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.
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