The maximum of a solution of a nonlinear differential equation

The maximum of a solution of a nonlinear differential equation

Citation for published version (APA):

Brands, J. J. A. M. (1981). The maximum of a solution of a nonlinear differential equation. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8105). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981

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Department of Mathematics Memorandum 1981-05 l1ei 1981 The maximum Technological Unive~ity Department of Mathematics PO Box 513, Eindhoven The Netherlands rential equation by J.J.1\.M. Brands )

THE MAXIMUM OF A SOLUTION OF A NONLINEAR DIFFERENTIAL EQUATION by

J.J.A.M. Brands

Department of Mathematics, Eindhoven University of Technology, The Netherlands

PBSTRACT

Parameters occur in a second order nonlinear differential equation and in the initial values. The solution of this initial value problem has a maximum M. An asymptotic expression is derived for M as a function of the parameters.

1. INTRODUCTION

A colleague*) of the author has posed the following problem:

(1) -y

=

-qy • + r y (.)2 - pe , y

(2) yeO)

=

0 ,

y(O)

=

r-1q,

where p, q, and r are positive real numbers, and r < 1. It is asked to determine

(3) M :={max yet)

I

t ~ O} .

This problem arose in the study of the stress-strain behaviour of polymers that deform by crazing. In the special case under

consideration the values of the parameters p and r are roughly p

=

0.01 , r

=

0.5 , and the values of q can be adjusted between

10.and 1000. It is unlikely that one can find an explicit solution. Therefore it is better to seek an expression M(p,q,r) which

approximates M with sufficient accuracy.

*)

S.D. Sjoerdsma, Laboratory of Polymer Technology, Eindhoven University of Technology.

2. RESULTS

The kind of formulas which we have derived are in fact asymptotic formulas. Instead of presenting them with the order symbols of Landau, we give explicit bounds for the error terms. We give two formulas for

M;

the first one is very simple, but not so precise as the second one. The simple formula reads as follows:

(4) _{M -} log(q !(pr» - Ct(r) 2 + RI '

where Cl(r) depends only on r. The error RI satisfies the condition:

i f

then

" 2 r-l

0< RI <.8(q !(pr» •

Explanations about the determination of Ct(r) will be given after formula (5).

A more complicated formula is

(5) _{M - log(q !(pr» - C1(r)} 2 _{+ C2(r)(q !Cpr»} 2 r-l _{+ R2 '}

where Ct(r) is the same function as in (4), and C

2(r) also depends only on r. The error R2 satisfies the condition:

i f

then

The determination of Ct(r) and C

2(r) can be done in several ways: (i) One can compute M numerically from (1), (2) for some large

values of q2/p and fixed r. Then (4) and (5) provide us some equations for C

1(r) and C2(r).

(ii) One can compute m, defined by the boundary value problem (11), 2

(12), for some large values of q /p, and then apply (9). (iii) One can compute Ct(r) and C

2(r) using their definitions (27)

and (37). Lemma's (16) and (38) provide some partial control.

3. THE BEHAVIOUR OF A SOLUTION

LEMMA. There are positive numbers T and TI with

a

< T <: T

I , such that

y is increasing on [O,T] with y' < 0, y is decreasing on [T,ot) with y <

a

on [T,T

t) and y :>

a

on (Tt ,.;). Moreover yet) -+

-co..

if t -+ 0:>. PROOF. Since yeo)

=

-p <: 0 we have that

a

< yet) <'r-1q for

a

<: t <: 0, 6 >:

a

and 0 sufficiently small. Since yet) ~ -p as long as

a

~

yet)

~

r-Iq, we see that yet) decreases to zero in a finite time T for the first time after t - O. From the fact that y - 0 implies y < 0, we deduce that yet) <

a

on (T,OIJ). The supposition that yet) <

a

on (T,~) leads to a contradiction since the right hand side of (1)

would become positive for t sufficiently large. The assumption that y has a lower bound leads also to a contradiction, for then yet) would decrease to a limit, say L, for t -+~, and yet) would increase to zero

t

for t -+~, and hence yet) would tend to -pe for t -+"(0. 0.

4. A FIRST APPROXIMATION OF T!1E MAXIMUM M.

Throughout the rest of this paper r is a fixed number between

a

and I, and P :

=

1/ r • Introducing

(6) a := (p ":1 2 -1 q r ) r ,

and transforming according to

we get the initial value problem 2 d u 2 + _du_ d't

=

d. ul-p. , u(O) = _{a ,}

The problem (3) is transformed into

du(O)

d1'.

=

-a .

Clearly m depends only on a (and r). By (3), (7) and (8) we have

(9) M

=

r -I log(a/m).

Considering u a function of v '" ,. 'd ";!:-d~ the problem becomes

u(a)

=

a , u(O)

=

m.

