Collocation method for numerical solution of two-dimensional Linear Volterra integral equations and prove its convergence

Document Type : Original Paper

Authors

Department of Mathematics, Faculty of Basic Science, Shahed University, Tehran, Iran

Abstract

In this paper, we extend the collocation method for the numerical solution of two-dimensional Volterra integral equations. For this purpose, we first prove the existence and uniqueness of the solution of these types of equations and present a resolvent kernel representation for their solution. Then, we extend the collocation method using piecewise polynomials to solve the mentioned equations and obtain the corresponding algebraic system of equations and show that the system has a unique solution. We also prove the convergence of the method and obtain the order of convergence of the method by proving a theorem. Finally, we present some numerical examples to show the efficiency of the method and confirm the obtained theoretical results.

Highlights

[1] Bechouat, T., 2023. A collocation method for Fredholm integral equations of the first kind via
iterative regularization scheme. Mathematical Modelling and Analysis, 28(2), pp. 237–254. doi:
https://doi.org/10.3846/mma.2023.16453.
[2] Bhrawya, A. H., Abdelkawy, M. A., Machadoc, J. T. and Amin, A. Z. M., 2016. Legendre–Gauss–
Lobatto collocation method for solving multi­dimensional Fredholm integral equations. Computers and Mathematics with Applications, in press. doi: https://doi.org/10.1016/j.camwa.2016.04.011.
[3] Brunner, H., 2004. Collocation methods for Volterra integral and related functional equations. Cambridge University Press.
[4] Brunner, H., 2017. Volterra integral equations: an introduction to theory and applications. Cambridge University Press, Cambridge.
[5] Ebrahimi, N. and Rashidinia, J., 2015. Collocation method for linear and nonlinear Fredholm
and Volterra integral equations. Applied Mathematics and Computation, 270, pp: 156–164. doi:
https://doi.org/10.1016/j.amc.2015.08.032.
[6] Guoqiang, H., Hayami, K., Sugihara, K. and Jiong, W., 2000. Extrapolation method of iterated collocation solution for two­dimensional nonlinear Volterra integral equations. Applied Mathematics and Computation, 112, pp: 49–69. doi: https://doi.org/10.1016/S00963003(99)00036­3.
[7] Hoseini, S. H., Tari, A. and Hassanpour­Ezatti, M., 2017. Introducing an Integro­Differential Equation Model for Spread of Addictive Drugs abuse Quarterly Journal of Research on Addiction, 10(40), pp:255–266. doi: http://etiadpajohi.ir/article­1­1002­fa.html, (In Persian).
[8] Hutson, V. and Pym, J. S., 2004. Applications of functional analysis and operator theory. Cambridge University Press.
[9] Jerri, A. J., 1999. Introduction to integral equations with applications. John Wiley and Sons, INC.
[10] Kasture, D. Y. and Deo, G., 1977. Inequalities of Gronwall type in two independent variables. Journal of Mathematical and Applications, 58, pp: 361–372.
[11] Kazemi, S. and Tari, A., 2022. Collocation Method for Solving Two­Dimensional Fractional Volterra Integro­Differential Equations. Iranian Journal of Science and Technology, Transactions A: Science, 46, pp: 1629–1639. doi: https://doi.org/10.1007/s40995­022­01346­x.
[12] Khan, F., Omar M. and Ullah, Z., 2018. Discretization method for the numerical solution of 2D
Volterra integral equation based on two­dimensional Bernstein polynomial. AIP Advances, 8, pp: 1–9.doi:10.1063/10.5051113

[13] Liang, H. and Brunner, H., 2019. The Convergence of collocation solution in continuous piecewisepolynomial spaces for weakly singular Volterra integral equations. Siam J. Numerical Analysis, 57(4),pp: 1875–1896. doi: https://doi.org/10.1137/19M1245062.
[14] Marasi, H.R., Soltani Joujehi, A. and Derakhshan, M.H., 2023. Solving Fractional Differential Equations Using Differential Transform Method Combined with Fractional Linear Multistep Methods. Journal of Advanced Mathematical Modeling (JAMM), 13(1), pp: 1–16. doi:10.22055/JAMM.2023.41315.2062, (In Persian).
[15] Rezazadeh, E., 2022. Investigation of a new method for the numerical solution of a system of hypersingular integral equations. Journal of Advanced Mathematical Modeling (JAMM), 12(3), pp: 448–468. doi: 10.22055/JAMM.2022.40893.2042, (In Persian).
[16] Tari, A. 2012. The differential transform method for solving the model describing biological species living together. Iranian Journal of Mathematical Sciences and Informatics, 7(2), pp: 55–66. doi:10.7508/ijmsi.2012.02.006.
[17] Tari, A. and Bildik, N., 2022. Numerical solution of Volterra series with error estimation. Applied and Computational Mathematics, 21(1), pp: 3–20. doi: 10.30546/1683­6154.21.1.2022.3.
[18] Wang, K. and Wang, Q., 2014. Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations. Journal of Computational and Applied Mathematics, 260, pp: 294–300.doi: https://doi.org/10.1016/j.cam.2013.09.050.

