The Numerical Solution of Singular Fredholm Integral Equations of the Second Kind

Transcription

1 WDS' Proceedigs of Cotributed Papers, Part I, 57 64, 2. ISBN MATFYZPRESS The Numerical Solutio of Sigular Fredholm Itegral Equatios of the Secod Kid J. Rak Charles Uiversity, Faculty of Mathematics ad Physics, Prague, Czech Republic., Departmet of Numerical Mathematics Abstract. This paper deals with umerical solutio of a sigular itegral equatio of the secod kid with special sigular kerel fuctio. The umerical solutio i this paper is based o Nystrom method. The Nystrom method is based o approximatio of the itegral i equatio by umerical itegratio rule. Covergece of the umerical solutio is show. The paper is completed with umerical examples. Itroductio Itegral equatios of the secod kid are ecoutered i various pieces of sciece (plasticity, plasticity, heat ad mass trasfer, oscillatio theory, fluid dyamics, filtratio theory, electrostatics, electrodyamics,..). Oe reaso for usig itegral equatios istead of differetial equatios is that all coditios specifyig boudary value problem or iitial value problem ca be reduced to oe itegral equatio. May methods for umerical solutio were developed. Most of them are described i [5]. The mai idea is to trasform itegral equatio ito system of liear equatios. Oe way is to fid solutio that satisfies itegral equatio approximately. We will obtai collocatio ad Galerki methods. Other way is to approximate itegral by umerical quadrature of cubature. We will obtai Nystrom methods. Problem is whe the kerel fuctio i itegral equatio has sigularity. Assume that λ. We will describe umerical method for the itegral equatio of the followig form λy(x) k(x, t)y(t)dt = f(x) () k(x, t) is called the kerel fuctio. For existece ad uiqueess of solutio of () let s rewrite () ito operator form (λi K)y = f (2) where K : C[, ] C[, ] is liear operator defied by Ky(x) = k(x, t)y(t)dt (3) For this paper assume a special case of sigularity called diagoal. It meas that there exist fuctios h C([, ]) C([, ]) ad g C(, ] L (, ) such that the kerel fuctio is of the form k(x, t) = g( x t )h(x, t) (4) ad fuctio g has sigularity at x =. Operator K is assumed to be compact itegral operator o C([, ]). The existece ad uiqueess of solutio of () ad (2) is described by Fredholm alterative theorem. See [4] for details. From Fredholm alterative we also have that if the operator (λi K) exists it is bouded operator. Numerical solutio - Nystrom method Now we will show umerical method for solutio of () with kerel fuctio satisfyig (4). Let s have umerical itegratio rule v(x)dx ω,j v(x,j ) (5) where x,j are called the ode poits ad satisfy x, < x,2 <... < x, < x, 57

2 Asume that umerical itegratio rule (5) coverges for all cotiuous fuctios. Here the itegratio rule ca t be applied directly to () because the kerel fuctio of form (4) is sigular. We will oly use itegratio rules that satisfy for itervals I = [a, b) ad (a, b], where b a / the coditio where c < is positive costat ad {j;x,j I } ω,j c (6) ω,j (7) Coditios (6) ad (7) are satisfied Romberg itegratio rules ad usual compoud itegratio rules. We have to formulate more coditios of fuctio g i (4). Assume that fuctio g(x) is o-icreasig fuctio o (, ] ad for all x (, ], g(x). Let {δ } be sequece δ = /. Let s defie approximatio g of g { g(r), r [δ, ] g : g (r) = (8) g(δ ), r [, δ ] Assume, that ad g () as (9) δ g(r) g (r) dr () The it easy to see that g is also o-icreasig fuctio, g C[, ] ad g (x) g(x) () g () g m (), m The sigularity i kerel fuctio i () ca be weake by followig steps. First let s rewrite () ito the form as i [2]. [ ] λ k(x, t)dt y(x) k(x, t)[y(t) y(x)] dt = f(x) (2) The the kerel fuctio i the right itegral o the left side of (2) ca be approximated by fuctio k (x, t) = g ( x t )h(x, t) (3) Note that k (x, t) = k(x, t), x t δ (4) We obtai [ λ ] k(x, t)dt y(x) ad we ca use umerical itegratio rule. We obtai [ λ ] k(x, t)dt y(x) k (x, t)[y(t) y(x)] dt = f(x) ω,j k (x, x,j )[y(x,j ) y(x)] = f(x) (5) Now let s ru x through the ode poits x,i ad we obtai system of liear equatios λ + ω,j k (x,i, x,j ) k(x i,, t)dt y(x,i ) ω,j k (x,i, x,j )y(x,j ) = f(x,i ),j i,j i (6) The itegral i (6) ca be calculated aalytically or by some special quadrature scheme. This system of liear equatios give us umerical solutio at the ode poits. I the followig we will write ω j istead of ω,j ad x j istead of x,j. 58

