Morgan-Voyce Polynomial Approach for Solution of High-Order Linear Differential-Difference Equations with Residual Error Estimation

Yıl 2016, Cilt: 4 Sayı: 1, 252 - 263, 31.01.2016

Bengü Türkyılmaz Burcu Gürbüz Mehmet Sezer

Öz

The main aim of this study is to apply the Morgan-Voyce polynomials for the solution of high-order linear differential difference equations with functional arguments under initial boundary conditions. The technique we have used is essentially based on the truncated Morgan-Voyce series and its matrix representations along with collocation points. Also, by using the Mean-Value Theorem and residual function, an efficient error estimation technique is proposed and some illustrative examples are presented to demonstrate the validity and applicability of the method.

Anahtar Kelimeler

Morgan- Voyce polynomials, Differential-difference equations, Collocation method, Matrix method, Error estimation, Residual function

Yüksek Mertebeden Lineer Diferansiyel Fark Denklemlerinin Hata Tahmini ile Morgan-Voyce Yaklaşımı

Öz

Bu çalışmanın amacı, yüksek mertebeden fonkisyonel argümanlı lineer diferansiyel fark denklemlerinin başlangıç sınır koşulları altında Morgan- Voyce polinomlarına bağlı çözümlerini araştırmaktır. Kullanılan metot esas olarak kesilmiş Morgan- Voyce serilerinin ve sıralama noktalarına bağlı matris gösterimlerine dayalıdır. Ayrıca, Ortalama Değer Teoremi ve rezidüel fonksiyonu ile etkin bir hata tahmini yöntemi verilmektedir. Bazı örneklerle metotun geçerlilik ve uygulanabilirliği gösterilmektedir

Anahtar Kelimeler

Morgan- Voyce polinomları, Diferansiyel-fark denklemleri, Kolokasyon metotu, Matris metotu, Hata tahmini, Rezidüel fonksiyon

Kaynakça

• M. L. Ablowitz, J. F. Ladik, A nonlinear difference scheme and inverse scattering, Stud. Appl. Math., 55 (1976) 213-229.
• X. B. Hu, W. X.Ma, Application of Hirota’s bilinear formalism to the teoplitz lattice some special soliton-like solitions, Phys.Lett:A 293 (2002) 161-165.
• E.Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KDV equation a coupled Mkdv equation, Phys.Lett.A 282 (2001) 18-22.
• C. Dai, J. Zhang, Jacobian elliptic function method for nonlinear differential difference equations, Caos, Soliton Frac. 27 (2006) 1042-1047.
• C. E. Elmer, E.S.Van Vleck, Traveling wave solutions for bistable differential difference equations with periodic diffusion, SIAM J.Appl.Math. 81 (2001) 1648-1657.
• C. E. Elmer, E. S. Van Vleck, A variant of Newton’s method for solution of traveling wave solutions of bistable differential difference equations, J. Dyn. Different.equat. 14 (2002) 493-517.
• A. Arıkoğlu, I. Özkal, Solution of differential equations by using differential transform method, Appl. Math. Comput. 174 (2006) 1216-1228.
• M. Sezer, M. Gülsu, Polynominal solution of the most general linear Fredholm integro- differential difference equation by means of Taylor matrix method, Complex variables, 50 (5) (2005), 367-382.
• M. Gülsu, M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Intern.J.Comput.Math. 82 (5) (2005) 629-641.
• T. L.Saaty, Modern nonlinear equations, Dover publications Inc., New York (1981),P.225.
• M. Sezer, A. Akyüz-Daşcıoğlu, Taylor polynominal solutions of general linear differential- difference equations with variable coefficients, Appl.Math.Comput.174 (2006) 1526-1538.
• M. Gülsu, M. Sezer, A Taylor polynomial approach for solvig differential difference equations, J.Comput.Appl.Math.186 (2006) 549-364.
• M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Intern. J. Math. Educ.Sci.Technol. 27 (4) (1996) 607-618.
• A. Akyüz, M. Sezer, A Chebyshev Collocation method for the solution of linear integro- differential equations, Intern.J.Comput.math. 72 (4) (1999) 491-507.
• A. Akyüz, M. Sezer, Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput. 144 (2003) 237-247.
• M. M. S.Swamy, Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 62 (1968) 165-175.
• Ö. İlhan, Morgan-Voyce polynomial solutions of linear differential and integro- differential equations, M.Sc. Thesis, Graduate School of Naturel and Applied Sciences, Muğla University, 2012.
• L. Fortuna, M. Frasca, Generating passive systems from recursively defined polynomials, Intem. J. Circuits system and signal processing, 6 (2) 2012 179-188.
• T. Koshy, Fibonocci and Lucas Numbers with Applications, Wiley, New York, 2001.
• B. Gürbüz, M. Gülsu, M. Sezer, Numerical Approach of high-order linear delay difference equations with variable coefficients in terms of Laguerre polynomials, Math. Comput. Appl. 16 (2011) 267-278.
• B. Gürbüz, M. Gülsu, M. Sezer, Numerical solution of differential difference equations by Laguerre collocation method, Erciyes Uni. Journ. Inst. Sci. Tech. 27 (2011) 75-87.
• Ö. İlhan, N. Şahin, M. Sezer, On Morgan- Voyce polynomial approximation for linear Fredholm integro- differential equations, International Sym posium on, Computing Science and Engineering, Gediz University, Proc. Number:700/62, İzmir.