Online Publication Date: 12 December, 2011 Publisher: Asian Economic and Social Society

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1 Ole Publcato Date: December, Publsher: Asa Ecoomc ad Socal Socety Soluto Of A System Of Two Partal Dfferetal Equatos Of The Secod Order Usg Two Seres Hayder Jabbar Abood (Departmet of Mathematcs, College of Educato, Babylo Uversty, Babylo,Iraq) Ctato: Hayder Jabbar Abood (): Soluto Of A System Of Two Partal Dfferetal Equatos Of The Secod Order Usg Two Seres Joural of Asa Scetfc Research, Vol., No.8,pp

2 Joural of Asa Scetfc Research, (8), pp Soluto Of A System Of Two Partal Dfferetal Equatos Of The Secod Order Usg Two Seres Abstract Author (s) Hayder Jabbar Abood Departmet of Mathematcs, College of Educato, Babylo Uversty, Babylo-Iraq. E-mal: drhayder_jabbar@yahoo.com For a system of two partal dfferetal equatos of secod order, we obta ad justfy two asymptotcs solutos the form of two seres wth respect to the small parameter. We have prove the soluto s uque ad uform the doma, ad, further, each the asymptotcs O. approxmatos are wth Key Words: Partal dfferetal equatos ; System ; Asymptotc 3. Itroducto Leveshtam (9), cosdered systems of ordary dfferetal equatos of the frst order whose terms oscllate wth a hgh frequecy ω.for the problem o perodc solutos, the author justfes the averagg method ad establshes a posteror bouds for the error of partal sums of the complete asymptotc expaso for the soluto. Asymptotc expasos of perodc solutos of secod ad thrd- order equatos ad formal asymptotcs of such solutos the case of equatos of arbtrary order were costructed Abood, (4). The frst-order asymptotc form was obtaed ad proved for the soluto of a system of two partal dfferetal equatos wth small parameters the dervatves for the regular part, two boudary-layer parts ad corer boudary part Vasl'eva ad Butuzob (99). I Leveshtam ad Abood (5), a algorthm of asymptotc tegrato of the talboudary-value problem for the heat-coducto equato wth mor terms (olear sources of heat) a th rod of thcess / oscllatg tme wth frequecy was proposed. I the preset paper, we cosder a system of partal dfferetal equatos of the secod order ad solve ths system wth the ad of two seres ad some codtos. Our paper cotues the le of research tated Leveshtam ad Abood (5). Examples of applcatos of systems of secod order partal dfferetal equatos modellg ca be foud elastcty Neratza ad Kadlas (8) ad paced-bed electrode Xao-Yg Q ad Ya-Pg Su (9). Statemet Of The Problem We study a system of secod order partal dfferetal equatos wth tal-boudary-value codtos u u u b x a x, tu f u, x, y, t, t y x v v v b x a x, tv f u, x, y, t, t y x the doma ( x, y, t) ( x) ( y) ( t T ). (.) The tal boudary codtos are 47

3 Soluto Of A System Of Two Partal Dfferetal Equatos.. u, u, v, v, t x, t x, u y v,. y y, y, (.) where s a small parameter, ad the fuctos f ( u, x, y, t, ), ad f (,,,, ) u x y t are cotuous ad ftely dfferetable wth respect to each of ther argumets. The costructo s made uder the followg codtos. b x a x t ad f,, I. The fuctos,,. II. We ca assume that b. Ifb b x b x have cotuous dervatves of order, the mag the chage of the varable shall assume that b ad b. x z b d, ad we Algorthm For Costructo Of The Asymptotcs We see a asymptotcs expaso of the soluto of problem (.) ad (.) as the followg two seres the powers of the form u( x, y, t, ) [ u ( x, y, t) p u( x, y, ) q u(, y, t)], v( x, y, t, ) [ v ( x, y, t) p v( x, y, ) q v(, y, t)], (3.) where u ad fuctos descrbg boudary layers ear the tal stat of tme The boudary-layer varables are t, ad v are coeffcets of the regular part of the asymptotc, pu, pv, qu ad pvare boudary-layer t ad the eds of the rod x ad x. x. Regular Parts Of The Asymptotcs u Ad v We substtute seres (3.) to equatos (.) - (.) ad the coeffcets of the same powers of the left ad rght-had sdes of the obtaed relatos are equated. If the varable s assumed to be depedet of t ad fuctos depedg o ε are represeted by the correspodg asymptotc seres the problem (.) ad (.) becomes 48

