bax~yy)] = p(x),uax~ y ) = q(y), (1.2) Solution as an F-Equation b4u(x,y) - Xu(x,y) = f(x,y) (1.1) Darboux Problem for a Polyvibrating Equation:

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1 Proceeding8 of the National Academy of Science8 Vol. 67, No. 3, pp , November 1970 Darboux Problem for a Polyvibrating Equation: Solution as an F-Equation D. Mangeron and M. N. Oguztoreli POLYTECHNIC INSTITUTE OF JASSY, ROMANIA; AND DEPT. OF MATHEMATICS UNIVERSITY OF ALBERTA, EDMONTON, ALBERTA, CANADA Communicated by Garrett Birkhoff, August 20, 1970 Abstract. In this paper we investigate a Darboux problem for a polyvibrating equation with analytic data. It is solved by the F-function approach of C. Truesdell, extending his F-equations to functions with several variables. 1. Many contributions have been devoted to the study of polyvibrating1-8 and generalized polyvibrating equations.9'2 After an elegant representation formula due to A+. Picone,'1",4 these equations proved useful in mathematical physics.'`'7 In recent papers on spline functions and the draftsman's equation, polyvibrating equations play an important role.'>20 Several boundary value problems, optimization problems, and integro-differential equations involving polyvibrating equations have been considered by the present authors and by L. E. Krivosheine.2126 Relevant references are presented in the bibliography. In this paper we establish the analytic solution of the simple polyvibrating equation: satisfying the conditions b4u(x,y) - Xu(x,y) = f(x,y) (1.1) u(x,y0) = g(x), u(x0,y) = h(y), bax~yy)] = p(x),uax~ y ) = q(y), (1.2) bjxb y =Y bxby 4= X where x and y are real variables, X is a real parameter such that X < 1, and f(x,y), g(x), h(y), p (x), and q(y) are certain given functions represented by the power series f(xy) = E fm nf X ) (y yo) m,n=0 mi n! {(X) = g(x -XO)m h(y) = Enh (1.)3) m=o m n=o n! 00 (X -X0)" M c (y - ye)" P(X) = E PM q(y) = Eq m=0 rm! n=o n 1488

2 VOL. 67, 1970 SOLUTION OF THE DARBOUX PROBLEM 1489 We assume that g(xo) = h(yo), p(xo) = q(yo), and Ifmn I < M, IgmI < M, Ih.I < M, PmIn < MIqgI < AM,m,n = 0,1,2,... (1.4) To solve the Darboux problem (1.1)-(1.2), we shall use the F-function approach of C. Truesdell,2728 extending his F-equations to functions with several variables. 2. F-Functions of Several Variables. We call F(x, y;a,3) an "F-function," if (ref. 27, p. 10) bf(zx8;a,#) = F(xy;a + 1,3), a (X8; :) = F(x,y;aO4 + 1). (2.1) ax by Suppose the function q(afl) is defined for a = ao + m,4 = flo + n,m,n = 0,1,2, where ao, #o are constants, and is such that 14(ao + m,3o + n)i < M1,mn = 0,1,2,... (2.2) Then, by a straight-forward extension of an existence theorem (ref. 27, p. 39), we can easily show that the system of functional equations (2.1) admits a unique solution satisfying the condition F(xo,yo;a,4) = 4(a4,), a = ao + m, = 1 #o + n, m,n = 0,1,2,..., (2.3) and this solution is given by the formula (I F(xy;a43) = 0(a + m,4 + n) (x-xo)m y-)n (2.4) mn=o n! The function F(x,y;a,4) defined by (2.4) is an analytic function of exponential type. Note: The concept of "F-function" may be extended to functions with more than two variables,29 in a similar manner. 3. Solution of the Darboux Problem (1.l)-(1.2). We now seek a solution to the Darboux problem (1.1)-(1.2) in the form of an F-function. To do so, assume that this F-function, say F(x,y;a,4), reduces to a function 4(a,4) for (xy) = (xo4yo). Hence, we have to determine the function 4(a,#), defined for a = ao + m = #o + n, m, n = 0,1,2,..., in such a way that the function F(x,y;a,43) given by the formula (2.4) will satisfy the equation (1.1), and the conditions (1.2). Since F(x,y;a,4) is supposed to be an F-function, we have at F(x,y;a,4) = F(x,y;a + 1,,3 + 1), F(x,y;oa,) = bxby 6~~~~~~~~~X26Jy2 F(x,y;a + 2,,8 + 2) (3.1) by Eqs. (2.1). Thus, from (1.1)-(1.2), (2.4), and (3.1), we obtain the following system of partial difference equations F(x,y;a + 2, # + 2) - XF(x,y;a,#) = f(x,y), F(x,yo;at,4) = g(x), F(xoy;cta) = h(y), (3.2) F(x,yo;a + 1,,3 + 1) = p(x), F(xoy;a + 1, f3 + 1) = q(y).

