Taylor series method with numerical derivatives for initial value problems

Implicit Extension of Taylor Series Method with Numerical Derivatives

“…The main idea of the construction of the method of Taylor algorithm with numerical derivatives is the numerical approximation of the derivatives Y (i) (x 0 ), i = 2, 3, 4, ... . In [2] we have given some construction for it. We get matrices (linear form for first derivatives), bilinear, threelinear etc.…”

“…One further advantageus property of such approximation is that its calculation can be made fully parallel. Summarizing the results from [2] following expression gives a fourth order explicit algorithm:…”

“…where the appropriate expressions giving the derivatives Y (i) (x k ), i = 2, 3, 4 are given in [2]. The explicit Taylor series methods can be used for the construction of implicit algorithm.…”

A New Generalized Taylor-Like Explicit Method for Stiff Ordinary Differential Equations

“…One of the main advantages of using Taylor-like methods is that the approximate solution is given as an arbitrary order piecewise analytical function defended on the sub-intervals of the whole integration interval. This property offers different facility for adaptive error control [19,20]. Moreover, the Taylor-like method [11] is an arbitrary high order A-stable method that avoids extremely small stepsizes during the integration procedure.…”

“…Moreover, the Taylor-like method [11] is an arbitrary high order A-stable method that avoids extremely small stepsizes during the integration procedure. To avoid the analytical computation of the successive derivatives involved in the Taylor-like methods, numerical differentiation [21,22], automatic differentiation [23], differential transformation [24][25][26] and Infinity Computer with a new numeral system [27][28][29][30][31][32] can be used. In fact, Taylor-like explicit methods [5,7,[9][10][11] have computational drawbacks with zero-component derivative or zero-vector norm in their component or vector forms, respectively.…”