Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets
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Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets S. Nemati1
· Pedro M. Lima2 · Delfim F. M. Torres3
Received: 4 October 2019 / Accepted: 12 February 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss–Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [−1, 1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss–Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme. Keywords Third-kind Volterra integral equations · Jacobi wavelets · Gauss–Jacobi quadrature · Collocation points Mathematics Subject Classification 2010 34D05 · 45E10 · 65T60
S. Nemati
[email protected] 1
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
2
Centro de Matem´atica Computacional e Estoc´astica, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
3
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Numerical Algorithms
1 Introduction In this paper, we consider the following Volterra integral equation (VIE) t β (t − x)−α κ(t, x)u(x)dx, t ∈ [0, T ], t u(t) = f (t) +
(1)
0
where β > 0, α ∈ [0, 1), α + β ≥ 1, f (t) = t β g(t) with a continuous function g, and κ is continuous on the domain := {(t, x) : 0 ≤ x ≤ t ≤ T } and is of the form κ(t, x) = x α+β−1 κ1 (t, x), where κ1 is continuous on . The existence of the term t β in the left-hand side of (1) gives it special properties, which are distinct from those of VIEs of the second kind (where the left-hand side is always different from zero), and also distinct from those of the first kind (where the left-hand side is constant and equal to zero). This is why in the literature they are often mentioned as VIEs of the third kind. This class of equations has attracted the attention of researchers in the last years. The existence, uniqueness, and regularity of solutions to (1) were discussed in [1]. In that paper, the authors have derived necessary conditions to convert the equation into a cordial VIE, a class of VIEs which was studied in detail in [2, 3]. This made possible to apply to (1) some res
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