ON SOLVING A FORMAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION IN THE COMPLEX FIELD

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1 A. Şt. Uiv. Ovidius Costaţa Vol. (), 003, ON SOLVING A FORMAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION IN THE COMPLEX FIELD Nicolae Popoviciu To Professor Silviu Sburla, at his 60 s aiversary Abstract Several papers of K. W. Tomatschger deals with the solvig of liear partial differetial equatios of secod order i complex field by usig the Bergma itegral operator. The Bergma operator cotais the Bergma kerel. To determie the Bergma kerel, is ot a easy work. This paper cotais a approach to costruct a algorithm to fid the Bergma kerel for a formal hyperbolic partial differetial equatio with two idepedet complex variables.. Mathematical Backgroud I [7] a list of itegral operators is give: J. Le Roux (895), I.N. Vekua (937), St. Bergma (937), M. Eichler (94), U. Stessel (99). The Stessel operator comprises all these operators as special cases. Nevertheless the most useful itegral operator is Bergma operator []. By this operator ca be obtaied a solutio of the liear partial differetial equatio of secod order i the complex field. I this paper we deal with the followig formal hyperbolic pde: v z ζ zp ζ q v = 0, v = v (z, ζ), p, q Z, z, ζ - complex variables. Notatio. Let L deote the differetial operator L = () z ζ zp ζ q, Lv (z, ζ) = 0. () Key Words: Bergma Itegral Operator, Bergma kerel, partial differetial equatio. Mathematical Reviews subject classificatio: 35C5 69

2 70 N. Popoviciu A solutio of the equatio Lv = 0 has the form [], [6] v (z, ζ) = (T f) (z, ζ) = where E (z, ζ, t) is a solutio of the pde ( z E (z, ζ, t) f ( t )) dt, (3) t ( t ) E z t t E + tle = 0 (4) ζ ad f (z) is ay holomorphic fuctio i a eighborhood of z = 0. Defiitio. E (z, ζ, t) is amed Bergma kerel ad (T f) (z, ζ) give by (3) is called the Bergma operator. Remark 3. The fuctio f is idepedet of the equatio Lv = 0 while the Bergma kerel depeds upo this equatio. The most useful Bergma kerels are polyomial kerels. The Bergma kerel of first type has the form: E (z, ζ, t) = + z P (z, ζ) t, E (0, ζ, t) =, (5) = where the fuctios P (z, ζ) have to determied. Remark 4. The first problem of solvig the pde () is to fid the kerel (5) i.e. to fid the fuctios P (z, ζ) which verify the equatio (4). The secod problem is to compute the itegral (3) for ay holomorfic fuctio f (z). Now we are iterested i fidig the Bergma kerel. Propositio 5. The fuctios P (z, ζ) satisfy the system of equatios + P + ζ + LP = 0, P o (z, ζ) =, 0. (6) Proof. We itroduce (5) i equatio (4) ad follow the coefficiet of t +. The system (5) is called a Bergma recurrece system. By solvig this system oe obtais the fuctios P (z, ζ) ad the kerel E (z, ζ, t).. Bergma Recurrece System Properties There are at least two methods to solve the Bergma recurrece system. Method. Oe uses the system (6) for = 0,,, 3, ad the geeral solutio P (z, ζ) is obtaied by iductio.

3 O solvig a formal hyperbolic partial differetial equatio 7 Method. We choose a particular form of the fuctios P (z, ζ). This form depeds o the pde we have to solve. I the case of equatio () we choose a polyomial form i ζ P (z, ζ) = ζ P (zζ), E (z, ζ, t) = + (zζ) P (zζ). (7) = If we deoted zζ = u, the the Bergma kerel becomes: E (z, ζ, t) = + u P (u), P o (u) = (8) = where P (u), are ukow fuctios. We deote N = {,, 3,...}. Remark 6. For the equatio () there are the followig cases: p = q = 0; p = q, p, q N; p q, p, q N; p = q, p, q Z {0} ; p q, p, q Z {0}. Now we deal with the case p = q N ad we deote p = q = m, m. The equatio () becomes Lv = 0, L = z ζ (zζ)m, v = v (z, ζ). Propositio 7. Usig (7), the Bergma recurrece system (6) becomes: + [ up + (u) + ( + ) P + (u) ] = u m P (u) ( + ) P (u) We call (9), Bergma particular system. Proof. The system (6) has the form: + P + ζ up (u), u = zζ, m N. (9) + P z ζ (zζ)m P = 0, P + (z, ζ) = ζ + P + (zζ). (0) From (0), by computig all partial derivatives, oe obtais (9).

