Abstract

A numerical method is developed for solving parabolic partial differential equations with integral boundary conditions. The method is moderately sixth-order accurate due to merging of sixth order finite difference scheme and fifth order Pade’s approximation. Simpson’s 1/3 rule is used to approximate integral conditions. The method does not involve the use of complex arithmetic and optimizes the results. It is observed that this numerical method can be easily coded on serial as well as parallel computers.

Keywords:

Integral boundary conditions, method of lines, Pade, parallel algorithm, Simpson,

References

1. Aug, W.T., 2002. A method of solution for the one dimensional heat equation subject to nonlocal boundary conditions. SEA Bull. Math., 26(2): 197-203.
   
2. Day, W.A., 1982. Extension of a property of the heat equation to linear thermo elasticity and other theories. Quart. Appl. Math., 40: 319-330.
   CrossRef    
3. Deghan, M., 2003. Numerical solution of a parabolic equation with nonlocal boundary specifications. Appl. Math. Comp., 145: 185-194.
   CrossRef    
4. Deghan, M., 2005. Efficient techniques for the second-order parabolic equation subject to nonlocal specifications. Appl. Numer. Math., 52: 39-62.
   CrossRef    
5. Liu, Y., 1999. Numerical solution of the heat equation with nonlocal boundary conditions. J. Comp. Appl. Math., 110: 115-127.
   CrossRef    
6. Muravei, L.A. and A.V. Philinovskii, 1982. On certain nonlocal boundary value problems for hyperbolic problems. DifferetsiatineUravnenia, 18: 280-285.
   
7. Nakhushev, A.M., 1982. On certain approximate method for boundary value problems for differential equations and its applications in ground water dynamics. Differetsiatine Uravnenia, 18: 72-81.
   
8. Rehman, M.A. and M.S.A. Taj, 2009. Fourth-order method for non-homogeneous heat equation with nonlocal boundary condition. Appl. Math. Sci., 3(37): 1811-1821.
   
9. Rehman, M.A., S.A. Mardan, M.S.A. Taj and A.A. Bhatti, 2012. Fusion higher -order parallel splitting methods for parabolic partial differential equations. Int. Math. Forum, 7(32): 1567-1580.
   
10. Vodakhova, V.A., 1982. A boundary value problem with Nakhushev non-local condition for certain psedo-parabolic water transfer equation. Differetsiatine Uravnenia, 18: 280-285.
    
11. Wang, S. andY. Lin, 1989. A finite difference solution to an inverse problem for determining a control function in parabolic partial differential equation. Inverse Problem, 5: 573-578.
    CrossRef    
12. Wang, S. and Y. Lin, 1990. A numerical method for the diffusion equation with nonlocal boundary specifications, Inter. J. Eng. Sci., 28: 543-546.
    CrossRef    

Competing interests

The authors have no competing interests.

Open Access Policy

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Copyright

The authors have no competing interests.