Yüksek Mertebeden Lineer Diferansiyel Fark Denklemlerinin Rezidüel Hata Tahminiyle Çözümü için Boubaker Polinom Yaklaşımı
Bu çalışmanın temel amacı başlangıç-sınır koşulları altında fonksiyonel argümentli yüksek mertebeden lineer diferansiyel-fark denklemlerinin çözümü için Boubaker polinomlarını uygulamaktır. Kullandığımız teknik, aslında sıralama noktaları ile birlikte kesilmiş Boubaker serisine ve bunların matris gösterimlerine dayandırılır. Ayrıca, Ortalama-Değer Teoremini ve rezidüel fonksiyonu kullanarak, etkili bir hata tahmin tekniği önerilir; metodun etkinliğini ve uygulanabilirliğini göstermek için bazı açıklayıcı örnekler sunulur.
Anahtar Kelimeler:
Boubaker Polinomları ve Serileri, Sıralama Noktaları, Diferansiyel Fark Denklemleri, Matris Metodu
Boubaker Polynomial Approach for Solving High-Order Linear Differential-Difference Equations with Residual Error Estimation
The main aim of this study is to apply the Boubaker polynomials for the solution of high-order linear differential-difference equations with functional arguments under the initial-boundary conditions. The technique we have used is essentially based on the truncated Boubaker series and its matrix representations together with collocation points. Also, by using the Mean-Volue 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.
Keywords:
Boubaker Polynomials and Series, Collocation Points, Differential-Difference Equations, Matrix Method,
___
- Ablowitz, M., L., Ladik, J., F. (1976). A nonlinear difference scheme and inverse scattering, Stud. Appl. Math., 55, 213-229.
- Hu, X., B., Ma, W., X. (2002). Application of Hirota’s bilinear formalism to the Toeplitz lattice some special soliton – like solutions, Phys. Lett. A 293, 161-165.
- Fan, E. (2001). Soliton solutions for a generalized Hirota-Sotsuma coupled KdV equation a Coupled MKdV equation, Phys. Lett. A 282, 18-22.
- Dai, C., Zhang, J. (2006). Jacobian elliptic function method for nonlinear differential difference equations, Chaos, Soliton Fract. 27, 1042-1047.
- Elmer, C., E., Van Vleck,, E., S. (2001). Traveling wave solutions for Bistable Differential- Difference Equations with Periodic Diffusion, SIAM J. Appl. Math. 61(5), 1648-1679.
- Elmer, C., E., Van Vleck, E., S. (2002). A Variant of Newton’s Method for the Computation of Traveling Waves of Bistable Differential-Difference Equation, J. Dyn. Different. Equat. 14, 493-517.
- Arıkoğlu, A., Özkol, I. (2006). Solution of difference equations by using differential transform method, Appl. Math. Comput. 174, 1216-1228.
- Sezer, M., Gülsu, M. (2005). Polynomial solution of the most general linear Fredholm İntegro- differntial-difference equation by means of Taylor matrix method, Complex variables, 50(5), 367-382.
- Gülsu, M., Sezer, M. (2005). A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Intern. J. Comput. Math. 82(5), 629-641.
- Saaty, TL. (1981). Modern nonlinear equations, Dover publications Inc., New York, P.225.
- Akkaya, T., Yalçınbaş, S., Sezer, M. (2013). Numeric solutions for the pantograph type delay differential equation using first Boubaker polynomials, Applied Mathematics and Computation 219, 9484–9492.
- Akgönüllü,N., Şahin, N., Sezer, M. (2011).A Hermite Collocation Method for the Approximate Solutions of High-Order Linear Fredholm Integro-Differential Equations, Numerical Methods Partial Differential Eq. 27: 1707–1721.
- Boubaker, K. (2007). Trends Appl. Sci. Res. On Modified Boubaker Polynomials: Some Differential and Analytical Properties of the New Polynomials Issued from an Attempt for Solving Bi-varied Heat Equation 2(6), (ss: 540–544).
- Akyuz-Dascioglu, A. (2006). A Chebyshev polynomial approach for linear Fredholm–Volterra Integro differential equations in the most general form, Appl Math Comput 181, 103–112.
- Yalçınbaş, S., and Sezer, M. (2006). A Taylor collocation method for the approximate solution of general linear Fredholm-Volterra integro-difference equations with mixed argument, Appl Math Comput 175, 675–690.
- Evans, D.J., Raslan, K.R, (2005). The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math. 82 (1), 49–54.
- Yalçınbas, S., Aynigül, M., Sezer, M. (2011). A collocation method using Hermite polynomials for approximate solution ofpantograph equations, J. Franklin Inst. 348 (6),1128–1139.
- Sezer, M., Akyuz-Dascioglu, A. (2006). Taylor polynomial solutions of general linear differential–difference equations with variable coefficients , Appl Math Comput 174, 1526–1538.
- Arıkoğlu, A.,, I. (2006). Solution of difference equations by using differential transform method, Appl. Matth. Comput. 174, 1216-1228.
- Başlangıç: 2016
- Yayıncı: Oğuzhan MALLI