New algorithm for problems with exponential solutions

In this paper, we consider the development of algorithms for the solution of first order initial value problems whose solution exhibits exponential behaviour. Method of interpolation and collocation of basis function to give system of nonlinear equations which is solved for the unknown parameters to give a continuous scheme. The discrete methods recovered from the continuous scheme are implemented in block form. The stability properties of the method is verified and numerical experiments show that our method is effi cient in handling these problems.


Introduction
In this paper, we develop an exponentially …tted two step, hybrid numerical integrator for initial value problems (IVPs) of …rst order di¤erential equations in the form y 0 = f (x; y) ; y (x 0 ) = y 0 ; a x b (1.1) where y : The Jacobian arising from (1:1)vary slowly and the eigenvalues have negative real part; moreover the solution is decaying or exhibit a pronuounced exponential behaviour.
Classical general purpose method developed using …nite power series basis function cannot produce satisfactory results due to the special nature of the problems.Such problems are found in the modeling of disease outbreak, war, radioactive decay, di¤usion process in biology and chemical reactions.Several scholars have developed exponentially …tted methods are the best methods for (1:1) among them are [1][2][3][4][5][6][7][8] .

Corresponding author
Email address: torlar10@yahoo.com,oziohumat@gmail.com,remoodekunle@gmail.com The paper is organize as follows: section 2 discusses mathematical background and spec-i…cation of the method.Section 3 discusses the stability properties of the developed block method which include convergence and stability region of the developed method.Numerical experiments are shown in section 4 and we concluded in section 5.

Methodology
We consider the approximate solution where a j and b' j s are constants to be determined.We seek approximation at an equidistant set of points de…ned by the the integration interval a Interpolating and collocating (2:1) at x n+i ; i = 0; 1; ; k 1; give where k = r + s 1: We then impose the following condition on y (x) in (2:1) where r and s are the numbers of interpolation and collocation points respectively.Solving (2:2)using Crammer's rule, substituting the result into (2:1) and after some algebraic simpli…cations gives the continuous method which when evaluated at selected grid points gives discrete methods.
In this paper, we consider interpolation at x = x n and collocation at x = x n+i ; i = 0; 1 2 ; 1; 2. Solving the resulting systems of equation gives the continuous method where y (x) is continuously di¤erentiable of [a:b] : Writing (3:1) as a Taylor series expansion about x to obtain where the constant coe¢ cient c p ; p = 0; 1: are given as  = ty 0 n (ii) it has order (p) 1 De…nition 3.3 A method is said to be zero stable if lim h!0 (y n+t ) ! y n De…nition 3.4 A Method is said to be convergent if it is consistent and zero stable.De…nition 3.5 The Region of absolute stability (RAS) of a method is the set f h : for that h where the root of the stability polynomial are absolute less than oneg In this paper, the boundary locus method is used to plot RAS.This is done by substituting the test equation y 0 = y into the equation to obtain where (r) and (r) are the …rst and second characteristic polynomials respectively.writing r = e i gives RAS as The e¢ ciency of the developed method is tested using …ve test problems with n = 300 Problem 4.1 We consider the linear system in the range 0 x 1 Source: [2].The eigenvalues of the Jacobian matrix are 1 = 2; 2 = 96 with the sti¤ness ratio 48.The exact solution is given as y (x) = 1 47 (95e 2x 48e 95x ; 48e 96x e 2x ) T .
The results is shown in

Conclusion
We have discussed the construction of order four hybrid method for sti¤ …rst order IVPs.The method is found suitable for problems whose solution exhibit exponential behaviour or decaying as demostrated in the numerical experiments.

Fig 1 :
Fig 1: Region of absolute stability of the method

Table 1 :
Order and LTE