the method of lines for the solutions of some

Comput. Math. Applic. Vol. 15, No. 6--8, pp. 635-658, 1988 0097-4943/88 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright 1988 Pergamon Press plc

THE METHOD OF LINES FOR THE SOLUTIONS OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

I. GYSlU Computing Centre of the Szeged University of Medicine, P~i u. 4/a, 6720 Szcged, Hungary

Abstract--In this paper we deal with the method of lines in the case of certain first order nonlinear partial differential equations as well as equations of the parabolic type. A new aspect, compared to earlier investigations, is that the system we consider may contain partial and ordinary differential equations alike and, in addition the r.h.s, of the equations may involve some nonlinear operators. The domain of definition of the solutions is bounded by time dependent functions. Our general results are demonstrated on a simple system which describes a biological model.

1. INTRODUCTION

It is a long-established method for the approximation of solutions of partial differential equations to substitute difference quotients for some of the derivatives in the considered partial differential equation and then to solve the generally obtained ordinary differential equations system in place of the original (see e.g. Refs [l, 2]). This procedure, mostly called "the method of (straight) lines", is very useful for the computer solution of partial differential equations [3, 4] and at the same time, it may help in setting up model equations for certain biological processes and estimating their parameters [5], as well as in the interpretation of their solutions [6].

There are two usual ways for the approximation of partial differential equations with ordinary ones which differ little from the theoretical, but more significantly from the practical point of view. In one case discretization is applied to the time variable, leaving space variables unchanged [2], in the other case space variables are discretized [1, 3, 4].

In this paper we deal with the latter method in case of certain first order nonlinear equations as well as equations of the parabolic type. A new aspect compared to earlier investigations is that we consider systems that may contain partial and ordinary differential equations alike and in addition, the r.h.s, of the equations may involve some nonlinear operators. The domain of definition of the solutions are bounded by time dependent functions which, along with the r.h.s, of the considered equations, are assumed to obey conditions similar to those used in Ref. [7].

Now let us illustrate the problems of approximation by the method of lines on the following simple scalar hyperbolic equation:

ut(t, s) = --us(t, s) +f[u(t, s)], to ~< t ~< T, 0 ~< s ~< y(t). (1)

Typical boundary conditions associated with equation (1) are of the form

u specified at t = 0, for 0 ~< s ~ 7(0),

uor us specified a ts=0, fo r0~

636 I. GORI

Then assuming that the function and the solution u of equation (1) are sufficiently smooth, we have

d("')(t)=u,[t,i~(-~tl)]+iY(-tl---~) u,[t,i'(-tl---~) ] and

where

- - = y -~ [a ' ( t ) - - a tl ' i - ')(t)] + o[ht] , l ~ + oo,

1 h i 1).

to ( t ) - - a ' >( t ) = u(t, 0), to

Solutions of nonlinear partial differential equations 637

Eliminating the o[h(], the corresponding approximating equation has the form

[ .7 '~' xt~d)(t)] "rTL =o J ' - ' -I g(l,'(t) = x(l,'(t)- x(l,I)(t)+ fl(t); .~tl?j~x(l"3(t)|. j -O J

The proof of our convergence theorems is based on the two propositions proved in Section 2. The first one (Proposition 2.1) assures the convergence

max ]a (/' (t, u) - x (t' i)(t) I --' O, l ~ + oo, O

638

Oi)

I. GVORI

there exists a matrix valued function A E C([t0, T] x R x x R K, R Kxx) for which

I Gt(t, x) - G2(t, y) - a(t, x, y)(x - Y) I ~< m(t, x) lx - y l + c(t),

if (t,x),(t,y)~[to, T )R to, wherem:[ t0 , T )Rx- - ,R+ (6)

is a continuous function and c: [to, T)---, R+ is locally integrable;

(iii) for any x, y ~ R ~,

# [A (t, x, y)] ~< 2, t0~


and expression (7) yields ] u(t, x, y) U- (s, x, y) 1 < exp[l (t - s)], to < s < t < T. But from expression (6) we obtain

Ix(t)-~(01 I,,) positive integers and suppose that (H.l) VEF(D, RN!), (D c RK, K > I), ace: [to, T) x Y+RNl are given in such a way that for

any w E V, the function a(O(t, w) is absolutely continuous on [to, T) and there exist functions cO(w) E C([t,,, T), RN/) and H): [to, T) x RN! x V -*R& such that H() satisfies the Charatheodory condition and a((& w) is a solution of initial-value problem

i)(t) = HcO[t, x)(t), w] + c(w)(t), t, < t < T,

~(~(t,) = a()(t,, w) > I > 1 . 0 09

(H.2) there are two functions

A()E C([t, T) x RN/ x RN! 9 7 RN/ xNj ) and m:[fo,T) x RN/ x V+R+

such that

(12)

