Modern Taylor Series Method Research Articles

Taylor Series Based Integration in Electric Circuits Simulations

This paper deals with the extremely precise, stable and fast solution of the ordinary differential equations. The solution of these is performed using a method based on the Taylor series - The Modern Taylor Series Method. The paper investigates two problems to demonstrate the positive properties of the method: linear problem - the behavior of signal transmission on the telegraph line and a non-linear problem - the Van der Pol oscillator. Both problems were analyzed and solved using newly implemented MATLAB Modern Taylor Series Method solvers. The results were then compared to the state-of-the-art MATLAB solvers.

Model of the telegraph line and its numerical solution

Abstract This paper deals with a model of the telegraph line that consists of system of ordinary differential equations, rather than partial differential telegraph equation. Numerical solution is then based on an original mathematical method. This method uses the Taylor series for solving ordinary differential equations with initial condition - initial value problems in a non-traditional way. Systems of ordinary differential equations are solved using variable order, variable step-size Modern Taylor Series Method. The Modern Taylor Series Method is based on a recurrent calculation of the Taylor series terms for each time interval. The second part of paper presents the solution of linear problems which comes from the model of telegraph line. All experiments were performed using MATLAB software, the newly developed linear solver that uses Modern Taylor Series Method. Linear solver was compared with the state of the art solvers in MATLAB and SPICE software.

Taylor Series Based Solution of Linear ODE Systems and MATLAB Solvers Comparison

The Modern Taylor Series Method (MTSM) is employed here to solve initial value problems of linear ordinary differential equations. An automatic computation of higher Taylor series terms and an efficient, vectorized coding of explicit and implicit schemes enables a very fast computation of the solution to specified accuracy. For a set of benchmark problems from literature, the MTSM significantly outperforms standard solvers. Finally, ideas of parallelizing the MTSM computations are discussed.

F and G Taylor series solutions to the Stark and Kepler problems with Sundman transformations

The classic $$F$$ and $$G$$ Taylor series of Keplerian motion are extended to solve the Stark problem and to use the generalized Sundman transformation. Exact recursion formulas for the series coefficients are derived, and the method is implemented to high order via a symbolic manipulator. The results lead to fast and accurate propagation models with efficient discretizations. The new $$F$$ and $$G$$ Stark series solutions are compared to the Modern Taylor Series (MTS) and 8th order Runge–Kutta–Fehlberg (RKF8) solutions. In terms of runtime, the $$F$$ and $$G$$ approach is shown to compare favorably to the MTS method up to order 20, and both Taylor series methods enjoy approximate order of magnitude speedups compared to RKF8 implementations. Actual runtime is shown to vary with eccentricity, perturbation size, prescribed accuracy, and the Sundman power law. The method and results are valid for both the Stark and the Kepler problems. The effects of the generalized Sundman transformation on the accuracy of the propagation are analyzed. The Taylor series solutions are shown to be exceptionally efficient when the unity power law from the classic Sundman transformation is applied. An example low-thrust trajectory propagation demonstrates the utility of the $$F$$ and $$G$$ Stark series solutions.

Application of the Modern Taylor Series Method to a multi-torsion chain

In this paper the application of a novel high accuracy numerical integration method is presented for a practical mechanical engineering application. It is based on the direct use of the Taylor series. The main idea is a dynamic automatic order setting, i.e. using as many Taylor series terms for computing as needed to achieve the required accuracy. Previous results have already proved that this numerical solver is both very accurate and fast. In this paper the performance is validated for a real engineering assembly and compared to a Jacobian power series method. The chosen experiment setup is a multi-torsional oscillator chain which reproduces typical dynamic behavior of industrial mechanical engineering problems. Its rotatory dynamics are described by linear differential equations. For the test series the system is operated in a closed-loop configuration. A reference solution of the linear differential equations of the closed-loop system for the output variable is obtained with the mathematical software tool Maple and validated by comparison to measurements from the experiment. The performance of the Modern Taylor Series Method is demonstrated by comparison to standard fixed-step numerical integration methods from the software tool Matlab/Simulink and to the Jacobian power series approximation. Furthermore, the improvement in numerical accuracy as well as stability is illustrated and CPU-times for the different methods are given.

Using differential equations in electrical circuits' simulation

Numerical methods for solving the system of linear algebraic equations as well as the system of differential equations have been known since the last century. Most numerical methods are very accurate and fast. However, some complicated problems can occur, such as stiff problems and bad-conditional equations, which could be computationally intensive. This paper describes the Modern Taylor Series Method for solving the system of algebraic equations using differential equations. The example of an electrical circuit with a parasitic capacity for demonstrating computational problems will be shown and the suggestion solution presented.