An efficient technique for solving systems of integral equations
محورهای موضوعی : Numerical Analysis
حمیده ابراهیمی
1
(Department of Applied Mathematics, Basic Science Faculty, Islamic Azad University, Rasht Branch, Rasht, Iran)
کلید واژه: Operational matrix, Mother wavelet, Multiresolution analysis, The second kind Chebyshev wavelet method, Systems of integral equations,
چکیده مقاله :
In this paper, the wavelet method based on the Chebyshev polynomials of the second kind is introduced and used to solve systems of integral equations. Operational matrices of integration, product, and derivative are obtained for the second kind Chebyshev wavelets which will be used to convert the system of integral equations into a system of algebraic equations. Also, the error is analyzed and at the end, some examples are presented to demonstrate the efficiency and the validity of the proposed method.
در این مقاله، روش موجک بر اساس چندجملهایهای چبیشف نوع دوم برای حل دستگاههای معادلات انتگرال معرفی و مورد استفاده قرار میگیرد. ماتریسهای عملیاتی انتگرال، حاصلضرب و مشتق برای موجکهای چبیشف نوع دوم به دست میآیند که برای تبدیل دستگاه معادلات انتگرال به یک دستگاه از معادلات جبری استفاده میشوند. همچنین، تحلیل خطا و در انتها برخی مثالها برای نشان دادن کارایی و اعتبار روش پیشنهادی ارائه میشوند.
Abbasbandy, S. & Taati, A. (2009). Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation, Journal of Computational and Applied Mathematics, 231, 106-113.
Biazar, J. & Aminikhah, H. (2009). A new technique for solving integro-differential equations, Computers and Mathematics with Applications, 58, 2084-2090
Biazar, J. & Ghazvini, H. (2009). He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind, Chaos, Solitons and Fractals, 39, 770-777.
Biazar, J., Babolian, E., & Islam, R. (2003). Solution of a system of Volterra integral equations of the first kind by Adomian method. Applied Mathematics and Computation, 139(2-3), 249-258.
Biazar, J., Eslami, M., & Aminikhah, H. (2009). Application of homotopy perturbation method for systems of Volterra integral equations of the first kind, Chaos, Solitons and Fractals, 42, 3020-3026.
Chau, F.T., Liang, Y.Z., Gao, J. & Shao, X.G. (2004). Chemometrics: From Basics to Wavelets Transform, Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Christensen O. & Christensen, K.L. (2004). Approximation Theory: from Taylor Polynomial to Wavelets. Birkhauser Boston.
Chui, C. K. (1997). Wavelets: A Mathematical Tool for Signal Analysis, SIAM, Philadelphia, PA.
Daubeches, I. (1992). Ten Lectures on Wavelets. CBMS-NSF.
Dehghan, M., Shakourifar, M., & Hamidi, A. (2009). The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique, Chaos, Solitons and Fractals, 39, 2509-2521.
Gautschi, W. (2004). Orthogonal Polynomials: Computation and Approximation, Oxford University Press.
Golbabai, A., Mammadov, M., & Seifollahi, S. (2009). Solving a system of nonlinear integral equations by an RBF network, Computers and Mathematics with Applications, 57, 1651-1658.
Hernandez E. & Weiss, G. (1996). A First Course on Wavelets. CRC Press LLC.
Maleknejad, K. & Shahrezaee, M. (2004). Using Runge-Kutta method for numerical solution of the system of Volterra integral equation, Applied Mathematics and Computation, 149, 399-410.
Nielsen, O. M. (1998). Wavelets in scientific computing, PH. D. Dissertation, Technical University of Denmark.
Pour-Mahmoud J. & Rahimi-Ardabili, M.Y., & Shamorad, S. (2005). Numerical solution of the system of Fredholm integro-differential equations by the Tau method, Applied Mathematics and Computation, 168, 465-478.
Walnut, D.F. (2002). An introduction to Wavelet Analysis, Birkhauser, Boston, Basel, Berlin.