3 1490 MATHEMATICS: MA NGERON AND 04UZTORELI PROC. N. A. S. We now put Omn = O(a + m, (3 + n)(m,n = 0,1,2,...). (3.3) By substituting the relevent power series on both sides of each of the equations in (3.2), and comparing the coefficients of similar terms, we obtain the following recurrence relationships for the 4m,as: Oom+2,n+2 - XkiOmn = fm.in 0m = gm) 400 n = hn, (m,n = 0,1,2,...) (3.4) Om+1,1 = Pm, O1,n+1 = qnn Note that go = ho and Po = qo by assumptions of section 1, and each 4m,n 4(a + m, (3 + n) is uniquely determined by the recurrent system (3.4), independently of the values of a and (3. Further, starting with the inequalities (1.4), we can show by induction that r(mn) lm,ni < M E V 1k, (3.5) k =o where, denoting by [I] the integer part of I, r(m,n) = min {f L']}- (3.6) Hence, for lxi < 1, we have ifm~nl All 1-' lx mn =0,1,2,... (3.7) Thus, for any fixed (aoflo), the function 4(a,3), defined for a = ao + m, (3 = (3o + n,mn = 0,1,2,... by (3.3), satisfies the condition (2.2). Consequently, the function F(x,y;as), given by formula (2.4), represents the unique F-function which is the unique analytic solution of the Darboux problem (1.1)-(1.2) for AII< 1. Remarks. 10. It is quite clear that the above method can be used in the solutions of Darboux problems for more general polyvibrating equations. The novelty of our approach lies in the determination of the function j(xa,#) which generates the F-function by formula (2.4), in such a way that F satisfies the differential equation as well as the data on the characteristics. 20. Further, in the above method, as observed by C. Truesdell (ref. 27, p. 75), we can use complex variables instead of real ones. For example, let z and r be complex variables (z = x + iy, r = t + in) and consider the polyvibrating equation -a 2>8z2) - Xv(zr) = f(zs) (4.1) subject to the conditions V(zS0) = g(z), v(zor) = = )l_(z), _= q (4.2)

4 VOL. 67, 1970 SOLUTION OF THE DARBOUX PROBLEM 1491 where f, g, h, p, and q are analytic functions given by power series similar to these which appear in (1.3). By the above method we can establish the unique analytic solution of the Darboux problem (4.1)-(4.2). Let v = v(z,~) be this solution. If we take = z = x - iy, the polyvibrating equation (4.1) reduces to the biharmonic equation A22v - Xv= ( == b2 + -". (4.3) -)X2 j2) In this way we find the solution of the biharmonic equation satisfying the conditions (4.2). The method of reduction of metaharmonic equations into polyvibrating equations with complex arguments has been used successfully by I. N. Vekua30'31 and others. 3. Since the data and the solution are of exponential type the method of Laplace and Laplace-Picone transforms32-35 can be applied in the solution of the problem (1.1)-(1.2) or (4.1)-(4.2). 