Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton Degeneratng nto a Parabolc Equaton n an Infnte Strp Mahr M. Sabzalev and Mahbuba E. Kermova Department of Mathematcs Azerbajan State Ol Academy Az,, Azadlıg av, Bau, Azerbajan Copyrght 4 Mahr M. Sabzalev and Mahbuba E. Kermova. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal wor s properly cted. Abstract In an nfnte strp a boundary value problem for a thrd order non-classc type equaton degeneratng nto a second order parabolc equaton s consdered. The total asymptotc expanson n small parameter of the soluton of the problem under consderaton s constructed and the remander term s estmated. Keywords: Asymptotcs, Boundary layer functon, Remader term. Introducton In studyng some real phenomena wth non-unform transtons from one physcal characterstcs to other ones, we have to nvestgate sngularly perturbed boundary value problems. A lot of mathematcans were nterested n such problems. But non-classcal sngular perturbed equatons n comparson wth classc equatons were studed not enough. M.. sh and L. A. Lustern n [5] ntroduced so-called one characterstc equatons. They called the equatons of odd order and of the form L A ( A u B u f () )
6 Mahr M. Sabzalev and Mahbuba E. Kermova one-characterstc ones f A s an operator of frst order, A s an ellptc operator of at most order. Obvously, only the characterstcs of the frst order operator A wll be real characterstcs of equaton (). In the same paper they nvestgated mutual degeneraton of ellptc and one-characterstc equatons n fnte domans, and constructed only the frst terms of the asymptotcs of soluton of the consdered problems. It should be noted that n references, manly asymptotcs of the soluton of boundary value problems for dfferent equatons was studed only n fnte domans. In the papers [], [], for one-characterstc equatons that were studed n [5] the boundary value problems were consdered n nfnte domans and the total asymptotcs of soluton of these problems were constructed. In the present paper, n an nfnte strp P {( t, t, x } we consder the followng boundary value problem: u u u L u ( u) au f t, x, () t t t x u u, u,, () t t t t lm u, (4) x where s a small parameter,, a s a constant, f ( t, x ) s a t x gven smooth functon. In [] the boundary value problem for equaton () n a rectangle s consdered and total asymptotcs of the soluton of consdered problem s constructed. Our goal s to construct the asymptotc expanson n small parameter of the soluton of boundary value problem ()-(4). In constructng the asymptotcs we are guded by the M.I. sh-l.a. Lustern method and conduct teratve processes.. The frst teratve process In the frst nteratve process, we ll loo for the approxmate soluton of equaton () n the form W W W n... Wn, (5) and the functons W ( t, ;,,..., n wll be chosen so that n L W ( ). (6) Substtutng (5) n (), and equatng the terms wth the same powers of, for determnng W ;,,..., n we get the followng recurrently connected equatons: W W aw f ( t, x ), (7) t x
Asymptotcs of the soluton of a BP 7 W W W aw, (8) t x t W W W aw ( W );,,..., n. t x t t (9) Equatons (7), (8), (9) dffer only by the rght sdes. Equaton (7) s obtaned from equaton () for, and s called a degenerate equaton correspondng to equaton (). Obvously, t s mpossble to use all boundary condtons of () for equatons (7), (8), (9). For the equatons (7), (8), (9) wth respect to t t should be used the frst condton from (), and wth respect to x the both condtons from (4),.e. W, ( x ); () t lm,( t );,,..., n. () W x Problem (7), (), () for s sad to be a degenerate problem correspondng to problem ()-(4). The followng lemma s vald Lemma. Let f ( t, be a functon gven n P, havng contnuous dervatves wth respect to t up to the ( n ) -th order nclusvely, be nfntely dfferentable wth respect to x and satsfy the condton l f ( t, () sup( x ) Cl, () x t x () where l s a nonnegatve number,, n, s arbtrary, C l. Then the functon W ( t, beng the soluton