Petr Veigend

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.

Modelling VLSI circuits using Taylor series

The paper introduces the capacitor substitution for CMOS logic gates, i.e. NANDs, NORs and inverters. It reveals the necessity of a very accurate and fast method for solving this problem. Therefore the Modern Taylor Series Method (MTSM) is used which provides an automatic choice of a higher order during the computation and a larger integration step size while keeping desired accuracy.

Solving wave equation using finite differences and Taylor series

The paper deals with the numerical solution of partial differential equations (PDEs), especially wave equation. Two methods are used to obtain numerical solution of the wave equation. The Finite Difference Method (FDM) is used for transformation of wave equation to the system of ordinary differential equations (ODEs), different types of difference formulas are used. The influence of arithmetic to higher order difference formulas is also presented. The Modern Taylor Series Method (MTSM) allows to solve ODEs numerically with extremely high precision. An important feature of this method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires.

Parallel solution of higher order differential equations

The paper focuses on a mathematical approach which uses Modern Taylor Series Method (MTSM) for solving differential equations in a parallel way. Even though this method is not much preferred in the literature, some experimental calculations have shown and verified that the accuracy and stability of the MTSM exceeds the currently used algorithms for solving differential equations. Further, the MTSM has properties suitable for parallel processing, i.e. many independent calculations. The MTSM allows these calculations to be performed independently on several processors using basic mathematical operations. Hardware representation of these operations and their principle are discussed in this paper. Generally, the MTSM can only solve systems of ordinary differential equations (ODEs) that are formed as initial value problems (IVPs). Therefore, this paper also presents methods for solving higher order differential equations, PDEs and their transformations to the corresponding systems of ODEs (IVPs). Effectiveness of hardware implementation of the MTSM is also discussed in this paper, e.g. implementation on FPGA. In many cases, the MTSM obtains results faster than the commonly used Runge-Kutta methods.

New trends in Taylor series based applications

The paper deals with the solution of large system of linear ODEs when minimal comunication among parallel processors is required. The Modern Taylor Series Method (MTSM) is used. The MTSM allows using a higher order during the computation that means a larger integration step size while keeping desired accuracy. As an example of complex systems we can take the Telegraph Equation Model. Symbolic and numeric solutions are compared when harmonic input signal is used.

Electronic representation of wave equation

The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations.This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.

Numerical Integration of Multiple Integrals using Taylor Polynomial

The paper concentrates on a new method of numerical computation of multiple integrals. Equations based on Taylor polynomial are derived. Multiple integral of a continuous function of n-variables is numerically integrated step by step by reducing its dimension. First, integration formulas for a function of two variables are derived. Formulas for function of n-variables are generalized using composition. Numerical derivatives for Taylor terms are repeatedly computed from simple integrals. Finally method is demonstrated on an exponential function of two-variables and a new approach to determine a number of Taylor terms is discussed.