4 7 N. Popoviciu Propositio 8. The Bergma particular system (9) yields the recurrece formula: ( P + (u) = + u + u + u m P + ) u P P du, 0, P o (u) = () Proof. Oe writes the geeral solutio differetial liear equatio (9) with ukow fuctio P Solutio of the Bergma Particular System We look for the solutio P (u) of the Bergma particular system, (). Oe starts with = 0,,, 3, 4 ad the we propose the geeral form of ukow fuctios P (u). a) The particular values of yield particular fuctios P 0 (u) =, P (u) = m u um =! ( )! u (m)!m um, P (u) =! ( )! P 3 (u) = 3 3! ( 3)! P 4 (u) = 4 4! ( 4)! u u 3 u 4 [ (m )!m u m + (m) [ (m ) 3!m 3 um ],!m u m + 3 (m )!m 3 u m (m) [ (m 3) 4 3!m 4 u3m ],!m u m + (m ) (7m )!m 3 u m ], () 6 (m ) 3!m 4 u 3m (m) 4!m 5 u4m where (C) = C (C + ) (C + )... (C + ) = A C+. All fuctios are writte i the same form i order to obtai the geeralizatio. The quatity i brackets is a polyomial i u. b) We propose the geeral form of ukow fuctios P (u) = ( )! ()! u k= b,k k!m k+ ukm, (3) where b,k are ukow coefficiets which deped o ad k. c) We ll fid some particular values for these coefficiets.

5 O solvig a formal hyperbolic partial differetial equatio 73 Propositio 9. The first coefficiets b,k, k have the values: =, b, = m; =, b, = (m ) m, b, = m; = 3, b 3, = (m ) (m ) m, b 3, = 3 (m ) m, b 3,3 = m; = 4, b 4, = (m 3) (m ) (m ) m, b 4, = (m ) m (7m ), b 4,3 = 6 (m ) m, b 4,4 = m. Proof. Oe idetifies the relatio () ad (3). d) We fid recurrece relatios for the coefficiets b,k. Propositio 0. For coefficiets b,k verify the recurrece relatios: b o,o =, b +, = (m ) b, (4) b +,+ = b,, (5) b +,k = (km ) b,k b,, k =,. (6) Proof. We itroduce the geeral form (3) ito Bergma particular system (9) ad we put u, u i the sum. I the first equality which we obtai, oe chages u (k+)m i u km by idex removals. Oe obtais a equality havig three sums. We isolate k =, k = + i the first sum, k = + i the secod sum, k = i the third sum. u m b +, m m = b, m (m ) m u (+)m b +,+ ( + )!m + ( + ) m = b,!m + u km b +,k km = b k!mk+, (k )!m k b,k km (km ), k =,. k!mk+ Corollary. By idex removal + for, m, oe obtais: ( ) (4) b, = (m + ) = A m =!Cm m =! (7) (5) b, = ( ) m (8) (6) b,k = (km + ) b,k b,,, k,. (9) e) Usig (9) we have to obtai the o-recurece relatio for coefficies b,k. We try to obtai the simplest form for each coefficiet. At the begiig we evaluate the particular cases k =, 3, 4 (because b, is kow).

6 74 N. Popoviciu Propositio. I.) k = ; for, b, has the equivalet expressios I.) The simplest form is b, = (m + ) j b j, (0) b, = A j m j Aj m,. () b, =! m ( ) C m Cm,. () Proof. From (9), we express b, with b, = m ad b j,. We obtai: b, = (m + ) b, (m + ) 3 b,... (m + ) b, b,. The result is (0). For (), oe takes ito accout that: (m + ) j = A j m j, b j, = A j m. For (), oe proves the equality: A j m j =! ( ) m C m Cm,. Propositio 3. II.). k = 3; for 3, b,3 has the equivalet expressios b,3 = (3m + ) j b j,, (3) i.e. b,3 = j= (m )! (m )! (3m )! II.) The simplest form is: C m 3m j C m 3m j j= j=. (4) b,3 = (m )! (m )! (3m )! [ C m 3m C m 3m C m 3m + C m 3m ], (5)

7 O solvig a formal hyperbolic partial differetial equatio 75 i.e. b,3 =! [ m 6 C 3m + C m ] C m. (6) Proof. From (9), we express b,3 with b 3,3 = b, ad b j, i the form: b,3 = (3m + ) 3 b 3,3 (3m + ) 4 b 3,... (3m + ) b, b,. The result is (3). For (4), i (3) oe uses (c) ad b j, give by (). The oe writes all the terms with k! ad we try to obtai C j i. Usig a trick, we get: b,3 = (m )! (m )! (3m j )! (3m )! (m j)! (m )! j= (m )! (3m )! j= (m )! (3m j )!. (m j)! (m )! Usig C j i ad C j i = C i j i ( ), oe obtai (4). For (5) we use C j C j i Cj i ( ). For (5), we simplify (5), ad the coefficiets become: (3m)! b,3 = 6m (3m )! + (m)! m (m )! m! m (m )!. By multiplicatio with!/! oe obtais the desired formula (6). i = Propositio 4. III.) k = 4; for 4, b,4 has the equivalet expressios b,4 = (4m + ) j b j,3 (7) j=3 b,4 = (3m )! (m )! C m 4m j m (4m )! m (m )! (m )! (4m )! j=3 j=3 C m 4m j + + (m )! (3m )! C 3m 4m j m (4m )!. (8) j=3

