Abstract
In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and ν-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.
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Apartsyn, A.S.: Nonclassical linear volterra equations of the first kind, VSP, Utrecht (2003)
Balakumar, V., Murugesan, K.: Numerical solution of Volterra integral-algebraic equations using block pulse functions. Appl. Math. Comput. 263, 165–170 (2015)
Banerjea, S., Dutta, B.: On a weakly singular integral equation and its application. Appl. Math. Lett. 1;21(7), 729–34 (2008)
Brunner, H.: Collocation methods for volterra integral and related functional equations. Cambridge University Press (2004)
Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge/UK (2017)
Brunner, H., Bulatov, M.V.: On singular systems of integral equations with weakly singular kernels. In: Proceeding 11-th Baikal International School Seminar, pp. 64–67 (1998)
Bulatov, M.V.: Regularization of singular system of Volterra integral equation. Comput. Math. Math. Phys. 42, 315–320 (2002)
Bulatov, M.V., Chistyakov, V.F.: The Properties of Differential-Algebraic Systems and Their Integral Analogs. Memorial University of Newfoundland, Newfoundland (1997)
Bulatov, M.V., Lima, P.M., Weinmuller, E.: Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations, Vienna Technical University, ASC Report No. 21 (2012)
Bulatov, M.V., Lima, P.M.: Two-dimensional integral-algebraic systems: analysis and computational methods. J. Comput. Appl. Math. 236, 132–140 (2011)
Budnikova, O.S., Bulatov, M.V.: Numerical solution of integral-algebraic equations for multistep methods. Comput. Math. Math. Phys. 52, 691–701 (2012)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag (2006)
Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comp. 79, 147–167 (2010)
Chistyakova, E.V., Chistyakov, V.F.: Solution of differential algebraic equations with the Fredholm operator by the least squares method, Applied Numerical Mathematics. https://doi.org/10.1016/j.apnum.2019.04.013 (2019)
Farahani, M.S., Hadizadeh, M.: Direct regularization for system of integral-algebraic equations of index-1. Inverse Probl. Sci. Eng. 26(5), 728–743 (2018)
Gear, C.W.: Differential-algebraic equations, indices, and integral-algebraic equations. SIAM J. Numer. Anal. 27, 1527–1534 (1990)
Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Number 88 in Teubner Texte Zur Mathematik, Teubner, Leipzig (1986)
Griepentrog, E., März, R.: Basic properties of some differential-algebraic equations. Z. Anal Anwendungen 8, 25–40 (1989)
Goldman, N.L.: Inverse Stefan Problems Mathematics and Its Applications, vol. 412. Kluwer Academic Publishers, Dordrecht (1997)
Hadizadeh, M., Ghoreishi, F., Pishbin, S.: Jacobi spectral solution for integral-algebraic equations of index-2. Appl. Numer. Math. 61, 131–148 (2011)
Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-dependent Problems. vol. 21, Cambridge University Press (2007)
Khater, A.H., Shamardan, A.B., Callebaut, D.K., Sakran, M.R.A.: Solving integral equations with logarithmic kernels by Chebyshev polynomials. Numer Algor. 47, 81–93 (2008)
Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods I. SIAM J. Numer. Anal. 51(4), 2238–2259 (2013)
Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods II. SIAM J. Numer. Anal. 54, 2640–2663 (2016)
Liang, H., Brunner, H.: Collocation methods for integro-differential algebraic equations with index 1. IMA J. Numer. Anal. 39, 36 (2019)
März, R.: The index of linear differential-algebraic equations with properly stated leading terms. Results Math. 42, 308–338 (2002)
März, R.: Solvability of linear differential algebraic equations with properly stated leading terms, Preprint Nr. 2002-12, Inst. für Mathematik, Humboldt-Universitätzu Berlin
Slodička, M., Schepper, H.D.: Determination of the heat-transfer coefficient during solidification of alloys. Comput. Methods Appl. Mech. Engrg. 194, 491–498 (2005)
Pishbin, S., Ghoreishi, F., Hadizadeh, M.: A posteriori error estimation for the Legendre collocation method applied to integral-algebraic Volterra equations. Electron. Trans. Numer. Anal. 38, 327–346 (2011)
Pishbin, S., Ghoreishi, F., Hadizadeh, M.: The semi-explicit Volterra integral algebraic equations with weakly singular kernels: the numerical treatments. J. Comput. Appl. Math. 245, 121–132 (2013)
Pishbin, S.: Numerical solution and structural analysis of two-dimensional integralalgebraic equations. Numer Algor. 73, 305–322 (2016)
Pishbin, S.: Operational Tau method for singular system of Volterra integro-differential equations. J. Comput. Appl. Math. 311, 205–214 (2017)
Pishbin, S.: Optimal convergence results of piecewise polynomial collocation solutions for integral-algebraic equations of index-3. J. Comput. Appl. Math. 279, 209–224 (2015)
Pishbin, S.: The numerical solution of the semi-explicit IDAEs by discontinuous piecewise polynomial approximation. Appl. Math. Comput. 339, 93–104 (2018)
Tao, L., Yong, H.: Extrapolation method for solving weakly singular nonlinear Volterra integral equations of second kind. J. Math. Anal. Appl. 324, 225–237 (2006)
Wazwaz, A.M.: Linear and Nonlinear Integral Equations, vol. 639. Springer, Heidelberg (2011)
Wazwaz, A.M., Khuri, S.A.: A reliable technique for solving the weakly singular second-kind Volterra-type integral equations. Appl. Math Comp. 80, 287–299 (1990)
Weiss, R.: Product integration for the generalized Abel equation. Math Comput. 26(117), 177–190 (1972)
Xianjuan, L., Tang, T.: Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind. Front. Math. China 7.1, 69–84 (2012)
Zabreiko, P.P., et al.: Integral’ nye uravneniya (Integral Equations), Moscow:Nauka (1968)
Zolfaghari, R., Nedialkov, N.: Structural analysis of linear integral-algebraic equations. Journal of Computational and Applied Mathematics. DOI10.1016/j.cam.2018.12.043 (2019)
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Sajjadi, S.A., Pishbin, S. Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels. Numer Algor 86, 25–54 (2021). https://doi.org/10.1007/s11075-020-00877-x
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DOI: https://doi.org/10.1007/s11075-020-00877-x
Keywords
- Numerical analysis
- Weakly singular integral-algebraic equations
- ν-Smoothing
- Product integration
- Convergence analysis
- Tractability index