Hybrid Numerical Method for Heat Equation with Nonlocal Boundary Conditions in Parallel Computing Environment

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.


INTRODUCTION
The need of both the scientific and the business communities' for ever growing computing supremacy led to vivid up grading in computer structural design.Most of the attempts concentrated on attaining high performance on a single processor, but recently it has been observed that attempts are being made to wrap multiple processors.Multiprocessor systems consist of a number of connected processors each of which is capable of performing compound tasks autonomously.In a sequential algorithm all tasks are carried out by a single processor but in a parallel algorithm autonomous components of the program are performed by varied processors simultaneously which save a lot of time.
In many developing countries, scientists and engineers are facing problems, when high computations and/or large memory storage is required.This is due to the lack of advance computing resources.In order to resolve such problems, the numerical method is proposed.Partial differential equations arise in many real life problems like thermo elasticity (Day, 1982), dynamics of ground water (Nakhushev, 1982) and pseudo-parabolic water transfer (Vodakhova, 1982).In the family of partial differential equations, one of the most important classes is parabolic partial differential equations with nonlocal boundary conditions.This class was studied by different authors (Wang and Liu, 1989;1990;Muravei and Philinovskii, 1982;Liu, 1999;Aug, 2002;Deghan, 2003Deghan, , 2005;;Rehman and Taj, 2009) in different ways to solve such model problems numerically.This study aims at exploring one dimensional non-homogeneous heat equation with integral boundary conditions.The idea of mixed order numerical method presented here was firstly introduced by Rehman et al. (2012) and now it is proposed to be the best candidate for numerical solution of nonlocal problems.
Actual concept behind the use of finite-difference methods for obtaining the approximate solution of a given PDE is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points.Consider one dimensional heat equation: Subject to the given initial condition: and the non-local boundary conditions: where, ‫,ܨ‬ ‫ܩ‬ ଵ , ‫ܩ‬ ଶ , ߬, ߮ and ܳ are known functions and are assumed to be sufficiently smooth to produce a smooth solution of ‫.ݑ‬ ܶ is given positive constant.

NUMERICAL EXAMPLES
Numerical method described in this study will be applied to four problems from the literature and results obtained will be compared with exact solutions as well as with the results existing in the literature.We select values of ܽ (݅ = 1, 2, 3, 4) such that stability conditions are satisfied (Rehman et al., 2012).
Example 3: Consider the problem (1)-( 4) with:     For the comparison purpose, in this problem is solved for ℎ = ݈ = 0.01 for different values of t.The errors obtained by new scheme are given in Table 3 and compared with fourth order scheme (Rehman and Taj, 2009).4 and compared with fourth order scheme (Rehman and Taj, 2009).

CONCLUSION
It is observed that the result obtained using hybrid scheme are highly precise in space and time.This technique can be coded easily on serial and parallel computers.The method involve only real domain and multiprocessor design, especially in nonlocal problems save significant computational time rather than the complex arithmetic based methods.This method is very flexible, user friendly and can be extended for multidimensional partial differential equations.

Table 1 :
Comparison of relative error for ‫ݐ‬ = 1

Table 2 :
Comparison of relative error for ‫ݐ‬ = 0.1