Abstract
In this paper, we discuss some Dirichlet problems for inhomogeneous complex mixed-partial differential equations of higher order in the unit disc. Using higher-order Pompeiu operators T m,n, we give some special solutions for the inhomogeneous equations. The solutions of homogeneous equations are given on the basis of decompositions of polyanalytic and polyharmonic functions. Combining the solutions of the homogeneous equations and special solutions, we obtain all solutions of the inhomogeneous equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
N. Aronszajn, T. Cresse and L. Lipkin, Polyharmonic Functions, Oxford University Press, 1983.
M. B. Balk, Polyanalytic Functions, Akademie Verlag, 1992.
H. Begehr, Boundary value problems in complex analysis I; II, Bol. Asco. Mat. Venez. 12 (2005), 65–85; 217–250.
H. Begehr, Dirichlet problems for the biharmonic equations, Gen. Math. 13(2005), 65–72.
H. Begehr, Six biharmonic Dirichlet problems in complex analysis, in Function Spaces in Complex and Clifford Analysis, Editors: H. S. Lee and W. Tutschke, National Univ. Publ. Hanoi, 2008, 243–252.
H. Begehr, Biharmonic Green functions, Le Matematiche 61(2006), 395–405.
H. Begehr, A particular polyharmonic Dirichlet problem, in Complex Analysis, Potential Theory, Editors: T. Aliev Azeroglu and P. M. Tamrazov, World Scientific, 2007, 84–115.
H. Begehr, J. Du and Y. Wang, A Dirichlet problem for polyharmonic functions, Ann. Mat. Pura Appl. 187(4) (2008), 435–457.
H. Begehr and E. Gaertner, A Dirichlet problem for the inhomogeneneous polyharmonic equations in the upper half plane, Georgian Math. J. 14(2007), 33–52.
H. Begehr and G. N. Hile, A hierarchy of integral operators, Rocky Mountains J. Math. 27(1997), 669–706.
H. Begehr, T. N. H. Vu and Z. Zhang, Polyharmonic Dirichlet problems, Proc. Steklov Inst. Math. 255(2006), 13–34.
H. Begehr and Y. Wang, A new approach for solving a Dirichlet problem for polyharmonic function, Complex Var. Elliptic Equ. 52(2007), 907–920.
P. Burgatti, Sulla funzionni analitiche d’ordini n, Boll. Unione Mat. Ital. 1 (1922), 8–12.
A. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88(1952), 85–139.
J. Du and Y. Wang, On boundary value problem of polyanalytic functions on the real axis, Complex Variables 48(2003), 527–542.
J. Du and Y. Wang, Riemann boundary value problem of polyanalytic function and metaanalytic functions on a closed curve, Complex Variables 50(2005), 521–533.
B. F. Fatulaev, The main Haseman type boundary value problem for metaanalytic function in the case of circular domains, Math. Model. Anal. 6(2001), 68–76.
E. Gaertner, Basic Complex Boundary Value Problems in The Upper Half Plane, Doctoral Dissertation, Freie Universität Berlin, 2006. www.diss.fu-berlin.de/diss/receive/FUDISS_thesis._000000002129
F. D. Gakhov, Boundary Value Problems, Second Edition, Dover, 1990.
E. Goursat, Sur l’équation ΔΔu=0, Bull. Soc. Math. France 26(1898), 236–237.
A. Kumar and R. Prakash, Iterated boundary value problems for the polyanalytic equation, Complex Var. Elliptic Equ. 52(2007), 921–932.
J. Lu, Boundary Value Problems for Analytic Functions, World Scientific, 1993.
I. N. Muskhelishvili, Singular Integral Equations, Second Edition, Dover, 1992.
E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
N. Teodorescu, La Derivée Aréolaire et ses Applications á la Physique Mathématique, Gauthier-Villars, Paris, 1931.
I. N. Vekua, Generalized Analytic Functions, Pergamon, 1962.
I. N. Vekua, On one method of solving the first biharmonic boundary value problem and the Dirichlet problem, Amer. Math. Soc. Transl. (2) 104(1976), 104–111.
Y. Wang and J. Du, On Riemann boundary value problem of polyanalytic functions on the real axis, Acta Math. Sci. 24B(2004), 663–671.
Y. Wang and J. Du, Hilbert boundary value problem of polyanalytic functions on the unit curcumference, Complex Var. Elliptic Equ. 51(2006), 923–943.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Begehr, H., Du, Z., Wang, N. (2009). Dirichlet Problems for Inhomogeneous Complex Mixed-Partial Differential Equations of Higher order in the Unit Disc: New View. In: Schulze, BW., Wong, M.W. (eds) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol 205. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0198-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0198-6_6
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0197-9
Online ISBN: 978-3-0346-0198-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)