equation is at least O(hm+1).

3 Analysis of the Deviation of the Error

3.1 Linear Case

In this subsection,we consider the following linear VID equation with linear z[·](t)defined in(1.5)

where a(t),b(t)are sufficiently smooth in I.

Theorem 3.1 Consider the VID equation(3.1)with initial condition(3.2).Assume that the VID problem has a unique and sufficiently smooth solution.Then the following estimate holds

where e is error,ε is the error estimate and θ is its deviation.

Proof According to the described procedure,we can write

Therefore we have

We write

Then from(3.6)and(3.7),we can get

In this step,we show that S1=O(hm+1)and S2=O(hm+1).We can consider

from Taylor expansion,we have

where ξi∈ [τi,τi+1].By using Theorem 2.4 and(3.11),we can say that S1=O(hm+1).Now we study S2.Similar to Lemma 2.8,we find

By using(3.12),we obtain

where ξi,k,ζk, ζk∈[τi,τi+1].By Theorem 2.4 and(3.13),we have

Therefore we can write(3.8)as

Stability requirements of forward Euler scheme yield the following result

which completes the proof.

3.2 Nonlinear Case

We consider the nonlinear VID equation(1.1)–(1.2).In the nonlinear case we assume that F(t,y,z),Ft(t,y,z),Fy(t,y,z)and Fz(t,y,z)are Lipschitz-continuous.Also when z[·](t)is nonlinear we assume that K(t,s,u)and Ku(t,s,u)are Lipschitz-continuous.

For nonlinear case,according to the presented method,we have

Lemma 3.2 For linear z[·](t)as given in(1.5),we have

Proof By using Lemma 2.8 and Lemma 2.3,we can write

Similarly we can prove(3.19).?

Lemma 3.3 For linear and nonlinear z[·](t),we have

Proof In the first step,we assume z[·](t)is linear.By using Lemma 2.3,Theorem 2.4 and the integral mean value theorem,we get

where ζi,j∈ [0,ti,j].For nonlinear z[·](t)by using Lemma 2.3 we obtain

by using the Lipschitz condition for K we find

which completes the proof.

Lemma 3.4 The defect defined in(2.33)has order O(hm).

Proof We can write

Since p′is a polynomial of degree m ? 1,therefore S1=0.Also according to the definition of collocation solution,we can say that S2=0 at all collocation grid points ti,j.For grid point τi,we have

For S3by using Lipschitz condition and Lemma 3.3,we get

This completes the proof of Lemma 3.4.?

The following lemma is a consequence of the above lemma.

Lemma 3.5 The π ? η has order O(hm).

First,we assume that z[·](t)is linear,i.e.,(1.5),for this case we have the following lemmas and theorem.

Lemma 3.7We have

Proof We can write

where

Similarly we can prove(3.30),(3.31).For(3.32),we have

Lemma 3.8We have

Proof By using Lemma 3.5 and Theorem 2.4,we can write

Theorem 3.9 Consider the VID equation(1.1)with initial condition(1.2),where F(t,y,z),Ft(t,y,z),Fy(t,y,z)and Fz(t,y,z)are Lipschitz-continuous.Also let z[·](t)is linear,i.e.,(1.3).Assume that the VID problem has a unique and sufficiently smooth solution.Then the following estimate holds

where e is error,ε is the error estimate and θ is the deviation of the error estimate.

Proof By using(3.17),we have

We rewrite I1as

where

Also we have

where

Analogously we can write

where

Also we get

where

Also we obtain

In this step by using the Lipschitz condition for Fy,Lemma 3.7 and Lemma 3.8,we find

From relations(3.50)–(3.51),one may readily deduce the expression

Analogously,we can write

By using the Lipschitz condition for Fzand Lemma 3.7,we get

Then by using(3.53)–(3.54),we have

Considering eqs.(3.44)and(3.49),we have

By using(3.12),we can say that

Having used the Lipschitz condition for Fzand(3.57),we get

Therefore we can write

By using Lemma 3.2,(3.57)and(3.59),we find

Based on the above discussion,we can rewrite(3.39)as

From the Lipschitz condition for F and Lemma 2.3,we obtain

Then by using(3.62)–(3.63),we rewrite(3.61)as

Now by using Taylor expansion,we have

We can find

From(1.5),we can get the following result for I5as

Then based on the above discussion,we get

Using stability of forward Euler scheme,we find

which completes the proof.

Now in this step we study nonlinear case with nonlinear z[·](t),i.e.,(1.3).

Lemma 3.10 When z[·](t)is nonlinear then,we have

ProofWe have

from the Lipschitz condition for K and Lemma 3.5,we can write

Similarly,we can prove(3.72).For(3.73)by Lemma 3.3,we get

which completes the proof.?

