Growth estimates for entire function solutions of the generalized bi-axially symmetric Helmholtz equation *∂*^{2}*u*/*∂x*^{2} + *∂*^{2}*u*/*∂y*^{2} + (2*µ*/*x*)·(∂*u*/∂*x*) + (2*ν*/*y*)·(∂*u*/∂*y*) +*k*^{2}*u* = 0, (*µ*,*ν* Є R^{+}), in terms of their Jacobi Bessel coefficients and ratio of these coefficients have been studied. Some relations for order and type also have been obtained in terms of Taylor and Neumann coefficients. Our results generalize and extend some results of Gilbert and Howard, McCoy, Kumar and Singh.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 8)

Pages:

12-26

Citation:

D. Kumar "Growth Estimates of Entire Function Solutions of Generalized Bi-Axially Symmetric Helmholtz Equation", International Journal of Advanced Research in Mathematics, Vol. 8, pp. 12-26, 2017

Online since:

Apr 2017

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

R.P. Boas, Entire functions, Academic Press, (1954).

Deepmala, A study on fixed point theorems for nonlinear contractions and its applications, Ph.D. Thesis, Pt. Ravishankar Shukla University, Raipur 492 010, Chhatisgarh, India, (2014).

A. Erd´elyi, Singularities of generalized axially symmetric potentials, Comm. Pure and Appl. Math. 9 (1956) 403-414.

R.P. Gilbert, On the singulariteis of generalized axially symmetric potentionals, Arch. Rat. Mech. Anal. 6 (1960) 171-176.

R.P. Gilbert, Function theoretic methods in partial differential equations, Academic Press, New York, NY, USA, (1969).

R.P. Gilbert, H.C. Howard, On a class of elliptic partial differential equations, TN BN-344, Fluid Dyn. Inst., Univ. of Maryland, (1963).

R.P. Gilbert, H.C. Howard, On solutions of generalized axially symmetric wave equation representated by Bergmann operators, Proc. London Math. Soc. 15 (1985) 346-360.

R.P. Gilbert, H.C. Howard, On solutions of the generalized bi-axially symmetric Helmholtz equation genreated by integral operators, Jurnal f¨ur Mathematik. Bd. 218 (1964) 109-120.

R.P. Gilbert, R.G. Newton (eds. ), Analytic methods in mathematical physics, Gordon and Breach Science Pub., New York, (1970).

P. Henrici, Zur funktionen theorie der wellingleichung, Commentarii Mathematici Helvetici. 27 (1953) 235-293.

P. Henrici, On the domain of regularity of generalized axially symmetric potentials, Proc. Amer. Math. Soc. 8 (1957) 29-31.

P. Henrici, Complete systems of solutions for a class of singular elliptic partial differential equations, Boundary Problems in Differential Equations, Univ. of Wis. Press, Madison, Wis. (1960) 19-34.

P. Henrici, A survey of I.N. Vekua's theory of elliptic partial differential equations with analytic coefficients, Z. Angew. Math. Phys. 8 (1957) 169-203.

O.P. Juneja, On the coefficients of an entire series of finite order, Archiv der Mathematik. 21(1) (1970) 374-378.

H.S. Kasana, D. Kumar, The Lp−approximation of generalized bi-axially symmetric potentials, Int. J. Diff. Eqs. Appl. 9(2) (2004) 127-142.

H.S. Kasana, D. Kumar, Lp−approximation of generalized biaxially symmetric potentials over Carath´eodory domains, Mathematica Slovaca. 55(5) (2005) 563-572.

R.B. Kelman, Axisymmetric potential problems suggested by biological considerations, Bull. Amer. Math. Soc. 69(6) (1963) 835-838.

D. Kumar, Ultra-spherical expansions of generalized bi-axially symmetric potentials and pseudoanalytic functions, Complex Variables and Elliptic Equations. 53(1) (2008) 53-64.

D. Kumar, Growth and Chebyshev approximation of entire function solutions of Helmholtz equation in R2, Europian Journal of Pure and Applied Mathematics. 3(6) (2010) 1062-1069.