4 Joural of Asa Scetfc Research, (8), pp u u u b x a x, tu f u, x, y, t, t y x v v v b x a x, tv f v, x, y, t, t y x u, u, v, v, t, x, t, x, u y v,. y y, y, (4.) The followg problems for the regular coeffcets u ad v are obtaed: u b x a x, tu f u, x, y, t, x v a x, tv f v, x, y, t, x u, y, t, v x, y,, u y y, v,, y y, (4.) By drect tegrato, t ca be see [V (), Va ad Room (8), Toovyy ad Che-Chg Ma,(9), Tosh, So Sh, Muraam ad Ngoc(7)] system (4.) s equvalet to system of tegral equatos x x u x t b p a p t dp b v t f y t d, exp,,,,, (4.3) t t v x t a x p dp u x s f x y t ds s, exp,,,,. (4.4) Substtutg (4.4) (4.3), we arrve at a tegral equato wth respect to u x t x t u x, t G x, t,, s u, s d ds g x, t, where G x, t,, s ad g (, ) of the resolvet x t s of the Kerel G x t s, : x t are ow fuctos. The soluto of ths tegral equato ca be expressed terms,,,,,, (as [V (), Va ad Room (8), Toovyy ad Che-Chg Ma,(9), Tosh, So Sh, Muraam ad Ngoc(7)]) u x, t g x, t x, t,, s g, s d ds. x t After that, the fucto v x, t ca be determed by (4.4). The system of equatos for u ad v s 49

5 Soluto Of A System Of Two Partal Dfferetal Equatos.. u b( x ) = a ( x, t ) u( x, t ) + f ( u x, y, t ) u + f ( u x, y, t )- x u u u -, t y v = a( x, t ) v ( x, t ) + f ( v x, y, t ) u + f ( v x, y, t )- t v u v - ( ) = - ( ) ( ) = - ( ) u, y, t q u, y, t, v x, y, p v x, y,, y x, (4.5) u v =, =. y y y =, y =, From system (4.5) ad drect tegrato, we have the system of tegral equatos x æx ö ç ò - - (, ) exp ç ( ) (, ) ( s )[ ( s, ) ( s, ) u x t = b p a p t dp b a t v t + ç è ø ò s u u f ( u x, y, t ) u + f ( u x, y, t )- - ] d s, u t y t æt ö ç ò (, ) exp ç (, ) [ (, ) (, ) v x t = a x p dp a x s u x s + ç è ø ò s u v ( ) ( ) + f v x, y, t v + f v x, y, t - - ] ds. v y x (4.6) (4.7) Substtutg (4.7) (4.6), we arrve at the tegral equato wth respect to u x, t : x t u x, t G x, y, t,, s u, s d ds g x, y, t, where G x, y, t,, s ad (,, ) terms of the resolvet x y t s of the Kerel G x y t s [V (), Va ad Room g x y t are ow fuctos. The soluto of ths tegral equato ca be expressed,,,,,,,, (8), Toovyy ad Che-Chg Ma,(9), Tosh, So Sh, Muraam ad Ngoc(7)] ad we have,,,,,,,,. After that, the fucto v,, x t u x y t g x y t x y s g s d ds (4.7). For regular terms u ad v we obta the problem x y t ca be determed by 4

6 Joural of Asa Scetfc Research, (8), pp u b( x ) = a( x, t ) u ( x, t ) + ( u x, y, t ), x u u (, y, t ) = - qu (, y, t ), =. v t y y =, ( ) ( ) ( ) = a x, t v x, t + v x, y, t, v v ( x, y,) = - pv ( x, y,), =. y y =, (4.8) (4.9) - - = å - - =! u t y u v ( v x, y, t) å f ( v x, y, t ) v ( x, t ) =! u y x where ( u x, y, t) f ( u x, y, t ) u ( x, t ) = - - u u problem (4.5) ad ca be solved by reducto to a system of tegral equatos. Boudary Parts Of The Asymptotcs pu Ad pv ad. Ths problem s qute smlar to We costruct the followg group of coeffcets of two seres (3.) that s the boudary-layer parts pu x, y,, ad pv x, y,,. We cosder problem (.) a eghbourhood of the upper boudary t of the doma ad perform the chage of varables t ; we obta the system b x a x t pu f pu x y pu pu pu y x,,,,, pv pv pv a x, t pv f pv, x, y,,, y x dpu dpv pu x, y, u x, y,, pv x, y, v x, y,,,. dy dy y, y, (5.) The boudary-layer fuctos are puad pv. I ths case pv. For pu we obta the problem pu pu b x a x, pu, x X,, x p u, y,, p u x, y, u x, y,, dp u dy dp v,. dy y, y, (5.) By drect tegrato, equato (5.) has a classcal soluto 4