of problem (7), (), () for, n P has contnuous dervatves wth respect to t up to the (n+4)-th order nclusvely, s nfntely dfferentable wth respect to x and satsfes the condton l W ( t, () sup( x ) Cl, () x t x () where n 4, C l. The proof of ths lemma s cted n the paper [4]. The remanng functons W, W,..., Wn, n expanson (5) wll be determned sequentally from boundary value problems (8), (9), (), () for,,..., n. From lemma t follows that the functons W beng the solutons of problem (8), (9), (), () for,,..., n wll have contnuous dervatves wth respect to х up to the ( n ) -th order, and condton () for the functon W wll be satsfed for n ;,,..., n. From (5) and (), () we get that the constructed functon W satsfes the followng boundary condtons: W,( x ); lm W,( t ). (4) t x
8 Mahr M. Sabzalev and Mahbuba E. Kermova The functon W doesn t satsfy, generally speang, boundary condtons for t from (). Therefore, conductng the second teratve process we should construct a boundary layer-functon near the boundary t so that the sum W could satsfy the boundary condtons W ), ( W ). (5) t t t (. The second teratve process-constructon of boundary layer functons The frst teratve process s conducted on the base of decomposton () of the operator L. For conductng other teratve process by means of whch a boundary layer functon wll be constructed near the boundary t, at frst t s necessary to wrte a new decomposton of the operator L near ths boundary. For that we mae change of varables: t, x x. The new decomposton of the operator L n the coordnates (, has the form form L,. a x x (6) We loo for a boundary layer functon near the boundary t n the... n n, (7) as the approxmate soluton of the equaton L,. (8) Substtutng the expresson for from (7) n (8), tang nto account (6) and mang comparson of the terms at the same powers of, for determnng the functon j (, ; j,,..., n we get the followng recurrently connected equatons:, x x a a, x ;,,..., n. (9) () ()
Asymptotcs of the soluton of a BP 9 For fndng boundary condtons for equatons (9), (), () t s necessary to substtute expansons (5), (7) for W and to (5) and mae comparson of the terms at the same powers of. Then we get: W ;,,..., ; n t n, () j W j, t ; j,,..., n. () Now construct the functons. From (9), () for and t,,..., n () we have that the functon s a boundary layer type soluton of equaton (9), satsfyng the boundary condtons W,. (4) t T The characterstc equaton correspondng to ordnary dfferental equaton (9) n addton to the zero root has two non-zero roots, wth negatve real parts. Ths fact provdes regularty of degeneraton of problem ()- (4) on the boundary t. The boundary layer type soluton of problem (9), (4) has the form W (, ( e e ). (5) By lemma, from (5) t follows that the functon (, ) and all ts even x dervatves wth respect to x vansh as x. Knowng the functon, we determne the functon as a boundary layer soluton of equaton (), satsfyng the boundary condtons: W W,. (6) tt t tt From (5) t follows that the rght sde of equaton () has the form f m ( e m( e, (7) where m, m ( ) are determned by the followng equaltes ( x m ( aw W, x W m ( aw. x (8)
Mahr M. Sabzalev and Mahbuba E. Kermova Followng (7) we get that equaton (9) has a partcular soluton n the form () c( c( e c ( c( e, (9) and the functons c (, c(, c(, c( are expressed by the functons W W ( T, and. They may be determned by the method of x undetermned coeffcents. () () Represent n the form layer type soluton of the followng problem: () () (). Then () wll be the boundary () ; (, (, where ( W c(, W () ( c( c ( c( c (. t Obvously, the boundary layer type soluton of problem () has the form () (, ) ( ) ( ) ( ) ( ) x x x e x x e. () () From (9) and () comes out that the functon s the sum of () and s determned by the formula a ( a ( e b ( b e, () () and (, () by a, a (, a (, a ( ) the functons ( x a b ( ( ( ( (, a ( ( c (, ( ( c (, b ( c ( are denoted. Accordng to (), (), (4) we have that the functon (, and all ts even dervatves wth respect to х vansh as x. By constructng the functons ;,,..., n we use the followng statement. Lemma. The functons beng the boundary layer type solutons of equatons () for,,..., n and satsfyng the correspondng condtons from (), () are determned by the formula c (4)