 

Keywords

Main Subjects

References

[1] Bechouat, T., 2023. A collocation method for Fredholm integral equations of the first kind via
iterative regularization scheme. Mathematical Modelling and Analysis, 28(2), pp. 237–254. doi:
https://doi.org/10.3846/mma.2023.16453.
[2] Bhrawya, A. H., Abdelkawy, M. A., Machadoc, J. T. and Amin, A. Z. M., 2016. Legendre–Gauss–
Lobatto collocation method for solving multi­dimensional Fredholm integral equations. Computers and Mathematics with Applications, in press. doi: https://doi.org/10.1016/j.camwa.2016.04.011.
[3] Brunner, H., 2004. Collocation methods for Volterra integral and related functional equations. Cambridge University Press.
[4] Brunner, H., 2017. Volterra integral equations: an introduction to theory and applications. Cambridge University Press, Cambridge.
[5] Ebrahimi, N. and Rashidinia, J., 2015. Collocation method for linear and nonlinear Fredholm
and Volterra integral equations. Applied Mathematics and Computation, 270, pp: 156–164. doi:
https://doi.org/10.1016/j.amc.2015.08.032.
[6] Guoqiang, H., Hayami, K., Sugihara, K. and Jiong, W., 2000. Extrapolation method of iterated collocation solution for two­dimensional nonlinear Volterra integral equations. Applied Mathematics and Computation, 112, pp: 49–69. doi: https://doi.org/10.1016/S00963003(99)00036­3.
[7] Hoseini, S. H., Tari, A. and Hassanpour­Ezatti, M., 2017. Introducing an Integro­Differential Equation Model for Spread of Addictive Drugs abuse Quarterly Journal of Research on Addiction, 10(40), pp:255–266. doi: http://etiadpajohi.ir/article­1­1002­fa.html, (In Persian).
[8] Hutson, V. and Pym, J. S., 2004. Applications of functional analysis and operator theory. Cambridge University Press.
[9] Jerri, A. J., 1999. Introduction to integral equations with applications. John Wiley and Sons, INC.
[10] Kasture, D. Y. and Deo, G., 1977. Inequalities of Gronwall type in two independent variables. Journal of Mathematical and Applications, 58, pp: 361–372.
[11] Kazemi, S. and Tari, A., 2022. Collocation Method for Solving Two­Dimensional Fractional Volterra Integro­Differential Equations. Iranian Journal of Science and Technology, Transactions A: Science, 46, pp: 1629–1639. doi: https://doi.org/10.1007/s40995­022­01346­x.
[12] Khan, F., Omar M. and Ullah, Z., 2018. Discretization method for the numerical solution of 2D
Volterra integral equation based on two­dimensional Bernstein polynomial. AIP Advances, 8, pp: 1–9.doi:10.1063/10.5051113
[13] Liang, H. and Brunner, H., 2019. The Convergence of collocation solution in continuous piecewisepolynomial spaces for weakly singular Volterra integral equations. Siam J. Numerical Analysis, 57(4),pp: 1875–1896. doi: https://doi.org/10.1137/19M1245062.
[14] Marasi, H.R., Soltani Joujehi, A. and Derakhshan, M.H., 2023. Solving Fractional Differential Equations Using Differential Transform Method Combined with Fractional Linear Multistep Methods. Journal of Advanced Mathematical Modeling (JAMM), 13(1), pp: 1–16. doi:10.22055/JAMM.2023.41315.2062, (In Persian).
[15] Rezazadeh, E., 2022. Investigation of a new method for the numerical solution of a system of hypersingular integral equations. Journal of Advanced Mathematical Modeling (JAMM), 12(3), pp: 448–468. doi: 10.22055/JAMM.2022.40893.2042, (In Persian).
[16] Tari, A. 2012. The differential transform method for solving the model describing biological species living together. Iranian Journal of Mathematical Sciences and Informatics, 7(2), pp: 55–66. doi:10.7508/ijmsi.2012.02.006.
[17] Tari, A. and Bildik, N., 2022. Numerical solution of Volterra series with error estimation. Applied and Computational Mathematics, 21(1), pp: 3–20. doi: 10.30546/1683­6154.21.1.2022.3.
[18] Wang, K. and Wang, Q., 2014. Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations. Journal of Computational and Applied Mathematics, 260, pp: 294–300.doi: https://doi.org/10.1016/j.cam.2013.09.050.
 
Journal of Advanced Mathematical Modeling
Volume 13, Issue 3 - Serial Number 31
December 2023
Pages 486-502

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History

  • Receive Date: 10 July 2023
  • Revise Date: 25 October 2023
  • Accept Date: 27 December 2023

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