3 Covergece of the umerical solutio Existece of umerical solutio i (6) is here doe by operator calculus. Let s defie operator K : C[, ] C[, ]. K y(x) = ω j k (x, x j )y(x j ) (7) We ca se that operator K : C[, ] C[, ] is for each bouded ad compact operator (it is cotiuous liear ad fiite raked operator - see [4] for details). For rewritig (5) ito operator form we eed to defie differet operator K : C[, ] C[, ]. K y(x) = ω j k (x, x j )[y(x j ) y(x)] + k(x, t)y(x)dt (8) The (5) ca be rewritte ito If we defie fuctio u we ca se that (λi K )y = f (9) K = K (K u Ku)I This operator is ot compact uless K u = Ku. First let s proof existece of (λi K ) for eough large. We will use followig theorem from []: Theorem Let X be a Baach space, let operators S, T be bouded o X ad let the operator S be compact. For give λ let s assume that λi T is bijectio o X (which meas (λi T ) exists, is bouded ad R(λI T ) = X). If (T S)S < λ (λi T ) (2) the (λi S) exists, is bouded ad (λi S) If (λi T )y = f ad (λ S)z = f, the + (λi T ) S λ (λi T ) (T S)S (2) If y z (λi S) T y Sy (22) (K K )K (23) the the assumptio (2) is satisfied ad theorem ca be used for T = K ad S = K. Usig theorem there exists such that for > (λi K ) exists, is bouded ad if (λi K)y = f ad (λi K )z = f the y z (λi K ) Ky K y We eed to proof (23) Let s defie aother operator K : C[, ] C[, ] Now where K y(x) = k (x, t)y(t)dt (24) (K K )K = (K K + K K )K (K K )K + K K K (25) e (x, ξ) = (K K )K max x [,] k (x, v)k (v, ξ)dv ω j e (x, x j ) ω j k (x, x j )k (x j, ξ) 59

4 This is the umerical itegratio error for cotiuous itegrad k (x,.)k (., ξ). I [] is show, that max e (x, ξ) x,ξ [,] ad the first term i (25) goes to zero. Fuctio h from (4) is cotiuous. The there exists costat M < M = max h(x, t) x,t [,] For the secod term i (25) hece (K K )y max x [,] [k(x, t) k (x, t)]y(t) dt y max x [,] k(x, t) k (x, t) dt = δ = y max k(x, t) k (x, t) dt M y 2 g(r) g (r) dr x [,] t [,], x t <δ K K 2M δ g(r) g (r) dr by assumptio (). Now we eed to boud K idepedetly of. We will use followig lemma Lemma 2 Let [a, b] be bouded iterval. Assume that v is cotiuous o-icreasig fuctio o [a, b] ad v(x) for every x [a, b]. Assume that umerical itegratio rule coverges for all cotiuous fuctios ad satisfies coditios (6), (7). The ω j v(x j ) c b v(a) + c v(t)dt (26) PROOF Let fuctio v be o-icreasig fuctio o [a, b] ad for all x [a, b], v(x). Let s defie umbers a j, = a + (j ) b a, b j, = a + j b a, j =,..., ad itervals I j, = [a j,, b j, ), j =,..., The ad Let s defie ω j v(x j ) = i= {j;x j I,i} I j, = [a, b) I j, I i, =, j i ω = ω j v(x j )+ ω v(b) a { ω if x = b if x < b v(a i ) i= where c is costat from (6). By other calculatios i= v(a i) + v(b) = v(a ) + v(a i) + v(b ) = v(a) i=2 From the defiitio of Rieamm itegral ad the proof is complete. i= v(b i) b a {j;x j I,i} + v(t)dt i= ω j +ω v(b) c ( i= ) v(a i) + v(b) v(b i) + v(b ) v(a) + i= v(b i) 6

