On Solving System of Linear Differential-Algebraic Equations Using Reduction Algorithm

Thota, Srinivasarao

doi:10.1155/2020/6671926

Cited by 5 publications

(2 citation statements)

References 20 publications

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“…As a consequence, many numerical methods were developed to solve higher index DAEs such as range-Kutta [13,17], projected Taylor series methods [18], hybrid block algorithms [19], stabilization methods [20][21][22], augmented Lagrangian method [23], sequential regularization methods [24], and the differential transform method (DTM) [25,26]. For the solution of DAEs, one can also nd in the literature the power series method combined with the Adomian polynomials [27,28] and other methods [29][30][31][32][33]. A very popular approach to treat higher index DAEs is rst to reduce the index by di erentiating the constraints one or more times with respect to time to obtain an ordinary di erential system or an index-1 DAE.…”

Section: Introductionmentioning

confidence: 99%

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Benhammouda

2023

Journal of Mathematics

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Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

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Section: Introductionmentioning

confidence: 99%

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Benhammouda

2023

Journal of Mathematics

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“…e reason is that these DAEs are in fact ill-posed problems in a certain sense and direct discretizations do not lead to satisfactory results in general [8][9][10][11]. ere are several methods in the literature for solving DAEs, which include Runge-Kutta methods [8,12], projected explicit and implicit Taylor series methods [13], hybrid block methods [14], variational methods [15], reduction methods [16,17], transformation methods [18], Adomian decomposition method [19], and the differential transform method [20]. One can find several techniques for solving DAEs of the form (1) [21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

A New Numerical Technique for Index‐3 DAEs Arising from Constrained Multibody Mechanical Systems

Benhammouda

2022

Discrete Dynamics in Nature and Society

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Index-3 differential-algebraic equations (DAEs) are mathematical models for the dynamics of constrained multibody mechanical systems that arise in many applications. These DAEs are known to pose a challenge to numerical methods. The purpose of this paper is to propose a new approach to efficiently solve these equations. This approach relies on an effective combination of the power series method (PSM) with the Adomian polynomials. Here, the PSM is directly applied to these DAEs without using the usual index reduction techniques, which are costly and often lead to nonphysical solutions. We expand the nonlinear terms in a series form using the Adomian polynomials to overcome the limitation of the PSM in collecting the coefficients of the power series solution. This technique has led to a simple and efficient algorithm. The domain of convergence of the power series solution is expanded by developing a multistage PSM (MSPSM). To demonstrate the efficiency of the MSPSM and show its applicability, an index-3 nonlinear DAE problem describing a two-link planar robot arm is solved. The numerical results show that the MSPSM is a powerful tool for solving the index-3 DAEs arising from constrained multibody mechanical systems.

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Asynchronous deployment scheme and multibody modeling of a ring-truss mesh reflector antenna

He,

Li,

Jia

et al. 2024

Acta Mech. Sin.

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On Solving System of Linear Differential-Algebraic Equations Using Reduction Algorithm

Cited by 5 publications

References 20 publications

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

A New Numerical Technique for Index‐3 DAEs Arising from Constrained Multibody Mechanical Systems

Asynchronous deployment scheme and multibody modeling of a ring-truss mesh reflector antenna

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