4. Finally, extending F-equations to functions with several variables, the authors prepared an essay toward a unified theory of special functions in the sense of C. Truesdell. * Supported in part by the National Research Council of Canada by Grant NRC-A Mangeron, D., "Problemi al contorno per le equazioni differenziali alle derivate parziali con caratteristiche reali multiple," Rend. Accad. Naz. Lincei, 16, (1932); Rend. Accad. Sci. Fis., Mat. Natur. Soc. Reale, Napoli, 2, (1932); Giorn. di Mat., 71, (1933). 2 Rosenblatt, A., "M6thode d'approximations successives dans la resolution de certains problemes a la frontiere pour les 6quations aux d6riv6es partielles aux caract6ristiques r6elles doubles," Comptes rendus, Paris, 197, (1933). 3 Salvadori, M., "Problemi variazionali sotto forma non parametrica per equazioni alle derivate parziali con caratteristiche reali doppie," Ann. Scuola Norm. Sup., Pisa, 5, (1936). 4Mangeron, D., "Nouveaux problemes A la frontiere pour une certaine classe d'6quations aux d6riv6es partielles d'ordre supdrieur," Comptes Rendus, Paris, 204, 94-96i ; (1937). 5 Stampacchia, G., "Problemi variazionali per le equazioni lineari alle derivate parziali con caratteristiche reali doppie," Giorn. di Mat., 78, ( ). 6 Mangeron, D., "Sulle equazioni di tipo "composito" nel senso di Hadamard," Boll. Un. Mat. Ital., 2, (1939). 7 Mangeron, D., and L. E. Krivosheine, "Valutazione dei moduli delle soluzioni dei problemi al contorno non lineari spettanti ai sistemi polivibranti con rimanenza ed argomenti ritardati," Rend. Accad. Naz. Lincei, 39, 1-2, (1965), and "New methods of numerical calculation for the solutions of various integro-differential systems with polyvibrating operators," Rom. J. A ppl. Mech., 9, 6, (1964) and 10, 1, 3-34 (1965). 8 Mangeron, D., and M. N. Oguzt6reli, "Programmazione dei calcoli con calcolatrici elettroniche per le soluzioni di una classe di sistemi polivibranti. I. Determinazione delle funzioni di Green spettanti ai problemi al contorno polivibranti," Rend. Accad. Naz. Lincei, 44, 6, (1968). 9 Mangeron, D., "On the generalized Bellman equations of dynamic programming concerning a new class of boundary value problems of non elliptic type," J. Math. Anal. and Appl., 9, 1, (1964), and "Nouveaux problemes a la frontiere pour differents systemes non elliptiques," I.C.M., Stockholm, Short Communicatiois, 195 (1962). 10 Mangeron, D., M. N. Okuztoreli, "Polynomes orthogonaux polyvibrants gen6ralises et quelques th6oremes qui s'y rattachent," Bull. Acad. Roy. Sci. Belg., in print. "1 Mangeron, D., M. N. Okuztoreli, "Problemes des spectres pour les 6quations polyvibrantes generalisees," Bull. Polytechn. Inst. Jassy, in print. 12 Mangeron, D., M. N. Okuztoreli, L. E. Krivosheine, and F. S. Rossi, "Calcolo tramite calcolatrici elettroniche delle funzioni speciali spettanti ai sistemi polivibranti e polivibranti generalizzati ed applicazioni. I," Rend. Accad. Naz. Lincei, in print.