Asymptotcs of the soluton of a BP (, a ( e b ( e ;,,..., n. (5) The coeffcents a (, b ( are expressed unformly by the functons W, W,..., W and ther dervatves of frst order wth respect to t, and wth respect to x only of even order. Proof. The lemma s proved by the mathematcal nducton method. It was shown above that the functons, are determned by formula (5). Now assume that the statement of the lemma s vald for,,,...,, and prove that t s vald for as well. Note that the functons, enter nto the rght sde of equaton () for and by assumpton these functons are determned by formula (5). Repeatng the reasonng conducted n determnng the functon, we can affrm that s also determned by formula (5). Lemma s proved. Multply all the functons j by the smoothng functon and the obtaned new functons agan denote by ; j,,..., n. As all the functons j j (, ; j,,..., n vansh as x, then from (4) and (7) t follows ~ that the sum U W constructed by us, n addton to (5) satsfes also the boundary condtons ( W ), lm ( W ). (6) t x Denote the dfference of the exact soluton of problem ()-(4) and U ~ by ~ n U U z, (7) and call n z a remander term. Now we should estmate the remander term. 4. Estmaton of remander term It holds the followng statement. Lemma. For the functon z t s vald the estmaton z z z c z c L ( ), (8) P t t x t L (, ) L ( P) L ( P) where c, c are the constants ndependent of.
Mahr M. Sabzalev and Mahbuba E. Kermova Proof. Actng on both sdes of equalty (7) by the approprate decompostons of the operator L, and tang nto account equatons (), (7)-(9) and (9)-(), we have L z F(, t,, (9) where F(, t, F (, t, F (,, s a unformly bounded functon. The functon F (, t, has the form Wn F ( W ) ( ) n Wn, t t t and F (,, near the boundary t T has the form n n n n n F a a n n. x x x x x Obvously, z wll satsfy the boundary condtons z z z,, z z. (4) t tt x x t tt Multplyng the both sdes of equaton (9) scalarly by z, and ntegratng the left sde of the obtaned equalty wth regard to boundary condtons (4), after some transformatons we get estmaton (8). Lemma s proved. 5. Concluson The obtaned results may be generalzed n the form of the followng statement. Theorem. Let f ( t, be a gven functon n P, havng contnuous dervatves wth respect to t to the ( n ) -th order nclusvely, wth respect to x be nfntely dfferentable, and satsfy condton (). Then for the soluton of problem ()-(4) t holds the asymptotc representaton n n j n u W z, j j where the functons W are determned by the frst teratve process, j are boundary layer type functons near the boundary t and are determned by the second teratve process, n z s a remander term, and estmaton (8) s vald for the functon z.
Asymptotcs of the soluton of a BP References [] Javadov M.G., Sabzalev M.M. On a boundary value problem for an onecharacterstc equaton degeneratng nto one-characterstc one. DAN SSSR, 5 (979), 4-46. [] Sabzalev M.M. On a boundary value problem for an one-characterstc equaton degeneratng nto ellptc one. DAN SSSR, 4 (979), 7 [] Sabzalev M.M., Kermova M. A. Asymptotcs of the soluton of a boundary value problem n a rectangle for an one-characterstc dfferental equaton degeneratng nto a parabolc equaton. Proceedngs of the Internatonal Scentfc Conference devoted to 85 years of acad. Azad Khall oglu Mrzajanzadeh, November -,, Bau, (), 7-9. [4] Sabzalev M.M. On a boundary value problem for a quas-lnear ellptc equaton degeneratng nto a parabolc equaton n an nfnte strp. Nonl. Analyss and Dfferental Equatons, ol.,, no., -4, HIKARI Ltd, www.m-har.com [5] sh M.I., Lustern L.A. Regular degeneraton and boundary layer for lnear dfferental equatons wth small parameter. UMN, 5 (957), -. Receved: May 5, 4