Skip to main content

A Computational Modeling Based on Trigonometric Cubic B-Spline Functions for the Approximate Solution of a Second Order Partial Integro-Differential Equation

  • Conference paper
  • First Online:
New Knowledge in Information Systems and Technologies (WorldCIST'19 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 930))

Included in the following conference series:

  • 3296 Accesses

Abstract

In this paper, the trigonometric cubic B-spline collocation method is extended for the solution of a second order partial integro-differential equations with a weakly singular kernel. The method is obtained by discretization of time derivative using backward finite difference formula while trigonometric cubic B-spline functions are used to approximate the spatial derivative. The scheme is validated through two benchmark test problems. Accuracy of the present approach is assessed in terms of \( L_{\infty } \), \( L_{2} \) error norms and pointwise error. Better accuracy is obtained and the results are compared with quasi wavelet method (QWM), quintic B-spline collocation method (QBCM) and sinc-collocation method using Linsolve Package (SMLP).

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Gurtin, M.E., Pipkin, A.C.: A general theory of heat conduction with finite wave speed. Arch. Ration. Mech. Anal. 31, 113–126 (1968)

    Article  MathSciNet  Google Scholar 

  2. Miller, R.K.: An integro-differential equation for rigid heat conductors with memory. J. Math. Anal. Appl. 66, 313–332 (1978)

    Article  MathSciNet  Google Scholar 

  3. Lodge, A.S., Renardy, M., Nohel, J.A.: Viscoelasticity and rheology. Academic Press, New York (1985)

    MATH  Google Scholar 

  4. Ortega, J.M., Davis, S.H., Rosemblat, S., Kath, W.L.: Bifurcation with memory. SIAM J. Appl. Math. 46, 171–188 (1986)

    Article  MathSciNet  Google Scholar 

  5. Chen, C., Thome, V., Wahlbin, L.: Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 58, 587–602 (1992)

    Article  MathSciNet  Google Scholar 

  6. Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. Numer. Math. 11(4), 309–319 (1993)

    Article  MathSciNet  Google Scholar 

  7. Dehghan, M.: Solution of a partial integro-differential equation arising from viscoelasticity. Int. J. Comp. Math. 83(1), 123–129 (2006)

    Article  MathSciNet  Google Scholar 

  8. Zarebnia, M.: Sinc numerical solution for the Volterra integro-differential equation. Comm. Nonlinear Sci. Num. Simul. 15(3), 700–706 (2010)

    Article  MathSciNet  Google Scholar 

  9. Fakhar-Izadi, F., Dehghan, M.: The spectral methods for parabolic Volterra integro-differential equations. J. Comput. Appl. Math. 235(14), 4032–4046 (2011)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Yang, X., Xu, D., Zhang, H.: Crank-Nicolson/quasi-wavelets method for solving fourth order partial integro-differential equation with a weakly singular kernel. J. Comput. Phys. 234, 317–329 (2013)

    Article  MathSciNet  Google Scholar 

  12. Zhang, H., Han, X., Yang, X.: Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel. Appl. Math. Comput. 219, 6565–6575 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Ali, A., Ahmad, S., Shah, S.I.A., Haq, F.I.: A quartic B-spline collocation technique for the solution of partial integro-differential equations with a weakly singular kernel. Sci. Int. 27(5), 3971–3976 (2015)

    Google Scholar 

  14. Ahmad, S., Ali, A., Shah, S.I.A., Haq, F.I.: A computational algorithm for the solution of second order partial integro-differential equations with a weakly singular kernel using quintic B-spline collocation method. SURJ. 47(4), 709–712 (2015)

    Google Scholar 

  15. Ali, A., Ahmad, S., Shah, S.I.A.: Fazal-i-Haq: A computational technique for the solution of parabolic type integro-differential equation with a weakly singular kernel. SURJ. 48(1), 71–74 (2016)

    Google Scholar 

  16. Fahim, A., Araghi, M.A.F., Rashidinia, J., Jalalvand, M.: Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method. Adv. Dif. Equ. 2017, 362 (2017)

    Article  MathSciNet  Google Scholar 

  17. Aziz, I., Khan, I.: Numerical Solution of Partial Integrodifferential Equations of Diffusion Type. Math. Prob. Eng. 2017, 11 (2017). Article ID 2853679

    MathSciNet  Google Scholar 

  18. Abbas, M., Majid, A.A., Ismail, AIMd, Rashid, A.: The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems. Appl. Math. Comput. 239, 74–88 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Zin, S.M., Majid, A.A., Ismail, AIMd, Abbas, M.: Cubic trigonometric B-spline approach to numerical solution of wave equation. Int. J. Math. Comput. Sci. 8(10), 1302–1306 (2014)

    Google Scholar 

  20. Abbas, M., Majid, A.A., Ismail, AIMd, Rashid, A.: Numerical method using cubic trigonometric B-Spline technique for nonclassical diffusion problems. Abstract Appl. Anal. 2014, 11 (2014). Article ID 849682

    MathSciNet  MATH  Google Scholar 

  21. Nazir, T., Abbas, M., Yaseen, M.: Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach. Cogent Math. 4, 1382061 (2015)

    MathSciNet  Google Scholar 

  22. Heilat, A.S., Ismail, AIMd: Hybrid cubic b-spline method for solving Non-linear two-point boundary value problems. Int. J. Pure Appl. Math. 110(2), 369–381 (2016)

    Article  Google Scholar 

  23. Hashmi, M.S., Awais, M., Waheed, A., Ali, Q.: Numerical treatment of Hunter Saxton equation using cubic trigonometric B-spline collocation method. AIP Adv. 7, 095124 (2017). https://doi.org/10.1063/1.4996740

    Article  Google Scholar 

  24. Yaseen, M., Abbas, M., Nazir, T., Bale, D.: A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation. Adv. Differ. Equ. 2017, 274 (2017)

    Article  MathSciNet  Google Scholar 

  25. Dag, I., Hepson, O.E., Kaçmaz, O.: The trigonometric cubic B-spline algorithm for burgers’ equation. Int. J. Nonlin. Sci. 24(2), 120–128 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Arora, G., Joshi, V.: A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers’ equation in one and two dimensions. Alex. Eng. J. 7(2), 1087–1098 (2018)

    Article  Google Scholar 

  27. Tamsir, M., Dhiman, N., Srivastava, V.K.: Cubic trigonometric B-spline differential quadrature method for numerical treatment of Fisher’s reaction-diffusion equations. Alex. Eng. J. 7(3), 2019–2026 (2018)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arshed Ali .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ali, A., Khan, K., Haq, F., Shah, S.I.A. (2019). A Computational Modeling Based on Trigonometric Cubic B-Spline Functions for the Approximate Solution of a Second Order Partial Integro-Differential Equation. In: Rocha, Á., Adeli, H., Reis, L., Costanzo, S. (eds) New Knowledge in Information Systems and Technologies. WorldCIST'19 2019. Advances in Intelligent Systems and Computing, vol 930. Springer, Cham. https://doi.org/10.1007/978-3-030-16181-1_79

Download citation

Publish with us

Policies and ethics