It is easily seen that u ~ v for 0 ~ v ~ a. This suggests the substitution

(10) w ;= u - v which leads to (11) (12) dw p-l -1 ( - = -(1 + v(v + w) ) =:!tv ,w) , dv w(O)

=

m , w(a)

=

O.

The problem (11),(12) is our starting point for finding approximations of m. To indicate that m depends on a we sometimes write mea) instead of m. A solution of (11),(12) is denoted by w(v,a). Obviously, for fixed v, w(v,a) and, hence, mea) '" w(O,a), are increasing functions of a.

Clearly w(v) > 0 on [O,a), hence w'(v)

~

-(1 + v P)-1 on [O,a]. Integrating over [v,a] we find

(13) w(v,a)'<

f~'(l

_{+ xP)-l dx} <

J""

(t + xP)-ldx < _{(p - 1)-l v-}p+l •

Formula (13) provides an upper bound for m when v == O.

(14) m <

J

r

p -I ~1-1

o

(1 + x) dx

=

x(psin(rc/p» < p(p - 1)

It follows that m(~) increases to a limit, say m(~), when ~ ~~.

We denote by W(V,cD) the solution w of (11) with initial value w(O) == m(~). Some properties of w(v,~) are summarized in the following

(15) LEMMA. The solution w(v,oo) is positive and decreasing, and w(v,m) ~ 0 if v ~ 00.

PROOF. w(v,oo) > 0 since w(v,oo) > w(v,a) for all ~ > O. w(v,o .. ) is decreasing since it satisfies (II). Let E> O. Let A:== (2y/E)Y, where

y == r

I

(I-r). Since a solution w of (11) depends continuous lyon the

initial value w(O), there exists a positive number, say ~, such that

w(v,cx.) -

w(v,~)

<.

h

for 0 ::;; v ::;; A. Hence, by (13), w(A,o:..) < E/2 + yA-1h == E. IT

We sample some useful properties of m(~) in the following

(I6) LEMMA.

(I 7)

( 18) (p ~~)

(19) e -I p(p - 1) - 1 . <m(oi» < p(p - 1) -1 (p> 1)

PROOF OF (17). Since u(v) == v + w(v,~) is increasing in v E [O,~] we have for all v E (O,~]

(20) u(v) < v + mea)

J

v -I -1

- 0 (1 + s(u(v»p ) ds ==

... v + m(a.) - (u(v» I-p log(I + v(u(v» p-I ).

Hence,

The righthand side of (21) is a decreasing function of v for fixed u; since v < u(v), it follows by substitution of v:= u, that

(22)

We will show that the inequality holds for all u € (O,a]. Let

f(u):= u1-Plog(1 + uP) for u >0. Let f(u

O)

=

max {feu)

I

°

< u ::; a} with O· <. u

o ::;

a. Suppose mea) ::; f(uO)' Then mea) <. U

o

since,

trivally, feu) <. u for all u > O. But by (22) we would have mea) > f(u

O)' a contradiction. So (23)

Of course (23) implies

(24) (u > 0).

If a ~ 1, then (17) follows by substitution of u

=

1 in (23).

PROOF OF (18). Obviously, v + w(v,~) ~ m(oo) for v € [O,m(oo)] ,

v + w(v,oo) > v'for v € (m(oo),oo). So

< mo(OO)

m(oo)

J

n (1 +

v(m(~»p-I)-ldv

+

J~

(I

m(oo)

=

(m(oo»I-Plog(1 + (m(oo»p) +

m(oo)J~

(I

Putting x

:=

(m(oo»p, we derive

1

<. x-

1

log(1 + x) +

~

fl

(tp/(p-t) + x)-ldt .,P-J

a

. -1 - 1 - 1 <. x log{i + x) + (p - 1) 10g(I + x ). P -1 + v) dv

Now (18) can be derived, using this latter inequality and (24),

PROOF OF (19). Substituting u

=

exp[.(p - I)-I] in (24) we get the first inequality. The second inequality follows from (14).

Since dd

(w(v,~)

- w(v,a)) >

°

on [O,a], we have, for v € (O,a),

v

-(25) m(OIt)' - mea) < W(V,07) - w(v,a) < w(a,oo).

From (13) it follows that

(26) w(a,oo) < (p - 1) -1 1-p a •

-1

We easily infer from (19), (25) and (26) that m(oo) - mea) ~ 3 e m(.,.,) if a> (3/p)I/(p-l)=: a

1• Using the fact that -log(l - x)' < 2.7x

if x ~ e/3 we infer

(]

-1

1

-]

R

1:= -p10g[ 1 - (m(oo)) (m(oo) - mea)) < 2. 7p (m(oo)) (m(oo)- mea))

if a >a

l• By (9), (19), (25) and (26) we infer (4) where

(27) C -]

1(r):=r 10gm(oo).

5. A SHARPER APPROXIMATION OF M.

We want to find a second term in the asymptotic expression for mea), a ~~. Therefore we need the following

(28) LEMMA.