5 Now let s choose some x [, ] ad y C[, ] such that y =. By lemma 2 K y(x) M ω j g( x j x ) 2Mc[ g () + By (9) g / ad is bouded for all large eough. g(x). By () g (t)dt g(t)dt g (t)dt] We assumed that g L (, ). so there exists a costat c such that K c. For proof of existece (I K ) ca be used followig theorem from [3]. Theorem 3 Let operators M, L be bouded liear operators o a Baach space X. Assume that (I M) exists as a bouded Liear operator o X ad The (I L) exists ad is bouded, ad = (I M) (L M)L < (I L) + (I M) L (I L) y (I M) y (I M) Ly My + (I M) y for y (I L)X. If L is compact the (I L)X = X Let s use theorem 3 for M = λ K ad L = λ K. We eed <. Note that for u the ad from (27) ad (23) K = K (K u Ku)I K K = K u Ku (27) (K K ) K (28) Sice (λi K) is bouded liear operator, (I M) exists ad is also bouded liear operator. From (28) there exists such that (I M) λ 2 (K K ) K < for Hece < for operators M = λ K ad L = λ K whe >. By theorem 3 operator (I λ K ) exists as bouded operator from (I λ K )C([, ]) to C([, ]). Hece operator (λi K ) exists. Eve from (27) for all sufficietly large is K K < λ ad by iverse theorem (see [4]) exists (λi K + K ) from C([, ]) to C([, ]). Operator (λi K ) ca be rewritte as λi K = [I K (λi K + K ) ](λi K + K ) Sice K (λi K + K ) is compact operator from C([, ]) to C([, ]) it follows that if (λi K ) exists the (λi K ) : C([, ]) C([, ]). Existece of (λi K ) was doe by theorem 3. So we have that (λi K ) : C([, ]) C([, ]). Now let s derive error estimate. Let y be solutio of (λi K)y = f ad y be solutio of (λi K )y = f. y y = (λi K ) f y = (λi K [ ) f (λi K ] )y = (λi K [ ) (λi K)y (λi K ] )y 6

6 hece y y (λi K ) K y Ky K y(x) Ky(x) = where ad ω j k (x, x j )[y(x j ) y(x)] E y(x) = E 2 y(x) = ω j k (x, x j )[y(x j ) y(x)] k (x, t)[y(t) y(x)] dt k(x, t)[y(t) y(x)] dt = E y(x) + E 2 y(x) (29) k (x, t)[y(t) y(x)] dt (3) k(x, t)[y(t) y(x)] dt (3) E is umerical itegratio error ad E 2 depeds o approximatio of sigular kerel fuctio k(x, t) by k (x, t). More estimates will be doe by followig example. Example Let s have a equatio y(t) y(x) dt = f(x) (32) x t γ where γ (, ) Here the fuctio g from (4) is g is oicreasig o (, ]. Let s defie δ = /. Approximatio (8) g(r) = r γ (33) g (r) = { r γ, r [δ, ] γ, r [, δ ] (34) We eed to verify compactess of K o o C[, ] Ky(x) = ad two assumptios (9) ad (). If we defie K y(x) = g ( x t )y(t)dt we ca se that the latter itegral operator K is compact for each. Ky(x) K y(x) ad g( x t ) g ( x t )y(t) dt 2 y K K y(t) dt (35) x t γ (r γ γ )dr = 2γ γ γ y 2γ (36) γ γ K is a sequece of compact operators o C[, ]. By (36) K K. Hece K is compact operator o C[, ]. This is a stadard result ad ca be foud i ay book of fuctioal aalysis (for example i [4].) We have eve proofed (). Sice g () = γ as is (9) also satisfied. Let s try compoud midpoit itegratio rule for (32) with γ = /2. ω j = /, x j = 2 + j, =, 2,..., ad f(x)dx ω j f(x j ) 62

