Solving system of DAEs by homotopy analysis method
Introduction
In the past decades, both mathematicians and physicists have devoted considerable effort to the study of explicit and numerical solutions to DAEs. Many powerful methods have been presented [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].
The subject of DAEs has researched and solidified only very recently (in the past 35 years). Through many exact solutions for linear DAEs has been found, in general, there exists no method that yields an exact solution for nonlinear DAEs.
Numerical approaches for approximating solutions of DAEs have been presented in the literature. However, these approaches approximate solutions at discrete points only thereby making it impossible to get continuous solutions. Recently there has been a growing interest in obtaining solutions to system of DAEs by analytical techniques.
It is well known that differential-algebraic equations (DAEs) can be difficult to solve when they have a higher index, i.e., an index greater than 1 [1] and an alternative treatment is the use of index reduction methods, whose essence is the repeated differentiation of the constraint equations until low-index problem (an index-1 DAE or ODE) is obtained. But repeated index reduction by direct differentiation leads to instability of the resulting ODE. Hence stabilized index reduction methods were used to overcome the difficulty.
Earlier in [1], [4], [11], [12], [13], for semi-explicit DAEs, an efficient reducing index method has been proposed that does not need the repeated differentiation of the constraint equations. This method is well applied to DAEs with and without singularities and then are numerically solved by pseudospectral method with and without domain decomposition.
This paper is devoted to find approximate solution to system of DAEs of the formwhere may be singular by using the homotopy analysis method (HAM) addressed in [16], [17].
HAM provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. The HAM is based on the homotopy, a basic concept in topology. The auxiliary parameter h is introduced to construct the so-called zero-order deformation equation. Thus, unlike all previous analytic techniques, the HAM provides us with a family of solution expressions in auxiliary parameter h. As a result, the convergence region and rate of solution series are dependent upon the auxiliary parameter h and thus can be greatly enlarged by means of choosing a proper value of h. This provides us with a convenient way to adjust and control convergence region and rate of solution series given by the HAM.
Section snippets
Basic ideas of the HAM
In HAM, system (1) is first written in the formwhere is nonlinear operator, is unknown function and t the independent variable. Let denote an initial guess of the exact solution , an auxiliary parameter, an auxiliary function, and an auxiliary linear operator with the property when . Then using as an embedding parameter, we construct such a homotopy
Numerical examples
The HAM provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution, and to obtain numerical values from the infinite power series. The consequent series truncation, and the practical procedure conducted to accomplish this task, together transforms the otherwise analytical results into an exact solution, which is evaluated to a finite degree of accuracy. In order to investigate the accuracy of the HAM solution with a finite
Discussion and conclusion
We have described and demonstrated the applicability of the HAM for solving system of DAEs. Our method is a direct method, further it is simple and accurate. It is a practical method and can easily be implemented on computer to solve such problems. We have used the method with three examples, and have tabulated the numerical results as well as the exact solutions. The tables show that the present method approximates the exact solution very well.
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