Numerical Solutions of Volterra Integral Equations of the Second Kind Using Barycentric Lagrange with Chebyshev Interpolation

Document Type : Original Article

Authors

1 Faculty of Electronic Engineering, Menoufia University Menouf, Egypt

2 Teaching Assistant of Eng. Math. Faculty of Engineering and Technology, FUE in Egypt Cairo, Egypt

Abstract

The Barycentric Lagrange interpolation with non-uniformly spaced Chebyshev interpolation nodes has been modified and re-configured in a new formula to solve Volterra integral equations of the second kind. To avoid the application of the collocation methods necessary to transform the solution into an equivalent linear system, the unknown function is interpolated using the given modified Barycentric Lagrange polynomials and is substituted twice into the integral equation, while the given data function is expanded into Maclaurin polynomial of the same degree as the interpolant unknown function. Thus, the presented technique yields substantially higher accuracy and faster convergence of the solutions. The obtained results of the five illustrated examples show that the numerical solutions converge to the exact solutions, particularly for unknown analytic functions, and strongly converge to the exact ones for all values of  including the endpoints of the integration domain regardless of whether the kernel and data functions are algebraic or not. 

Keywords

References

[1]     S. Hatamzadeh-Varmazyar, M. Naser-Moghadasi, “An Integral Equation Modeling of Electromagnetic Scattering from the surfaces of arbitrary resistance distribution,” Progress In Electromagnetics Research B, vol.3,  pp. 157–172, 2008.
[2]     E. S. Shoukralla and M. A. Markos, “The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind,” Asian-European Journal of Mathematics, vol.12, no. 1, pp. 1-10,  2018.
[3]     P. Kumar and G. C. Dubey, “ An application of Volterra integral equation by expansion of Taylor’s series in the problem of heat transfer and electrostatics,” IOSR Journal of Mathematics (IOSR-JM), vol.11, no. 5, pp. 59-62,  2015.
[4]     E. S. Shoukralla, M. Kamel, and M. A. Markos., “A new computational method for solving weakly singular Fredholm integral equations of the first kind,” in IEEE International Conf. on Computer Engineering and Systems (ICCES 2018), Cairo, Egypt, pp.202-207, 2018.
[5]     K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind,” Cambridge University Press, 2010.
[6]     A. M. Wazwaz, “ A First Course in Integral Equations - Solutions Manual,” 2nd ed., World Scientific Publishing Co. Pte. Ltd, (2015).
[7]     Z. Masouri, “Numerical expansion-iterative method for solving second kind Volterra and Fredholm integral equations using block -pulse functions,” Advanced Computational Techniques in Electromagnetics, pp.1-7, 2012 Pages.
[8]     J. E. Mamadu, Ignatius N. Njoseh, “ Numerical Solutions of Volterra Equations Using Galerkin Method with Certain Orthogonal Polynomials,” Journal of Applied Mathematics and Physics, vol.4, no.2, pp. 376-382,  2016.
[9]     W. Wang, “A mechanical algorithm for solving the Volterra integral equation,” Applied Mathematics and Computation, vol.172, no. 2, pp. 1323-1341,  2006.
[10]  G. Uma, V. Prabhakar and S. Hariharan. “Numerical integration using hybrid of block-pulse functions and Lagrange polynomial,” International Journal of Pure and Applied mathematics, vol.106, no.5, pp. 33-44,  2016.
[1]     M. M. Muna and I. N. Ghanim, “Numerical Solution of Linear Volterra-Fredholm Integral Equations Using Lagrange Polynomials,” Mathematical Theory and Modeling, vol.4, no.5, pp. 137- 146,  2014.
[2]     R. Meyer, B. Cai, F. Perron, “Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2,” Computational Statistics and Data Analysis, vol.52, pp. 3408–3423,  2008.
[3]     J. P. Berrut, L. N. Trefethen, “Barycentric Lagrange Interpolation,” SIAM REVIEW, Society for Industrial and Applied Mathematics, vol.46, no.3, pp. 501–517,  2004.
[4]     J. H. Nicholas, “The numerical stability of Barycentric Lagrange interpolation,”  IMA Journal of Numerical Analysis, vol.24, pp. 547-556 , 2004.
[5]     O. Runborg, “Notes on Polynomial Interpolation, 2D1250, Till¨ampade numeriska metoder II, ” March 19, 2003.
[6]     W. Gander,  “Change of basis in polynomial interpolation,” Numerical Linear Algebra with Applications , vol.12, no. 2005, pp.769– 778.
[7]     K. Maleknejad, M. Tavassoli Kajani and Y. Mahmoudi, “Numerical solution of linear Fredholm and volterra integral equation of the second kind by using Legendre wavelets,” Journal of Sciences, Islamic Republic of Iran,  vol.13, no.2, pp.161-166, 2002.
[8]     A. Tahmasbi, “A New Approach to the Numerical Solution of Linear Volterra Integral Equations of the Second Kind,” Int. J. Contemp. Math. Sciences, vol.3, no. 32, pp. 1607 -1610,  2008.
Menoufia Journal of Electronic Engineering Research
Volume 28, ICEEM2019-Special Issue
ICEEM2019-Special Issue: 1st International Conference on Electronic Eng., Faculty of Electronic Eng., Menouf, Egypt, 7-8 Dec.
2019
Pages 275-279

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