Skip to main content
Top
Published in: Numerical Algorithms 1/2021

04-03-2020 | Original Paper

Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

Authors: Sayed Arsalan Sajjadi, Saeed Pishbin

Published in: Numerical Algorithms | Issue 1/2021

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Apartsyn, A.S.: Nonclassical linear volterra equations of the first kind, VSP, Utrecht (2003) Apartsyn, A.S.: Nonclassical linear volterra equations of the first kind, VSP, Utrecht (2003)
2.
go back to reference Balakumar, V., Murugesan, K.: Numerical solution of Volterra integral-algebraic equations using block pulse functions. Appl. Math. Comput. 263, 165–170 (2015)MathSciNetMATH Balakumar, V., Murugesan, K.: Numerical solution of Volterra integral-algebraic equations using block pulse functions. Appl. Math. Comput. 263, 165–170 (2015)MathSciNetMATH
3.
go back to reference Banerjea, S., Dutta, B.: On a weakly singular integral equation and its application. Appl. Math. Lett. 1;21(7), 729–34 (2008)MathSciNetCrossRef Banerjea, S., Dutta, B.: On a weakly singular integral equation and its application. Appl. Math. Lett. 1;21(7), 729–34 (2008)MathSciNetCrossRef
4.
go back to reference Brunner, H.: Collocation methods for volterra integral and related functional equations. Cambridge University Press (2004) Brunner, H.: Collocation methods for volterra integral and related functional equations. Cambridge University Press (2004)
5.
go back to reference Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge/UK (2017)CrossRef Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge/UK (2017)CrossRef
6.
go back to reference 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) 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)
7.
go back to reference Bulatov, M.V.: Regularization of singular system of Volterra integral equation. Comput. Math. Math. Phys. 42, 315–320 (2002)MathSciNet Bulatov, M.V.: Regularization of singular system of Volterra integral equation. Comput. Math. Math. Phys. 42, 315–320 (2002)MathSciNet
8.
go back to reference 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., Chistyakov, V.F.: The Properties of Differential-Algebraic Systems and Their Integral Analogs. Memorial University of Newfoundland, Newfoundland (1997)
9.
go back to reference 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., Weinmuller, E.: Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations, Vienna Technical University, ASC Report No. 21 (2012)
10.
go back to reference Bulatov, M.V., Lima, P.M.: Two-dimensional integral-algebraic systems: analysis and computational methods. J. Comput. Appl. Math. 236, 132–140 (2011)MathSciNetCrossRef Bulatov, M.V., Lima, P.M.: Two-dimensional integral-algebraic systems: analysis and computational methods. J. Comput. Appl. Math. 236, 132–140 (2011)MathSciNetCrossRef
11.
go back to reference Budnikova, O.S., Bulatov, M.V.: Numerical solution of integral-algebraic equations for multistep methods. Comput. Math. Math. Phys. 52, 691–701 (2012)MathSciNetCrossRef Budnikova, O.S., Bulatov, M.V.: Numerical solution of integral-algebraic equations for multistep methods. Comput. Math. Math. Phys. 52, 691–701 (2012)MathSciNetCrossRef
12.
go back to reference Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag (2006) Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods Fundamentals in Single Domains. Springer-Verlag (2006)
13.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
15.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
16.
go back to reference Gear, C.W.: Differential-algebraic equations, indices, and integral-algebraic equations. SIAM J. Numer. Anal. 27, 1527–1534 (1990)MathSciNetCrossRef Gear, C.W.: Differential-algebraic equations, indices, and integral-algebraic equations. SIAM J. Numer. Anal. 27, 1527–1534 (1990)MathSciNetCrossRef
17.
go back to reference 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.: Differential-Algebraic Equations and Their Numerical Treatment. Number 88 in Teubner Texte Zur Mathematik, Teubner, Leipzig (1986)
18.
go back to reference Griepentrog, E., März, R.: Basic properties of some differential-algebraic equations. Z. Anal Anwendungen 8, 25–40 (1989)MathSciNetCrossRef Griepentrog, E., März, R.: Basic properties of some differential-algebraic equations. Z. Anal Anwendungen 8, 25–40 (1989)MathSciNetCrossRef
19.
go back to reference Goldman, N.L.: Inverse Stefan Problems Mathematics and Its Applications, vol. 412. Kluwer Academic Publishers, Dordrecht (1997)CrossRef Goldman, N.L.: Inverse Stefan Problems Mathematics and Its Applications, vol. 412. Kluwer Academic Publishers, Dordrecht (1997)CrossRef
20.
go back to reference Hadizadeh, M., Ghoreishi, F., Pishbin, S.: Jacobi spectral solution for integral-algebraic equations of index-2. Appl. Numer. Math. 61, 131–148 (2011)MathSciNetCrossRef Hadizadeh, M., Ghoreishi, F., Pishbin, S.: Jacobi spectral solution for integral-algebraic equations of index-2. Appl. Numer. Math. 61, 131–148 (2011)MathSciNetCrossRef
