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

Year 2019, Volume: 9 Issue: 2, 314 - 328, 30.12.2019
https://doi.org/10.37094/adyujsci.508986

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. 

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.

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

Year 2019, Volume: 9 Issue: 2, 314 - 328, 30.12.2019
https://doi.org/10.37094/adyujsci.508986

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.

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.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences, Applied Mathematics
Journal Section Mathematics
Authors

Yalçın Öztürk

Publication Date December 30, 2019
Submission Date January 6, 2019
Acceptance Date December 18, 2019
Published in Issue Year 2019 Volume: 9 Issue: 2

Cite

APA Öztürk, Y. (2019). Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel. Adıyaman University Journal of Science, 9(2), 314-328. https://doi.org/10.37094/adyujsci.508986
AMA Öztürk Y. Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel. ADYU J SCI. December 2019;9(2):314-328. doi:10.37094/adyujsci.508986
Chicago Öztürk, Yalçın. “Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations With Weakly Singular Kernel”. Adıyaman University Journal of Science 9, no. 2 (December 2019): 314-28. https://doi.org/10.37094/adyujsci.508986.
EndNote Öztürk Y (December 1, 2019) Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel. Adıyaman University Journal of Science 9 2 314–328.
IEEE Y. Öztürk, “Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel”, ADYU J SCI, vol. 9, no. 2, pp. 314–328, 2019, doi: 10.37094/adyujsci.508986.
ISNAD Öztürk, Yalçın. “Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations With Weakly Singular Kernel”. Adıyaman University Journal of Science 9/2 (December 2019), 314-328. https://doi.org/10.37094/adyujsci.508986.
JAMA Öztürk Y. Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel. ADYU J SCI. 2019;9:314–328.
MLA Öztürk, Yalçın. “Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations With Weakly Singular Kernel”. Adıyaman University Journal of Science, vol. 9, no. 2, 2019, pp. 314-28, doi:10.37094/adyujsci.508986.
Vancouver Öztürk Y. Chebyshev Series Solutions for a Class of System of Linear Integro-Differential Equations with Weakly Singular Kernel. ADYU J SCI. 2019;9(2):314-28.

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