Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 16, 779 - 787

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijcms.2013.3791

Approximate Solution of Nonlinear Ordinary

Differential Equation for Cauchy Problem

Based on Linearization

K. S. Al-Basyouni

Department of Mathematics, Science Faculty

King Abdulaziz University, Saudi Arabia

kalbasyouni@kau.edu.sa

Copyright © 2013 K. S. Al-Basyouni. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract

This paper aims to find an approximate solution for the nonlinear differential equation

of first order Cauchy problem. To solve this problem we are going to study one

variant of linearization method, in which the nonlinear terms are discarded or are

linearzed with the help of special construction to find solution at the initial moment of

time, or at any bounded point. This method is proved by the help of the estimation of

closeness of the exact and linearized solutions by (uniform, integral and step by step

procedures). This estimation has important value in this work.

Keywords: Linearization, Cauchy problem, Lipchitz constant l, Initial value,

Estimation

780 K. S. Al-Basyouni

1. Introduction

This study is devoted to an exposition and justification of a new method for solving

nonlinear differential equations. This method is based on the replacement of the

original equation by linear equation with the same initial and boundary conditions.

Moreover the originality of this method is in the method of constructing linearization.

Existing methods of linearization, omit nonlinear terms, replacing them by linear

segments Taylor series in the neighborhood of the initial value, lead to approach the

desired solution the upper side or lower side. Therefore within the framework of

linear models it is difficult to a priori take this solution into the fork. The linearization

proposed effectively realizes in practice in the case of first order equations because of

the presence known formula for the solution of the linear Cauchy problem. In the

concrete examples shown, it is possible to consider the global properties of the

desired solution in the presence of special points, the periodicity and so on. In this

paper the estimation of proximities of that desired solution to its linearized

counterpart, depending on the deviation of the linearized solution from the value at

initial point were obtained. This mater have many applications and attracted the

attention of many researchers such as [1-3]. Lions [4] investigated some methods for

the solution of nonlinear boundry problems. Boyce and. Dipriman [5] investigated

elementary differential equations and boundary value problems. Drainville

and.Bedient [6] discussed elementary differintal equations. Goristski and Kaujkove

[7] discussed first order quazilinear equations with partial derivatives. Durikovich [8]

studied on the solution of nonlinear initial-boundary value problems. Yuki

and.Satoshi [9] studied on the existence of multiple solutions of the boundary value

problems for nonlinear second-order differential equations. Adomain and Roch [10]

investigated on the solution of nonlinear differential equations with convolution

product nonlinearities. He. Huan [11] investigated approximate solution of nonlinear

differential equations with convolution product nonlinearities. Kuzenkov [12] studied

the Cauchy problem for a class of nonlinear differential equations in a Banach space.

Yaming and Rivera [13] studied blow-Up of solutions to the Cauchy problem in

nonlinear one-dimensional thermelasticity. Bulychev [14] studied method of the

reference integral curves of the solution of the problems of Cauchy for the ordinary

differential equations.

In this paper, we discuss an approximate solution for the nonlinear differential

equation of first order Cauchy problem. This method is proved by the help of the

estimation of closeness of the exact and linearized solutions by (uniform, integral and

step by step procedures). This estimation has important value in this work.

Approximate solution of nonlinear ODE 781

2. The linearization of the nonlinear Cauchy problem

Let D be a bounded domain in the plane X Y , 0 0,x y - is an interior point.

Consider the following Cauchy problem in D : , , ,y f x y x y D

0 0y x y , (1)

where ,f x y is continuous and bounded in D and uniformly with respect to x

and satisfies the condition of Lipchitz with constant :

, , , , , ,f x y f x y y y x y D x y D (2)

Under these assumptions, the problem (1) has a unique solution on the segment of

Piano , which can be extended into a broader area, and the problem (1) equivalent to

solving a nonlinear integral equation: 0

0 ,x

xy x y f y d (3)

3. The solution of the problem

On the basis of Picard method of successive approximations the desired solution

y x of the problem (1) and (3) is the uniform limit ny x with respect to x :

lim nn

y x y x

, (4)

where:

0

0 1, , 1

x

n n

x

y x y f y d n

(5)

0 0y x y Const

From (5) results 0 0y x , that is 0y x which is obtained as a result of neglecting

in the original equation (1) nonlinear term (classical linearization). We turn now to the

problem (1). Assume that the initial value of the solution 0y x is different from zero

0 0y . This restriction does not have a significant character, because we can always

come to this condition by introducing new unknown functions y x x A ,

where 0A . x

1' , ,x f x x f x x , 0 0x

782 K. S. Al-Basyouni

Linearization of the original problem (1) will be implemented on the basis of such a

problem:

y k x y

, (6)

where: 0 0

0

0

,f x yk k x

y

(7)

By the method of separation of variables we have:

0

0 exp

x

x

y y k x dx

(8)

Now we estimate the proximity of the original and the linearized solution of the

Cauchy problems, for which we denote by Z x the difference between them:

(9)

From (1) and (6) we have:

,f x y k x y ,

0 0x

Therefore:

(10)

Which yields:

(11)

For the integrand (11), we have this identity

0

0 0

0

,, , , , ,

f x yf x y k x y f x y f x y f x y f x y y y

y

(12)

Then from (12), (2) imply the inequality:

0,f x y k x y y y k y y (13)

where: | ( )| |

( )

| (14)

Introducing (13) in (11), we get:

……………………………

…………………………..(15)

Where:

0 0

0

,

xt x t x

x

z x f t y t k t y t e e dt

0

,

x

x

x f t y t k t y t dt

Z x y x y x

0c c cz h z h k y y

Approximate solution of nonlinear ODE 783

maxc

z z x (16)

0 0maxc

y y y x y

Assume that the constants and h satisfy:

1h (17)

Then from (15) immediately implies the following inequality:

0 , 1

1c c

h kz y y h

h

(18)

Inequality (18) presents the desired evaluation of the proximity of solutions of

problems (1) and (6).

