Space-time Muntz spectral collocation approach for parabolic Volterra integro-differential equations with a singular kernel

Document Type : Research Paper

Authors

Department of Mathematics, Isfahan (Khorasgan) Branch, Islamic Azad University, Isfahan, Iran

Abstract

We consider a type of Volterra integro-differential equations of the parabolic type that arise naturally in the study of heat flow in materials with memory. We present a simple and accurate numerical method for problems with a weakly singular kernel subject to an initial condition and given boundary conditions. In this method, both the space and time discretizations are based on the Muntz-Legendre collocation method that converts the problem to a system of algebraic equations. For numerical stability purposes, the Muntz-Legendre polynomials and their partial derivatives are stated in terms of Jacobi polynomials. Moreover, to deal with the weakly singular integral term of the problem, two efficient schemes based on the integration by parts and nonclassical Gaussian quadrature are derived. Comparisons between the two proposed schemes and other methods in the literature are made to demonstrate the efficiency, convergence and superiority of our method in the space and time directions.

Keywords

[1] M.L. Heard, An abstract parabolic volterra integrodifferential equation, SIAM J. Math. Anal. 13 (1982), no. 1, 81–105.
[2] T. Hayat, R. Naz and S. Abbasbandy, On flow of a fourth-grade fluid with heat transfer, Int. J. Numer. methods Fluids. 67 (2011), no. 12, 2043–2053.
[3] M. Esmailzadeh, J. Alavi and H. Saberi-Najafi, A numerical scheme for solving nonlinear parabolic partial differential equations with piecewise constant arguments, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 783–798.
[4] M. Rcnardy, Mathematical analysis of viscoelastic flows, Ann. Rev. Fluid Mech. 21 (1989), 21–36.
[5] M.E. Gurtin and A.C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Ration. Mech. Anal. 31 (1968), 113–126.
[6] R.K. Miller, An integro-differential equation for grid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), 313–332.
[7] J.C.Lopez-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal. 31 (1968), 113–126.
[8] W. Long, D. Xu and X. Zeng, Quasi wavelet based numerical method for a class of partial integro-differential equation, Appl. Math. Comput. 218 (2012), no. 24, 11842–11850.
[9] M. Gholamian, J. Saberi-Nadjafi and A.R. Soheili, Cubic B-splines collocation method for solving a partial integrodifferential equation with a weakly singular kernel, Comput. Meth. Diff. Equ. 7 (2019), no. 3, 497–510.
[10] T. Tang, A finite difference scheme for a partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11 (1993), 309–319.
[11] M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83 (2006), no. 1, 123–129.
[12] F. Fakhar-Izadi and M. Dehghan, Space-time spectral method for a weakly singular parabolic partial integrodifferential equation on irregular domains, Comput. Math. Appl. 67 (2014), 1884–1904.
[13] M. Luo, D. Xu and L. Li, A compact difference scheme for a partial integro-differential equation with a weakly singular kernel, Appl. Math. Modell. 39 (2015), 947–954.
[14] J. Biazar, A. Aasaraai and M.B. Mehrlatifan, A compact scheme for a partial integro-differential equation with weakly singular kernel, J. Sci. Islam. Repub. 28 (2017), no. 4, 359–367.
[15] A. Mohebbi, Crank-Nicolson and Legendre spectral collocation methods for a partial integro-differential equation with a singular kernel, J. Comput. Appl. Math. 349 (2019), 197–206.
[16] Y.-M. Wang and Y.-J. Zhang, A Crank-Nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations, Appl. Numer. Math. 172 (2022), 566–590.
[17] E. Qahremani, T. Allahviranloo, S. Abbasbandy and N. Ahmady, A study on the fuzzy parabolic Volterra partial integro-differential equations, J. Intell. Fuzzy Syst. 40 (2021), no. 1, 1639–1654.
[18] M. Hajiketabi and S. Abbasbandy, The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation, Eng. Anal. Bound. Elem. 87 (2018), 36–46.
[19] A.A. Ahmad, Solving partial differential equations via a hybrid method between homotopy analytical method and Harris hawks optimization algorithm, Int. J. Nonlinear Anal. Appl. 13 (2022), no. 1, 663–671.
[20] W. Gautschi, Orthogonal polynomials and quadrature, Electron. Trans. Numer. Anal. 9 (1999), 65–76.
[21] G.H. Golub, Some modified matrix eigenvalue problems, SIAM. Rev. 15 (1973), 318–334.
[22] J. Almira, M¨untz type theorems I, Surv. Approx. Theory 3 (2007), 152–194.
[23] GV. Milovanovi´c, M¨untz orthogonal polynomials and their numerical evaluation, in: Applications and Computation of Orthogonal Polynomials, Internat. Ser. Numer. Math., vol.131, Birkh¨auser, Basel, 1999, pp.179–194.
[24] S. Esmaeili, M. Shamsi and Y. Luchko, Numerical solution of fractional differential equations with a collocation method based on M¨untz polynomials, Comput. Math. Appl. 62 (2011), no. 3, 918–929.
[25] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
Volume 14, Issue 3
March 2023
Pages 153-162
  • Receive Date: 16 January 2022
  • Accept Date: 13 March 2022
  • First Publish Date: 15 September 2022