Boundary value problems for periodic analytic functions. (English) Zbl 1338.30027
Summary: In this paper boundary value problems for periodic analytic functions are discussed. We first introduce definitions of principal part and order at \(\pm\infty i\) for periodic analytic functions through detailed analysis. Then Riemann boundary value problems for periodic analytic functions with finite order at \(\pm\infty i\) are formulated. Based on those, by using the exponential conformal mapping, Riemann boundary value problems for periodic sectionally holomorphic functions with periodic closed and periodic quasi-closed contours as their jump curves are solved. The method that we use here has computational advantages compared with the tangent mapping one used in solving the classical problems. Several types of Hilbert boundary value problems on the real axis and the circumferences for periodic analytic functions are also solved.
MSC:
| 30E25 | Boundary value problems in the complex plane |
| 45E05 | Integral equations with kernels of Cauchy type |
References:
| [1] | Lu, JK: Complex Variable Methods in Plane Elasticity. World Scientific, Singapore (1995) · Zbl 0845.73002 |
| [2] | Muskhelishvili, NI: Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen (1963) · Zbl 0124.17404 |
| [3] | Al-Smadi, M, Abu Arqub, O, Momani, S: A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations. Math. Probl. Eng. 2013, Article ID 832074 (2013) · Zbl 1299.65297 · doi:10.1155/2013/832074 |
| [4] | Abu-Gdairi, R, Al-Smadi, M: An efficient computational method for 4th-order boundary value problems of Fredholm IDEs. Appl. Math. Sci. (Ruse) 7(93-96), 4761-4774 (2013) |
| [5] | Al-Smadi, M, Abu Arqub, O, El-Ajou, A: A numerical iterative method for solving systems of first-order periodic boundary value problems. J. Appl. Math. 2014, Article ID 135465 (2014) · Zbl 1406.65050 · doi:10.1155/2014/135465 |
| [6] | Abu Arqub, O, Al-Smadi, M: Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations. Appl. Math. Comput. 243, 911-922 (2014) · Zbl 1337.65083 · doi:10.1016/j.amc.2014.06.063 |
| [7] | Lu, JK: Boundary Value Problems for Analytic Functions. World Scientific, Singapore (1993) · Zbl 0818.30027 |
| [8] | Muskhelishvili, NI: Singular Integral Equations, 2nd edn. Noordhoff, Groningen (1968) · Zbl 0174.16202 |
| [9] | Gakhov, FD: Boundary Problems. Nauka, Moscow (1977) · Zbl 0449.30030 |
| [10] | Chibrikova, LI: On Riemann boundary problem for automorphic functions. J. Kazan Univ. 116(4), 59-109 (1956) |
| [11] | Lu, JK: Periodic Riemann boundary value problems and their applications to elasticity. Chin. Math. 4, 372-422 (1964) |
| [12] | Lu, JK: On fundamental problems of plane elasticity with periodic stresses. Acta Mech. Sin. 7(4), 316-327 (1964) |
| [13] | Cai, HT, Lu, JK: Mathematical Theory in Periodic Plane Elasticity. Gordon & Breach, Singapore (2000) · Zbl 0959.74002 |
| [14] | Du, JY: Some systems of orthogonal trigonometric polynomials associated with singular integrals with Hilbert kernel. J. Wuhan Univ. Natur. Sci. Ed. 2(2), 17-35 (1988) · Zbl 0695.45006 |
| [15] | Du, JY: On the collocation methods for singular integral equations with Hilbert kernel. Math. Comput. 78(266), 891-928 (2009) · Zbl 0993.54040 · doi:10.1090/S0025-5718-08-02182-0 |
| [16] | Du, JY, Han, HL, Jin, QX: On trigonometric and paratrigonometric Hermite interpolation. J. Approx. Theory 131(1), 74-99 (2004) · Zbl 1064.42002 · doi:10.1016/j.jat.2004.09.005 |
| [17] | Wang, XY, Du, JY: Periodic Riemann boundary value problems. China Science Paper Online. http://www.paper.edu.cn |
| [18] | Du, JY, Wang, YF: On boundary value problem of polyanalytic function on the real axis. Complex Var. Theory Appl. 48(6), 527-542 (2003) · Zbl 1146.30307 · doi:10.1080/0278107031000103412 |
| [19] | Wang, YF, Du, JY: Hilbert boundary value problems of polyanalytic functions on the unit circumference. Complex Var. Elliptic Equ. 51(8-11), 923-943 (2006) · Zbl 1105.30029 · doi:10.1080/17476930600667692 |
| [20] | Mshimba, AS: On the Hilbert boundary value problem for holomorphic function in the Sobolev spaces. Appl. Anal. 30, 87-99 (1988) · Zbl 0631.35015 · doi:10.1080/00036818808839794 |
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