(13)

(14)

I H() [t, xc) , w) - H[t, y(I), w) - A ()(t, x(I), yo] [xc0 - y()] I < m [t, xc), w] Ix() - y) 1,

for any [t, x(O, w], [t, y(I), w] E [to, T] x RN! x V, and

m [t a(O(t w) w] G m, (t, w) 7 7, 3

where m,(t, w) is continuous function on [to, T); (H.3) for any x(O, y() E RN! ,

p {A[t, xc), ~~1) < I, co < t < T,

where 1 is a given constant. Then

(a) for any w E V, the solution x()(w) of

1

P(t) = H([t, x((t), w] 3

~(~(f,) = aco(to, w) 9

exists on [to. T) and for any I > I,,

Ia(t,w)-xX([)(w)(t)1 Sexp(+At) f

exp(-Is)]c(/)(w)(s)]ds.exp 10

[)-d.c w) W]; (b) if

II @W II[to,Tpo9 l+m, T~[fo, T),

then

max la(O(t, w) - x(O(w) (t) I + 0, loCf

640 I. GYORI

and

then

(c) if

L sup exp(+2t) lexp(-2s)E(O(w)(s)lds--,O, to~t 1o, moreover we define G~ and G2 by

and

G,(t, x) = H(t, x, w) + E(0(w)(t)

G2(t , x) = H(t, x, w),

for any (t, x)~ [to, T) x RNt. Then a(0(t, w) and x((w) are solutions of equations (9) and (10), respectively. Furthermore, the

conditions of I . ,mma 2.1 are satisfied by the definitions of Gi (i = 1, 2) and the conditions of the present proposition.

Thus, by Lemma 2.1 we obtain

I a((t, w) -- x((w)(t) I ~< I I a((to, w) -- x((w)(to) I exp[+R(t -- to)] 6

~[ xp(--As)] E((w)(s)]#I t } +exp(+2t ) Ids exp m[s ,a ( (s ,w) ,w]ds , (23) ._l t.J to

on [to, T). But, from system (16) we have a((to, w) = x((w)(t0), therefore inequalities (14) and (23) imply the required inequality (17).

The proof of the proposition is complete.

Proposition 2.2

Suppose that lo and Nt, (1 >t 1o), are positive integers and (H.4) 17 cF(D, RS~), (D c R ~, K >I 1), its subset V c 17 and a(: [to, T) x V~R s,

are given so that condition (H.1) is satisfied and in addition to this, for any we I 7 there exists a function sequence {w, }~=j c V such that

sup [ a((t0, w) -- a((t0, wk)[ ~ 0, k ~ + oo (24) t ;~ l o

and

SU t;.]~ ,o,/~sup ri P(a)(t' w) - P(a((t, wk)l ~0, k ~ + oo, (25)

where p(o e R #, x ~, and

IP(ol ~ 1o; (26)

(H.5) there are functions A ( E C ([ to, T) x R Nt x R NI, R N~ X #, ), m ~ C ([ to, T], R + ) and ~! : [to, T) x V x 17 --, R+ such that for any w ~ 17 and sequence {wk}F. 1 c V satisfying expressions (24) and (25)

lira f r / ( t , wk, w) dt = 0 (27) k--+ +oo J ,o

641

and

Solutions of nonlinear partial differential equations

I~~~~,~,~,~-~~~,~,~~-~~~~,~,~)(~-y)l~m(t)I~-yl~(t,w,,w), (28)

for any (2, x), (t, y) E [to, T) x RN!, (I 3 &), where H is given in condition (H.l), and condition (H.3) holds for this A().

