Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line

Author(s)
Lixia Cao

Affiliation(s)

Department of Information and Computing Sciences, Mathematics College, Northeast Petroleum University, Daqing, China.

Department of Information and Computing Sciences, Mathematics College, Northeast Petroleum University, Daqing, China.

ABSTRACT

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12].

In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.

Cite this paper

L. Cao, "Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 1226-1230. doi: 10.4236/am.2013.48165.

L. Cao, "Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line,"

References

[1] M. B. Balk, “Polyanalytic Functions,” Akademie Verlag, Berlin, 1991.

[2] H. Begehr and A. Kumar, “Boundary Value Problems for the Inhomogeneous Polyanalytic Equation I,” Analysis: International Mathematical Journal of Analysis and Its Application, Vol. 25, No. 1, 2005, pp. 55-71.

[3] D. Jinyuan and W. Yufeng, “On Boundary Value Prob lems of Polyanalytic Functions on the Real Axis,” Com plex Variables, Vol. 48, No. 6, 2003, pp. 527-542. doi:10.1080/0278107031000103412

[4] B. F. Fatulaev, “The Main Haseman Type Boundary Value Problem for Metaanalytic Function in the Case of Circular Domains,” Mathematical Modelling and Analy sis, Vol. 6, No. 1, 2001, pp. 68-76.

[5] J. K. Lu, “Boundary Value Problems for Analytic Func tions,” World Scientific, Singapore City, 1993.

[6] A. S. Mshimba, “A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn, p(D),” Complex Variables, Vol. 47, No. 12, 2002, pp. 278-1077.

[7] N. I. Muskhelishvili, “Singular Integral Equations,” World Scientific, Singapore City, 1993.

[8] W. Yufeng and D. Jinyuan, “Hilbert Boundary Value Problems of Polyanalytic Functions on the Unit Circum ference,” Complex Variables and Elliptic Equations, Vol. 51, No. 8-11, 2006, pp. 923-943. doi:10.1080/17476930600667692

[9] L. Xing, “A Class of Periodic Riemann Boundary Value Inverse Problems,” Proceedings of the Second Asian Mathematical Conference, Nakhon Ratchasima, 17-20 October 1995, pp. 397-400.

[10] M. H. Wang, “Inverse Riemann Boundary Value Prob lems for Generalized Analytic Functions,” Journal of Ningxia University of Natural Resources and Life Sci ences Education, Vol. 27, No. 1, 2006, pp. 18-24.

[11] X. Q. Wen and M. Z. Li, “A Class of Inverse Riemann Boundary Value Problems for Generalized Holomorphic Functions,” Journal of Mathematical, Vol. 24, No. 4, 2004, pp. 457-464.

[12] L. X. Cao, P.-R. Li and P. Sun, “The Hilbert Boundary Value Problem With Parametric Unknown Function on Upper Half-Plane,” Mathematics in Practice and Theory, Vol. 42, No. 2, 2012, pp. 189-194.

[1] M. B. Balk, “Polyanalytic Functions,” Akademie Verlag, Berlin, 1991.

[2] H. Begehr and A. Kumar, “Boundary Value Problems for the Inhomogeneous Polyanalytic Equation I,” Analysis: International Mathematical Journal of Analysis and Its Application, Vol. 25, No. 1, 2005, pp. 55-71.

[3] D. Jinyuan and W. Yufeng, “On Boundary Value Prob lems of Polyanalytic Functions on the Real Axis,” Com plex Variables, Vol. 48, No. 6, 2003, pp. 527-542. doi:10.1080/0278107031000103412

[4] B. F. Fatulaev, “The Main Haseman Type Boundary Value Problem for Metaanalytic Function in the Case of Circular Domains,” Mathematical Modelling and Analy sis, Vol. 6, No. 1, 2001, pp. 68-76.

[5] J. K. Lu, “Boundary Value Problems for Analytic Func tions,” World Scientific, Singapore City, 1993.

[6] A. S. Mshimba, “A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn, p(D),” Complex Variables, Vol. 47, No. 12, 2002, pp. 278-1077.

[7] N. I. Muskhelishvili, “Singular Integral Equations,” World Scientific, Singapore City, 1993.

[8] W. Yufeng and D. Jinyuan, “Hilbert Boundary Value Problems of Polyanalytic Functions on the Unit Circum ference,” Complex Variables and Elliptic Equations, Vol. 51, No. 8-11, 2006, pp. 923-943. doi:10.1080/17476930600667692

[9] L. Xing, “A Class of Periodic Riemann Boundary Value Inverse Problems,” Proceedings of the Second Asian Mathematical Conference, Nakhon Ratchasima, 17-20 October 1995, pp. 397-400.

[10] M. H. Wang, “Inverse Riemann Boundary Value Prob lems for Generalized Analytic Functions,” Journal of Ningxia University of Natural Resources and Life Sci ences Education, Vol. 27, No. 1, 2006, pp. 18-24.

[11] X. Q. Wen and M. Z. Li, “A Class of Inverse Riemann Boundary Value Problems for Generalized Holomorphic Functions,” Journal of Mathematical, Vol. 24, No. 4, 2004, pp. 457-464.

[12] L. X. Cao, P.-R. Li and P. Sun, “The Hilbert Boundary Value Problem With Parametric Unknown Function on Upper Half-Plane,” Mathematics in Practice and Theory, Vol. 42, No. 2, 2012, pp. 189-194.