Skip to main content
SpringerLink
Log in

Orthogonal cubic spline basis and its applications to a partial integro-differential equation with a weakly singular kernel

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, a new orthogonal basis for the space of cubic splines has been introduced. This basis is obtained based on B-splines and using the Gram–Schmidt orthogonalization process. A linear combination of these basis is used to estimate functions and numerical solutions of a partial integro-differential equation with a weakly singular kernel. The convergence analysis in the approximate scheme is investigated. The accuracy of the proposed method is demonstrated by three test problems. The results of numerical experiments are compared with analytical solutions by calculating errors \(L_2\)-norm, \(L_\infty \)-norm and \(H^1\)-norm. The numerical results are found to be in good agreement with the exact solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Aguilar M, Brunner H (1988) Collocation methods for second-order Volterra integro-differential equations. Appl Numer Math 4:455–470

    Article  MathSciNet  Google Scholar 

  • Aminikhah H, Alavi J (2017) B-spline collocation and quasi-interpolation methods for boundary layer flow and convection heat transfer over a flat plate. Calcolo 54(1):299–317

    Article  MathSciNet  Google Scholar 

  • Biazar J, Asadi MA (2015) FD-RBF for partial integro-differential equations with a weakly singular kernel. Appl Comput Math 4(6):445–451

    Article  Google Scholar 

  • Brunner H (1987) Implicit Runge–Kutta–Nyström methods for general second-order Volterra integro-differential equations. Comput Math Appl 14:549–559

    Article  MathSciNet  Google Scholar 

  • Brunner H, Makroglou A, Miller RK (1997) Mixed interpolation collocation methods for first and second order Volterra integro-differential equations with periodic solution. Appl Numer Math 23:381–402

    Article  MathSciNet  Google Scholar 

  • Christensen RM (1971) Theory of viscoelasticity. Academic Press, New York

    Google Scholar 

  • De Boor C (2001) A practical guide to splines. Springer, New York

    MATH  Google Scholar 

  • Dehghan M (2006) Solution of a partial integro–differential equation arising from viscoelasticity. Int J Comput Math 83:123–129

    Article  MathSciNet  Google Scholar 

  • Devore RA, Lorentz GG (1993) Constructive approximation. Springer, New York

    Book  Google Scholar 

  • Elnagar GN, Kazemi M (1996) Chebyshev spectral solution of nonlinear Volterra–Hammerstein integral equations. J Comput Appl Math 76:147–158

    Article  MathSciNet  Google Scholar 

  • Fahim A, Araghi MAF, Rashidinia1 J, Jalalvand M (2017) Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method. Adv Differ Equations. https://doi.org/10.1186/s13662-017-1416-7

  • Fujiwara H (2003) High-accurate numerical method for integral equations of the first kind under multiple-precision arithmetic. Theor Appl Mech Jpn 52:193–203

    Google Scholar 

  • Gholamian M, Saberi-Nadjafi J, Soheili AR (2019) Cubic B-splines collocation method for solving a partial integrodifferential equation with a weakly singular kernel. Comput Methods Differ Equations 7(3):497–510

    MathSciNet  MATH  Google Scholar 

  • Gurtin ME, Pipkin AC (1968) A general theory of heat conduction with finite wave speed. Arch Ration Mech Anal 31:113–126

    Article  MathSciNet  Google Scholar 

  • Khuri SA, Sayfy A (2010) A spline collocation approach for a generalized wave equation subject to non-local conservation condition. Appl Math Comput 217(8):3993–4001

    MathSciNet  MATH  Google Scholar 

  • Liu Q, Gu Y, Zhuang P, Liu F, Nie Y (2011) An implicit RBF meshless approach for time fractional diffusion equations. Comput Mech 48:1–12

    Article  MathSciNet  Google Scholar 

  • Long W, Xu D, Zeng X (2012) Quasi wavelet based numerical method for a class of partial integro–differential equation. Appl Math Comput 218:11842–11850

    MathSciNet  MATH  Google Scholar 

  • Luo M, Xu D, Li L (2015) A compact difference scheme for a partial integro–differential equation with a weakly singular kernel. Appl Math Model 39(2):947–954

    Article  MathSciNet  Google Scholar 

  • Mason JC, Rodriguez G, Seatzu S (1993) Orthogonal splines based on B-splines with applications to least squares, smoothing and regularisation problems. Numer Algorithms 5:25–40

    Article  MathSciNet  Google Scholar 

  • Mirzaee F, Alipour S (2020) Cubic B-spline approximation for linear stochastic integro-differential equation of fractional order. J Comput Appl Math 366:112440

    Article  MathSciNet  Google Scholar 

  • Mohebbi A (2017) Compact finite difference scheme for the solution of a time fractional partial integro-differential equation with a weakly singular kernel. Math Methods Appl Sci. https://doi.org/10.1002/mma.4549

  • Qiao L, Wang Z, Xu D (2020a) An alternating direction implicit orthogonal spline collocation method for the two dimensional multi-term time fractional integro-differential equation. Appl Numer Math 151:199–212

    Article  MathSciNet  Google Scholar 

  • Qiao L, Xu D, Yan Y (2020b) High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem. Math Methods Appl Sci. https://doi.org/10.1002/mma.6258.

  • Renardy M (2000) Mathematical analysis of viscoelastic flows. SIAM, Philadelphia

    Book  Google Scholar 

  • Tang T (1993) A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl Numer Math 11:309–319

    Article  MathSciNet  Google Scholar 

  • Tang T (2008) On spectral methods for Volterra integral equations and the convergence analysis. J Comput Math 26:825–837

    MathSciNet  MATH  Google Scholar 

  • Vasudeva Murthy AS, Verwer JG (1992) Solving parabolic integro-differential equations by an explicit integration method. J Comput Appl Math 39:121–132

    Article  MathSciNet  Google Scholar 

  • Yanik EG, Fairweather G (1988) Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal 12:785–809

    Article  MathSciNet  Google Scholar 

  • Zhang H, Yang X (2018) The BDF orthogonal spline collocation method for the two dimensional evolution equation with memory. Int J Comput Math 95(10):2011–2025

    Article  MathSciNet  Google Scholar 

  • Zhang H, Yang X, Tang Q (2020) Discrete-time orthogonal spline collocation method for a modified anomalous diffusion equation. Int J Comput Math. https://doi.org/10.1080/00207160.2020.1741556

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    J. Alavi & H. Aminikhah

  2. Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran

    H. Aminikhah

Authors
  1. J. Alavi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Aminikhah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Aminikhah.

Additional information

Communicated by Hui Liang.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alavi, J., Aminikhah, H. Orthogonal cubic spline basis and its applications to a partial integro-differential equation with a weakly singular kernel. Comp. Appl. Math. 40, 55 (2021). https://doi.org/10.1007/s40314-021-01442-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01442-5

Keywords

Mathematics Subject Classification

Search

Navigation