International Journal of Nonlinear Analysis and Applications
Legendre spectral projection methods for linear second kind Volterra integral equations with weakly singular kernels
Document Type : Research Paper
Authors
1 Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India
2 Department of Mathematics \& Statistics, Indian Institute of Technology Kanpur, Kanpur 208 016, India
Abstract
In this paper, Galerkin and iterated Galerkin methods are applied to approximate the linear second kind Volterra integral equations with weakly singular algebraic kernels using Legendre polynomial basis functions. We discuss the convergence results in both $L^{2}$ and infinity norms in two cases: when the exact solution is sufficiently smooth and non-smooth. We also apply Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods and derive the superconvergence rates. Numerical results are given to verify the theoretical results.
Keywords
[1] M. Ahues, A. Largillier and B. Limaye, Spectral Computations for Boundary Operator, CRC Press, 2001.
[2] P. Assari, Solving weakly singular integral equations utilizing the meshless local discrete collocation technique,
Alexandria Engineering J. 57 (2018), no. 4, 2497–2507.
[3] P. Assari, Thin plate spline Galerkin scheme for numerically solvingnonlinear weakly singular Fredholm integral
equations, Appl. Anal. 98 (2019), no. 11, 2064-2084.
[4] P. Assari, H. Adibi and M. Dehghan, A meshless discrete Galerkin (MDG) method for the numerical solution of
integral equations with logarithmic kernels, J. Comput. Appl. Math. 267 (2014), 160–181.
[5] P. Assari and M. Dehghan, A meshless local Galerkin method for solving Volterra integral equations deduced from
nonlinear fractional differential equations using the moving least squares technique, Appl. Numer. Math. 143
(2019), 276–299.
[6] P. Assari, F. Asadi-Mehregana and S. Cuomob, A numerical scheme for solving a class of logarithmic integral
equations arisen from two-dimensional Helmholtz equations using local thin plate splines, Appl. Math. Comput.
356 (2019), no. 1, 157–172.
[7] K.E. Atkinson, The Numerical Solutions of Integral Equations of Second Kind, Vol-4, Cambridge University
Press, 1997.
[8] Ben-yu Guo and Pen-yu Kuo, Spectral Methods and their Applications, World Scientific, Singapore, 1998.
[9] H. Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer.
Anal. 20 (1983), no. 6, 1106–1119.
[10] H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocationon graded meshes,
Math. Comput. 45 (1985), no. 172, 417–437.
[11] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Vol-15, Cambridge
University Press, 2004.
[12] H. Brunner, Volterra Integral Equations: An Introduction to Theory and Applications, Cambridge: Cambridge
University Press, 2017.
[13] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods, Fundamentals in single domains,
Springer, 2006.
[14] Y. Cao, T. Herdman and A. Xu, A hybrid collocation method for Volterra integral equations with weakly singular
kernels, SIAM J. Numer. Anal. 41 (2003), 364–381.
[15] F. Chatelin, Spectral Approximation for Linear Operators, Academic Press, New York, 1983.
[16] Z. Chen, G. Long and G. Nelakanti, The discrete multi-projectionmethod for Fredholm integral equations of the
second kind, J. Integral Equations Appl. 19 (2007), no. 2, 143–162.
[17] E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
[18] P. Das, M.M. Sahani and G. Nelakanti, Legendre spectral projection methods for Urysohn integral equations, J.
Comput. Appl. Math. 263 (2014), 88–102.
[19] P. Das, M.M. Sahani, G. Nelakanti and G. Long, Legendre spectral projection methods for Fredholm-Hammersteinintegral equations, J. Sci. Comput. 68 (2016), no. 1, 213–230.
[20] De Hoog, Frank, and R. Weiss, On the solution of a Volterra integral equation with a weakly singular kernel, Siam
J. Math. Anal. 4 (1973), no. 4, 561–573.
[21] I.G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comput. 39 (1982),
no. 160, 519–533.
[22] C. Huang and M. Stynes, A spectral collocation method for a weakly singular Volterra integral equation of the
second kind, Adv. Comput. Math. 42 (2016), 1015–1030.
[23] K. Kant and G. Nelakanti, Approximation methods for second kind weakly singular Volterra integral equations,
J. Comput. Appl. Math. 368 (2020), 1–16.
[24] G. Long, M.M. Sahani and G. Nelakanti, Polynomially based multi-projection methods of Fredholm integral equations of the second kind, Appl. Math. Comput. 215 (2009), no. 1, 147–155.
[25] M. Mandal and G. Nelakanti, Superconvergence results of Legendre spectral projection methods for FredholmHammerstein integral equations, J. Comput. Appl. Math. 319 (2017), 423–439.
[26] M. Mandal and G. Nelakanti, Superconvergence results of Legendre spectral projection methods for Volterra integral
equations of second kind, J. Comput. Appl. Math. 37 (2018), no. 4, 4007–4022.
[27] M. Mandal and G. Nelakanti, Superconvergence results for weakly singular Fredholm-Hammerstein integral equations, Numer. Funct. Anal. Optim. 40 (2019), no. 5, 548–570.
[28] B.L. Panigrahi and G. Nelakanti, Legendre Galerkin method for weakly singular Fredholm integral equations and
the corresponding eigenvalue problem, J. Appl. Math. Comput. 43 (2013), 175–197.
[29] B.L. Panigrahi, G. Long and G. Nelakanti, Legendre multi-projection methods for solving eigen value problems for
a compact integral operator, J. Comput. Appl. Math. 239 (2013), 135–151.
[30] L. Schumaker, Spline Functions: Basic Theory, Wiley, Newyork, 1981.
