On approximate solution of integral equations in the space-time

ANNALES SOCIETAT1S MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXII (1981)

Lechoslaw Hacia (Poznan)

On approximate solution of integral equations in the space-time

1. Introduction. Some initial-boundary value problems for a number of differential partial equations of physics, mechanics and technology are reduced to the following integral equations

t

(1) f ( x , t ) + J f N ( x , t ; y,s)f{y,s)dfi{y)ds = g (x ,t), 0 M

where M is sufficiently regular and compact manifold in Rn, n ^ 2 (see [1], p. 90), ju the measure on M, N (..;..) the continuous function on W x W, W = M x [0 , T~\.

In this paper we shall give the approximate method for solving equations for the form (1) on the compact manifold IPin the space-time. The constructed algorithm is based on a partition of the unity and the approximation of continuous functions.

2. Definitions, notations and lemmas. Let W = M x [0, T ] be the compact manifold in the space-time Rnx R , where M is compact manifold in Rn and [0, Г ] an interval in R.

By C{W) we denote the space of continuous bounded functions in W with the norm

(2) Ц/l! = sup {\f(x, t)| : (x , t ) e W } .

For /, C (W ) and positive 0,y we define the modulus of continuity:

(3) co(f\ Ô, y)

= sup {\f(x, t ) - f { y , s)\: |x-y| < ô, \t s| < y, {x, t), (y, s)e W}.

In the space C (!F ) we consider the operator N of the form (4) N f ( x , t ) = J J iV (x,t; y,s)f(y,s)dn(y)ds,

0 M

with the continuous kernel N (..;..) on Wx W. It is clear that this operator is a compact operator in C{W).

For the operator defined by (4) we introduce the modulus of continuity:

(5) co(N; S,y) = sup{ffl(JV/; <5,y): / s C ((T ), ||/|| « 1 }.

2 4 0 L. H q cia

Now we give some properties of the function defined above.

Lemma 1. Let W = A f x [ 0 , T ] be a compact manifold and N an operator of the form (4). Then co(N; is the non-negative and non-decreasing function on (0, oo)x(0, oo) that has the following properties:

Г w{Nf; <5,y) ^ \\f\\(o{N; 0,y) for a l l f e C ( W ) and positive Ô, y, 2° (o ( N ; <5,y)->0 if, <5,y->0+.

P roof. From definition (5) it follows that the function o)(N; .,.) is a non-negative and non-decreasing function. Inequality 1° is clear for / = 0, and for / Ф 0 it follows from the equality

œ(Nf; ô,y) = «(ll/ ll ll/ll~1 Nf; S , y ) = \\f\\ «(Ц / 1 Г 1 Nf; 0,y)

and definition (5). Since the operator N defined by (4) is a compact operator in C (W ), therefore for every e > 0 there exist ô, у > 0 such that to(Nf; Ô, y)

< e for К/ll ^ 1. Hence co(N; S,y)~*0 if <5, y-+0+.

A set of the functions Z = (wk)k==12.../ is called a partition of the unity on the compact manifold V if:

1° wke C ( V ), wk(x) ^ 0 for к = 1,2,...,/;

i

2° £ wk (x) = 1 for every xeV . k= 1

Now we define a diameter of the partition of the unity Z = (wk)k=

on the compact manifold V:

(6) q(Z ) = sup {й/(supp wk): wke Z } .

It is know that for every tj > 0 there exists on the manifold V the partition of the unity with the diameter q(Z) ^ r\.

Lemma 2. Let Z = (Uj)j=lt2,...,H be the partition of the unity on the compact manifold M with the diameter q(Z ) < Ô, and = (vk)k = l i 2 the partition of the unity on interval [0, T ] with the diameter q{2?) ^ у for positive S,y.

Then operator T6>y: C (W ) -*■ C (W ), W = M x [0, T ] defined for arbitrary Xj e supp Uj, tk e supp vk and for f e C ( W ) by

n m

(7) Tôtyf ( x , t ) = £ £ f { xj , tk)u j(x)vk{t) j - 1 Jlc=l

has the properties:

(a ) | | 7 i . y || < 1 ,

(b) I I W - / H < © (/ ; S,y).

P ro of. In virtue of definition (7) we get

IIW II < ll/ll and ll^.vll < 1,

Integral equations in the space-time 241

Is easy to see that

f t m

I

O K X X I /(*» 0-/(*/, tk)K (x)M0-

J=1k=t

Moreover, for every (x, t )e W from definition (3) it follows that

\ f ( x , t ) - f (Xj, tk)\ Uj(x) vk(0 ^ Uj(x) vk(t) O)(/; <5, y)

since d(supp u,) ^ S and d(supp vk) ^ y.

