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Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets

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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.

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Acknowledgment

Pedro M. Lima acknowledges support from Fundação para a Ciência e a Tecnologia (FCT, the Portuguese Foundation for Science and Technology) through the grant UIDB/04621/2020 (CEMAT); Delfim F. M. Torres was supported by FCT within project UIDB/04106/2020 (CIDMA).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

    S. Nemati

  2. Centro de Matemática Computacional e Estocástica, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Pedro M. Lima

  3. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Delfim F. M. Torres

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  1. S. Nemati
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  2. Pedro M. Lima
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  3. Delfim F. M. Torres
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Correspondence to S. Nemati.

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Nemati, S., Lima, P.M. & Torres, D.F.M. Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets. Numer Algor 86, 675–691 (2021). https://doi.org/10.1007/s11075-020-00906-9

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  • DOI: https://doi.org/10.1007/s11075-020-00906-9

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