Numerical solving initial value problem for Fredholm type linear integro-differential equation system

https://doi.org/10.1016/j.jfranklin.2009.03.003Get rights and content

Abstract

In this paper, we introduce a numerical method for solving initial value problems for a system of linear integro-differential equations. The main idea is based on the interpolations of unknown functions at distinct interpolation points. We next use Clenshaw–Curtis quadrature formulae required in the approximation of the integral equations. The technique is very effective and simple. In the end, to show the efficiency of this method, we present some numerical examples.

Introduction

Integro-differential equations (IDE) arise in many branches of science, for example in control theory and financial mathematics. IDEs are important, but they are hard to solve even numerically, so the progress on how to solve them is slow. IDEs are equations of the form where the unknown function appears under the sign of integration and they also contain the derivatives of the unknown function.

The books edited by Green [14] and Kanwal [21] contain many different methods to solve the integral equations analytically. Numerical methods also take an important place in solving the integral equations (see [1], [12], [15], [42]).

In [27], Legendre wavelets on the interval [0,1) are introduced which reduced the solution of linear Fredholm and Volterra integral equation of the second kind to the solution of linear algebraic equations. Implicit and explicit forms of the Pouzet Volterra Runge–Kutta methods to obtain the numerical solution of a system of Volterra integral equation have been introduced in [29].

Chebyshev series solutions of linear integral and integro-differential equations and their systems have been considered in [3], [4], [38].

Compact finite difference method for second- and first-order IDEs as well as for systems of IDEs has been considered by Zhao and Corless [40].

Two methods for the numerical solution of a Fredholm IDE modelling neural networks have been proposed in [19]. The first approach is based on approximating the integral term by Gauss–Hermite or Gauss Laguerre quadrature formula with a linear interpolation to compute the approximation to the solution between the grid points in time. The second approach is based on representing the solutions in a linear combination of Lagrange's fundamental polynomials with coefficients depending on time.

Tau method to find numerical solutions of the Fredholm, Volterra and Fredholm–Volterra IDEs with standard, Chebyshev and Legendre bases is considered in [16], [17], [18], [32], [37].

A mechanization of solving Fredholm–Volterra IDEs has been proposed in [41]. The bases of mathematical mechanization are algorithm establishing and programming techniques. The procedure presented in this work provides Taylor polynomial solutions of the IDEs. Also, Taylor polynomial solutions of high-order nonlinear Fredholm–Volterra IDEs have been considered in [28].

In paper [22], [39], a matrix method called the Taylor collocation method is presented for a numerical solution of the linear IDEs by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation into a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients.

Wavelet–Galerkin method for IDEs is presented in [2]. Rationalized Haar functions method for IDEs is considered in [24], [25].

Hybrid Legendre and block-pulse functions on the interval [0,1) to solve a linear IDE system is considered in [26]. Tau method for the numerical solution of systems of IDEs is considered in [33].

Adomian decomposition method for solving a system of integral equations is considered in the literature, for example [5], [6], [7], [9], [43], [44]. The initial value problem for an IDE has been solved in [43].

In papers [30], [31], [34], [35], [36] methods for the numerical solution of integral and IDEs have been proposed. The suggested methods are based on representing the solutions in the linear combination of Lagrange's fundamental polynomials and on approximating the integral term by the Clenshaw–Curtis quadrature formula.

In this paper, we introduce numerical expansion methods for solving a system of linear IDEs by interpolation and Clenshaw–Curtis quadrature rules (cf. [10], [11], [12], [13], [30]). Chebyshev polynomials are important in approximation theory, and Gauss quadrature appears in the theory of numerical integration. This method transforms the integral system into a matrix equation with the help of interpolation points. They are easily acquired on using the transformed matrix equation, which corresponds to the system of linear algebraic equations. As a result, the Lagrange interpolation solution is obtained.

Our focus here is on the system of Fredholm type IDEs. The basic ideas of the previous works are developed and applied to a system of high-order linear Fredholm type IDEsk=1mj=0npik(j)(x)yk(j)(x)+k=1mj=0nabqik(j)(x,t)yk(j)(t)dt=fi(x)with the initial conditionsyk(0)(a)=βk0,yk(1)(a)=βk1,,yk(n-1)(a)=βk(n-1),where pik(j)(x) (i,k=1,2,,m; j=0,1,,n), fi(x) (i=1,2,,m) and qik(j)(x,t) (i,k=1,2,,m; j=0,1,,n) are known functions having nth (nm) derivatives on an interval ax, tb, and a, b and βki (k=1,2,,m;j=0,1,,n-1) are appropriate constants, yk(j)(x) (k=1,2,,m;j=0,1,,n) denote the express of the jth derivatives of the unknown functions yk(x), respectively.