Definition 3.11For nonlinear z[·](t),let us define χ[ε]i,jand bχ[ bε]i,jby

Lemma 3.12 We have

Proof By using the Lipschitz condition for Ku,we get

Therefore we can see that

then we get the following identity

Lemma 3.13 When z[·](t)is nonlinear then we have

Proof From Lemma 3.12 and Lemma 3.8,one may readily deduce the following result

Theorem 3.14 Consider the VID equation(1.1)with initial condition(1.2),where F(t,y,z),Ft(t,y,z),Fy(t,y,z)and Fz(t,y,z)are Lipschitz-continuous.Also let z[·](t)is nonlinear,i.e,(1.3),where K(t,s,u)and Ku(t,s,u)are Lipschitz-continuous.Assume that the VID problem has a unique and sufficiently smooth solution,then the following estimate holds

where e is error,ε is the error estimate and θ is its deviation.

Proof Similar to Theorem 3.9,we get

by using Lemma 3.10 and the Lipschitz condition for Fy,we have

therefore we can write

Also we get

Having used Lemmas 3.10–3.13 and the Lipschitz condition for Fz,we have

Therefore we write

As Theorem 3.9,we have

where

From the above equations we can obtain

When z[·]is nonlinear,similar to(3.57)we can find

Then by using Lemmas 3.3,3.10–3.13,eq.(3.106)and the Lipschitz condition for Fz,we can get

therefore we can say that

Then we find

Also by using Lemma 3.12 we obtain

We may rewrite eq.(3.88)as

Similar to Theorem 3.9,we can complete the proof.

Similar to the above theorem,we can prove the following theorem.

Theorem 3.15 Consider the VID equation(3.1)with initial condition(3.2).Also let z[·](t)is nonlinear,i.e.,(1.3),where K(t,s,u)and Ku(t,s,u)are Lipschitz-continuous.Assume that the VID problem has a unique and sufficiently smooth solution.Then the following estimate holds

where e is error,ε is the error estimate and θ is its deviation.

4 Numerical Illustration

In this section,in order to illustrate the theoretical results,we consider some test problems.Note that we compute the numerical results by Mathematica-9 programming.

Example 1 Consider the Volterra integro-differential problem

with exact solution y(t)=exp(t2).This example serve to illustrate Theorem 3.1.The numerical results are shown in Table 1 and 2.For this example we choose n collocation subintervals of length 1/n.In Table 1 for this example we choose m=2 and assume that ρi(i=0,···,m+1)are equidistant point.Also,numerical results are shown in Table 2 for m=3 and{ρ0,ρ1ρ2,ρ3,ρ4}={0,0.1,0.55,0.8,1}.

Table 1 Numerical results for Example 1

Table 2 Numerical results for Example 1

with b(t)=1/4+t/2 and exact solution y(t)=exp(t).The numerical results reveal Theorem 3.9.The numerical results are tabulated in Table 3.For this example,we choose n collocation intervals of length 1/n and assume that ρi(i=0,···,4)are equidistant points.

Example 2 In this example,we consider the Volterra integro-differential problem

Table 3 Numerical results for Example 2 with m=3

Example 3 By using this example,we reveal Theorem 3.14.Consider the Volterra integro-differential problem

with b(t)=2exp(2t)+t/16+t2/4 and exact solution y(t)=exp(2t).We can see that F and z[·]are nonlinear.In Table 4,we consider m=4 and assume that τiand ρi(i=0,···,5)are equidistant points.

Table 4 Numerical results for Example 3 with m=4

Example 4 In this example,we study the following VID equation

Table 5 Numerical results for Example 4

Remark 4.1 According to numerical results for Examples 2 and 3,we can see that the rate of convergence is quite slow.For accelerating the rate of convergence,we can choose Chebyshev nodes for τiand ρi.The comparison results for the rate of convergence for different nodes are given in Tables 6–9 for m=4.We solve Examples 2 and 3 for equidistant points τi(i=0,···,n)and ρi(i=0,···,m+1)where numerical results are given in Tables 6 and 8.Also by choosing Chebyshev nodes for τi(i=0,···,n)and ρi(i=0,···,m+1),numerical results for Examples 2 and 3,are tabulated in Tables 7 and 9.

Table 6 Numerical results for Example 2 with m=4

Table 7 Numerical results for Example 2 with m=4 and Chebyshev nodes

Table 8 Numerical results for Example 3 with m=4

Table 9 Numerical results for Example 3 with m=4 and Chebyshev nodes

5 Conclusion

In this paper,the deviation of the error estimation by using piecewise polynomial collocation method for Volterra integro-differential equations is studied.Also we indicated that the order of the deviation of the error estimation is O(hm+1),where m is degree of the piecewise polynomial.In addition,the numerical results confirmed the analytical results.