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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
[1] S. R. Mahmoud, M. Marin, S. I. Ali, and K. S. Al-Basyouni, “On free vibrations
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).
[4] G. L. Lions "Some methods for the solution of nonlinear boundry problems"
Paris:Dunod,(1969).Translated to Russian, Mir publesher, Moscow, (1972).
[5] W. E. Boyce, R. C. Dipriman "Elementary differential equations and boundary
value problems " John willey pub., New York, (1986).
[6] E. D. Drainville,P.E.Bedient "Elementary differintal equations" Makmillan pub.,
New yourk, (1981).
[7] A. Y. Goristski, S.N,Kaujkove "First order quazilinear equations with partial
derivatives " Moscow university press, Moscow, (1997).
[8] D. R. Durikovich "On the solution of nonlinear initial-boundary value problems"
Applied Mathematics & Analysis, Vol.12, pp. 407-424, (2004).
[9] N. R. Yuki,T.A.Satoshi "On the existence of multiple solutions of the boundary
value problems for nonlinear second-order differential equations" Nonlinear Analysis,
Vol.56, pp. 919-935, (2004).
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,
pp.171-175, (1986).
[11] J. He. Huan "Approximate solution of nonlinear differential equations with
convolution product nonlinearities" Comput. & Mech.energ., Vol.167, pp. 69-73
(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
one-dimensional thermelasticity" Applied Mathematics & Analysis, Vol.292, pp. 160-
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 &
Analysis, Vol. 28, No.10pp. 167-195 (1988).
Received: July 11, 2013
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