K-Step Rational Runge-Kutta Method for Solution of Stiff System of Ordinary Differential Equations

Abstract: This study described the development, analysis and implementation of K-step implicit rational Runge-Kutta schemes for solution of stiff system of ordinary differential equations. Its development adopted taylor and binomial series expansion techniques to generate its parameters. The analysis of its basic properties adopted dalhquist a-stability model test equation and the results showed that the scheme was a-stable, consistent and convergent. Numerical results showed that the method was accurate and effective.

where, E(x) is continuously differentiable, λ is a complex constant with Re(λ) <<0, with the exact solution: x o y(x) = E(x) +y e λ Whose transitory phase is the entire interval of integration 0 ≤ ×≤10π with 50π complete oscillation per unit interval is an ODEs possessing these types of properties are called stiff oscillating ODEs.
Most of the conventional Runge-Kutta schemes cannot effectively solve them because they have small region of absolute stability. This perhaps motivated [8] to introduce a rationalized Runge-Kutta scheme of the form confirm existing phases in samples according to Emmanualson [4] . Where: Since method possesses adequate stability property for solution of stiff ODEs, the papers consider the extension of the scheme to a general step process so that is can serve as a general purpose predictor for multistep schemes: Where: In the spirit of Ademiluyi and Babatola [1] the scheme is classified into: • Explicit if the constraints (10) is such that a ij = 0 for j ≥ i • Semi-implicit if a ij = 0 for j > i • Implicit if a ij ≠ 0 for at least one j > i

MATERIALS AND METHODS
Derivation of the Method: In this research, the parameters Vi, Wi, Ci, di, a ij , b ij are to be determined from the system of non-linear equations generated by adopting the following steps: • Obtained the Taylor series expansion of Ki's and Hi's about point (x n , y n ) for i = 1(1)R • Insert the series expansion into (10) • Compare the final expansion with the Taylor series expansion of y n+1 about (x n ,y n ) in the power series of h The number of parameters normally exceeds the numbers of equations, but in the spirit of [9] , Gill [7] and Blum [3] , these parameters are chosen as to ensure that (the resultant computation method has: • Adequate order of accuracy of the scheme is achieved • Minimum bound of local truncation error • Large maximize interval of absolute stability • Minimum computer storage facilities One-step one-stage schemes: By setting M = 1 and R = 1, in Eq. 11 the general one-step one-stage scheme is of the form:  The binomial expansion theorem of order one on the right hand side of (10) yields: s) order term (higher H V y k W y y Similarly the Taylor series expansion of K 1 about (x n y n ) is: Collecting coefficients of equal powers of h, Eq. 23 can be rewritten in the form: where, Where: Substitute (28) into (26), we obtained: Using (25) and (26) Comparing the coefficients of the powers of h in Eq. 22 and 30, we obtained: Since A 1 = f n N 1 = 2 n n y f − , Eq. 31 yields: Similarly from coefficients of h 2 in Eq. 22 and 30, we have: Also from (22) and (31) we obtained: Putting Eq. 35 and 32 together, we have a system of non linear simultaneous equations: Substituting these values in Eq. 14 we obtain a family of one-step, one stage schemes of the form: Where: K 1 = hf (x n + 1/3 h, y n + 1/3 K 1 ) H 1 = hg(x n + 7/12 h, x n + 7/12 H 1 ) The basic properties of the method: The basic properties required of a good computational method for stiff ODEs includes consistency, convergence and stability and A-stability.

Consistency:
A scheme is said to be consistent, if the difference equation of the computation formula exactly approximate the differential equation it intends to solve [2] . To prove that Eq. 11 is consistent. Hence the method is consistent.

Convergence:
Since the proposed scheme is one -step, the numerical scheme (11) for solving ODEs (1) is said to be convergent, and it is consistent, by Lambert (1973) when it is applied to initial value problem (1) generated a corresponding approximation y n which tend to the exact solution y(x n ) as n approaches infinity, that is: y n → y(x n ) as n→ ∞ Let e n+k and T n+k denote the discretization and Truncation errors generated by (1) where, ψ 2 (x n (y(x n );h) are assumed to be continuous functions in the domain: a ≤ x ≤b, y < ∞, 0≤ h ≤ h o defined as: This Eq. 54 modifies into: where, L and K are the Lipschitz constant for φ 1 (x,y;h) and ψ 2 (x,y;h) respectively and: By adopting this theorem on convergence of sequence of real numbers quoted without proof from [7] , that is: If {e j , j = o(1) n+k } be set of real number. If there exist finite constants R and S such that: By taking the absolute value of (55) on (66) both sides Eq. 46 into consideration, we have: Stability Properties: To analyse the stability property of this schemes, apply scheme (1) to Dalhquist [4] stability scalar test initial value problem: Where: To illustrate, this we consider the one-step, onestage scheme: with stability function: For example, to analyze scheme (74) will introduce a convergent and stable approximation to the solution of stability function if: That is (-∞ <x < o), the scheme is A-stable because the interval of absolute stability is (-∞, o).

RESULTS AND DISCUSSION
In order to access performance of the schemes the following sample problems were solved, with the schemes adopting Fehlberg [6] approach. In the theoretical interval 0≤x≤1. Its general solution is: x ( y Its numerical solution is found in Table 1.

Problem 2:
The second sample problem considered is the stiff system of initial values problems of ODEs below: The results are shown in Table 2. Table 1: Numerical result of k-step implicit rational runge-kutta schemes for solving stiff systems of ordinary differential equations