Abstract and Applied Analysis

Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

Muhammad Abbas, Ahmad Abd. Majid, Ahmad Izani Md. Ismail, and Abdur Rashid

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of a nonclassical diffusion problem with nonlocal boundary constraints. The standard finite difference approach is applied to discretize the time derivative while cubic trigonometric B-spline is utilized as an interpolating function in the space dimension. The technique is shown to be unconditionally stable using the von Neumann method. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. The L 2 and L error norms are also computed at different times for different space size steps to assess the performance of the proposed technique. The technique requires smaller computational time than several other methods and the numerical results are found to be in good agreement with known solutions and with existing schemes in the literature.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 849682, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277009

Digital Object Identifier
doi:10.1155/2014/849682

Mathematical Reviews number (MathSciNet)
MR3214457

Zentralblatt MATH identifier
07023192

Citation

Abbas, Muhammad; Majid, Ahmad Abd.; Ismail, Ahmad Izani Md.; Rashid, Abdur. Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems. Abstr. Appl. Anal. 2014 (2014), Article ID 849682, 11 pages. doi:10.1155/2014/849682. https://projecteuclid.org/euclid.aaa/1412277009


Export citation

References

  • Y. S. Choi and K.-Y. Chan, “A parabolic equation with nonlocal boundary conditions arising from electrochemistry,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 4, pp. 317–331, 1992.
  • A. Bouziani, “On a class of parabolic equations with a nonlocal boundary condition,” Académie Royale de Belgique. Bulletin de la Classe des Sciences. 6e Série, vol. 10, no. 1–6, pp. 61–77, 1999.
  • W. A. Day, “Parabolic equations and thermodynamics,” Quarterly of Applied Mathematics, vol. 50, no. 3, pp. 523–533, 1992.
  • A. A. Samarskiĭ, “Some problems of the theory of differential equations,” Differential Equations, vol. 16, no. 11, pp. 1925–1935, 1980.
  • A. R. Bahad\ir, “Application of cubic B-spline finite element technique to the termistor problem,” Applied Mathematics and Computation, vol. 149, no. 2, pp. 379–387, 2004.
  • J. H. Cushman, B. X. Hu, and F. Deng, “Nonlocal reactive transport with physical and chemical heterogeneity: localization errors,” Water Resources Research, vol. 31, no. 9, pp. 2219–2237, 1995.
  • F. Kanca, “The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions,” Journal of Inequalities and Applications, vol. 2013, article 108, 2013.
  • M. Dehghan, “Efficient techniques for the second-order parabolic equation subject to nonlocal specifications,” Applied Numerical Mathematics, vol. 52, no. 1, pp. 39–62, 2005.
  • J. Martín-Vaquero and J. Vigo-Aguiar, “A note on efficient techniques for the second-order parabolic equation subject to non-local conditions,” Applied Numerical Mathematics, vol. 59, no. 6, pp. 1258–1264, 2009.
  • X. Li and B. Wu, “New algorithm for nonclassical parabolic problems based on the reproducing kernel method,” Mathematical Sciences, vol. 7, article 4, 2013.
  • A. Golbabai and M. Javidi, “A numerical solution for non-classical parabolic problem based on Chebyshev spectral collocation method,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 179–185, 2007.
  • W. A. Day, “A decreasing property of solutions of parabolic equations with applications to thermoelasticity,” Quarterly of Applied Mathematics, vol. 40, no. 4, pp. 468–475, 1983.
  • W. A. Day, “Extensions of a property of the heat equation to linear thermoelasticity and other theories,” Quarterly of Applied Mathematics, vol. 40, no. 3, pp. 319–330, 1982.
  • W. A. Day, Heat Conduction within Linear Thermoelasticity, vol. 30 of Springer Tracts in Natural Philosophy, Springer, New York, NY, USA, 1985.
  • J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 2004.
  • Du. V. Rosenberg, Methods for Solution of Partial Differential Equations, vol. 113, American Elsevier, New York, NY, USA, 1969.
  • W. T. Ang, “A method of solution for the one-dimensional heat equation subject to nonlocal conditions,” Southeast Asian Bulletin of Mathematics, vol. 26, no. 2, pp. 197–203, 2002.
  • J. Martín-Vaquero and J. Vigo-Aguiar, “On the numerical solution of the heat conduction equations subject to nonlocal conditions,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2507–2514, 2009.
  • M. Dehghan, “On the numerical solution of the diffusion equation with a nonlocal boundary condition,” Mathematical Problems in Engineering, vol. 2, pp. 81–92, 2003.
  • M. Dehghan, “A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications,” Numerical Methods for Partial Differential Equations, vol. 22, no. 1, pp. 220–257, 2006.
  • M. Tatari and M. Dehghan, “On the solution of the non-local parabolic partial differential equations via radial basis functions,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1729–1738, 2009.
