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Legendre spectral projection methods for linear second kind Volterra integral equations with weakly singular kernels | ||
International Journal of Nonlinear Analysis and Applications | ||
مقاله 113، دوره 13، شماره 2، مهر 2022، صفحه 1377-1397 اصل مقاله (473.87 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22075/ijnaa.2020.20716.2195 | ||
نویسندگان | ||
Samiran Chakraborty1؛ Kapil Kant* 2؛ Gnaneshwar Nelakanti1 | ||
1Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India | ||
2Department of Mathematics \& Statistics, Indian Institute of Technology Kanpur, Kanpur 208 016, India | ||
تاریخ دریافت: 05 تیر 1399، تاریخ پذیرش: 18 مرداد 1399 | ||
چکیده | ||
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. | ||
کلیدواژهها | ||
Volterra integral equations؛ Galerkin method؛ Multi-Galerkin method؛ Weakly singular kernels؛ Legendre polynomials | ||
مراجع | ||
[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. | ||
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