Using Runge–Kutta method for numerical solution of the system of Volterra integral equation
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:
- (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]:According to initial condition f(0)=γ we have
Section snippets
Runge–Kutta method for second kind of integral equations
Suppose that interval [0,T] has divided into subinterval as long asIntegral equations (ii) can be written as the following form:That Fn called lag-term, it can be written as the following form:And soThen 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 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 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 is orthogonal to the column-subspace spanned by 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 . In this case, we need the vectors of to be linearly independent.
If the formula is to use no more than six stages, then rank (e)=2. Lemma 5 A Bel’tyukov pair of orders 3 and 4 must have at least six stages. 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 aswhere 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 thatis 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
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