We show how to get rid of condition (17). To this

purpose, we rewrite (10) in this form:

(19)

Where is arbitrary as long as parameter, 0const .

We denote:

0maxx x

z x Z x e

(20)

Taking into account (13) and (20), we have from (19):

0

0

x xe

z x z k y y

Whence:

0z k y y

(21)

Let's choose now parameter so that the condition was satisfied:

Then:

0 ,k

z y y

(22)

It is also a required inequality, which is free from restriction on h in the interval of

change independent variable x .

Since: 0 0max maxx x x x hz x z x e e z e

So the norms (16) and (20) hold such a relationship: h

cz z e

(23)

Linearization (6) can be considered as a zero approximation to the exact solution of

the problem (1) and can be improved by using the following iterative procedure:

0c c cz h z h k y y

784 K. S. Al-Basyouni

0y x y x

1, , 1n ny f x y x n

0 0ny x y

(24)

For evaluating the convergence of the iterative process of (24), let us denote nZ x

the difference between the solutions of problems (1) and (24):

n nZ x y x y x (25)

From (1) and (24) we have:

1, ,n nZ x f x y x f x y x

0 0nZ x

Hence:

0

1, ,

x

n n

x

Z x f t y t f t y t dt ,

So: 0

1

x

n n

x

Z x z t dt (26)

As usual, from (26) we obtain:

0

, 1!

n

n c

x xZ x z n

n

(27)

Consequently:

, , 1

!

n

n c c

hz z z y y n

n (28)

From the obtained inequality (28) immediately results the possibility of approaching

to the desired solution of problem (1) with any degree of accuracy in the entire

domain of its existence.

In practice it is often necessary to solve such Cauchy problems:

,y a x y f x y g x

0 0 0, 0y x y y (29)

It is easy to carry out the linearization of this equation on the basis:

y a x y ky g x

0 0 0, 0y x y y

(30)

Where: 0

0

,f x yk k x

y (31)

Approximate solution of nonlinear ODE 785

Or: 0 0

0

0

,f x yk k x

y (32)

Estimating of the closeness of the resulting approximate solutions to the exact

solution of the problem (29) is easy to be established by means of previously used

calculations. For this purpose it is enough to notice that the equation (29) can be

converted to the form (1) by using the replacement:

expy x v x a x dx (33)

Since: expy v av a x dx ,

then the equation (29) is converted to a form of a new unknown function v x :

1 1,v f x v g x , (34)

where:

1 , ,a x dx a x dx

f x v f x v x e e

1

a x dx

g x g x e . (35)

Believing absolutely in a similar way:

expy x v x a x dx . (36)

We obtain for v x the equation:

1v x kv g x (37)

Now the applicability of the estimates (18) and (22) for the establishment of the

closeness of the solutions of the Cauchy problems for the equations (34) and (37) and

consequently for equations (29) and (30) is obvious. As in the case of the (6)

linearized Cauchy problem (30) can be solved by quadrature's on the basis of well-

known formula.

4. Conclusion

A new approach has been used to construct an approximate analytical method for

solving ordinary differential equations. This method is based on replacing the

nonlinear scalar differential equations by linear or more simple nonlinear equations of

a special design.Accurate and simple assessments have been obtaned which are close

to the exat and approximate solutions.This suggested method can be used to solve the

786 K. S. Al-Basyouni

Cauchy problem and boundary value problems of the different types of the ordinary

differential equations.

References

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of elastodynamic problem in rotating non- homogeneous orthotropic hollow sphere”

Mathematical Problems in Engineering, ID Article 250567, (2013).

[2] S. R. Mahmoud , A.M.Abd-Alla, K.S.Al-Basyouni, A. T. Ali “On problem of the

radial vibrations in non-homogeneous orthotropic hollow sphere subject to the initial

stress and rotation” Journal of Computational and Theoretical Nanoscience, Vol.11,

No. 2, (2014).

[3] K.S.Al-Basyouni “A model for an approximate solution of nonlinear boundary

differential equations of second order” journal of Teachers college,vol.2,Jeddah,

(2007).

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Approximate solution of nonlinear ODE 787

[10] G. Adomain,R.Roch "On the solution of nonlinear differential equations with

convolution product nonlinearities" Applied Mathematics & Analysis, Vol.114,

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[11] J. He. Huan "Approximate solution of nonlinear differential equations with

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(1988).

[12] O.A.Kuzenkov " The Cauchy problem for a class of nonlinear differential

equations in a Banach space" Differential Equations, Vol.40,No.1 pp. 23-32 (2004).

[13] Q. Yaming, J.Rivera "Blow-Up of solutions to the Cauchy problem in nonlinear

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193 (2004).

[14] Yu.G. Bulychev "Method of the reference integral curves of the solution of the

problems of Cauchy for the ordinary differential equations" Applied Mathematics &

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Received: July 11, 2013