Then

I aco((t, w) - xco(w)(t) I < exp(r2t) s

t exp( -As) I @(w)(s) I ds exp [llm(s)ds], to < t 6 T, (29)

'0

where C()(W) is defined in condition (H.l); (b) for any w c P and sequence {wk}pS 1 c V satisfying expressions (24) and (25),

I xW(~) - xh)(t) I G Ia(9(zo,w)-aa(9(fo,w,)I+ [ I Ttf(s,wk,w)ds 1 x exi[~~mO) h],

(c) if

s

T

m(s)ds < cc 0

and

sup c(w)(t)1 +o, 1+ + 00 toSr$T

for any w E F, then

lim sup Ia'yt,W)- x'yt)l = 0, WE v I++m ro

642 I. GvOm

where

and

a~ = sup [P(a((t, w)- P(Oa(O(t, wk)l to~t/1. j~+c

From expressions (24), (26) and (30) we obtain

sup?~ ~< c, "sup sup Ix(O(wk)(t) - x(t)(w)(t)l ~ m o 1>10 I>~l 0 to~ t o

Using these relations and expression (35), we have

0 ~< lim sup M ( .10 1>1 I 0 l>il 0

Thus

lim M ( = O, I~+oo

which means that equation (34) is valid and the proof of the proposition is complete.

if k--+ + ~.

as k--+ + oo.

Remark 2.1 In the applications of Proposition 2.2, the matrix P(E RN~ N~ used in condition (H.I) will be

chosen only in one of the following two ways: either p(0 = ist( l > lo), or p(0 is the matrix of the orthogonal projection of R ~ on R"(l >I/0), where N~ > n. The first definition of p(0 is used in this paper to investigate partial differential equations and the second one will be used for functional differential equations in Ref. [9].

3. CONVERGENCE OF THE METHOD OF LINES FOR SOME NONLINEAR PARTIAL D IFFERENTIAL EQUATIONS

First of all, we investigate the following scalar partial differential equation:

u,(t, s) =f i t , s, u(t, s), us(t, s), uss(t, s)], (t, s) e F, (36)

where f :F x R3--,R is a given continuous scalar function and there are functions Yl, 7 ~ C([t0, T], R) such that ?(t) > 0, to


Let u ~ V. Then we have the approximate equalities

1 {u[t,s~O(t)] - u[t,s~O (t)]}, us[t, s~t)(t)] ~ ~ l

(1 ) 2 t ' /+1 -- i--I( )]} uss[t,s~t)(t)]~, ~-~ {uft s cO (/)] 2u[t,s~O(t)]+u[t,s ~0 t

i= l , l - l , where l1> 2 and

s~t)(t) = ~l(t) + i ~(-~tl), i = 1, 1 - 1.

Let us define a~(t, u) = [a~(t, u) . . . . . alt)(t, u)] x by

a~(t, u) = u[t, s~O(t)], i = O, 1, (37)

then

d d~o(t, u) = dt u (t, 7,(t)], (380)

d~(t, u) = ut[t, s~(t)] + us[t, s~(t )]~(t )

=~(t)us[t,s~(t)]+f{t,s~(t),us[t,s~(t)],usdt, sl')(t)]}, to

644 I. GYM

Ignoring the term ~(0 in equation (39), we get the line approximation equations

1

kCO(t) = H([t , x()(t) U] 9 , to < t < T,

x(O(t,) = a()(to, u) 9 12 2 9

the solution xc0 of which will be used to approximate the solution u of equation (36)

Theorem 3. I

Suppose that

(i) f: T x R3 + R is a continuous function and it possesses continuous derivatives f,,(t, s, uI, u2, u3) and f.,(t, s, ui, u2, u3) which are nonpositive and nonnegative, respectively, moreover

(ii) y,, y : [to, T] --) R are continuously differentiable, r(t) > 0, t E [to, T] and

l;l(t)+j(t)~inf(--f,z(t,s,ul,u2,u3):y,(t)~s~y(t)+y,(t), (ui,u2,ug)~R3).

(44) Then, for any solution u E V of equation (36) the following holds:

(42)

I[ u x exp[m,(u)(t - to)] +O, I + 00, uniformly on [to, T], (45)

where x()(u) = [X&~(U), . . . , xj(u)lT is a solution of equation (42), c()(u) is given in equation (41) and m, = m,(u) > 0 is a constant.

Proof. To prove the theorem it suffices to show that the hypotheses of Proposition 2.1 are satisfied when the functions a(): [to, T] x Y+R+, H(O: [to, T] x R+ x V-+R+ and @u): [to, T] -+ R+ are defined by equations (37), (40) and (41), respectively, and N, = I+ 1.