[31] C. Wagner, On the numerical solution of Volterra integral equations, J Math. Phys. 32 (1953), 289–301.
[32] Zhang Xiao-yong, Jacobi spectral method for the second-kind Volterra integral equations with a weakly singular
kernel, Appl. Math. Modell. 39 (2015), no. 15, 4421–4431.
[2] P. Assari, Solving weakly singular integral equations utilizing the meshless local discrete collocation technique,
Alexandria Engineering J. 57 (2018), no. 4, 2497–2507.
[3] P. Assari, Thin plate spline Galerkin scheme for numerically solvingnonlinear weakly singular Fredholm integral
equations, Appl. Anal. 98 (2019), no. 11, 2064-2084.
[4] P. Assari, H. Adibi and M. Dehghan, A meshless discrete Galerkin (MDG) method for the numerical solution of
integral equations with logarithmic kernels, J. Comput. Appl. Math. 267 (2014), 160–181.
[5] P. Assari and M. Dehghan, A meshless local Galerkin method for solving Volterra integral equations deduced from
nonlinear fractional differential equations using the moving least squares technique, Appl. Numer. Math. 143
(2019), 276–299.
[6] P. Assari, F. Asadi-Mehregana and S. Cuomob, A numerical scheme for solving a class of logarithmic integral
equations arisen from two-dimensional Helmholtz equations using local thin plate splines, Appl. Math. Comput.
356 (2019), no. 1, 157–172.
[7] K.E. Atkinson, The Numerical Solutions of Integral Equations of Second Kind, Vol-4, Cambridge University
Press, 1997.
[8] Ben-yu Guo and Pen-yu Kuo, Spectral Methods and their Applications, World Scientific, Singapore, 1998.
[9] H. Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer.
Anal. 20 (1983), no. 6, 1106–1119.
[10] H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocationon graded meshes,
Math. Comput. 45 (1985), no. 172, 417–437.
[11] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Vol-15, Cambridge
University Press, 2004.
[12] H. Brunner, Volterra Integral Equations: An Introduction to Theory and Applications, Cambridge: Cambridge
University Press, 2017.
[13] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods, Fundamentals in single domains,
Springer, 2006.
[14] Y. Cao, T. Herdman and A. Xu, A hybrid collocation method for Volterra integral equations with weakly singular
kernels, SIAM J. Numer. Anal. 41 (2003), 364–381.
[15] F. Chatelin, Spectral Approximation for Linear Operators, Academic Press, New York, 1983.
[16] Z. Chen, G. Long and G. Nelakanti, The discrete multi-projectionmethod for Fredholm integral equations of the
second kind, J. Integral Equations Appl. 19 (2007), no. 2, 143–162.
[17] E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
[18] P. Das, M.M. Sahani and G. Nelakanti, Legendre spectral projection methods for Urysohn integral equations, J.
Comput. Appl. Math. 263 (2014), 88–102.
[19] P. Das, M.M. Sahani, G. Nelakanti and G. Long, Legendre spectral projection methods for Fredholm-Hammersteinintegral equations, J. Sci. Comput. 68 (2016), no. 1, 213–230.
[20] De Hoog, Frank, and R. Weiss, On the solution of a Volterra integral equation with a weakly singular kernel, Siam
J. Math. Anal. 4 (1973), no. 4, 561–573.
[21] I.G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comput. 39 (1982),
no. 160, 519–533.
[22] C. Huang and M. Stynes, A spectral collocation method for a weakly singular Volterra integral equation of the
second kind, Adv. Comput. Math. 42 (2016), 1015–1030.
[23] K. Kant and G. Nelakanti, Approximation methods for second kind weakly singular Volterra integral equations,
J. Comput. Appl. Math. 368 (2020), 1–16.
[24] G. Long, M.M. Sahani and G. Nelakanti, Polynomially based multi-projection methods of Fredholm integral equations of the second kind, Appl. Math. Comput. 215 (2009), no. 1, 147–155.
[25] M. Mandal and G. Nelakanti, Superconvergence results of Legendre spectral projection methods for FredholmHammerstein integral equations, J. Comput. Appl. Math. 319 (2017), 423–439.
[26] M. Mandal and G. Nelakanti, Superconvergence results of Legendre spectral projection methods for Volterra integral
equations of second kind, J. Comput. Appl. Math. 37 (2018), no. 4, 4007–4022.
[27] M. Mandal and G. Nelakanti, Superconvergence results for weakly singular Fredholm-Hammerstein integral equations, Numer. Funct. Anal. Optim. 40 (2019), no. 5, 548–570.
[28] B.L. Panigrahi and G. Nelakanti, Legendre Galerkin method for weakly singular Fredholm integral equations and
the corresponding eigenvalue problem, J. Appl. Math. Comput. 43 (2013), 175–197.
[29] B.L. Panigrahi, G. Long and G. Nelakanti, Legendre multi-projection methods for solving eigen value problems for
a compact integral operator, J. Comput. Appl. Math. 239 (2013), 135–151.
[30] L. Schumaker, Spline Functions: Basic Theory, Wiley, Newyork, 1981.
[31] C. Wagner, On the numerical solution of Volterra integral equations, J Math. Phys. 32 (1953), 289–301.
[32] Zhang Xiao-yong, Jacobi spectral method for the second-kind Volterra integral equations with a weakly singular
kernel, Appl. Math. Modell. 39 (2015), no. 15, 4421–4431.
)
Volume 13, Issue 2
July 2022Pages 1377-1397
- Receive Date: 25 June 2020
- Accept Date: 08 August 2020