Abstract
In this paper, we present an hp-version Jacobi spectral collocation method for the third-kind VIEs. We establish several new approximation results of the Jacobi polynomial interpolations. In addition, we also derive the convergence of the hp-version of Jacobi spectral collocation method under the \(L^{\infty }\)-norm. Numerical experiments confirm the theoretical expectations.
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Allaei, S., Yang, Z., Brunner, H.: Collocation methods for third-kind VIEs. IMA J. Numer. Anal. 37, 1104–1124 (2017)
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15. Cambridge University Press, Cambridge (2004)
Brunner, H., Schötzau, D.: \(hp\)-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J. Numer. Anal. 44, 224–245 (2006)
Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)
Chen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)
Jiang, Y., Ma, J.: Spectral collocation methods for Volterra-integro differential equations with noncompact kernels. J. Comput. Appl. Math. 244, 115–124 (2013)
Lighthill, M.J.: Contributions to the theory of the heat transfer trough a laminar boundary layer. Proc. R. Soc. Lond. A 202, 359–377 (1950)
Mustapha, K., Brunner, H., Mustapha, H., Schötzau, D.: An \(hp\)-version discontinuous Galerkin method for integro-differential equations of parabolic type. SIAM J. Numer. Anal. 49, 1369–1396 (2011)
Nemati, S., Lima, P.M.: Numerical Solution of a Third-Kind Volterra Integral Equation using an Operational Matrix Technique. In: European Control Conference (ECC). Limassol 2018, 3215–3220 (2018)
Seyed, A., Diogo, T., Rebelo, M.: Analytical and numerical results for nonlinear singular Volterra integral equations. Appl. Numer. Math. 114, 2–17 (2017)
Seyed, A., Yang, Z., Brunner, H.: Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations. J. Integral Equ. Appl. 27, 325–342 (2015)
Shayanfard, F., Dastjerdi, H.L., Ghaini, F.M.: A numerical method for solving Volterra integral equations of the third kind by multistep collocation method. Comput. Appl. Math. 38, 174 (2019)
Shen, J., Sheng, C., Wang, Z.: Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. J. Math. Study 48, 315–329 (2015)
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, vol. 41. Springer, Berlin (2011)
Sheng, C., Wang, Z., Guo, B.: A multistep Legendre-Gauss spectral collocation method for nonlinear Volterra integral equations. SIAM J. Numer. Anal. 52, 1953–1980 (2014)
Song, H., Yang, Z., Brunner, H.: Analysis of collocation methods for nonlinear Volterra integral equations of the third kind. Calcolo 56, 7–29 (2019)
Szegö, G.: Orthogonal Polynomials. AMS, Providence (1959)
Tang, T., Xu, X., Chen, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)
Wang, C., Wang, Z., Jia, H.: An hp-version spectral collocation method for nonlinear Volterra integro-differential equation with weakly singular kernels. J. Sci. Comput. 72, 647–678 (2017)
Wang, Z., Guo, Y., Yi, L.: An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. Math. Comput. 86, 2285–2324 (2017)
Yi, L.: An \(h\)-\(p\) version of the continuous Petrov-Galerkin finite element method for nonlinear Volterra integro-differential equations. J. Sci. Comput. 65, 715–734 (2015)
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The work was supported by National Natural Science Foundation of China (Grant No. 12071294) and China Postdoctoral Science Foundation (Grant No. 2020M681345).
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Wang, Z., Zhou, M. & Guo, Y. An hp-Version Jacobi Spectral Collocation Method for the Third-Kind VIEs. J Sci Comput 87, 19 (2021). https://doi.org/10.1007/s10915-021-01426-x
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DOI: https://doi.org/10.1007/s10915-021-01426-x