D. Kumar, On the (p, q)−growth of entire function solutions of Helmholtz equation, Journal of Nonlinear Science and Applications. 4(1) (2011) 5-14.

D. Kumar, Growth and approximation of solutions to a class of certain linear partial differential equations in RN, Mathematica Slovaca. 64(1) (2014) 139-154.

D. Kumar, K.N. Arora, Growth and approximation properties of generalized axisymmetric potentials, Demons. Math. 43(1) (2010) 107-116.

D. Kumar, R. Singh, Measures of growth of entire solutions of generalized axially symmetric Helmholtz equation, J. of Complex Analysis. 2013 (2013) 1-6.

A. Mackie, Contour integral solutions of a class of differential equations, J. Rat. Mech. and Anal. 4 (1955) 733-750.

P.A. McCoy, Polynomial approximation of generalized biaxisymmetric potentials, Journal of Approx. Theory. 25(2) (1979) 153-168.

P.A. McCoy, Approximation of pseudoanalytic functions on the unit disk, Complex Variables and Elliptic Equations. 6 (1986) 123-133.

P.A. McCoy, Near-best approximate solutions for a class of elliptic partial differential equations, J. Approx. Theory. 55 (1988) 248-263.

P.A. McCoy, Optimal approximation and growth of solutions to a class of elliptic partial differential equations, J. Math. Anal. Appl. 154 (1991) 203-211.

P.A. McCoy, Interpolation and approximation of solutions to a class of linear partial differential equations in several real variables, Complex Variables and Elliptic Equations. 2 (1994) 213-223.

L.N. Mishra, On existance and behaviour of solutions to some nonlinear integral equations with applications, Ph.D. Thesis, National Institute of Technology, Silcher 788 010, Assam, India, (2017).

L.N. Mishra, R.P. Agarwal, M. Sen, Solvability and asymptotic behaviour for some nonlinear quadratic integral equation involving Erd´elyi-Kober fractional integrals on the unbounded interval, Progress in Fractional Differentiation and Applications. 2(3) (2016).

V.N. Mishra, Some problems on approximations of functions in Banach spaces, Ph.D. Thesis, Indian Institute of Technology, Roorkee-247 667, Uttarakhand, India, (2007).

V.N. Mishra et al., Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications. 2013 (2013) 586.

A. Nautiyal, On the growth of entire solutions of generalized axially symmetric Helmholtz equation, Indian J. Pure Appl. Math. 14(6)(1983) 718-721.

K. Ranger, Some integral transformation formulae for the Stokes-Beltrami equations, J. math. and Mech. 12 (1963) 663-674.

S.M. Shah, On the lower order of integral functions, Bull. Amer. Math. Soc. 52 (1946) 1046- 1052.

S.M. Shah, On the coefficients of an entire series of finite order, J. London. Math. Soc. 26 (1952) 45-46.

G.S. Srivastava, Approximation and growth of generalized axisymmetric potentials, Approximation Theory and Applications. 12(4) (1996) 96-104.

G.S. Srivastava, On the growth and approximation of generalized axisymmetric potentials, Functiones et Approximatio. XXIV (1996) 113-123.

G.S. Srivastava, On the growth and polynomial approximation of generalized biaxisymmetric potentials, Soochow J. Math. 23(4) (1997) 347-358.

G. Szeg¨o, Orthogonal polynomials, Colloquim Publications, Vol. 23, Amer. Math. Soc. Providence, R.I., (1967).

I. Vekua, Novye metody reˇsenija elliptiˇceskikh uravnenji, OGIZ, Moskow and Leningrad, USSR, 1948. (in Russian).

G. Watson, A treatise on the theory of Bessel functions, 2nd Eds., New York, (1944).

A. Weinstein, Generalized axially symmetric potential theory, Bull. Amer. Math. Soc. 59 (1953) 20-38.

A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc. 63 (1948) 342-354.

A. Weinstein, The method of axial symmetry in partial differential equations, Dagli Atti del convegnointernazionale sulle Equazioni alle derivate parziali, Trieste, (1954).

A. Weinstein, Transonic flow and generalized axially symmetric potential theory, Proc. Naval Ord. Lab. Aeroballistics Symp., White Oak, Md. 78 (1949).