Journal of Applied Mathematics and Physics
Vol.05 No.11(2017), Article ID:80336,9 pages
10.4236/jamp.2017.511179
Stability Analysis of a Numerical Integrator for Solving First Order Ordinary Differential Equation
Samuel Olukayode Ayinde1*, Adesoji Abraham Obayomi1, Funmilayo Sarah Adebayo2
1Department of Mathematics, Faculty of Science, Ekiti State University, Ado Ekiti, Nigeria
2Federal Polytechnic, Ado Ekiti, Nigeria
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: October 6, 2017; Accepted: November 13, 2017; Published: November 16, 2017
ABSTRACT
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.
Keywords:
Numerical Integrator, Autonomous and Non-Autonomous, Ordinary Differential Equation, Initial Value Problems, Stability Analysis
1. 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]
Consider the nth-order ordinary differential equation
(1)
where F is a real function of its (n + 2) arguments
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 . The function f is called an explicit solution of the differential Equation (1) on interval I if it fulfills the following two requirements.
(2)
is defined for all , and
(3)
for all .
That is, the substitution of and its various derivatives for y and its corresponding derivatives.
2) A relation 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 in the xy plane and that the value of the derivative of F at x, , may be interpreted as the slope of the curve at x.
2. Formulation of the Interpolating Function
Consider the initial value problem of the IVP
(4)
where is a discrete variables in the interval . In this we consider the method based on local representation of the theoretical solution .
Let us assume that the theoretical solution to the initial value problem 4) can be locally represented in the interval by the non-polynomial interpolating function given by:
(5)
where and are real undetermined coefficients, and is a constant.
3. Derivation of the Integrator
We assumed that the theoretical solution to the initial value problem (5) can be locally represented in the interval by the non-polynomial interpolating function;
(6)
where and are real undetermined coefficients, and is a constant.
We shall assume is a numerical estimate to the theoretical solution and .
We define mesh points as follows:
(7)
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 and . Hence we required that
(8)
(9)
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 .
We denote the i-th derivatives of with respect to x with such that
(10)
This implies that,
(11)
(12)
(13)
Solving for and from Equations (11) (12) and (13), we have
(14)
(15)
and
(16)
Since and
Implies that and
(17)
Then we shall have from (8) and (9) into (17)
(18)
Recall that , with
Substitute (14) (15) (16), into (18), and simplify we have the integrator
(19)
for solution of the first order differential equation.
4. Properties of the Integration Method
4.1. Qualitative Properties of the Scheme
4.1.1. Definition 2 [6]
Define any algorithm for solving differential equations in which the approximation to the solution at the point can be calculated if only and h are known as one-step method. It is a common practice to write the functional dependence, , on the quantities and h in the form:
(20)
where is the increment function.
The numerical integrator can be expressed as a one-step method in the form (20) above thus:
From (19) i.e.
Expanding into the fourth term, we have
(21)
Put (21) into (19), then expand
(22)
(23)
Let and (24)
Thus our integrator (19) can be written compactly as
(25)
Which is in the form
(26)
where (27)
4.1.2. Theorem 1. [7]
Let the increment function of the method defined by (25) be continuous as a function of its arguments in the region defined by
where , and let there exists a constant such that
(28)
for all and 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.
4.1.3. Proof of Convegence of the Integrator
The increment function can be written in the form
(29)
where A and B are constants defined below.
and
Consider Equation (29), we can also write
(30)
Let be defined as a point in the interior of the interval whose points are and , applying mean value theorem, we have
(31)
We define
Therefore
(32)
Taking the absolute value of both sides
(33)
If we let
then our Equation (33) turns to
(34)
which is the condition for convergence.
4.2. Consistence of the Integrator
Definition 3 [8]
The integration scheme: is said to be consistent with the initial-value problem , , , provided the increment function satisfies the following relationship
(35)
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
(36)
where then
and
If , then (36) reduced to
(37)
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 .
4.3. Stability Analysis of the Integration Method
We shall establish the stability analysis of the integrator by considering the theorem established by Lambert 1972.
Let and denote two different numerical solutions of initial value problem of ordinary differential Equation (35) with the initial conditions specified as and respectively, such that , . If the two numerical estimates are generated by the integrator (19). From the increment function (26), we have
(38)
(39)
The condition that
(40)
is the necessary and sufficient condition that our new method (19) be stable and convergent.
Proof
From (27) we have
(41)
Then let
(42)
and
(43)
Therefore,
(44)
Applying the mean value theorem as before, we have
(45)
(46)
(47)
Taking absolute value of both sides of (47) gives
(48)
Let and , , given , then
(49)
and
, for every (50)
Then we conclude that our method (19) is stable and hence convergent.
5. The Implementation of the Integrator
Example 1
Using the Integrator (19) to solve the initial value problem
The analytical solution
Example 2
Consider the initial value problem
The analytical solution
6. Summary and Conclusion
In this paper, we have proposed a new integration for the solution of standard initial value problem of first order ordinary differential equations. The new method was found to be convergence, consistence, and stable.
Cite this paper
Ayinde, S.O., Obayomi, A.A. and Adebayo, F.S. (2017) Stability Analysis of a Numerical Integrator for Solving First Order Ordinary Differential Equation. Journal of Applied Mathematics and Physics, 5, 2196-2204. https://doi.org/10.4236/jamp.2017.511179
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