5 1492 MATHEMATICS: MANGERON AND O0( UZTOREII PROC. N. A. S. 13 Picone, M., "Nuovi metodi per il calcolo delle soluzioni delle equazioni alle derivate parziali della fisica matematica," Ann. Sci. Univ., Jassy, 26, 1, (1940). 14 Mangeron, D., M. N. Oguztoreli, "Estensione dei metodi di maggiorazione di Picone alle soluzioni di sistemi di equazioni polivibranti," Rend. Accad. Naz. Lincei, 44, 5, (1968). 11 Beresanski, Ju. M., "Expansions in Eigenfunctions of Selfadjoint Operators," Amer. Math. Soc. Transl. Monogr. (Providence 1968), vol. 17, chap. 4, p Rossi, F. S., "Metodi di Mangeron in certi problemi non lineari di fisica matematica," Bull. Polytechn. Inst. Jassy, 9(13), 3-4, (1963). 17 Birkhoff, G., "Piecewise Bicubic Interpolation and Approximation in Polygons," in Approximations with Special Emphasis on Spline Functions, ed. I. J. Schoenberg (Academic Press, 1969), p Gordon, W. J., "Blending-function methods of bivariate and multivariate interpolation and approximation," J. Approx. Theory, 1, (1968). 19 Birkhoff, G., W. J. Gordon, "The Draftsman's and Related Equations," SIAM J. on Numer. Anal., in press 20 Gordon, W. J., "Distributive Lattices and the Approximation of Multivariate Functions," in Approximations with Special Emphasis on Spline Functions, ed. I. J. Schoenberg (Academic Press, 1969), pp , See p Okuztoreli, M. N., and D. Mangeron, "Problemi ottimali spettanti ai sistemi di controllo con parametri distribuiti," Rend. Accad. Naz. Lincei, 44, 1-6; 45, and (1968). 22 Mangeron, D., and M. N. Okuztoreli, "Sopra alcuni problemi concernenti la stabilita e la valutazione dell'errore commesso nelle approssimazioni spettanti a certi sistemi non lineari con operatori di rimanenza," Rend. Ist. Lomb. Sci. Lett., 102A, (1968). 23 Mangeron, D., and M. N. Okuzt6reli, "Sur quelques problemes de calcul des variations concernant les equations polyvibrantes," Math. Notae, 21, 7, 1-2, 9-16 ( ). 24 Mangeron, D., "Equations fonctionnelles. Etudes concernant les equations polyvibrantes," Comptes Rendus, 266, and (1968). 25 Oguzt6reli, M. N., "A boundary value problem for a polyvibrating equation," Rend. Accad. Sci. Torino, in print. 26 Mangeron, D., and L. E. Krivosheine, "Sulla risoluzione di una classe di problemi al contorno per una classe di equazioni integro-differenziali," Rend. Sem. Mat. Univ. Padova, 34, (1964). 27 Truesdell, C., "An essay toward a unified Theory of special functions," Ann. of Mathematics Stud., 18, 182 (1948). 28 Truesdell, C., "On the addition and multiplication theorems for special functions," Proc. Nat. Acad. Sci. USA, 36, (1950). 29 Mangeron, D., and M. N. Okuzt6reli, "Alcuni problemi concernenti funzioni speciali in put variabili indipendenti," Rend. Accad. Sci. Fis., Mat. Natur. Soc. Reale, Napoli, in press. 30 Vekua, I. N., New Methods for Solving Elliptic Equations (North-Holland Pub. Co., 1967). 31 Vekua, I. N., "The IVth Romanian Congress of Mathematicians," in Russian, Izv. Akad. Nauk SSSR, Ser. Math., (1956). 32 Picone, M., "Mauro Picone," Annuario dell'accademia dei XL, Rome, 1-29 (1968). 33 Poli, L., and P. Delerue, Le Calcul symbolique a deux variables (Paris, Gauthier-Villars, 1954). See pp Mangeron, D., "Applicazione del metodo di Picone spettante alla trasformata Laplace ad intervallo d'integrazione finito allo studio delle equazioni alle derivate parziali d'ordine superiore," Rend. Accad. d'italia, 1, 1-9 (1939). 35 Mangeron, D., and M. N. Oguzt6relli, "Application of the Laplace-Picone transforms with finite integration interval to the study of mathematical physics differential systems," Romanian J. Appl. Mech., 13, 5, (1968).