  • J. H. Cushman, “Diffusion in fractal porous media,” Water Resources Research, vol. 27, no. 4, pp. 643–644, 1991.
  • V. V. Shelukhin, “A non-local in time model for radionuclides propagation in Stokes fluid,” Siberian Branch of Russian Academy of Sciences, Institute of Hydrodynamics, no. 107, pp. 180–193, 1993.
  • A. S. Vasudeva Murthy and J. G. Verwer, “Solving parabolic integro-differential equations by an explicit integration method,” Journal of Computational and Applied Mathematics, vol. 39, no. 1, pp. 121–132, 1992.
  • C. V. Pao, “Numerical methods for nonlinear integro-parabolic equations of Fredholm type,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 857–877, 2001.
  • C. V. Pao, “Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions,” Journal of Mathematical Analysis and Applications, vol. 195, no. 3, pp. 702–718, 1995.
  • J. R. Cannon and A. L. Matheson, “A numerical procedure for diffusion subject to the specification of mass,” International Journal of Engineering Science, vol. 31, no. 3, pp. 347–355, 1993.
  • M. Dehghan, “Numerical solution of a parabolic equation with non-local boundary specifications,” Applied Mathematics and Computation, vol. 145, no. 1, pp. 185–194, 2003.
  • G. Ekolin, “Finite difference methods for a nonlocal boundary value problem for the heat equation,” BIT. Numerical Mathematics, vol. 31, no. 2, pp. 245–261, 1991.
  • G. Fairweather and J. C. López-Marcos, “Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions,” Advances in Computational Mathematics, vol. 6, no. 3-4, pp. 243–262, 1996.
  • Y. Liu, “Numerical solution of the heat equation with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 110, no. 1, pp. 115–127, 1999.
  • G. E. Farin, Curves and Surfaces for Computer-Aided Geometric Design: A Practical Code, Academic Press, 1996.
  • J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A K Peters, Boston, Mass, USA, 1993.
  • Y. S. Lai, W. P. Du, and R. H. Wang, “The viro method for construction of piecewise algebraic hypersurfaces,” Abstract and Applied Analysis, vol. 2013, Article ID 690341, 7 pages, 2013.
  • R. H. Wang, X. Q. Shi, Z. X. Luo, and Z. X. Su, Multivariate Spline and Its Applications, Science Press, Beijing, China; Kluwer Academic Publishers, New York, NY, USA, 2001.
  • P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1989.
  • C. de Boor, A Practical Guide to Splines, vol. 27 of Applied Mathematical Sciences, Springer, 1978.
  • H. N. Caglar, S. H. Caglar, and E. H. Twizell, “The numerical solution of third-order boundary-value problems with fourth-degree $B$-spline functions,” International Journal of Computer Mathematics, vol. 71, no. 3, pp. 373–381, 1999.
  • H. N. Çaglar, S. H. Çaglar, and E. H. Twizell, “The numerical solution of fifth-order boundary value problems with sixth-degree $B$-spline functions,” Applied Mathematics Letters, vol. 12, no. 5, pp. 25–30, 1999.
  • I. Da\vg, D. Irk, and B. Saka, “A numerical solution of the Burgers' equation using cubic B-splines,” Applied Mathematics and Computation, vol. 163, no. 1, pp. 199–211, 2005.
  • J. Rashidinia, M. Ghasemi, and R. Jalilian, “A collocation method for the solution of nonlinear one-dimensional para-bolic equations,” Mathematical Sciences Quarterly Journal, vol. 4, no. 1, pp. 87–104, 2010.
  • K. Qu, Z. Wang, and B. Jiang, “A finite element method by using bivariate splines for one dimensional heat equations,” Journal of Information & Computational Science, vol. 10, no. 12, pp. 3659–3666, 2013.
  • J. Goh, A. A. Majid, and A. I. M. Ismail, “Numerical method using cubic B-spline for the heat and wave equation,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4492–4498, 2011.
  • J. Goh, A. A. Majid, and A. I. M. Ismail, “Cubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equations,” Journal of Applied Mathematics, vol. 2012, Article ID 458701, 8 pages, 2012.
  • J. Goh, A. A. Majid, and A. I. M. Ismail, “A comparison of some splines-based methods for the one-dimensional heat equation,” Proceedings of World Academy of Science, Engineering and Technology, vol. 70, pp. 858–861, 2010.
  • R. C. Mittal and G. Arora, “Quintic B-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2798–2808, 2010.
  • R. C. Mittal and G. Arora, “Numerical solution of the coupled viscous Burgers' equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1304–1313, 2011.
  • M. Abbas, A. A. Majid, A. I. M. Ismail, and A. Rashid, “Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system,” PLoS ONE, vol. 9, no. 1, article e83265, 2014.
  • M. Abbas, A. A. Majid, A. I. M. Ismail, and A. Rashid, “The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems,” Applied Mathematics and Computation, vol. 239, pp. 74–88, 2014.
  • A. Nikolis, “Numerical solutions of ordinary differential equations with quadratic trigonometric splines,” Applied Mathematics E-Notes, vol. 4, pp. 142–149, 1995.
  • N. N. Abd Hamid, A. A. Majid, and A. I. M. Ismail, “Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two,” World Academy of Science, Engine-ering and Technology, vol. 70, pp. 798–803, 2010. \endinput