Stability Analysis of a Numerical Integrator for Solving First Order Ordinary Differential Equation

In this paper, we used an interpolation function to derive a Numerical Integrator that can be used for solving first order Initial Value Problems in Ordinary Differential Equation. The numerical quality of the Integrator has been analyzed to authenticate the reliability of the new method. The numerical test showed that the finite difference methods developed possess the same monotonic properties with the analytic solution of the sampled Initial Value Problems.


Introduction
Many Scholars have derived various Numerical Integrators using various techniques including interpolating functions that include the work of [1] [2] [3] [4] among others.All these authors have employed some analytically continuous functions to create numerically stable Integrators that can be used for ordinary differential equations.In this work we use an analytically differentiable interpolating function to create a one-step Finite Difference scheme for solving Initial Value Problems of first order Ordinary Differential Equations, and we are considering the concept of Nature, of Solutions of first order Ordinary Differential Equations to assume a theoretical solution and use that assumption to derive a discrete model that can be applied to some Ordinary differential equations.
Definition 1 [5] ( ) where F is a real function of its (n + 2) arguments 1 , , , ., n x y y y  1) Let f be a real function defined for all x in a real interval I and having an nth derivative (and hence also all lower ordered derivatives) for all I x ∈ .The function f is called an explicit solution of the differential Equation (1) on interval I if it fulfills the following two requirements.
is defined for all

I
x ∈ , and for all

I
x ∈ .That is, the substitution of ( ) f x and its various derivatives for y and its corresponding derivatives.
2) A relation ( ) , 0 g x y = is called an implicit solution of (1) if this relation defines at least one real function f of the variable x on an internal I such that this function is an explicit solution of (1) on this interval.
3) Both explicit solutions and implicit solutions will usually be called simply Solutions.
We now consider the geometric significance of differential equations and their solutions.We first recall that a real function F(x) may be represented geometrically by a Curve ( ) y F x = in the xy plane and that the value of the derivative of F at x, ( ) , may be interpreted as the slope of the curve ( )

Formulation of the Interpolating Function
Consider the initial value problem of the IVP where η is a discrete variables in the interval [ ] ( ) where 1 2 , α α and 3 α are real undetermined coefficients, and 4 α is a con- stant.

Derivation of the Integrator
We assumed that the theoretical solution ( ) ( ) where 1 2 , α α and 3 α are real undetermined coefficients, and 4 α is a con- stant.
We shall assume n y is a numerical estimate to the theoretical solution ( ) We define mesh points as follows: We impose the following constraints on the interpolating function ( 6) in order to get the undetermined coefficients: 1) The interpolating function must coincide with the theoretical solution at n x x = and Hence we required that ( ) ( ) 2) The derivatives of the interpolating function are required to coincide with the differential equation as well as its first, second, and third derivatives with respect to x at n x x = .
We denote the i-th derivatives of ( ) This implies that, Solving for 1 2 , α α and 3 α from Equations (11) (12) and (13), we have and Then we shall have from ( 8) and ( 9) into (17 [ ] Recall that n x a nh = + , ( )  Journal of Applied Mathematics and Physics Substitute (14) (15) (16), into (18), and simplify we have the integrator ( ) for solution of the first order differential equation.x y h ∅ is the increment function.

Properties of the Integration Method
The numerical integrator can be expressed as a one-step method in the form (20) above thus: ( ) Put ( 21) into (19), then expand

n n y y f h f x h f h f x h f h
Thus our integrator (19) can be written compactly as Which is in the form ( ) x y h in the region just defined.Then the relation (28) is the Lipscitz condition and it is the necessary and sufficient condition for the convergence of our method (19).
We shall proof that (19) satisfies (28) in line with the established Fatunla's theorem.

Proof of Convegence of the Integrator
The increment function ( ) , ; n n x y h ∅ can be written in the form , ; , , , where A and B are constants defined below.
Consider Equation (29), we can also write Let y be defined as a point in the interior of the interval whose points are y and * y , applying mean value theorem, we have We define , and sup Taking the absolute value of both sides

Consistence of the Integrator Definition 3 [8]
The integration scheme: ( ) The significance of the consistency of a formula is that it ensures that the method approximates the ordinary differential equation in its place.
Therefore from It is a known fact that a consistent method has order of at least one [9].Therefore, the new numerical integrator is consistent since Equation (36) can be reduced to (37) when 0 h = .

+≥
x to the initial value problem(5) can be locally represented in the interval [ ] by the non-polynomial interpolating function; ,