7 Soluto Of A System Of Two Partal Dfferetal Equatos.. u ( x ),exp a ( x s), ds, pu x,,, x where x b d ad z s the fucto verse to z x The system of equatos for puad pv s pu pu b x a x, t pu x, x p u x, y, u x, y,, p u, y,, dpu dy y,, pv a x, t pv x,, x p v x y v x y,,,, dpv dy y,. a pu t y Here x, x, p u x, f p u, x, y,, a pv x, x, pv x, f pv, x, y,, t y same way we obta,, x. ad s smooth. I the a x, p v x, s ds, x p v a x p v x s ds x,. u ( x ),exp a ( x s), pu ( x s), ds, x, x. 4

8 Joural of Asa Scetfc Research, (8), pp The fucto puca be defed as the soluto of the problems pu pu bx a x, t pu x, y,, x p u x, y, u x, y,, p u, y,, dpu dy y,, (5.3) (5.4) where a x, y, [ x, p u x, f p u, x, y,,! t p u p u,,,, ], y pu x y sce (5.3) s a partal dfferetal equato of the frst order ad frst degree. Here the smooth fucto xy,, s ow. Usg the tal ad boudary codtos (5.4), we obta u ( x ), exp a ( x s), pu ( x s), ds, x, x. For the fucto where pv, t s the soluto of the problem pv p v x y,, vx, y, dpv dy y, xy,,,, a x, y, a x, t p v [ x, p vx, f p v, x, y,,! t f p v p v f pv, x, y,, ], p v x y 43

9 Soluto Of A System Of Two Partal Dfferetal Equatos.. that satsfes the codto p x, y,, where xy,, for x ad wth partal dervatve mag jump o the characterstc le x x, y, r dr, x, pv x, y, x, x, where the fucto,, p v x y s smooth everywhere. Boudary Parts Of The Asymptotcs qu Ad qv Now, we fd the coeffcets q u, y, t ad q v, y, t s a ow cotuous fucto vashg. Itegratg, we obta. Cosder problem (.) ad the codtos (.) a eghbourhood of the left boudary of the doma ad perform the chage of varable system qu qu qu b a t qu f qu y t t y! qv qv qv t y x,,,,, a, t qv, y, t, f qu,, y, t,. For q u y t we obta the equato b,, qu.. We obta the From ths equato, tag to accout the stadard codto for boudary parts at fty, that meas q u, t, ad so we have q u, y, t. Now, for q v y t,, we have the problem qv qv a, tqv, y, t,, t T, t q v, y, t v, y, t, q v, y,, qv y y, Sce ths s a partal dfferetal of frst order ad frst degree, t s soluto have the form q v, y, t v, y, t exp a, y, s t ds, t T, t. 44

10 Joural of Asa Scetfc Research, (8), pp It s possble to fd quad qv the same way. Fally, we wll cotue to fd For quwe have the equato quad qv. qu b, y, t, (6.) where a, y, t [, y, tq u f q u,, y, t,! t s ow ad cotuous everywhere. The fucto yt the form q u q u f qu, x, y,, ], q u t y,, for t (s evdet from our fdgs the above), so we ca see ths fucto, y, t z y t, t, for t. (6.) The fucto, t s smooth everywhere. By the codto q u y t q u, y, t We ca defe the part q v, y, t b z y t s s, tds, t T t, t. as the soluto of the problem,, ad tegratg (6.), we have qv qv t, yt,, q v, y, t v, y, t, q v, y,, qv y y, (6.3) (6.4) here 45

