numerical solutions of nonstandard first order initial value problems

Journal of Applied Mathematics and Stochastic Analysis 5, Number 1, Spring 1992, 69-82

NUMERICAL SOLUTIONS OF NONSTANDARD FIRST ORDER INITIAL VALUE PROBLEMS ’ M. VENKATESULU and P.D.N. SRINIVASU

Department of Mathematics Sri Saihya Sat Institute of Higher Learning Prasanthinilayam 515 134 A ndhra Pradesh INDIA ABSTRACT Differential equations of the form y’= f(t,y,y’), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut’s and Chrystal’s equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IVP) y’ = f(t, y, y’), Y(to) = Yo under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NSTD IVPs.

Key words:

Nonstandard initial value problem, existence, unique solution, real valued, continuous, differentiable, numerical scheme, inequality, partition, approximate, algorithm, influence, round-off error, estimate, truncation.

AMS (MOS) subject classification: 1.

34A99, 34A50.

INTRODUCTION

Differential equations of the form

y’= f(t,y,y’),

where

f

is not necessarily linear in its

arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut’s and Chrystal’s equations fall into this category

[2]. A few authors, notably E.L.

Ince [3], H.T. Davis [2], et. al. have given some methods for finding solutions of equations of the above type. In fact these methods are best described as follows.

y’ = f(t, y, y’) can be solved for y’ as a single valued function of (t,y) in a neighborhood of (t0,Yo) say y’= g(t,y), then the solution of the initial value problem (IVP) y’= g(t,y),y(to) = Yo, if it exists, is also a solution of the original equation y’ = f(t, y, y’) (and satisfies the initial condition y(to) = Y0)- Or, if there exists (to, Y0) such that the equation y’= f(t, y, y’) can be solved for y’ as a multivalued function of (t, y) in a neighborhood of If there exists

(to, Yo)

1Received: February,

such that the equation

1990.

Revised: February, 1991.

Printed in the U.S.A. (C) 1992 The Society of Applied Mathematics, Modeling and Simulation

69

M. VENKATESULU and F.D.N. SRINIVASU

70

(t0,Yo)

then a (nonunique) solution of the IVP

y’= f(t,y, Vt),y(to)= Y0

is given by certain

(not

necessarily convergent) infinite series.

In our earlier paper [7], we have established the existence of a (unique) solution of the nonstandard IVP = f(t,y,y’),y(to)= 0 under certain natural hypotheses on f, and have shown

’

the continuous dependence of the solution on initial conditions, and parameters.

result that we established in

[7]

The existence

is of theoretical nature and it does not provide methods of finding

Our aim in this paper is to present numerical methods for finding the approximate solutions of the nonstandard IVPs. the solutions explicitly either analytically or numerically.

Before proceeding to numerical methods, we introduce few notations and the nonstandard

IVP. Let R denote the real line, and let R n denote the n-dimensional real space where ’n’ is a Let (to, Yo) ER 2 and let D be a convex subset of R3 defined by positive integer. D-{(t,,z)R3[ It-to[ 0 is a constant. Let r/i denote the approximate value of y at ti, = 0,1,...,n, and let e denote tolerance limit.

Scheme I (Contraction-Euler Method):

We define

a)

% = Yo for

= 0,1, 2,..., n

r/lio-0; b)

1

j=0

repeat j-j+l

c)

rltij f(ti, rli, rltij 1) until k < hl and IY’ij-Y’ij-1[ 0 is a constant, -< k4 for all (t,y,O) E D, where k4 > 0 is a constant such that 0"

Of(t,y,O)]-I

k3k4c < 2/3, and

M. VENKATESULU and P.D.N. SRINYVASU

76

vz)

If(t, y, O) 0 ck4 < (1

is a constant such that

k3k4c)c.

Scheme m (Newt0n-Kantorovich-Euler Method):

We define

a)

r/O = YO for = 0, 1,2,..., n- 1,

r/’i0 b)

0; j = 0

repeat

j=j+l

U’ij- 1 f(ti, rli, rltij_ 1)

rltij rltij- 1

1 until

(k3k4c/6(1 k3k4c))

(18)

Ol(ti, rli, rl’ij 1) Oz

2j- 1