Using Runge–Kutta method for numerical solution of the system of Volterra integral equation

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Abstract

In this paper we introduce some of general methods of Runge–Kutta, they are using for numerical solution of Volterra integral equations of the second kind [H. Brunner, P.J. Vander Houwen, The numerical solution of Volterra Equation, CWI monograph 3, North-Holland, Amsterdam, 1986; P.W. Sharp, J.H. Verner, Extended explicit Bel’tyukov pairs of orders 4 and Volterra integral equations of the second kind, Appl. Numer. Math. (from author’s Homepage)]. We first introduce Pouzet Volterra Runge–Kutta methods (implicit and explicit forms) PVRK, and then explain about some extended explicit Bel’tyukov pairs, EBVRK.

Each pair uses six stages and consists of on order 3 formula completely embedded in order 4 formula. In numerical examples this method uses for solving a system of Volterra integral equations and comparison with iterative methods (as [Comput. Math. Appl. 37 (1999) 1; Nonlinear Stud. 6 (1) (1999)]).

Introduction

Volterra integral equation is an equation which integrates on interval [a,s]. The general forms of Volterra integral equation of the first, second and homogenous kind, defined as the following:h(s)f(s)=g(s)+λ∫ask(s,t)f(t)dt.

  • (i)

    If h(s)=0 then we have the first kind of Volterra integral equations.

  • (ii)

    If h(s)=1 then we have the second kind of Volterra integral equations.

  • (iii)

    If h(s)=1 and g(s)=0 we have the homogenous kind of integral equations.


In this paper we discuss about the second kind of Volterra integral equations.

Note. We can easily get the second kind of Volterra integral equations if the differential equation is as the following form [5]:f(x)=F(x,f(x))(a⩽x⩽s).According to initial condition f(0)=γ we havef(x)=∫asF(y,f(y))dy+γ.

Section snippets

Runge–Kutta method for second kind of integral equations

Suppose that interval [0,T] has divided into subinterval as long astn=nh(n=0,1,…,N).Integral equations (ii) can be written as the following form:y(t)=Fn(t)+∫tntk(t,s,y(s))ds,t∈[tn,T].That Fn called lag-term, it can be written as the following form:Fn(t)=g(t)+∫0tnk(t,s,y(s))ds,n=0,…,N−1.And son(t)=∫0tnk(t,s,y(s))ds,t∈[tn,T],n=0,…,N−1.Then ψn(t) is called exponential function. Runge–Kutta method for Eq. (1) is based on two approximation, of course they stand independently.

  • (i)

    One approximation for

Order condition for RK method

p-Series theorem is used to find the order condition therefore, it is necessary to go through the main results of this theorem. The coefficients are included in Butcher modified in which A is an strictly lower-triangular matrix, and each of b,b̂,c,d is an s-component column vector. For ease, can D which are diagonal matrices c=Ce and d=De are used (e=[1,…,1]t).

It is known that two formulas of a Bel’tyukov pair will have order p−1 and p if A,b,b̂,c,d are selected to satisfy a set of algebraic

Order 4 formula

Theorem 1

An s-stage Bel’tyukov method for solving (1) which satisfies Ae=Ce, has order 4 if and only if the row-subspace spanned by l=bt,bt(2c−I),bt(2A−I),bt(D−I) is orthogonal to the column-subspace spanned by R=(D−I)(2D−I)e,(D−I)(2c−D)e,q[2,0],q[2,1],(6c2−6c+I)e and le=(1,0,0,0).

For proof of this theorem see [3].

In developing the results, we use L to denote the matrix whose rows are the vectors of G and we use R to denote the matrix whose columns are the vectors of R.

Lemma 3

Suppose a Bel’tyukov method has

Pairs of orders 3 and 4

Let we want to construct pair of methods of orders 3 and 4 for which b̂≠b. In this case, we need the vectors of λ̂={bt,bt(2c−I),bt(2A−I),b̂t} to be linearly independent.

If the formula is to use no more than six stages, then rank (e)=2.

Lemma 5

  • (a)

    A Bel’tyukov pair of orders 3 and 4 must have at least six stages.

  • (b)

    For each six-stage Bel’tyukov method of order 4 with R of rank 2, there is an embedded method of order 3.

According to this lemma exist a group of at least six-stage of orders 3 and 4.

Theorem 2

One parametric

Selection of an individual pair

The performance of a pair from the new family of Theorem 2 depends primarily on the size of the principal error coefficients and of the stability region of order 4 formula.

The principal error coefficients are defined ase(ti)=β(ti)qiγ(ti)∑k=1s(bkϕk(ti))−1,tiTV5,where TV5 is the set of Volterra trees order 5, β(ti) and γ(ti) are positive integers, and ϕk(ti) is an elementary weight involving A, C, D, c and d.

We want to choose values of the parameters so thatT5=maxtiTV5{|e(ti)|}is as small as

Numerical experimentation

Here, we present some numerical results for two system of Volterra integral equations. This systems solved by Runge–Kutta method and Adomain decomposition method. we consider two test problems. The first one is linear and the second is nonlinear. The numerical results shown in Table 1, Table 2 are the approximate solution of Example 1, Example 2. The column heading eRK and eAd have the following meaning:

  • eRK: Actual error for the Runge–Kutta method.

  • eAd: Actual error for the Adomains method.


All

Conclusions

The numerical results of Runge–Kutta method of solving linear and non-linear Volterra equation system of the second kind indicate that this method is appropriate one for solving such systems.

Comparison of the conclusion obtained from this method with such iterative methods as Adomain (see [6], [7], [8]) indicate that after a few repetitions. Iterative methods present comparatively better results. Since such method are very sensitive to initial values it recommended that the methods proposed in

References (8)

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