11 Soluto Of A System Of Two Partal Dfferetal Equatos.. a, y, t a, y, t p v [ x, p v, y, t f p v,, y, t,! t p v p v f pv,, y, t, ]. p v x y, yt, s a ow cotuous fucto that satsfes the codto p v y t ad wth partal dervatve mag jump o the characterstc le q v, y, t where the fucto q v, y, t Also,,, vashg for t t. Solvg (6.3) ad usg (6.4), we obta v, y, t exp, y, s t ds, t T, t. s smooth everywhere. 7. JUSTIFICATION OF THE ASYMOTOTICS By U x, y, t, ad,,, V x y t we deote the - th partal sums of the seres (3.). Theorem. The soluto ux, y, t,, vx, y, t, of problem (.) ad (.) admts the asymptotc soluto u( x, y, t, ) U ( x, y, t, ) O, v( x, y, t, ) V ( x, y, t, ) O, (7.) uformly the doma x y t T. Proof. We set u U w ad v V w. Substtutg ths (.) ad (.) we obta the followg problem for the remader terms w ad w w w w b x a x, t w x, y, t,, t y x w w w b x a x, t w x, y, t,, t y x w, w, w, w, t x, t x, w y w,. y y, y, Obvously, the homogeeous terms ad ca be estmated as x, y, t, O shall prove that the same estmate s vald for w ad w. I vew of the equaltes, u U u U U U w O v V w O ths wll drectly mply the asserto of the theorem., uformly. We 46

12 Joural of Asa Scetfc Research, (8), pp xyt We mae the chage of the varables w re,, where cost >. We obta the followg system of equatos for r ad r : r r r b x c x, t r ˆ x, y, t,, t y x r r r b x c x, tr ˆ x, y, t,, t y x (7.) wth the homogeeous boudary codtos, as the case of r,,. Here ˆ xt c a x, t b x, c a x, t, e O( ). Assume that r has a maxmum at ( ) a pot x, t r has a maxmum at a pot x, t, ad assume that r r of ad cosder the frst equato of (7.) at at ). We rewrte ths equato as, (If r r. We shall, the we shall cosder the frst equato of (7.) r r r b x c x, tr ˆ x, y, t,, t y x (7.3) Assume that the fucto r s egatve ad has a mmum at. (The case of a postve maxmum ca be r r cosdered the same way.) The ad at ths pot (the equalty sg s possble oly f les o t x the boudary of ). Hece the left-had sde of (7.3) s egatve at ad s ot larger tha r, whle the rght- had sde s of order. Cosequetly, r max r x, t O.Sce r r that r max r x, t O.Therefore r x, t O xt w r e O uformly G. The proof of Theorem s complete. Refereces, t follows uformly. Hece we also have Abood, H. J. (4) Asymptotc Itegrato Problem of Perodc Solutos of Ordary Dfferetal Equatos Cotag a Large Hgh-Frequecy Terms, Dep. VINITI, o., 338-B4, 8 Pages, Moscow, Russa, Abood, H. J. (4) Asymptotc Itegrato Problem of Perodc Solutos of Ordary Dfferetal Equato of Thrd Order wth Rapdly Oscllatg Terms, Dep. VINITI, o. 357-B4, 3 pages, Moscow, Russa, Leveshtam, V. B. (9) Katorovch Theorem ad Justfcato of Asymptotcs of Solutos of Dfferetal Equatos wth Large Hgh-Frequecy Summads Joural of Mathematcal Sceces Vol.58, No., pp Leveshtam, V. B. ad Abood, H. J. (5) Asymptotc tegrato of the problem o heat dstrbuto a th rod wth rapdly varyg sources of heat Joural of Mathematcal Sceces, Vol.9, No., pp Mara S. Neratza ad Chrs B. Kadlas (8) A boudary elemet method soluto for asotropc ohomogeeous elastcty Acta. Mech., Vol., pp

13 Soluto Of A System Of Two Partal Dfferetal Equatos.. Tosh Nato; Jog So Sh; Satoru Muraam ad Pham Huu Ah Ngoc(7) Characterzatos of lear Volterra tegral equatos wth oegatve erels J. Math. Aal. Appl., Vol.335, pp Vasl'eva, A.B. ad Butuzob, V.F.(99). Asymptotc Methods Theory of Sgular Perturbatos, Moscow: Hay. Room, R.(8) Radato feld a sem-fte homogeeous atmosphere wth teral Va, T. ad sources Joural of Quattatve Spectroscopy & Radatve Trasfer, Vol.9, pp V, T. () Radato feld of a optcally fte homogeeous atmosphere wth teral sources Joural of Quattatve Spectroscopy & Radatve Trasfer, Vol., pp Xao-Yg Q ad Ya-Pg Su(9) Approxmate aalytc solutos for a two- dmesoal mathematcal model of a paced-bed electrode usg the Adoma decomposto method Appled Mathematcs ad Computato Vol. 5, pp Yury Toovyy ad Che-Chg Ma,(9) A explct-form soluto to the plae elastcty ad thermoelastcty problems for asotropc ad homogeeous solds Iteratoal Joural of Solds ad Structures, Vol.46, pp