BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS. Aysegul Akyuz Dascioglu and Nese Isler

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1 Mathematcal ad Computatoal Applcatos, Vol. 8, No. 3, pp , 203 BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Aysegul Ayuz Dascoglu ad Nese Isler Departmet of Mathematcs, Pamuale Uversty, 20070, Dezl, Turey 2 Departmet of Mathematcs, Mehmet Af Ersoy Uversty, 5000, Burdur, Turey aayuz@pau.edu.tr, sler@mehmetaf.edu.tr Abstract- I ths study, a collocato method based o Berste polyomals s developed for soluto of the olear ordary dfferetal equatos wth varable coeffcets, uder the mxed codtos. These equatos are expressed as lear ordary dfferetal equatos va quaslearzato method teratvely. By usg the Berste collocato method, solutos of these lear equatos are approxmated. Combg the quaslearzato ad the Berste collocato methods, the approxmato soluto of olear dfferetal equatos s obtaed. Moreover, some umercal solutos are gve to llustrate the accuracy ad mplemetato of the method. Key Words- Berste polyomal approxmato, Quaslearzato techque, Nolear dfferetal equatos, Collocato method. INTRODUCTION The quaslearzato method [2, 4, 2] based o the Newto-Raphso method s a effectve approxmato techque for soluto of the olear dfferetal equatos ad partal dfferetal equatos. Am of ths method s to solve a olear th order ordary or partal dfferetal equato N dmesos as a lmt of a sequece of lear dfferetal equatos. So t s a powerful tool that olear dfferetal equatos are expressed as a sequece of lear dfferetal equatos. Ths method also provdes a sequece of fuctos whch coverges rather rapdly to the solutos of the orgal olear equatos. Moreover, ths method has bee appled to a varety of problems volvg dfferet equatos le olear tal ad boudary value problems volvg fuctoal dfferetal equatos [], fuctoal dfferetal equatos wth retardato ad atcpato [5], sgular boudary value problems [], olear Volterra tegral equatos [8,0], mx tegral equatos [3], tegro-dfferetal equato [3]. The Berste polyomals ad ther bass form that ca be geeralzed o the terval [a,b],are defed as follows: Defto. Geeralzed Berste bass polyomals ca be defed o the terval [a, b]; by p, x x a b x ; 0,,,. b a For coveece, we set p, (x) = 0, f < 0 or >.

2 294 A. Ayuz Dascoglu ad N. Isler We gve the propertes of the geeralzed Berste bass polyomals the followg lst: (a) Postvty property: x a, b. p x s hold for all =0,,..., ad all, 0 (b) Uty partto property: 0, x p, x p,x 0 0 p. (c) Recurso's relato property: p, x b x p, x ( x a) p, x b a. (d) Frst dervatves of the geeralzed Berste bass polyomals: d p, x p, x p, x. dx b a Defto.2 Let y :, a b be cotuous fucto o the terval [a, b]. Berste polyomals of th-degree are defed by ( b a) B ( y; x) ya p, x. 0 Theorem. If y C a, b coverges uformly., for some teger m 0, the ( ) ( ) lm ( ; ) ; 0,,, B y x y x m For more formato about Berste polyomals, see [6, 7]. Cosder the mth-order olear dfferetal equato ( m),,,..., m y x f x y x y x y x, a x b uder the tal codtos m, () y c ; 0,,..., m, c[ a, b] ; (2) 0 or boudary codtos m y a y b ; 0,,..., m. (3) 0 Here f s olear fucto ad f f s fuctoal dervatves of the m f x, y x, y( x),..., y ( x) o the terval [a, b],,,, y y ad are ow costats, ad y(x) s uow fucto. I ths paper, frst purpose s to express the olear equato () wth codtos (2) or (3) as a sequece of mth-order lear dfferetal equatos by usg quaslearzato method [9] teratvely:

3 Berste Collocato Method for Solvg Nolear Dfferetal Equatos 295 m y f x, y, y,, y y y f x, y, y,, y. (4) ( m) ( m ) ( ) ( m ) r r r r r r y r r r 0 uder the tal codtos m ( ) yr c ; 0,,..., m, ca, b (5) 0 or boudary codtos m yr a yr b ; 0,,, m. (6) 0 Secod purpose s to approxmate the solutos of lear dfferetal equatos (4) wth Berste polyomals: ( ) ( ) r( ) r; r, 0 ( ba) y x B y x y a p x (7) The paper s orgazed as follows. I Secto 2, some fudametal relatos are gve for the geeralzed Berste bass polyomals ad ts dervatves. Combg the quaslearzato ad the Berste collocato methods, the approxmato solutos of the olear dfferetal equatos are troduced Secto 3. I Secto 4, some umercal examples are preseted for exhbtg the accuracy ad applcablty of the proposed method. The Secto 5 s eded wth the coclusos. 2. FUNDAMENTAL RELATIONS Theorem 2. Ay geeralzed Berste bass polyomals of degree ca be wrtte as a lear combato of the geeralzed Berste bass polyomals of degree + : p,, x p x p, x. Proof. We ca easly prove ths theorem va defto of the geeralzed Berste polyomals. For more formato, see [6]. Theorem 2.2 The frst dervatves of th degree geeralzed Berste bass polyomals ca be wrtte as a lear combato of the geeralzed Berste bass polyomals of degree : d p, x p, x 2 p, x p, x. dx b a Proof. By utlzg Theorem 2., the followg fuctos ca be wrtte as p, x p, x p, x, p, x p, x p, x. Substtutg these relatos to the rght had sde of the property (d), the desred relato s obtaed. Theorem 2.3 There s a relato betwee geeralzed Berste bass polyomals matrx ad ther dervatves the form P () (x) = P (x) N ; =,...,.

4 296 A. Ayuz Dascoglu ad N. Isler Here the elemets of ( + ) ( +) matrx N m,, = 0,,, are defed by:, f 2, f m. ba, f 0, otherwse Proof. From Theorem 2.2 ad codto, p x f < 0 or >, we have, 0 ba ba (2 ) 2 ( ) (4 ) 3 p x p x p x 0, 0,, p x p x p x p x, 0,, 2, p 2, x p b a, x p2, x p3, x 2 ( 2) ba. p, x p2, x p, x p, x p, x p, x p, x ba Hece we obta the matrx relato P (x) =P(x)N such that P ( x) p 0, x p, x p, x P ( x) p 0, x p, x p, x N I a smlar way, the secod dervatves P (x)= P (x)n= P(x)N². Thus we get dervatves of the uow fucto the form P () (x) = P (-) (x)n= P(x)N. Ths completes the proof. Theorem 3. Let x a, b 3. METHOD OF THE SOLUTION ; = 0,,, be collocato pots. Geeral mth-order olear dfferetal equato () ca be wrtte as the matrx form of a sequece of lear dfferetal equatos: m m PN Hr PN Yr G r; r 0,... (8) 0

5 Berste Collocato Method for Solvg Nolear Dfferetal Equatos 297 Here the matrces are H dag h x, P x, x r r p, G ad r gr Y ba r yr a ;, 0,...,. Proof. Let x be chose fucto that provde gve tal or boudary codtos. y0 Cosder the sequece of lear dfferetal equatos (4) for olear dfferetal equato () as follows: m ( m) ( m ) ( ) ( m ) ( ) ( m ) r ( ) r r r r y y r r r r r r r 0 y f x, y, y,..., y y f x, y, y,..., y y f x, y, y,..., y. We use the Berste collocato method for solvg these seres of lear equatos. The expresso (7) ca be deoted by the matrx form ( ) ( ) ( ) y ( x) B y ; x P Y. r r r By utlzg Theorem 2.3, the dervatves of the uow fuctos ca also be wrtte by ( ) y ( x) P x N Y ; 0,,..., m. (9) r r Substtutg the collocato pots ad relato (9) to equatos (4), we obta the lear algebrac equato systems m m P x N Y h x P x N Y g x ; 0,, (0) r r r r 0 ( ) ( ) r( ) r; ; 0,,, hr such that y x B y x m. Here x ad gr x by ( m) hr x f x, yr x, yr ' x,..., yr x, y ( m),, ',..., g x f x y x y x y x r r r r m 0 ( ) y ( m) ( ) r r r r f x, y x, y ' x,..., y x y x. s deoted Cosderg the matrces P( x0 ) hr x0 0 0 g rx0 ( x ) P P, 0 hr x 0 H r, g r x G r, P x 0 0 h r x g r x the lear equato systems (0) ca be deoted by the matrx form (8) ad the proof s completed. We ca solve the dfferetal equato wth varable coeffcets uder the codtos gve the followg steps: Step. The equato (8) ca be wrtte the compact form W ; G, W Y G or r r r r r