8 76 N. Popoviciu III.) The simplest form is: b,4 = (3m )! (m )! ( C m m (4m )! 4m 3 C4m m ) m [(m )!] (4m )! + (m )! (3m )! m (4m )! ( C m 4m 3 C m 4m ) + ( C 3m 4m 3 C4m 3m ) (9) b,4 =! [ m 3 4 C 4m 6 C 3m + 4 C m ] 6 C m. (30) Proof is doe i the same way as for the previous Propositio. Remark 5. The simplest form determied for each particular coefficiet was obtaied i may steps, geerally four steps. Our mai purpose is to fid the o-recurrece form of b.k from (9), usig the particular coefficiets b.k, k =,, 3, 4 give by their simplest form (7), (), (6), (30) respectively. We ll do it i the ext propositio. Propositio 6. The geeral form of coefficiets b,k from (3) is: b,k =! k!m k ( ) j C j k C jm, k, m, 0, m. (3) Proof. We itroduce (3) i (9) ad obtai k ( ) j C j k C jm = km + k ( ) j C j k C jm km ( ) j C j C jm. We isolate terms for j = k ad it results: ( ) j C j k C jm = km + ( ) j C j k C jm km ( ) j C j C jm. Here we use some kow formulas as: C j = jm + C m, ad we get = k j k Cj k, ad C jm =

9 O solvig a formal hyperbolic partial differetial equatio 77 ( ) j C j k C jm = km + therefore, we get: ( ) j C j k jm + C jm km ( ) j k j k Cj k jm + C jm, ( ) j C j k C jm = ( ) j C j km + k jm + C jm ( ) j C j m (k j) k jm + C jm. The last equality is a idetity. Hece (3) satisfies (9). Remark 7. The whole work was doe to fid the relatio (3). The fuctio P (u) becomes P (u) = ( )! ()! u P (u) = ( )!! ()!! k!m k+ ukm k!m ( ) j C j k C jm k= k ( ) j C j k C jm k!k!m k ukm. (3) u k= 4. Algorithm to Fid Bergma Kerel of First Type ) Choose the Bergma kerel of first type E (z, ζ, t) havig the form (5). The fuctios P (z, ζ) are ukow fuctios. ) Choose the fuctios P (z, ζ) havig the form (7). The P (z, ζ) are ukow fuctios. 3) Use the Bergma particular recurrece system () ad fid several fuctios P for the particular values = 0,,, 3, 4. All fuctios P should be writte i the same aspect, to be possible the geeralizatio (3). 4) Usig the particular fuctios P oe fids the particular values of the coefficiets b,k, k. 5) Fid b, ad the recurrece relatio for b,k, (9).

10 78 N. Popoviciu 6) Fid the o-recurrece relatio for b,k, ad obtai the simplest form for each coefficiet. 6.). Fid the equivalet expressios for b, ad obtai the simplest forms. 6.). Fid the equivalet expressios for b,3 ad obtai the simplest forms. 6.3). Fid the equivalet expressios for b,4 ad obtai the simplest forms. 7) Write the coefficiets b,k, k =,, 3, 4 i the same simplest form (3). 8) Write the coefficiets form of the coefficiets b,k (3). 9) Write the geeral form of the fuctio P (3). 0) Write the geeral form of the fuctio P ad the Bergma kerel E (z, ζ, t) (7). Refereces [] BERGMAN St., Itegral Operators i the Theory of Liear Partial Differetial Equatios, Spriger, Berli, 97. [] KRACHT M., KREYSZIG E., Methods of Complex Aalysis i Partial Differetial Equatios with Applicatios, A. Wiley-Itersciece Publicatio, New-York, Chichester, Brisbae, Toroto, Sigapore, 998. [3] POPOVICIU N., Ecuatii cu Derivate Partiale de Ordiul Doi, Muzeul Natioal de Arta, Bucuresti, 996. [4] STESSEL Ursula M. Ch., Lösugsoperatore bei elliptische Differetialgleichuge, Complex Variables, 9, 49-6, 99. [5] TOMANTSCHGER K. W., Bergma Kerels of First Kid for w zζ (s + zζ) w = 0, s Z, Bulletis for Applied Mathematics, 94, XXX, 60-74, 983. [6] TOMANTSCHGER K. W., Itegral Operators for w zζ z M ζ M w = 0, M, M Z, Bulletis for Applied Mathematics, 98, XXX, 8-4, 983. [7] TOMANTSCHGER K. W., Itegral Operators for Elliptic resp. Formal Hyperbolic Differetial Equatios, Proceedigs of The 8-th Symposium of Mathematics ad its Applicatios, Politehica Uiversity of Timisoara, November, 999, pag [8] TOMANTSCHGER K. W., Solutio of Self-Adjoit Partial Differetial Equatios by a Itegral Operator Method, Proceedigs of The 8-th Symposium of Mathematics ad its Applicatios, Politehica Uiversity of Timisoara, November, 999, pag [9] VEKUA I. N., Geeralized Aalytic Fuctios, Pergamo Press, Lodo, 96. [0] VEKUA I. N., New Methods for Solvig Elliptic Equatios, Nort-Hollad Publ., Amsterdam, 967. Military Techical Academy, Departmet of Mathematics, Bucharest, Romaia popoviciu@myet.ro