(29) d _{da mea)}

₌

_{-F(a,O)exp[ -}

'Ja

₀

aw

SF _(v,w(v,a))dv

_J.

PROOF. Let a be a given positive number. Then, using (I]), we have for every

°

~ v ~ a and every h>

°

that there is a number n

=

n(v,h) between w(v,a) and w(v,a + h) such that

d ~

Dividing both sides by w(v,a + h) - w(v,a), integrating over

[O,a],

and exponentiating we find

(30) m(a + h) - m(a) .. w(a,a + h) exp[

-fa

F (v, n)dv ].

o

w Furthermore we have, for all h > :0,

~(a,a

+ h)

=

_fa:h

F(v,w(v,a + h)dv •

a

Since -F(v,w) is decreasing in both v and w, we have

-hF(a + h,w(a,a + h» < w(a,a + h) < -hF(a,O).

It follows that

-J

lim h w(a,a + h) .. -F(a,O). h4-0

Dividing both sides of (30) by h and taking limits for h

+

°

we arrive at (29) for the righthand derivative of mea). In a

similar way we can prove (29) for the lefthand derivative.

We define a function g by

(31) _{g(a) ,} '=

fa

₀

_oF

_{dW (v,w(v,a»dv} (0 " a < <Xl). As we shall see

J

<Xl

a

(32)

g(~):=

a

F (v,w(v,<Xl»dv

o

w

is the limit value of g(a) for a ~ <Xl, The integral in (32) exists "'3F

since the integrand is continuous on [0,<Xl) and 3w (v,w(v,<Xl» ==

=

O(v-P- J) (v

~

(0).

We need an estimate of g(oo)- g(a). We have

where (00 aF f o o " : 1 :=

I

;.!....(v,w(v,oo»dv~ (p -l)v

.0

)a. aw' a. p)-I d .-+ v v_~ ::;; (I - r) a. -p , and

f

a. 3F - .. ~F I₂ := 0 ~aw(v,w(v,oo») - '§W(v,w(v,a.»]dv.

Denoting the integrand of I2 by A, we have

I

a2F

IAI

~ (w(v,oo) - w(v,a.» ~(v,n)

3w

::;; pa.l-P(I + vP)-I(v + w(v,a.»-2,

where, besides (25) and (26), we used that

a

2F 2

-

=

(p - l)(v + w) - F(I + F)(p + 2(p - 1 )F), aw2 and

11

+ F

I I

p + 2(p - ])F

I

~ p • It follows that

f

a. -2 - 2 - ' ]-p [(m(a.» + v Pdv ]pa. . ]

So, using (17), we have, for a. ~ 1,

I-p

Integrating both sides of (29) over the interval [0.,<») we get

m(~)

-

mea)

=

-f~

F(a,O)e-g(S)dB

=

J<» s-Pe-g(<»)dS + R ,

a a

where R is given by

where we used that lex -

11

S e lxl - 1 for all x E R

It follows that

where

(34)

I I

R

'

< lJe -g(~) pep - 1) -1 a 2-2p (a ;:: e).

As before, we easily infer from (19), (25) and (26) that

(m(~»-l(m(a)

-

mea»~

S e/3 if

a ;::

(3/p)1/(P-l). Further, using that

J

-2 - -1

I

" x l~og (I + x) + x < -2 ~f x s~~/3 ,-.and (19), we deduce that ~ for a ;:: (3/p)I/(p-J), .

°

< log

m (a) -

log m (co) + (m(co»

-I

(m(co) -

mea)

>'

where

(36)

I

R21 < 4Spa 2-2p

Using (35) and (36) in (9) we arrive at (5) with

(37) C -1 -1 -g(o» 2(r) == (1 - r)

(m(m»

e .

Finally~ we sample some properties of g(oo) and C

2(r) in the following lemma. (38) LEMMA. (39) e -g(co) < (t + (m(oo»-p)-l+r. (40) e -g(oo) ... 0 _{(r '" 0)} (41) e -g(w) == O(l)(r t 1) (42) C 2(r) ... 0 (r '" 0) (43) C_{2(r) - O«r -} 1) ) 2 (r t 1) . aF -1

PROOF OF (39). We have

a-

== -(p - I)u (I + F)F, where u is defined by

w d p -1 du

(10). Furthermore, 1 + F{v,w) -

d~

< uP(I + u) and -F(v,w)

=

1 - dv' Hence,

g(<o) == {p -

1)f:·[:~·

-

(:~)2Ju-ldV

•

. -1 du 2 P-I P -t du

Sl.nce u (dv) < u (I + u) dv we have

g(oo) :>(p - 1)

foo

[u-1 du _ up-1

0

+ u )-1 du1dv

o

dv dvJ

== (p - 1) log [ (m(oo»-lO + (m(oo»p)l/p].

PROOF OF (40), (41), (42) and (43). Now using (18) and (19) it is a routine