From the above inequality we obtain

|/(x, 0 - Ts.yfix, 0| ^ co(f; Ô, y) for every (x , 0 e lF and also for sup. The proof is complete.

3. The approximate method based on the partition of the unity. The above definitions and lemmas form a basis of certain approximate method for solving integral equations of the form (1) on the compact manifold W in the space-time. That method is based on the partition of the unity and approximation of continuous function.

Equation (1) we write in the form

(8)

=

where the operator N is defined as follows:

(9) N f (x , t) = f f N ( x , t ; y , s ) f (y,s)dn(y)ds.

0 M

We will find an approximate solution of the equation of the form (8)-{9), which has an unique solution for g € C {W ) and N e C (W x W) in space C(W).

Consider the equation

(10) f + N 0J = g ,

where the operator N is defined as follows:

(H ) N it1 = NTôty,

the operator Tô y being of the form (7).

We treat as the solution of the equation an approximate solution of equation (8). Equation (10) can be written in the form:

n m

(12) f { x , t)+ X X /(*/> h) (Nujvk) (x, t) = g(x, t), j= 1 к = 1

where

t

(13) (NujVk) ( x , t ) = J J N {x , t; y, s)uj (y)vk(s)dg(y)ds.

suppu; SUppt)*

[0,Л M

242 L. H^ci a

Setting x = xp (p = 1 , 2 , n), t = tq (q = 1 , 2 , m) in (12), we obtain a system of m n linear algebraic equations of the form

n m

(14) f ( x p, t q) + £ Z Т (Ъ > * к )(Щ » к )(х р, ^ ) = д(хр,Гя).

j= 1 k= 1

The approximate solution of equation (8), denoted by / (x ,t), is of the form (12), where the numbers f ( x p, t q) are the solutions of the system (14),

Now we can state the following theorem.

Theorem. Let Z — (iifi, j = 1,2,..., и, be the partition of the unity on the compact manifold M with the diameter q( Z ) ^ Ô and let Ж — (vk), к = l,2 ,...,m be the partition of the unity on interval [0, T ] with the diameter д(Ж) ^ y.

Suppose that:

Г (x j, tk) g supp Uj x supp vk,

2° the operator N defined by (9) is the operator with a continuous function /V(..,..) on Wx W, W — М у. [0, T ],

3° a){N; Ô,y)-\\R(N)N\\ < 1.

Then the system (14) has a unique solution and an estimation of the error of the approximate solution of equation (8) is the following

(15) II/-/II « --- «.V)+II/H

»(JV;

И,

1 - c o ( N ; ô,y)\\R(N)N\\

where f = R (N )g is the solution of equation (8) and R ( N ) is the resolvent of the operator N .

Proof. Suppose that the system of linear equations (14) has a solution for given g e C (W ) . Then the approximate solution of equation (8), denoted by /, is defined by (12) and/G C (lT). From (10) and (11) we get

f N f = g + N (T 0J - f ) .

Hence

/ = R (N )g + R ( N ) N ( T 0J - f ) . In virtue of Lemma 2 we obtain

ll/ll < \\R(N)g\\ + \\R(N)N\\œ(f; Ô,y).

Since for the function

f(x,t) = g(x,t) + NT0,,f(x,t)

we have

co(f; ô, у) ^ co(g; ô, y)+\\f |j co(N; S, y),

Integral equations in the space-time 2 4 3

therefore from (15) we get

co(/; S,y) ^ co(g; Ô, y)+U\R(N)g\\+ \\R(N) N\\ co(f; 3,yJ]co(N; S,y) and hence

(6) <»(/; s,y) ^

co(g; ô,y)+\\R(N)g\\(o(N; 0,y) 1 ~co(N; ô,y)\\R(N)N\\

If g = 0, then w(g; <5,y) = 0. From this in the presence of (16) and 3°

we have co(f; 6,y) = 0. In virtue of inequality (15) we get / = 0 . In this way it was proved that the system of linear equations (14) has only the trivial solution for g = 0, so for every g e C ( W ) it has a unique solution.

The proof of estimation (15) is based on Lemma 2 and inequality (16).

In this way the proof of theorem is complete.

Remark. If the diameter of the partition of the unity gets smaller, then the solution of the equation becomes more precise.

References

[1 ] K. J o rg en s , L in eare In tegraloperatoren , Stuttgart 1970.

[2 ] A. P is k o r e k , R ôw nania calkow e, Warszawa 1971.

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