First we consider the function E()(U). Since fu2 and f,, are continuous functions, for any bounded B c i; x R3 there exists a constant KB 2 0 such that for any (t, s, u), (t, s, ii) E F x R3,

If (t, ~3 U) -f (t, 3, ii) I G Ks(f, S) ,zy3 I ui - 41. , . On the other hand, there is a bounded set B, c T x R3 such that [t, s, u(t, s), u,(t, s), u,(t, s)] and

( t, SW), uk #+)I, --$ {a, 4%)1- UP, dl,(01),

(u[t, s;!,(t)] - 2u[t, syyt)] + u(t, sy! ,)) ( >) -$) *

belong to B,,, for any i = 1, I - 1. But this yields

I #(u)(t) I G Max (

I U,[G s?(t)1 - & {u [t, sl(t)l - u [t, sl?, ON}, I u,,[t, s%t)l

-- ( ) y fi, 2{~[t, sl? , (tll - 24 k #W)l + u it, sll , (Ol}), (46)

here

K(u)=,~~~~{I*j,(t)I+lj~(~)l}+K~~. 0

Since u is twice continuously differentiable in s, it follows from inequality (46) that

max (do(u)(t) I = max max ] #)(u)(t) I 40, as I + + co. G)GLT t~~t


Thus, we have got that in equation (39), ~C(u) and a

646 I. Gv6RI

Using inequalities (44) and (48), it is easy to verify that

i . .~t>(t) + ~, = yt(t) + )y(t) + 0t i ~< 0

and thus it follows from inequality (49) that / /

#[ Act)] =a~ + E la~J)l =a~ )+ ~ a~


Theorem 3.2 Suppose that f : _r x R 3__)R is a continuous function and it possesses continuous derivatives fu2

and fu3 which are nonpositive and nonnegative functions, respectively, and

If(t, s, ut, us, u3) - f ( t , s, ul, u2, u3)l ~< K lul - ~ I, (56)

for any (t, s, ul, u2, u3), (t, s, Ul, u2, u3) e F x R 3, where K > 0 is a given constant. If),, ~: [to, T] ~ R satisfy condition (ii) of Theorem 3.1, then for any generalized solution u e V.

maxlu l t , ) '~(t )+iT( -~t , ) ] -x~(u) ( t ) ~0, as l~+oo, (57) O

648 I. GYORI

Remark 3.3 Consider a function f : r R 3 ---, R of the form

f (t, s, ul, u2, U3) = - - 0~1 (t, S, u,) u2 + ~2 (t, S, ul ) u3 + g (t, s, U 1 ) ,

where ~, ~2, g : r x R --, R are continuous functions such that

~l(t,s, ul)>~O,~2(t,S, Ul)>>-O, (t,S, Ul)eF R

and

(60)

is solution of

1 Yc~ t) = {s!(t) - ~, [t, s~(t), x~t)]} [x~ t) - x[_, ] ~(t )

+ ~2[t,s~O(t), " 0, t e [to, T] and

~l(t)+ ~(t)


It is evident that v (t, s) = ~ (t + s), (t, s) E [0, T] x [0, 1],

is a continuously differentiable solution of equation (64) on [0, T] x [0, 1] if ~b ~ C~([O, T + 1], R), and the approximate initial-value problem is

"~0( t ) = t" [x}(t) -- x!_l (t)], 0 ~< t ~< T,

x~(0)=v 0, =qb , i= l , l ,

where x~(t) = ok(t), 0

650 I. GY6RI

where

ylO(t)=-v(t, 7) to


Systems of this kind are interesting for example in the modeling of some biological processes [6, 8].

Following the technique of the proof of Theorem 3.1 we can prove two approximation theorems which are significant generalizations of some known results (see for example Ref. [1]), one of them gives an answer to a problem raised in Ref. [6]. Here, similarly as before, the norm of a vector x = (x~ . . . . . x,) r e R" is defined by

Ixl-- max Ixil and IAI-- max ~ la;jl for any A =(aij)eR "~". I

652 I. GY6RI

Using these notations, we give the assumptions which we will use to treat our system (71)-(72) (A.1) f=( f~ . . . . . f"~)EC(r x fl x R ~, x R%R"t)

and

I f ' ( t ,s ,u ,p,v) - f~(t ,s , fi, P,)I ~/1,

where the functions

H (t) = [H(l'),... ,H(t'l+l)]: [to, T] x R u, .-*R Nt and

are defined by

(76)

(77)

E(0 = (E(l.o) . . . . . ~(t.t+ t)): [to, T] ~ R u'

l H(t'J)(t, x, w) =j ~t) y(t)

H(t')(t, x, w) = zi(t, 0), (78)