6 298 A. Ayuz Dascoglu ad N. Isler m m so that W PN H PN. Ths matrx equato () correspods to a lear r 0 r algebrac systems wth uow coeffcets y ; 0,, r r teratvely. Step 2. From the expresso (9), the matrx forms of the codtos (5) ad (6) ca be wrtte respectvely m c v,0 v, v, 0 V P N m [ a b ] u,0 u, u, 0 U P N P N or compactly V Y or [ V ; ], r U Y or [U ; ]. r Step 3. To obta the solutos of equatos (4) uder the codtos (5) or (6), we add the elemets of the row matrces (2) or (3) to the ed of the matrx (). I ths way, we have the ew augmeted matrx r; r W G. Here the augmeted matrx s a (+m+) (+) rectagular matrx. Ths ew matrx equato shortly ca be deoted by W r Y G r. r Step 4. If ra W r ra r; r W G, the uow coeffcets y ; 0,, r r are uquely determed teratvely. These ds of systems ca be solved by the Gauss Elmato, Geeralzed Iverse ad QR factorzato methods., 4. NUMERICAL RESULTS Two umercal examples are cosdered by usg the preseted method o ba collocato pots x a ; 0,,. Numercal results are wrtte Matlab 7.. Example 4. Cosder the followg olear boudary value problem [2]: y y e ; 0 x ; y 0 y 0 The exact soluto of the above equato s yx c cx where c = Let be y ( x) 0 0 Table. Maxmum errors of Example 4.. E E 2 E 3 2.0e e e e e e e e e e e e e e e l 2 2l sec / 2 / 2,

7 Berste Collocato Method for Solvg Nolear Dfferetal Equatos 299 Usg the Berste collocato method, the maxmum errors are gve the Table. The umercal results show that the proposed method ca be applcable to the olear dfferetal equatos ad have effectve results for creasg. Example 4.2 Cosder the followg olear boudary value problem: 3 y 2 y ; x 0, ; y 0, y / 2 The aalytc soluto of the above equato s y( x) ( x). Let be Table 2. Absolute errors of Example 4.2. E E 2 E 3 E e e e e e e e 006.4e e e e e e e e e - 05 y0 x x 2. I Table 2, the maxmum errors are computed wth creasg. We show that the preseted method coverges rapdly to the exact soluto of the olear dfferetal equatos for creasg. 5. CONCLUSIONS I ths study, by usg quaslearzato techque, olear dfferetal equatos uder the tal or boudary codtos are expressed as a sequece of the lear dfferetal equatos teratvely. The, a collocato method based o geeralzed Berste polyomals s developed for solvg these equatos. If y(x) ad ts dervatves are cotuous fuctos o bouded terval [a,b], the the Berste collocato method ca be appled to ay tal or boudary value problems. Moreover, ths method has bee tested o two olear boudary problems, ad umercal results have bee preseted for showg applcablty, accuracy of the proposed method. Cosequetly, all of the reasos are revealed that the proposed method s ecouragg for solutos of the other problems volvg dfferet equatos. Acowledgmet- Ths wor s supported by Scetfc Research Proect Coordato Ut of Pamuale Uversty, No:202FBE REFERENCES. B. Ahmad, R. A. Kha ad S. Svasudaram, Geeralzed Quaslearzato Method for Nolear Fuctoal Dfferetal Equatos, Joural of Appled Mathematcs ad Stochastc Aalyss 6, 33-43, R. E. Belma ad R. E. Kalaba, Quaslearzato ad Nolear Boudary Value Problems, Amerca Elsever Publshg Compay, New Yor, F. Calo, F. Muoz ad E. Marchett, Drect ad teratve methods for the umercal soluto of mxed tegral equatos, Appled Mathematcs ad Computato 26, , 200.

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