Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Affiliation(s)

School of Mathematical Sciences, Peking University, Beijing, China.

Mathematics Department, Beijing Technology and Business University, Beijing, China.

Uniformed Services University of the Health Sciences, Bethesda, USA.

School of Mathematical Sciences, Peking University, Beijing, China.

Mathematics Department, Beijing Technology and Business University, Beijing, China.

Uniformed Services University of the Health Sciences, Bethesda, USA.

ABSTRACT

Several
approximate methods have been used to find approximate solutions of elliptic
systems of first order equations. One common method is the Newton imbedding
approach, *i.e.* the parameter
extension method. In this article, we discuss approximate solutions to
discontinuous Riemann-Hilbert boundary value problems, which have various
applications in mechanics and physics. We first formulate the discontinuous
Riemann-Hilbert problem for elliptic systems of first order complex equations
in multiply connected domains and its modified well-posedness, then use the parameter
extensional method to find approximate solutions to the modified boundary value
problem for elliptic complex systems of first order equations, and then provide
the error estimate of approximate solutions for the discontinuous boundary
value problem.

KEYWORDS

Discontinuous Riemann-Hilbert Problem, Elliptic Systems of First Order Complex Equations, Esti-mates and Existence of Solutions, Multiply Connected Domains

Discontinuous Riemann-Hilbert Problem, Elliptic Systems of First Order Complex Equations, Esti-mates and Existence of Solutions, Multiply Connected Domains

Cite this paper

Wen, G. , Zhang, Y. and Chen, D. (2014) Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations.*Applied Mathematics*, **5**, 1546-1556. doi: 10.4236/am.2014.510148.

Wen, G. , Zhang, Y. and Chen, D. (2014) Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations.

References

[1] Wen, G.C. and Begehr, H. (1990) Boundary Value Problems for Elliptic Equations and Systems. Longman Scientific and Technical Company, Harlow.

[2] Wen, G.C. (1992) Conformal Mappings and Boundary Value Problems, Translations of Mathematics Monographs 106. American Mathematical Society, Providence.

[3] Wen, G.C., Tai, C.W. and Tian, M.Y. (1996) Function Theoretic Methods of Free Boundary Problems and Their Applications to Mechanics. Higher Education Press, Beijing (Chinese).

[4] Wen, G.C. (1986) Linear and Nonlinear Elliptic Complex Equations. Shanghai Scientific and Technical Publishers, Shanghai (Chinese).

[5] Wen, G.C. (1999) Approximate Methods and Numerical Analysis for Elliptic Complex Equations. Gordon and Breach, Amsterdam.

[6] Wen, G.C. (1999 Linear and Nonlinear Parabolic Complex Equations. World Scientific Publishing Co., Singapore City.

[7] Wen, G.C. and Zou, B.T. (2002) Initial-Boundary Value Problems for Nonlinear Parabolic Equations in Higher Dimensional Domains. Science Press, Beijing.

[8] Wen, G.C. (2002) Linear and Quasilinear Complex Equations of Hyperbolic and Mixed Type. Taylor & Francis, London. http://dx.doi.org/10.4324/9780203166581

[9] Huang, S., Qiao, Y.Y. and Wen, G.C. (2005) Real and Complex Clifford Analysis. Springer Verlag, Heidelberg.

[10] Wen, G.C. (2008) Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy. World Scientific, Singapore City.

[11] Wen, G.C., Chen, D.C. and Xu, Z.L. (2008) Nonlinear Complex Analysis and its Applications, Mathematics Monograph Series 12. Science Press, Beijing.

[12] Wen, G.C. (2010) Recent Progress in Theory and Applications of Modern Complex Analysis. Science Press, Beijing.

[1] Wen, G.C. and Begehr, H. (1990) Boundary Value Problems for Elliptic Equations and Systems. Longman Scientific and Technical Company, Harlow.

[2] Wen, G.C. (1992) Conformal Mappings and Boundary Value Problems, Translations of Mathematics Monographs 106. American Mathematical Society, Providence.

[3] Wen, G.C., Tai, C.W. and Tian, M.Y. (1996) Function Theoretic Methods of Free Boundary Problems and Their Applications to Mechanics. Higher Education Press, Beijing (Chinese).

[4] Wen, G.C. (1986) Linear and Nonlinear Elliptic Complex Equations. Shanghai Scientific and Technical Publishers, Shanghai (Chinese).

[5] Wen, G.C. (1999) Approximate Methods and Numerical Analysis for Elliptic Complex Equations. Gordon and Breach, Amsterdam.

[6] Wen, G.C. (1999 Linear and Nonlinear Parabolic Complex Equations. World Scientific Publishing Co., Singapore City.

[7] Wen, G.C. and Zou, B.T. (2002) Initial-Boundary Value Problems for Nonlinear Parabolic Equations in Higher Dimensional Domains. Science Press, Beijing.

[8] Wen, G.C. (2002) Linear and Quasilinear Complex Equations of Hyperbolic and Mixed Type. Taylor & Francis, London. http://dx.doi.org/10.4324/9780203166581

[9] Huang, S., Qiao, Y.Y. and Wen, G.C. (2005) Real and Complex Clifford Analysis. Springer Verlag, Heidelberg.

[10] Wen, G.C. (2008) Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy. World Scientific, Singapore City.

[11] Wen, G.C., Chen, D.C. and Xu, Z.L. (2008) Nonlinear Complex Analysis and its Applications, Mathematics Monograph Series 12. Science Press, Beijing.

[12] Wen, G.C. (2010) Recent Progress in Theory and Applications of Modern Complex Analysis. Science Press, Beijing.