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

Abstract

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.

A Novel Computational Approach for Solving Fully Implicit Singular Systems of Ordinary Differential Equations

“…The numerical results show that the proposed method has successfully solved these examples by providing the exact solutions in a convergent power series form. To solve these implicit classes of differential equations over large intervals, a multi-stage algorithm can be used [28]. We emphasize that this new technique can be also applied to solve other fully implicit systems of ordinary differential equations, delay differential equations, differential-algebraic equations, fractional differential equations, and time-dependent algebraic equations.…”

“…However, a literature review revealed that no previous attempts were made to solve fully implicit ordinary differential systems using of semi-analytical methods in general and the DTM in particular. The DTM was recently used to solve implicit differential algebraic systems [28]. The DTM has found a large application in solving explicit differential equations [15,16,17].…”

A Proposed Analytical and Numerical Treatment for the Nonlinear SIR Model via a Hybrid Approach

“…In this context, mathematicians have invented and developed many ways to solve mathematical and physical models. For example, the Adomian decomposition method (ADM) [1][2][3][4][5][6][7][8][9][10][11] is one of the most popular methods which has been widely used during recent decades; the same applies to the homotopy perturbation method (HPM) [12][13][14][15], the homotopy analysis method (HAM) [16,17], and the differential transform method (DTM) [18][19][20][21]. However, each of these methods has its own advantages, but at the same time has some defects that must be faced and overcome.…”

Nonclassical Parametric Variational Technique to Manipulability Control of a Serial-Link Robot That Is Used in Treatment of Femoral Shaft Fractures

“…The concept of DAC-SYS has been introduced [13][14][15], and in recent years, tools for simulation, identification, analysis, and design of these systems have been developed. The algorithms for numerical simulation are addressed [16,17], while for mechanical DAC-SYS, a recent study focuses on the design and manipulability analysis of a serial-link robot (S-LR) [18][19][20]. This paper is aimed at investigating the kinematics of the robotic DAC-SYS, specifically under the influence of algebraic kinematic constraints at the joints.…”