Differential transform method applied to high index differential-algebraic equations. (English) Zbl 1115.65089

Liu, Hong; Song, Yongzhong

Appl. Math. Comput. 184, No. 2, 748-753 (2007).

Summary: F. Ayaz [Appl. Math. Comput. 152, No. 3, 649–657 (2004; Zbl 1077.65088)] had considered the numerical solution of linear differential-algebraic equations (DAEs) by the differential transform method. Two index-1 DAEs have been solved by using the method and the numerical solutions coincide with the exact solution very well.
In this paper, high index DAEs are considered. Index-2 DAEs can be done in a similar way. A counter example is proposed to explain that the method is not suitable for all index-3 DAEs.

MSC:

65L80	Numerical methods for differential-algebraic equations
34A09	Implicit ordinary differential equations, differential-algebraic equations

Keywords:

differential; algebraic equations (DAES); differential transform method; series solution; numerical example; Index-2 DAEs; counter example; index-3 DAEs

Citations:

Zbl 1077.65088

References:

[1]	Ascher, U. M., On symmetric schemes and differential-algebraic equations, SIAM J. Sci. Stat. Comput., 10, 937-949 (1989) · Zbl 0687.65084
[2]	Ascher, U. M.; Petzold, L. R., Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal., 28, 1097-1120 (1991) · Zbl 0732.65067
[3]	Ayaz, F., Applications of differential transform method to differential-algebraic equations, Appl. Math. Comput., 152, 649-657 (2004) · Zbl 1077.65088
[4]	Ayaz, F., Solutions of the systems of differential equations by differential transform method, Appl. Math. Comput., 147, 547-567 (2004) · Zbl 1032.35011
[5]	Babolian, E.; Hosseini, M. M., Reducing index, and pseudospectral methods for differential-algebraic equations, Appl. Math. Comput., 140, 77-90 (2003) · Zbl 1042.65067
[6]	Brenan, K. E.; Campell, S. L.; Petzold, L. R., Numerical Solution of Initial Value Problems in Differential-Algebraic Equations (1989), Elsevier: Elsevier New York · Zbl 0699.65057
[7]	Campell, S. L., Singular Systems of Differential Equations ⨿ (1982), Pitman: Pitman San Francisco, CA
[8]	Chen, C. K.; Ho, S. H., Solving partial differential equations by two dimensional differential transform method, Appl. Math. Comput., 106, 171-179 (1999) · Zbl 1028.35008
[9]	Gear, C. W.; Petold, L. R., ODE systems for the solution of differential-algebraic systems, SIAM J. Numer. Anal., 21, 716-728 (1984) · Zbl 0557.65053
[10]	Griepentrog, E.; März, R., Basic properties of some differential-algebraic equations, Z. Anal. Anwend., 8, 25-40 (1989) · Zbl 0666.34016
[11]	Higueras, I.; Garcya-Celayeta, B., Runge-Kutta methods for DAEs. A new approach, J. Comput. Appl. Math., 111, 49-61 (1999) · Zbl 0948.65081
[12]	Song, Y., Solvability of higher index time-varying linear differential-algebraic equations, Acta Math. Sci., 21, B, 77-92 (2001) · Zbl 0992.65087
[13]	Zhou, J. K., Differential Transformation and its Application for Electrical Circuits (1986), Huazhong University Press: Huazhong University Press Wuhan, China, (in Chinese)