An index reduction method for linear Hessenberg systems
Introduction
Consider a linear (or linearized) semi-explicit DAEswhere Aj and C are smooth functions t, t0 ⩽ t ⩽ tf, Aj(t) ∈ Rn×n, j = 1, … , m, B(t) ∈ Rn×k, C(t) ∈ Rk×n, n > k, and CB is nonsingular (the DAEs has index m + 1). The homogeneities are q(t) ∈ Rn and r(t) ∈ R. It is well-known that the DAEs (1a), (1b) can be difficult to solve when it has a higher index (index greater than one, [1]). In this case an alternative treatment is the use of index reduction methods (see, e.g., [4], [5], [7], [9]), until a well-posed problem (index-1 DAEs or ordinary differential equations) is obtained. Early in [3] and [8], for k = 1, the index of problem (1a), (1b) has been reduced by introducing a simple formulation. In this paper, we will reduce the index of (1a), (1b) when k > 1. For this reason, we putand by substituting (2) in (1a), we obtain an implicit DAE which has index m, as follows:where Ej(t) ∈ Rn×n, j = 0,1, … , m, and except E0(t), others are singular matrices. Note that system (3) has k equations less than system (1a), (1b).
It is known that the eigenfunctions of certain singular Sturm–Liouville problems allow the approximation of functions C∞[a, b] where truncation error approaches zero faster than any negative power of the number of basic functions used in the approximation, as that number (order of truncation N) tends to infinity [6]. This phenomenon is usually referred to as “spectral accuracy” [6]. The accuracy of derivatives obtained by direct, term-by-term differentiation of such truncated expansion naturally deteriorates [2], but for low-order derivatives and sufficiently high order truncations this deterioration is negligible, compared to the restrictions in accuracy introduced by typical difference approximations (for more details, refer to [3], [8]). Throughout, we are using first kind orthogonal Chebyshev polynomials which are eigenfunctions of singular Sturm–Liouville problem
Section snippets
A simple formulation for index reduction
In this section, DAEs (1a), (1b) is considered when m = 1 and k = 2. To extend it to general case (1a), (1b) is easy. Now consider the Hessenberg index-2 system,where A = (aij)n×n, B = (bij)n×2, C = (cij)2×n, n ⩾ 3 andFrom (4a) and (5), we can writeand substituting (6) into (4a), implies,So, problem (4a), (4b) transforms to the systemHere, the overdetermined system (7a),
Implementation of numerical method
Here, the implementation of pseudospectral method is presented for DAEs systems (4a), (4b) and (8). This discussion can simply be extended to general forms and (3). Now consider the DAEs systems,with initial condition,For an arbitrary natural number ν, we suppose that the approximate solution of DAEs systems (14a), (14b) is as below,wherewhere a = (a0, a1, … , a3
Numerical example
Here, we use “ex” and “ey” to denote the maximum absolute error in vectors X = (x1, x2, x3) and y = (y1, y2). These values are approximately obtained through their graphs. Results show the advantages of techniques, mentioned in Sections 2 A simple formulation for index reduction, 3 Implementation of numerical method. Also, the presented algorithm in Section 3, is performed by using Maple 8 with 25 digits precision. Example Consider for −1 ⩽ t ⩽ 1,where
References (9)
- et al.
A modified spectral method for numerical solution of ordinary differential equations with non-analytic solution
Appl. Math. Comput.
(2002) - et al.
Reducing index, and pseudospectral methods for differential–algebraic equations
Appl. Math. Comput.
(2003) Reducing index method for differential–algebraic equations with constraint singularities
Appl. Math. Comput.
(2004)- et al.
Regularization methods for solving differential–algebraic equations
Appl. Math. Comput.
(2001)
Cited by (12)
Solving system of DAEs by homotopy analysis method
2009, Chaos, Solitons and FractalsApplication of He's variational iteration method for solution of differential-algebraic equations
2009, Chaos, Solitons and FractalsAdomian decomposition method for solution of differential-algebraic equations
2006, Journal of Computational and Applied MathematicsCitation Excerpt :Hence stabilized index reduction methods were used to overcome the difficulty. Earlier in [3,5–7], 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.
Adomian decomposition method for solution of nonlinear differential algebraic equations
2006, Applied Mathematics and ComputationNumerical algorithms for the determinant evaluation of general Hessenberg matrices
2018, Journal of Mathematical ChemistryOn sinc discretization method and block-tridiagonal preconditioning for second-order differential-algebraic equations
2017, Computational and Applied Mathematics