Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 1012-2354 Erciyes Üniversitesi Kompleks düzlemin dairesel bölgesindeki lineer diferansiyel denklemlerin çözümleri için bir polinom yaklaşımı A polynomial approximation for solutions of linear differential equations in circular domains of the complex plane Sezer Mehmet Department of Mathematics, Faculty of Science, Muğla University Akyüz Daşcıoğlu Ayşegül 02 01 2012 28 1 60 64 02 01 2012 Copyright © 1985, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 1985 Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi

Bu makalede, dairesel bölgelerde yüksek mertebeden lineer kompleks diferansiyel denklemlerin çözümü için bir polinom yaklaşımı verilmektedir. Kullanılan bu sıralama yöntemi esas olarak denklemdeki bilinmeyen fonksiyon ve türev ifadelerinin kesilmiş Taylor seri temsillerinin matris gösterimlerine dayanır ki bunlar verilen bölgede tanımlanan sıralama noktalarını içerir. Yöntemin özelliklerini göstermek için karışık koşullu bazı sayısal örnekler verilmiştir.

In this paper we give a polynomial approach to the solution of higher order linear complex differential equations in the circular domains. The used collocation method is essentially based on the matrix representations of the truncated Taylor series of the expressions in equation and their derivatives, which consist of collocation points defined in the given domains. Some numerical examples with the mixed conditions are given to show the properties of the technique.

 Taylor Collocation method Polynomial approximation Complex differential equations Taylor sıralama yöntemi Polinom yaklaşımı Kompleks diferansiyel denklemler Cveticanin, L., Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type, Physica A, 297, 348-360, 2001. Cveticanin, L., Free vibration of a strong non-linear system described with complex functions, J. Sound and Vibration, 277, 815-824, 2004. Cveticanin, L., Approximate solution of strongly nonlinear complex differential equation, J. Sound and Vibration, 284, 503-512, 2005. Barsegian, G., Gamma-Lines: On the Geometry of Real and Complex Functions, Taylor and Francis, London-New York, 2002. Barsegian, G., Le, D.T., On a topological description of solutions of complex differential equations, Complex Variables, 50, 5, 307-318, 2005. Ishizaki, K., Tohge, K., On the complex oscillation of some linear differential equations, J. Math. Anal. Appl., 206, 503-517, 1997. Heittokangas, J., Korhonen, R., Rattya, J., Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn. Math., 29, 233-246, 2004. Andrievskii, V., Polynomial approximation of analytic functions on a finite number of continua in the complex plane, J. Approx. Theory, 133, 2, 238-244, 2005. Prokhorov, V.A., On best rational approximation of analytic functions, J. Approx. Theory, 133, 284-296, 2005. Akyüz, A., Sezer, M., A Chebyshev collocation method for the solution of linear integro-differential equations, Intern. J. Comput. Math., 72, 4, 491-507, 1999. Akyüz, A., Sezer, M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Applied Math. and Comp., 144, 237-247, 2003. Gülsu, M., Sezer, M., The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. and Comp., 168, 76-83, 2005. Nas, Ş., Yalçınbaş, S., Sezer, M., A Taylor polynomial approach for solving high- order linear Fredholm integro-differential equations, Int. J. Math. Educ. Sci. Technol., 31, 2, 213-225, 2000. Sezer, M., A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol., 27, 6, 821-834, 1996. Sezer, M., Akyüz-Daşcıoğlu, A., Taylor polynomial solutions of general linear differential-difference equations with variable coefficients, Applied Math. and Computation, 174, 2, 1526-1538, 2006. Sezer, M., Kaynak, M., Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol., 27, 4, 607-618, 1996. Ahlfors, L.V., Complex Analysis, McGraw-Hill Inc., Tokyo, 1966. Chiang, Y.M., Wang, S., Oscillation results of certain higher-order linear differential equations with periodic coefficients in the complex plane, J. Math. Anal. Appl., 215, 560-576, 1997. Spiegel, M. R., Theory and Problems of Complex Variables, McGraw-Hill Inc., New York, 1972. Sezer, M., Gülsu, M., Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method, Applied Math. and Computation, 177, 2, 844-851, 2006.