Abstract
Hypercomplex analysis is useful for treating elliptic systems in plane domains. A modified hypercomplex Pompeiu operator is introduced leading to a singular hypercomplex integral operator. It serves to solve the Schwarz problem for the inhomogeneous hypercomplex Cauchy-Riemann equation. A hypercomplex approach is also used to solve some boundary value problem for a linear first order system in two complex variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Begehr, H.: Complex analytic methods for partial differential equations. An introductory text. World Scientific, Singapore, 1994.
Begehr, H., Dzhuraev, A.: An introduction to several complex variables and partial differential equations. Addison Wesley Longman, Harlow, 1997.
Begehr, H., Gilbert, R.P.: Randwertaufgaben ganzzahliger Charakteristik für verallgemeinerte hyperanalytische Funktionen. Appl. Anal. 6 (1977), 189–205.
Begehr, H., Gilbert, R.P.: Boundary value problems associated with first order elliptic systems in the plane. Contemporary Math. 11 (1982), 13–48.
Begehr, H., Gilbert, R.P.: Pseudohyperanalytic functions. Complex Variables, Theory Appl. 9 (1988), 343–357.
Begehr, H., Gilbert, R.P.: Transformations, transmutations, and kernel functions; I, II. Longman, Harlow, 1992, 1993.
Begehr, H., Wen, G.C.: Nonlinear elliptic boundary value problems and their applications. Addison Wesley Longman, Harlow, 1996.
Begehr, H., Wen, G.C.: Some second order systems in the complex plane. Preprint, FU Berlin, 1997.
Douglis, A.: A function-theoretic approach to elliptic systems of equations in two variables. Comm. Pure Appl. Math. 6 (1953), 259–289.
Dzhuraev, A., Begehr, H.: On a boundary value problem for a first order holomorphic system in ℂ2. Ross. Akad. Nauk Doklady 339 (1994), 297–300
Dzhuraev, A., Begehr, H.: Russ. Acad. Sci. Dokl. Math. 50 (1995), 418–422.
Gilbert, R.P.: Constructive methods for elliptic equations. Lecture Notes in Math. 365, Springer-Verlag, Berlin, 1974.
Gilbert, R.P., Hile, G.N.: Generalized hypercomplex function theory. Trans. Amer. Math. Soc. 195 (1974), 1–29.
Gilbert, R.P., Buchanan, J.L.: First order elliptic systems. A function the-oretic approach. Acad. Press, New York, 1983.
Hile, G.N.: Hypercomplex function theory applied to partial differential equations. Ph.D. thesis, Indiana University, Bloomington, Indiana, 1972.
Huang, S.X.: On properties of some operators in Douglis albegra and their applications to pde. J. Part. Diff. Eq. 1 (1988), 21–30.
Kühn, E.: Über die Funktionentheorie und das Ähnlichkeitsprinzip einer Klasse elliptischer Differentialgleichungen in der Ebene. Dissertation, Univ. Dortmund, Dortmund, 1974.
Vekua, I.N.: Generalized analytic functions. Pergamon Press, Oxford, 1962.
Wen, G.C., Begehr, H.: Boundary value problems for elliptic equations and systems. Longman, Harlow, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Kluwer Academic Publishers
About this chapter
Cite this chapter
Begehr, H. (1999). Systems of First Order Partial Differential Equations — A Hypercomplex Approach. In: Begehr, H.G.W., Gilbert, R.P., Wen, GC. (eds) Partial Differential and Integral Equations. International Society for Analysis, Applications and Computation, vol 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3276-3_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3276-3_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3278-7
Online ISBN: 978-1-4613-3276-3
eBook Packages: Springer Book Archive