Numerical Solution of Interval Volterra-Fredholm-Hammerstein Integral Equations via Interval Legendre Wavelets Method
محورهای موضوعی : مجله بین المللی ریاضیات صنعتیN. khorrami 1 , A. Salimi Shamloo 2 , B. Parsa Moghaddam 3
1 - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
2 - Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran.
3 - Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.
کلید واژه: Interval Legendre wavelet method, Interval Shifted Legendre Polynomial, Interval Legendre Polynomial, Interval System of Equation, Interval Volterra-Fredholm-Hammerstein integral equation,
چکیده مقاله :
In this paper, interval Legendre wavelet method is investigated to approximated the solution of the interval Volterra-Fredholm-Hammerstein integral equation. The shifted interval Legendre polynomials are introduced and based on interval Legendre wavelet method is defined. The existence and uniqueness theorem for the interval Volterra-Fredholm-Hammerstein integral equations is proved. Some examples show the effectiveness and efficiency of the approach.
در این مقاله، روش موجک بازه ای لژاندر برای تقریب جواب معادله انتگرال بازه ای فردهلم –ولترا از نوع همرشتاین مورد بررسی قرار گرفته است. چند جمله ای های انتقال یافته ی بازه ای لژاندر بر اساس موجک بازه ای لژاندر تعریف می شوند. قضیه وجود و یکتایی برای معادله انتگرال بازه ای فردهلم-ولترا از نوع همرشتاین اثبات می شود. چند مثال دقت و کارایی روش را نشان می دهند.
[1] G. Alefeld, G. Mayer, Interval analysis: Theory and applications,J. Comput. Appl. Math. 121 (2000) 421-464.
[2] R. Baker Kearfott, V. Kreinovich (Eds.), Applications of Interval Computations, Kluwer Academic Publishers, 1996.
[3] J. A. Ferreira, F. Patrcio, F. Oliveira, On the computation of the zeros of interval polynomials, J. Comput. Appl. Math. 136 (2001) 271-281.
[4] E. R. Hansen, On solving systems of equations using interval arithmetic, Mathematics of Computation 22 (1968) 374-384.
[5] E. R. Hansen, S. Sengupta. Bounding solutions of systems of equations using interval analysis, BIT. 21 (1981) 203-211.
[6] T. C. Hales, A proof of the Kepler conjecture, Annals of Mathematics 162 (2005) 1065-1185.
[7] M. Hukuhara, Integration des applications mesurables dont la valeur est un compact convex, Funkcial Ekvac 10 (1967) 205-229.
[8] L. Jaulin, M. Kie er, O. Didrit, E. Walter, Applied Interval Analysis With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer-Verlag Lon-don, 2001.
[9] T. Johnson, W. Tucker, Rigorous parameter reconstruction for di erential equations with noisy data, Automatica 44 (2008) 2422-2426.
[10] L. V. Kolev, Interval Methods for Circuit Analysis, World Scienti c, 1993.
[11] V. Lakshmikantham, T. Bhaskar, J. Devi, Theory of Set Di erential Equations in Metric Spaces, Cambridge Scienti c Publishers, 2006.
[12] M. T. Malinowski, Interval di erential equations with a second type Hukuhara derivative, Appl. Math. Lett. 24 (2011) 2118-2123.
[13] R. E. Moore, Methods and Applications of the Interval Analysis, SIAM, Philadelphia, 1979.
[14] R. Moore, Interval Arithmetic. Prentice Hall, Englewood CliMs, NJ, USA, 1966.
[15] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009.
[16] D. Moens, D. Vandepitte, A survey of non-probabilistic uncertainty treatment in nite element analysis, Comput. Methods Appl. Mech. Engrg. 194 (2005) 1527-1555.
[17] A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press,
1990.
[18] N. S. Nedialkov, K. R. Jackson, J. D. Pryce, An e ective high order interval method for validating existence and uniqueness of the solution of an IVP for an ODE, Reliab. Comput.
7 (2001) 449-465.
[19] A. Neumaier, Rigorous sensitivity analysis for parameter dependent systems of equations,
J. Math. Anal. Appl. 114 (1989) 16-25.
[20] F. Patrcio, J. A. Ferreira, F. Oliveira, On the interval Legendre polynomials, Journal of Computational and Applied Mathematics 154 (2003) 215-227.
[21] J. Rohn, Solvability of systems of interva linear equations and inequalities, Springer, New York, 2006.
[22] A. Salimi Samloo, E. Babolian, Numerical solution of Fractional Di erential, integral and integro-di erential Equations by using piecewise constant orthogonal functions, Journal of Computational and Applied Mathematics 214 (2007) 495-508.
[23] A. Salimi Shamloo, Parisa Hajagharezalou, Interval Interpolation by Newton's Divided Di ferences, Journal of mathematics and computer science 13 (2014) 231-237.
[24] A. Salimi Shamloo, Sanam Shahkar, Alieh Madadi, Numerical Solution of the Fredholme-Volterra Integral Equation by the Sinc Function, American Journal of Computational Mathematics 2 (2012) 136-142.
[25] M. Seifollahi, A.Salimi Samloo, Numerical Solution of Nonlinear Multi-Order Fractional Diferential Equations by Operational Matrix Of Chebyshev Polynomails , World Applied Programming 3 (2013) 85-92.
[26] I. Skalna, M. V. Rama Rao, A. Pownuk, Systems of fuzzy equations in structural mechanics, J. Comput. Appl. Math. 218 (2008) 149-156.
[27] L. Stefanini, B. Bede, Generalized Hukuhara di erentiability of interval-valued functions and interval di erential equations, Nonlinear Analysis: Theory, Methods and Applications 71 (2009) 1311-1328.
[28] P. Sevastjanov, L. Dymova, A new method for solving interval and fuzzy equations: Linear
case, Information Sciences 179 (2009) 925-937.
[29] A. Truong Vinh, P. Nguyen Dinh, H. Ngo Van, A note on solutions of interval-valued Volterra integral equations, J. Integral Equations Applications 26 (2014) 1-14. http:// dx.doi.org/10.1216/JIE-2014-26-1-1/
[30] C. Wu, Z. Gong, On Henstock integrals of interval-valued functions and fuzzy-valued
functions, Fuzzy Sets and Systems 115 (2000) 377-391.