{ ~-~ ~ l [xJ--x(J- ')],xU+')} (79) - - [xU)- -x( J - l ) ]+f t, j ~x(et.,,

and

H (t't+ I)(t, x, w) = g x(Oel.i ' x(l+ t) , (80)

for any t ~ [to, T], x -- [x (), x ) . . . . , x C~+ ~)] e R Nt and w ~ ~, moreover c(~')(w)(t) = 0, to ~< t ~ T,

~(t~j)w(t)=u*[t~j~-~]j~(-~t~)+fIt~j~(~t~)~u~u~[t~j~(-~t~)]~v(t)~-H(t~j)[t~a(t)(t~w)~w]~ j= ITI, (81)

and

(t.l+ I)(w)(t) = g[t, u, v(t)] -- I'I (t't+ l)[t, a(I)(t, w), w], to ~< t ~< T. (82)


I_.emma 4.1 Suppose that conditions (A.1)-(A.3) hold. Then for any fixed solution (u, v)e V of system

(71)-(72), we have

It(ZJ)(w)(t)l~Mo(w)to(u,,l)+M,(W)l-*O, toI 0 are suitable w dependent constants,

to(u,,l)=max{lu,(t, s2)-u,(t,s,)l:(t,s,),(t, s2)eI'ls2-s,l

654 I. GYORI

where re(u)1> 0 is some given constant, then inequalities (83) yields (N, j)

~(w)(t) ~< [m(u).M0(w ) + M,(w)] Y/, t o ~< t ~< T, j = 0,----I + 1, (88)

since in this case, we have

('7) m us , - - ~1. Now we prove the following convergence result.

Theorem 4.1 Assume that conditions (A.1)-(A.3) hold. Then for any generalized solution (u, v) of systems

(71)-(72),

max u[t,i~(t--~)]-x~'o(t)[-+O, 1~+oo, (89) O


where 0., is the zero matrix in R ", ~"' (i = 1, 2) and the elements of the diagonal matrix

1 diag[bt~J)(t ' x ' y) , . . h (l'j)(t x y)] B (l'y) = B(l'J)(t, x, y) = -~ ., ~,,,, , , ,

are defined by

~

656 I. GYOal

thus using inequality (94) it is easy to verify that the hypothesis (H.5) of Proposition 2.2 is satisfied.

Thus, all conditions of Proposition 2.2 are satisfied, that is, this proposition implies the required results (89) and (90). The proof of theorem is complete.

Proposition 4. I If the conditions of Theorem 3.1 hold and the solution (u, v)~ V of problem (71)-(72) is such

that equation (81) is satisfied, then

max{ ult, i~-(-tl!l-x("i)(t)l,lv(t)-x(t"+')(t)[}


Finally, let us consider an application of Theorem 4.1 to a system which describes a biological model.

Example 4.1 For the mathematical description of the model of a capillary bed, Jacquez [6] used the equations

f u,(t,s)--aus(t,s)+b[vl(t)--u(t,s)], O

658 I. (3YORI

like to express my personal appreciation to academidan G. I. Marchukand the members of his research group, particularly to candidates N. V. Pertsev and L. N. Belykh for the maintenance of our fruitful scientific cooperation.

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0952). 3. P. M. Dew and J. E. Walsh, A set of bibrary routines for solving parabolic equations in one space variable. A.C.M.

Trans. Math. Softw. 7, 295--315 (1981). 4. V. Vemuri and W. J. Karplus, Digital Computer Treatment of Partial Differential Equations. Prentice-Hall, Englewood

Cliffs, New Jersey (1981). 5. H. T. Banks and P. Kareiva, Parameter estimation techniques for transport equations with application to population

dispersal and tissue bulk flow models. J. Theoret. Biol. 17, 253-275 (1983). 6. J. A. Jaequez, Compartmental Analysis in Biology and Medicine. Elsevier, Amsterdam (1972). 7. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Theory and Applications. Vol. II. Academic

Press, New York (1969). 8. I. Gy6ri, J. Eller, M. Z611ei and F. Krizsa, Properties of a mathematical model of thrombopoiesis based on the Von

Focrstor approach, Mathematical Modeling in Immunology and Medicine (Eds (3. I. Marehuk and L. N. Belykh). North-Holland, Amsterdam 0983).

9. I. (3y6ri, Two approximation techniques for functional differential equations. Comput. Math. Applic. (in press). 10. I. (3y6ri, On a connection between classical and pipe compartmental systems using an approximation technique (to be

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the method of lines for the solutions of some

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