Skip to main content
SpringerLink
Log in

A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.

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

Similar content being viewed by others

References

  1. Mesloub, S., Bouziani, A.: On a class of singular hyperbolic equation with a weighted integral condition. Int. J. Math. Math. Sci. 22, 511–520 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouziani, A.: Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations. Hiroshima Math. J. 27(3), 373–390 (1997)

    MATH  MathSciNet  Google Scholar 

  3. Bouziani, A.: Mixed problem for certain nonclassical equations containing a small parameter. Acad. R. Belg. Cl. Lett. Sci. 6(5), 389–400 (1994)

    MathSciNet  Google Scholar 

  4. Bouziani, A.: Initial-boundary value problem with a nonlocal condition for a viscosity equation. Int. J. Math. Math. Sci. 30(6), 327–338 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beilin, S.A.: Existence of solutions for one-dimensional wave equations with non-local conditions. Electr. J. Differ. Equ. 76, 1–8 (2001)

    MathSciNet  Google Scholar 

  6. Dehghan, M.: On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numer. Methods Partial Differ. Equ. 21, 24–40 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dehghan, M.: Numerical solution of a parabolic equation with non-local boundary specifications. Appl. Math. Comput. 145, 185–194 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dehghan, M.: Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math. 52, 39–62 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dehghan, M.: Implicit collocation technique for heat equation with non-classic initial condition. Int. J. Nonlinear Sci. Numer. Simul. 7, 461–466 (2006)

    Google Scholar 

  10. Dehghan, M., Shokri, A.: A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn. 50, 111–120 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dehghan, M.: A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numer. Methods Partial Differ. Equ. 22, 220–257 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golberg, M.A., Chen, C.S., Ganesh, M.: Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions. Eng. Anal. Bound. Elem. 24, 539–547 (2000)

    Article  Google Scholar 

  13. Hon, Y.C., Wu, Z.: Additive Schwarz domain decomposition with radial basis approximation. Int. J. Appl. Math. 4(1), 599–606 (2000)

    MathSciNet  Google Scholar 

  14. Kansa, E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics—I. Comput. Math. Appl. 19, 127–145 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kansa, E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics—II. Comput. Math. Appl. 19, 147–161 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kavalloris, N.I., Tzanetis, D.S.: Behaviour of critical solutions of a nonlocal hyperbolic problem in ohmic heating of foods. Appl. Math. E-Notes 2, 5965 (2002)

    Google Scholar 

  17. Pulkina, L.S.: A nonlocal problem with integral conditions for hyperbolic equations. Electr. J. Differ. Equ. 45, 1–6 (1999)

    MathSciNet  Google Scholar 

  18. Ramezani, M., Dehghan, M., Razzaghi, M.: Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition. Numer. Methods Partial Differ. Equ. 24, 1–8 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Saadatmandi, A., Dehghan, M.: Numerical solution of the one-dimensional wave equation with an integral condition. Numer. Methods Partial Differ. Equ. 23, 282–292 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ang, W.T.: A numerical method for the wave equation subject to a non-local conservation condition. Appl. Numer. Math. 56, 1054–1060 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16–30 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Dehghan, M., Lakestani, M.: The use of cubic B-spline scaling functions for solving the one-dimensional hyperbolic equation with a nonlocal conservation condition. Numer. Methods Partial Differ. Equ. 23, 1277–1289 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang, Y.: Solving partial differential equations by meshless methods using radial basis functions. Appl. Math. Comput. 185, 614–627 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Karageorghis, A., Chen, C.S., Smyrlis, Y.S.: A matrix decomposition RBF algorithm: approximation of functions and their derivatives. Appl. Numer. Math. 57(3), 304–319 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Chantasiriwan, S.: Solutions of partial differential equations with random Dirichlet boundary conditions by multiquadric collocation method. Eng. Anal. Bound. Elem. 29, 1124–1129 (2005)

    Article  Google Scholar 

  27. Cheng, A.H.-D, Golberg, M.A., Kansa, E.J., Zammito, G.: Exponential convergence and h-c multiquadric collocation method for partial differential equations. Numer. Methods Partial Differ. Equ. 19, 571–594 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Leitão, V.M.A.: RBF-based meshless methods for 2D elastostatic problems. Eng. Anal. Bound. Elem. 28, 1271–1281 (2004)

    Article  MATH  Google Scholar 

  29. Chen, C.S., Ganesh, M., Golberg, M.A., Cheng, A.H.-D.: Multilevel compact radial functions based computational schemes for some elliptic problems. Comput. Math. Appl. 43, 359–378 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Golberg, M.A., Chen, C.S., Bowman, H.: Some recent results and proposals for the use of radial basis functions in the BEM. Eng. Anal. Bound. Elem. 23, 285–296 (1999)

    Article  MATH  Google Scholar 

  31. Wong, S.M., Hon, Y.C., Golberg, M.A.: Compactly supported radial basis functions for shallow water equations. Appl. Math. Comput. 127, 79–101 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tatari, M., Dehghan, M.: On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl. Math. Model. 33, 1729–1738 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  33. Dehghan, M., Tatari, M.: Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions. Math. Comput. Model. 44, 11–12 (2006)

    Article  MathSciNet  Google Scholar 

  34. Dehghan, M., Tatari, M.: The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data. Numer. Methods Partial Differ. Equ. 23, 984–997 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Muleshkov, A.S., Golberg, M.A., Chen, C.S.: Particular solutions of Helmholtz–type operators using higher order splines. Comput. Mech. 23, 411–419 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. Li, J., Cheng, A.H.-D., Chen, C.S.: A comparison of efficiency and error convergence of multiquadric collocation method and finite element method. Eng. Anal. Bound. Elem. 27, 251–257 (2003)

    Article  MATH  Google Scholar 

  37. Golberg, M.A., Muleshkov, A.S., Chen, C.S., Cheng, A.H.-D.: Polynomial particular solutions for certain partial differential operators. Numer. Methods Partial Differ. Equ. 19, 112–133 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Dehghan, M.: The one-dimensional heat equation subject to a boundary integral specification. Chaos Solitons Fractals 32, 661–675 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Shakeri, F., Dehghan, M.: The method of lines for solution of the one-dimensional wave equation subject to an integral conservation condition. Comput. Math. Appl. 56, 2175–2188 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  40. Dehghan, M.: Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement. Numer. Methods Partial Differ. Equ. 21, 611–622 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

    Mehdi Dehghan & Ali Shokri

Authors
  1. Mehdi Dehghan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Shokri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehdi Dehghan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dehghan, M., Shokri, A. A meshless method for numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions. Numer Algor 52, 461–477 (2009). https://doi.org/10.1007/s11075-009-9293-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-009-9293-0

Keywords

Search

Navigation