On the Numerical Solution of Differential‐Algebraic Equations with Hessenberg Index‐3

Abstract

Numerical solution of differential-algebraic equations with Hessenberg index-3 is considered by variational iteration method. We applied this method to two examples, and solutions have been compared with those obtained by exact solutions.

Block Method for Numerical Solution of Differential-Algebraic Equations With Hessenberg Index-3

“…The results for the two problems are tabulated for h = 0.1 and compared with the results in [12] Table 1: Exact y1(t) and Numerical solution y * 1 (t) and h = 0.1 for Example5.1 Exact y 2 (t) = Numerical solution y * 2 (t) = t. Exact y 3 (t) = Numerical solution y * 3 (t) = 1. …”

“…These include mechanical or multibody systems, chemical processes, optimal control, electric circuit design, analytical surveys and dynamical systems. In the literature, some numerical methods have been developed for the solution of DAEs such as the BDF (see [2], [3], [9]), implicit Runge-Kutta methods [3], Pade and Chebysev approximation methods (see [4] [5] [6]) and variational iterative method [12]. These methods are only directly suitable for low-index problems and often require that the problem, have special structure.…”

Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism

“…In recent years, approximation methods have been developed to solve DAEs. Among such approaches we can find Adomian decomposition method (ADM) [28,29], homotopy perturbation method (HPM) [30,31], variational iteration method (VIM) [32], homotopy analysis method (HAM) [33], Padé method [34], and the differential transform method (DTM) [35].…”

An Algorithm for the Approximation of Fractional Differential-Algebraic Equations with Caputo-type Derivatives

“…This has further generated further and more intense interest in finding numerical methods which accurately and efficiently solve nonlinear fractional differential equations. Some of the methods include Adomian Decomposition Method (ADM) [7][8][9][10][11][12][13], the Variational Iteration Method (VIM) [13][14][15][16], Homotopy Analysis Method (HAM) [17][18][19], Homotopy perturbation Method (HPM) [13,20]. Many physical problems are governed by a system of differential-Algebraic Equations (DAE's), and the solution of these equations has been a subject of many investigations in recent years [7][8][9][14][15][16]18,19,21,22].…”

“…In [17,19], the Homotopy Analysis Method (HAM) was applied for Fractional Differential-Algebraic Equations (FDAE's). The Adomian Decomposition Method (ADM) was applied in [7,[9][10][11]20,23] while in [16] the Variational Iteration Method was applied and the Differential Transform was applied in [14,21].…”