Numerical solution of volterra integral equations with weakly singular kernel using legendre wavelet method

Document Type : Original Article

Authors

1 ‎Department of Mathematics‎, Azarbaijan Shahid Madani University‎, Tabriz‎, ‎Iran‎‎

2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Abstract
The presented paper investigates a new numerical method based on the characteristics of Legendre wavelet for solving Volterra Integral equations in this method,With the help of block-pulse functions and their characteristics, we obtain the fractional integral operational matrices corresponding to these wavelets. Then, by introducing collocation points, we use them to convert the desired equations into a system of algebraic equations. After solving the device, the approximate solution of the equation is easily calculated. Finally, we provide some numerical examples to demonstrate the accuracy and efficiency of the proposed methods.

Keywords

Subjects


[1] Babayar-Razlighi, B. (2019). Numerical solution of a free boundary problem from heat transfer by the second kind chebyshev wavelets. Journal of Sciences, Islamic Republic of Iran, 30(4), 355-362.
[2] Boggess, A., Narcowich, F. J. (2009). A first course in wavelets with Fourier analysis. John Wiley Sons.
[3] Brunner, H. (2004). Collocation methods for Volterra integral and related functional differential equations (Vol. 15). Cambridge university press.
[4] Chen, Y., Tang, T. (2010). Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Mathematics of computation, 79(269), 147-167.
[5] Cheng, A. H. D., Cheng, D. T. (2005). Heritage and early history of the boundary element method. Engineering analysis with boundary elements, 29(3), 268-302.
[6] Christensen, O., Christensen, K. L. (2004). Approximation theory: from Taylor polynomials to wavelets. Springer Science Business Media.
[7] Chui, C. K. (1997). Wavelets: a mathematical tool for signal analysis. Society for Industrial and Applied Mathematics.
[8] Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Communications on pure and applied mathematics, 41(7), 909-996.
[9] Daubechies, I. (1992). Ten lectures on wavelets. Society for industrial and applied mathematics.
[10] Diogo, T. (2009). Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations. Journal of computational and applied mathematics, 229(2), 363-372.
[11] Diogo, T., Franco, N. B., Lima, P. (2004). High order product integration methods for a Volterra integral equation with logarithmic singular kernel. Communications on Pure and Applied Analysis, 3(2), 217-236.
[12] Diogo, T., Lima, P. (2008). Superconvergence of collocation methods for a class of weakly singular Volterra integral equations. Journal of Computational and Applied Mathematics, 218(2), 307-316.
[13] Jerri, A. J. (1999). Introduction to integral equations with applications. John Wiley and Sons.
[14] Juræv, D. A., Agarwal, P., Shokri, A., Elsayed, E. E. (2024). Integral formula for matrix factorizations of Helmholtz equation. In Recent Trends in Fractional Calculus and Its Applications (pp. 123-146). Academic Press.
[15] Katunin, A., Korczak, A. (2009). The possibility of application of B-spline family wavelets in diagnostic signal processing. acta mechanica et automatica, 3(4), 43-48.
[16] Khakzad, P., Moradi, A., Hojjati, G., Khalsaræi, M. M., Shokri, A. (2023). Strong Stability Preserving Integrating Factor General Linear Methods. Computational and Applied Mathematics, 42(5), 214.
[17] Lakestani, M., Saray, B. N., Dehghan, M. (2011). Numerical solution for the weakly singular Fredholm integro differential equations using Legendre multiwavelets. Journal of Computational and Applied Mathematics, 235(11), 3291-3303.
[18] Li, X., Tang, T. (2012). Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind. Frontiers of Mathematics in China, 7, 69-84.
[19] Maleknejad, K., Khademi, A., Lotfi, T. (2011). Convergence and condition number of multi-projection operators by Legendre wavelets. Computers and Mathematics with Applications, 62(9), 3538-3550.
[20] Mallat, S. G. (1989). Multiresolution approximations and wavelet orthonormal bases of ￿² (￿). Transactions of the American mathematical society, 315(1), 69-87.
[21] Meng, Z., Wang, L., Li, H., Zhang, W. (2015). Legendre wavelets method for solving fractional integro-differential equations. International Journal of Computer Mathematics, 92(6), 1275-1291.
[22] Meyer, Y. (1993). Wavelets: algorithms and applications. Philadelphia: SIAM (Society for Industrial and Applied Mathematics.
[23] Resnikoff, H. L., Raymond Jr, O. (2012). Wavelet analysis: the scalable structure of information. Springer Science and Business Media.
[24] Rivlin, T. J. (1981). An introduction to the approximation of functions. Courier Corporation.
[25] Sahu, P. K., Ray, S. S. (2015). Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Applied mathematics and computation, 256, 715-723.
[26] Sedaghat, S., Ordokhani, Y., Dehghan, M. (2014). On spectral method for Volterra functional integro-differential equations of neutral type. Numerical Functional Analysis and Optimization, 35(2), 223-239.
[27] Sunday, J., Shokri, A., Akinola, R. O., Joshua, K. V., Nonlaopon, K. (2022). A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations. Results in Physics, 42,106031.
[28] Tao, X., Xie, Z., Zhou, X. (2011). Spectral Petrov-Galerkin methods for the second kind Volterra type integro- differential equations. Numerical Mathematics: Theory, Methods and Applications, 4(2), 216-236.
[29] Walnut, D. F. (2013). An introduction to wavelet analysis. Springer Science and Business Media.
[30] Yang, C. (2014). An efficient numerical method for solving Abel integral equation. Applied Mathematics and Computation, 227, 656-661.
[31] Zhang, R., Zhu, B., Xie, H. (2013). Spectral methods for weakly singular Volterra integral equations with panto-graph delays. Frontiers of Mathematics in China, 8, 281-299.
Volume 6, Issue 1
Winter 2025
Pages 160-169

  • Receive Date 07 November 2024
  • Revise Date 23 February 2025
  • Accept Date 24 February 2025