Explicit and Implicit Crandall's Scheme for the Heat Equation with Nonlocal Nonlinear Conditions

Main Article Content

Bensaid Souad
Dehilis Sofiane
Bouziani Abdelfatah


In this paper, explicit and implicit Crandall's formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. The integrals in the boundary equations are approximated by the composite Simpson quadrature rule. Here nonlinear terms are approximated by Richtmyer's linearization method. Finally, some numerical examples are given to show the effectiveness of the proposed method.

Article Details


  1. W.T. Ang, A method of solution for the one-dimensional heat equation subject to nonlocal conditions, Southeast Asian Bull. Math. 26 (2003), 185-191.
  2. S. Bensaid, A. Bouziani, M. Zereg, Backward Euler method for the diffusion equation with integral boundary specifications, J. Pure Appl. Math.: Adv. Appl. 2 (2009), 169-185.
  3. A. Borhanifar, M.M. kabir, A.H. Pour A numerical method for solution of the heat equation with nonlocal nonlinear condition, World Appl. Sci. J. 13 (2011), 2405-2409.
  4. A. Borhanifar, S. Shahmorad, E. Feizi, A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions, Numer. Algorithms, 74 (2017), 1203-1221.
  5. A. Bouziani, Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stoch. Anal. 9 (1996), 323-330.
  6. J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math. 21 (1963), 155-160.
  7. W.A. Day, Extension of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982), 319-330.
  8. W.A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 41 (1983), 468-475.
  9. W.A. Day, Parabolic equations and thermodynamics, Quart. Appl. Math. 50 (1992), 523-533.
  10. G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resour. Res. 30 (1994), 3327-3336.
  11. S. Delihis, A. Bouziani, T.E. Oussaeif, Study of solution for a parabolic integrodifferential equation with the second kind integral condition, Int. J. Anal. Appl. 16 (2018), 569-593.
  12. M. Dehghan, Numerical solution of a parabolic equation with non-local boundary specifications, Appl. Math. Comput. 145 (2003), 185-194.
  13. M. Dehghan, Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, Appl. Math. Comput. 157 (2004), 549-560.
  14. M. Dehghan, Efficient techniques for the second-odrer parabolic equation subject to nonlocal specifications, Appl. Numer. Math. 52 (2005), 39-62.
  15. G. Ekolin, Finite difference methods for a nonlocal boundary value problem for the heat equation, BIT Numer. Math. 31 (1991), 245-261.
  16. M. Javidi, The MOL solution for the one-dimentional heat equation subject to nonlocal conditions, Int. Math. Forum, 12 (2006), 597-602.
  17. B. Kawohl, Remark on a paper by W.A. Day on a maximum principle under nonlocal boundary conditions, Quart. Appl. Math. 44 (1987), 751-752.
  18. J. Martin-Vaquero, J. Vigo-Aguiar, A note on efficient techniques for the second-order parabolic equation subject to non-local conditions, Appl. Numer. Math. 59 (2009), 1258-1264.
  19. J. Martin-Vaquero, A. Queiruga-Dios, A.H. Encinas, Numerical algorithms for diffusion-reaction problems with nonclassical conditions, Appl. Math. Comput. 218 (2012), 5487-5495.
  20. J. Martin-Vaquero, S. Sajavicius, The two-level finite difference schemes for the heat equation with nonlocal initial condition, Appl. Math. Comput. 342 (2019), 166-177.
  21. K.W. Morton, D.F. Mayers, Numerical solution of partial differential equations, Cambridge University Press, Cambridge, 1994.
  22. C.V. Pao, Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 136 (2001), 227-243.
  23. L.S. Pulkina, A.B. Beylin, Nonlocal approach to problems on longitudinal vibration in a short bar, Electron. J. Differ. Equ. 2019 (2019), 29.
  24. M. Slodicka, and S.Dehilis, A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition, J. Comput. Appl. Math. 231 (2009), 715-724.
  25. G.D. Smith, Numerical solution of partial differential equations: Finite difference methods, Clarendon Press, Oxford, 1985
  26. S. Wang, The numerical method for the conduction subject to moving boundary energy specification, Numer. Heat Transfer, 130 (1990), 35-38.
  27. S. Wang, Y. Lin, A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation, Inverse Probl. 5 (1989), 631-640.
  28. Z.-Z. Sun, A high-order difference scheme for a nonlocal boundary-value problem for the heat equation, Comput. Methods Appl. Math. 1 (2001), 398-414.
  29. A. Bouziani, S. Bensaid, S. Dehilis, A second order accurate difference scheme for the diffusion equation with nonlocal nonlinear boundary conditions, J. Phys. Math. 11 (2020), Art. ID 2, 1-7.