Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves

ABSTRACT

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

Cite this paper

Cao, L. and Zheng, H. (2015) Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves.*Advances in Pure Mathematics*, **5**, 51-58. doi: 10.4236/apm.2015.51006.

Cao, L. and Zheng, H. (2015) Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves.

References

[1] Balk, M.B. (1991) Polyanalytic Functions. Akademie Verlag, Berlin.

[2] Begehr, H. and Kumar, A. (2005) Boundary Value Problems for the Inhomogeneous Polyanalytic Equation I. Analysis: International Mathematical Journal of Analysis and its Application, 25, 55-71.

[3] Du, J.Y. and Wang, Y.F. (2003) On Boundary Value Problems of Polyanalytic Functions on the Real Axis. Complex Variables, 48, 527-542. http://dx.doi.org/10.1080/0278107031000103412

[4] Fatulaev, B.F. (2001) The Main Haseman Type Boundary Value Problem for Metaanalytic Function in the Case of Circular Domains. Mathematical Modelling and Analysis, 6, 68-76.

[5] Lu, J.K. (1993) Boundary Value Problems for Analytic Functions. World Scientific, Singapore.

[6] Mshimba, A.S. (2002) A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn, p(D). Complex Variables, 47, 278-1077.

[7] Muskhelishvili, N.I. (1993) Singular Integral Equations. World Scientific, Singapore.

[8] Wanf, Y.F. and Du, J.Y. (2006) Hilbert Boundary Value Problems of Polyanalytic Functions on the Unit Circumference. Complex Variables and Elliptic Equations, 51, 923-943.

http://dx.doi.org/10.1080/17476930600667692

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

[10] Wang, M.H. (2006) Inverse Riemann Boundary Value Problems for Generalized Analytic Functions. Journal of Ningxia University of Natural Resources and Life Sciences Education, 27, 18-24.

[11] Wen, X.Q. and Li, M.Z. (2004) A Class of Inverse Riemann Boundary Value Problems for Generalized Holomorphic Functions. Journal of Mathematical, 24, 457-464.

[12] Cao, L.X., Li, P.-R. and Sun, P. (2012) The Hilbert Boundary Value Problem With Parametric Unknown Function on Upper Half-Plane. Mathematics in Practice and Theory, 42, 189-194.

[13] Cao, L.X. (2013) Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line. Applied Mathematics, 4, 1126-1230.

[14] Cao, L.X., Li, X.W. and Lin, C.X. (2014) A Kind of Doubly Periodic Riemann Boundary Value Problem on Two Parallel Curves. Advances in Pure Mathematics, 4, 627-634.

[1] Balk, M.B. (1991) Polyanalytic Functions. Akademie Verlag, Berlin.

[2] Begehr, H. and Kumar, A. (2005) Boundary Value Problems for the Inhomogeneous Polyanalytic Equation I. Analysis: International Mathematical Journal of Analysis and its Application, 25, 55-71.

[3] Du, J.Y. and Wang, Y.F. (2003) On Boundary Value Problems of Polyanalytic Functions on the Real Axis. Complex Variables, 48, 527-542. http://dx.doi.org/10.1080/0278107031000103412

[4] Fatulaev, B.F. (2001) The Main Haseman Type Boundary Value Problem for Metaanalytic Function in the Case of Circular Domains. Mathematical Modelling and Analysis, 6, 68-76.

[5] Lu, J.K. (1993) Boundary Value Problems for Analytic Functions. World Scientific, Singapore.

[6] Mshimba, A.S. (2002) A Mixed Boundary Value Problem for Polyanalytic Function of Order n in the Sobolev Space Wn, p(D). Complex Variables, 47, 278-1077.

[7] Muskhelishvili, N.I. (1993) Singular Integral Equations. World Scientific, Singapore.

[8] Wanf, Y.F. and Du, J.Y. (2006) Hilbert Boundary Value Problems of Polyanalytic Functions on the Unit Circumference. Complex Variables and Elliptic Equations, 51, 923-943.

http://dx.doi.org/10.1080/17476930600667692

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

[10] Wang, M.H. (2006) Inverse Riemann Boundary Value Problems for Generalized Analytic Functions. Journal of Ningxia University of Natural Resources and Life Sciences Education, 27, 18-24.

[11] Wen, X.Q. and Li, M.Z. (2004) A Class of Inverse Riemann Boundary Value Problems for Generalized Holomorphic Functions. Journal of Mathematical, 24, 457-464.

[12] Cao, L.X., Li, P.-R. and Sun, P. (2012) The Hilbert Boundary Value Problem With Parametric Unknown Function on Upper Half-Plane. Mathematics in Practice and Theory, 42, 189-194.

[13] Cao, L.X. (2013) Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line. Applied Mathematics, 4, 1126-1230.

[14] Cao, L.X., Li, X.W. and Lin, C.X. (2014) A Kind of Doubly Periodic Riemann Boundary Value Problem on Two Parallel Curves. Advances in Pure Mathematics, 4, 627-634.