Benhammouda Brahim
Higher Colleges of Technology, Abu Dhabi Men's College, P.O. Box 25035, Abu Dhabi, United Arab Emirates.
Springerplus. 2015 Oct 29;4:648. doi: 10.1186/s40064-015-1443-3. eCollection 2015.
The solution of higher-index Hessenberg differential-algebraic equations (DAEs) is of great importance since this type of DAEs often arises in applications. Higher-index DAEs are known to be numerically and analytically difficult to solve. In this paper, we present a new analytical method for the solution of two classes of higher-index Hessenberg DAEs. The method is based on Adomian polynomials and the differential transform method (DTM). First, the DTM is applied to the DAE where the differential transforms of nonlinear terms are calculated using Adomian polynomials. Then, based on the index condition, the resulting recursion system is transformed into a nonsingular linear algebraic system. This system is then solved to obtain the coefficients of the power series solution. The main advantage of the proposed technique is that it does not require an index reduction nor a linearization. Two test problems are solved to demonstrate the effectiveness of the method. In addition, to extend the domain of convergence of the approximate series solution, we propose a post-treatment with Laplace-Padé resummation method.
高指标 Hessenberg 微分代数方程（DAEs）的求解具有重要意义，因为这类方程在实际应用中经常出现。众所周知，高指标 DAE 在数值求解和解析求解方面都存在困难。在本文中，我们提出了一种新的解析方法来求解两类高指标 Hessenberg DAE。该方法基于 Adomian 多项式和微分变换法（DTM）。首先，将 DTM 应用于 DAE，其中非线性项的微分变换使用 Adomian 多项式来计算。然后，根据指标条件，将得到的递归系统转化为一个非奇异线性代数系统。接着求解该系统以获得幂级数解的系数。所提出技术的主要优点是它不需要进行指标约简或线性化。通过求解两个测试问题来证明该方法的有效性。此外，为了扩展近似级数解的收敛域，我们提出了一种拉普拉斯 - 帕德求和法的后处理方法。