21.
go back to reference Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-dependent Problems. vol. 21, Cambridge University Press (2007) Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-dependent Problems. vol. 21, Cambridge University Press (2007)
22.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
23.
go back to reference Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods I. SIAM J. Numer. Anal. 51(4), 2238–2259 (2013)MathSciNetCrossRef Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods I. SIAM J. Numer. Anal. 51(4), 2238–2259 (2013)MathSciNetCrossRef
24.
go back to reference Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods II. SIAM J. Numer. Anal. 54, 2640–2663 (2016)MathSciNetCrossRef Liang, H., Brunner, H.: Integral-algebraic equations: theory of collocation methods II. SIAM J. Numer. Anal. 54, 2640–2663 (2016)MathSciNetCrossRef
25.
go back to reference Liang, H., Brunner, H.: Collocation methods for integro-differential algebraic equations with index 1. IMA J. Numer. Anal. 39, 36 (2019) Liang, H., Brunner, H.: Collocation methods for integro-differential algebraic equations with index 1. IMA J. Numer. Anal. 39, 36 (2019)
26.
go back to reference März, R.: The index of linear differential-algebraic equations with properly stated leading terms. Results Math. 42, 308–338 (2002)MathSciNetCrossRef März, R.: The index of linear differential-algebraic equations with properly stated leading terms. Results Math. 42, 308–338 (2002)MathSciNetCrossRef
27.
go back to reference März, R.: Solvability of linear differential algebraic equations with properly stated leading terms, Preprint Nr. 2002-12, Inst. für Mathematik, Humboldt-Universitätzu Berlin März, R.: Solvability of linear differential algebraic equations with properly stated leading terms, Preprint Nr. 2002-12, Inst. für Mathematik, Humboldt-Universitätzu Berlin
28.
go back to reference Slodička, M., Schepper, H.D.: Determination of the heat-transfer coefficient during solidification of alloys. Comput. Methods Appl. Mech. Engrg. 194, 491–498 (2005)MathSciNetCrossRef Slodička, M., Schepper, H.D.: Determination of the heat-transfer coefficient during solidification of alloys. Comput. Methods Appl. Mech. Engrg. 194, 491–498 (2005)MathSciNetCrossRef
29.
go back to reference 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)MathSciNetMATH 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)MathSciNetMATH
30.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
31.
go back to reference Pishbin, S.: Numerical solution and structural analysis of two-dimensional integralalgebraic equations. Numer Algor. 73, 305–322 (2016)MathSciNetCrossRef Pishbin, S.: Numerical solution and structural analysis of two-dimensional integralalgebraic equations. Numer Algor. 73, 305–322 (2016)MathSciNetCrossRef
32.
go back to reference Pishbin, S.: Operational Tau method for singular system of Volterra integro-differential equations. J. Comput. Appl. Math. 311, 205–214 (2017)MathSciNetCrossRef Pishbin, S.: Operational Tau method for singular system of Volterra integro-differential equations. J. Comput. Appl. Math. 311, 205–214 (2017)MathSciNetCrossRef
33.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
34.
go back to reference Pishbin, S.: The numerical solution of the semi-explicit IDAEs by discontinuous piecewise polynomial approximation. Appl. Math. Comput. 339, 93–104 (2018)MathSciNetMATH Pishbin, S.: The numerical solution of the semi-explicit IDAEs by discontinuous piecewise polynomial approximation. Appl. Math. Comput. 339, 93–104 (2018)MathSciNetMATH
35.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
36.
go back to reference Wazwaz, A.M.: Linear and Nonlinear Integral Equations, vol. 639. Springer, Heidelberg (2011)CrossRef Wazwaz, A.M.: Linear and Nonlinear Integral Equations, vol. 639. Springer, Heidelberg (2011)CrossRef
37.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
38.
39.
go back to reference 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)MathSciNetMATH 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)MathSciNetMATH
40.
go back to reference Zabreiko, P.P., et al.: Integral’ nye uravneniya (Integral Equations), Moscow:Nauka (1968) Zabreiko, P.P., et al.: Integral’ nye uravneniya (Integral Equations), Moscow:Nauka (1968)
41.
go back to reference 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) 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)
Metadata
Title
Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels
Authors
Sayed Arsalan Sajjadi
Saeed Pishbin
Publication date
04-03-2020
Publisher
Springer US
Published in
Numerical Algorithms / Issue 1/2021
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-00877-x

Other articles of this Issue 1/2021

Numerical Algorithms 1/2021 Go to the issue

Premium Partner