Zayıf Tekil Çekirdekli Lineer İntegro Diferansiyel Denklemlerin Bir Sınıfının Chebyshev Seri Çözümleri

Year 2019, Volume: 9 Issue: 2 , 314 - 328 , 30.12.2019

Yalçın Öztürk

https://doi.org/10.37094/adyujsci.508986

Cited By: 1

https://izlik.org/JA42RJ45FD

Abstract

Bu çalışmada, zayıf tekil çekirdekli lineer integro diferansiyel denklemlerin bir sınıfı için bir nümerik algoritma sunulacaktır. Bu algoritma birinci tip Chebyshev polinom bazı yardımıyla polinom yaklaşımı ve sıralama metodunu temel almaktadır. Bu metot verilen denklem ve koşulları bir matris denklemine dönüştürür. Nümerik metodun uygulanabilirliğini ve doğruluğunu göstermek amacıyla bazı örnekler incelenecektir. Sunulan metot diğer metotlar ile kıyaslanmıştır. 

Keywords

İntegro-diferansiyel denklemlerin singular sistemleri , Zayıf tekil çekirdek , Abel denklemi , Sıralama metodu , Chebyshev polinomları

Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel

https://izlik.org/JA42RJ45FD

Abstract

  In this study, a numerical algorithm for solving a class of system of linear integro differential equations with weakly singular kernel is presented. This algorithm is based on polynomial approximation and collocation method, using the first kind Chebyshev polynomial basis. This method transforms the equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. To show the validity and applicability of the numerical method some experiments are examined. Present method is compared some numerical methods.

Keywords

Singular system of integro-differential equations , weakly singular kernel , Chebyshev polynomials , Abel's equation , Collocation method , Chebyshev polynomials

References

• [1] Frankel, J., A Galerkin solution to a regularized Cauchy singular integro-differential equation, Quarterly of Applied Mathematics, L11 (2), 245–258, 1995.
• [2] Green, C.D., Integral Equation Methods, Nelsson, New York, 1969.
• [3] Hori, M., Nasser, N., Asymptotic solution of a class of strongly singular integral equations, SIAM Journal on Applied Mathematics (SIAP), 50 (3), 716–725, 1990.
• [4] Muskhelishvili, N.I., Singular Integral Equations, Noordhoff, Leiden, 1953.
• [5] Brunner, H., Pedas, A., Vainikko, G., Piecewise polynomial collocation methods for linear Volterra integrodifferential equations with weakly singular kernels, SIAM Journal on Numerical Analysis, 39, 957–982, 2001.
• [6] Gülsu, M., Öztürk, Y., Numerical approach for the solution of hypersingular integro differential equations, Applied Mathematics and Computation, 230, 701-710, 2014.
• [7] Işık, O.R., Sezer, M., Güney, Z., Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Applied Mathematics and Computation, 217, 7009-7020, 2011.
• [8] Karimi, S., Soleymani, F., Tau approximate solution of weakly singular Volterra integral equations, Mathematical and Computer Modelling, 57(3-4), 494-502, 2013.
• [9] Koya, A.C., Erdoğan, F., On the solution of integral equations with strongly singular kernels, Quarterly of Applied Mathematics, 45, 105–122, 1987.
• [10] Lakestani, M., Saray, B.N., Dehghan, M., Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets, Journal of Computational and Applied Mathematics, 235, 3291-3303, 2011.
• [11] Maleknejad, K., Arzhang, A., Numerical solution of the Fredholm singular integro differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method, Applied Mathematics and Computation, 182, 888-897, 2006.
• [12] Öztürk, Y., Gülsu, M., A collocation method for solving system of Volterra-differential-difference equations with terms of Chebyshev polynomials, British Journal of Applied Science & Technology, 14(4), 1-20, 2016.
• [13] Parts, I., Pedas, A., Collocation approximations for weakly singular Volterra integro-differential equations, Mathematical Modelling and Analysis, 8, 315–328, 2003.
• [14] Pedas, A., Tamme E., Spline collocation method for integro-differential equations with weakly singular kernels, Journal of Computational and Applied Mathematics, 197, 253-269, 2006.
• [15] Razlighi, B.B., Soltanalizadeh, B., Numerical solution for system of singular nonlinear Volterra integro-differential equations by Newton-Product method, Applied Mathematics and Computation, 219, 8375-8383, 2013.
• [16] Singh, V.K., Postnikov, E.B., Operational matrix approach for solution of integro-differential equations arising in theory ofanomalous relaxation processes in vicinity of singular point, Applied Mathematical Modelling, 37, 6609-6616, 2013.
• [17] Turhan, İ., Oğuz, H., Yusufoğlu, E., Chebyshev polynomial solution of the system of Cauchy type singular integral equations of first kind, Journal of Computational Mathematics, 90(5), 944-954, 2013.
• [18] Fox, L., Parker, I.B., Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
• [19] Mason, J.C., Handscomb, D.C., Chebyshev Polynomials, Chapman and Hall/CRC, New York, 2003.