7 Assume that v C 2 [, ]. The compoud midpoit rule satisfies (see [6] for details) v(x) ω j v(x j ) 24 2 max x [a,b] v (x) Hece For E 2 E 2 y(x) max x [,] E y(x) D 2,where D = v (x) 24 k(x, t) k (x, t) y(t) y(x) dt 2 y k(x, t) k (x, t) dt = = 2 y g( x t ) g ( x t ) dt = 4γ γ γ y If we add other assumptios o fuctio y(x) we ca have better results. Assume that The E 2 y(x) (37) y(t) y(x) A t x α for < α ad A < (38) k(x, t) k (x, t) y(t) y(x) dt A k(x, t) k (x, t) t x α dt t [,], t x < A α k(x, t) k (x, t) dt = 2 A γ α+ γ (39) To demostrate error behavior let s choose five differet fuctio f(x). The system of liear equatios (6) is solved by Gauss elimiatio. All itegratio is doe by software Maple [7]. The error developmet is described by ratio fuctio r where is umber of ode poits I first example let s choose r = y(x) y /2(x) y(x) y (x) f(x) = x ( 4 3 x x x x x) The exact solutio is y(x) = x ad satisfies (38) with α =. Followig two table shows umerical solutio y (x) ad error i the ode poits. Followig table shows error evolutio. 5 ode poits x y(x) y (x) y(x) y (x),,,966474,335926,3,3, ,743679,5,5,5,,7,7, ,743679,9,9, , odes y(x) y (x) r odes y(x) y (x) r 5, ,4444 3,84,234 3,2 6,384 3,2 2,6383 3,63 32,57 2,73 4,78 3,74 64,8 2,8 From (37) ad (39) the theoretical value of r should be approximately 2, 83. For other examples oly error evolutio tables will be show. Let s ow choose f(x) = x 2 ( 6 5 x2 x x2 x x x + 2 x) 5 The exact solutio is y(x) = x 2 ad also satisfies (38) with α =. So we are expectig r 2, 83. Followig table shows error evolutio. 63

8 odes y(x) y (x) r odes y(x) y (x) r 5, ,2399 3,, ,69 6,45 3,9 2,4254 3,89 32,36 3,5 4,3857 3,69 64,435 3,3 Let s ow choose f(x) such that exact solutio is y(x) = e x. (38) is also satisfied with α =. we expect r Error evolutio is show by followig table. odes y(x) y (x) r odes y(x) y (x) r 5, ,426 3,8, ,58 6,482 2,95 2,68 3,83 32,636 2,95 4, ,84 64,549 2,98 Now let s choose f(x) so that the exact solutio is y(x) = x. The solutio satisfies (38) with α = /2. The r is expected approximately 2. odes y(x) y (x) r odes y(x) y (x) r 5, ,564 2,8, ,2 6, ,7 2, ,7 32, ,5 4, ,28 64,5234 2,2 For last example let s choose f(x) so that the exact solutio is y(x) = 4 x. The solutio satisfies (38) with α = /4. r is expected approximately, 68. Coclusio odes y(x) y (x) r odes y(x) y (x) r 5, ,48252,87, ,8 6,227937,83 2, ,32 32,25873,8 4,77643,9 64,7393,79 From the tables we ca se that results correspods to theory. Other methods - projectio methods or product itegratio methods (see [5]) have better error behavior. But here oly diagoal members of the matrix are itegrals (this is ot true for projectio ad product itegratio methods). Others are easy to compute from the kerel fuctio. This is the advatage of this method whe usig some iterative scheme for solvig liear systems of equatios, because we ca reduce memory usage. Disadvatage is that matrixes are fully populated. Refereces [] Kedall E. Atkiso, Weimi Ha: Theoretical Numerical Aalysis, Spriger-Verlag, ISBN: , 2. [2] L. V. Katorovich, V. I. Krylov: Approximate Methods of Higher Aalysis, Itersciece, New York, 958. [3] P. M. Aseloe: Collectively compact operator approximatio theory, Pretice-Hall, Egelwood Cliffs, NJ, 97. [4] Jaroslav Lukes: Zapisky z fukcioali aalyzy (Karolium), ISBN [5] Kedall E. Atkiso: The Numerical Solutio of Itegral Equatios of the Secod Kid (Cambridge Uiversity Press), ISBN: , 997. [6] Prem K. Kythe, Michael R. Schaferkotter: Hadbook of Computatioal Methods for Itegratio (Chapma ad Hall/CRC), ISBN: , 24. [7] Maplesoft: Maple, 64