We suppose that systems (1.1), (1.2) have a unique solution. However, the necessary and sufficient for a unique solution for Eqs. (1.1), (1.2) could be found in [1], [8], [11], [14], [20], [21], [23].

Section snippets

Clenshaw–Curtis quadrature

Clenshaw–Curtis quadrature rule is represented by means of the relation-11g(t)dt=k=0Nwkg(tk),where the double prime on the summation symbol here and elsewhere indicates that the terms with suffixes k=0 and N to be halved, and the points {tk} are (see [10], [11], [12], [30], [34], [35]) Chebyshev collocation pointstk=coskπN,wk=4Ns=0NvscosskπN,vs=11-s2if s even,0if s odd.

Cauchy formula

Theorem 1

If F(x,y) and F/y are continuous in x and y on the closed interval [a,b], and A(x) and B(x) are continuous, thenddxA(

General technique by using interpolation

We know that the Lagrange interpolation in points xi,i=0(1)n, for the function f(x) isf(x)i=0nLi(0)(x)f(xi),whereLi(0)(x)=k=0ksnx-xkxs-xk,x[a,b],fi=f(xi),i=0(1)n.Therefore we haveaxf(t)dtaxi=0nk=0kint-xkxs-xkf(xi)dt=i=0nf(xi)axk=0kint-xkxi-xkdt=i=0nf(xi)Li(1)(x),where Li(1)(x)=axk=0kint-xkxi-xkdt(i=0,1,,n).Following the previous subsection, we can writeLi(1)(x)=x-a2-11k=0kinx-a2t+x+a2-xkxi-xkdt=x-a2s=0Nwsk=0kinx-a2ts+x+a2-xkxi-xk.

Now, from Eq. (2.12) we can writeaxax

Method of solution for the system of Fredholm type integro-differential equations

Let yk(n)(x) has the Lagrange expansionsyk(n)(x)s=0nyk(n)(xs)ls(x)=s=0nyks(n)ls(0)(x),axb,yk(n-1)(x)yk(n-1)(a)+s=0nyks(n)axls(0)(t)dt=yk(n-1)(a)+s=0nyks(n)ls(1)(x),axb,ls(1)(x)=axls(0)(t)dt,yk(n-2)(x)yk(n-2)(a)+yk(n-1)(a)(x-a)+s=0nyks(n)axaxls(0)(t)dtdt=yk(n-2)(a)+yk(n-1)(a)(x-a)+s=0nyks(n)ls(2)(x),axb,ls(2)(x)=ax(x-t)ls(0)(t)dt.Next, we haveyk(j)(x)yk(j)(a)+yk(j+1)(a)(x-a)++yk(n-1)(a)(x-a)n-j-1(n-j-1)!+s=0nyks(n)ls(n-j)(x)=r=jn-1yk(r)(a)(x-a)r-j(r-j)!+s=0nyks(n)ls(n-j)(

Numerical experimentation

In order to show the numerical efficiency, we consider three problems.

Example 2

Consider the following initial value problem of linear Fredholm type integro-differential equations with the exact solutions (see, [33]): y1(x)=3x2+1 and y2(x)=x3+2x-1.y1(x)+y2(x)+012xt(y1(t)-3y2(t))dt=3x2+3x10+8,y1(x)+y2(x)+013(2x+t2)(y1(t)-2y2(t))dt=21x+45,y1(0)=1,y1(0)=0,y2(0)=-1,y2(0)=2and for n=2 approximate the solution in the form (4.8).

We take the interpolation points as x0=0, x1=12 and x2=1. We then haveyi0=y

Conclusions

  • 1.

    The presented method gives the solution at the points xj=(b-a)cos(jπ/n)+(b+a)/2 , j=0(1)n or xj=a+jh, j=0(1)n, h=b-a/n.

  • 2.

    The suggested methods can be applied to a system of Fredholm type integro-differential equations (see Examples 1–3).

  • 3.

    The suggested methods can be applied to a system of Volterra type integro-differential equations.

  • 4.

    Comparison of the results obtained from this method with the Tau method (see Examples 1 and 2) and rationalized Haar functions method (see Example 3) indicate that the

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