An analytic approximation to the solution of heat equation by Adomian decomposition method and restrictions of the method

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Abstract

Adomian decomposition has been applied to solve many functional equations so far. In this article, we have used this method to solve the heat equation, which governs on numerous scientific and engineering experimentations. Some special cases of the equation are solved as examples to illustrate ability and reliability of the method. Restrictions on applying Adomian decomposition method for these equations are discussed.

Introduction

Since the governing equation in many experiments in engineering as well as science leads to the heat equation, this equation has attracted much attention, and solving the equation has been one of the interesting tasks for mathematicians. Analytical methods commonly used for solving the heat equation are very restricted and can be used in very special cases so they cannot be used to solve equations of numerous realistic scenarios. Numerical techniques, which are commonly used, encounter difficulties in terms of the size of computational works needed and usually the round-off error causes the loss of accuracy. Adomian decomposition method well addressed in [1], [2], [3] has been applied to solve many functional equations and systems of functional equations [3], [4], [5], [6]. The method has a useful feature in that it provides the solution in a rapid convergent power series with elegantly computable convergence of the solution. The decomposition method has proven to be very effective and results in considerable savings in computation time.

Section snippets

Solution of the heat equation by Adomian method

Consider the following general form of heat equation, with the indicated initial conditions:pt=A(x,y,z,t)2px2+B(x,y,z,t)2py2+C(x,y,z,t)2pz2+D(x,y,z,t),P(x,y,z,0)=f(x,y,z).For solving this equation by Adomian decomposition method the equation should be in canonical form which can be derived by rewriting Eq. (1) as follows:Ltp=A(x,y,z,t)2px2+B(x,y,z,t)2py2+C(x,y,z,t)2pz2+D(x,y,z,t),where Lt = ∂/∂t, with the inverse operator Lt-1=0t(.)dt.

Applying the inverse operator, we get:p(x,y,z,t)

Restrictions of the method

Adomian method is very sensitive to the initial conditions, when one is looking for the t-solution. If the initial conditions are zero and there is no D(x, y, z, t) term or it is a constant, the method could not be applied. In this case, one should resort to the x, y or z solution, for which the corresponding boundary conditions are suitable.

Conclusions

The main goal of this article has been to derive an analytical solution for the heat equation. We have achieved this goal by applying Adomian decomposition method.

The small size of computations in comparison with the computational size required in numerical methods and the rapid convergence shows that the Adomian decomposition method is reliable, and introduces a significant improvement in solving the heat equation over existing methods.

It is worth mentioning that looking for some modifications

References (7)

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