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__C++ Template Programs for ODE and DAE by Taylor Series__

H. Hirayama

**Abstract**

The arithmetic operations and functions of Taylor series can be defined by C++ language easily.
It is shown that Taylor series solutions of the following
ordinary differential equations and differential algebraic equations
$
{F}({y},{y}',x)=0 with initial conditions {y}(0)={y}_0
$
can be computed by similar methods as above Taylor series methods.
The solutions can be expanded up to arbitrary order,
so they can be used instead of higher order Runge-Kutta formula.
Taylor series can be used for the evaluations of the errors and the optimal step size within
given error allowance easily.
We can also transform Taylor series into Padé series, which give A-stable methods for solving
differential algebraic equations numerically.

**Bibliography**

[1] Y. F. Chang and G. Corliss, ATOMFT: Solving ODEs and DAE Using Taylor Series, Computers Math.
Applic., 28 (1994), 209-233

[2] G. Corliss and Y. F. Chang, Solving Ordinary Differential Equations Using Taylor Series, ACM Trans.
Math. Soft., 8 (1982), 114-144

[3] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, 1991

[4] H. Hirayama, S. Komiya and S. Sato, Solving Ordinary Differential Equations by Taylor Series,
JSIAM, 12 (2002), 1-8 (Japanese)

[5] N.S. Nedialkov and J.D. Pryce, Solving differential-algebraic equations by Taylor
series (III): The DAETS code, J. Numerical Analysis, Industrial and Applied Mathematics 3 (1-2) (2008), pp. 61-80.