Filip Kocina

New Trends in Taylor Series Based Computations

The paper is a part of student cooperation in AKTION project (Austria-Czech) and concentrates on the numerical solution of partial differential equations (PDEs) using high-order forward, backward and symmetrical formulas. As an example, the hyperbolic PDE is analyzed. The paper is based on the numerical solution of ordinary differential equations by the Taylor series method and on the simulation language TKSL that has been created to test the properties of the technical initial problems and to test an algorithm for Taylor series method. The idea of parallel computations using special integrators is also a part of the paper.

Adaptive Solution of the Wave Equation

The paper focuses on the adaptive solution of two-dimensional wave equation using an adaptive triangulation update based on a posteriori error estimation. The a posteriori error estimation is based on the Gradient super-approximation method which is based on works of J. Dalik et al that is briefly explained. The Modern Taylor Series Method (MTSM) used for solving a set of ordinary differential equations is also explained. The MTSM adapts to the required accuracy by using a variable number of Taylor Series terms. It possible to use the MTSM to solve wave equation in conjunction with Finite Difference Method (FDM).

Taylor series based computations and MATLAB ODE solvers comparisons

The paper is a part of student cooperation in AKTION project (Austria-Czech). 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. The paper deals with possibilities of numerical solution of Initial Value Problems of Ordinary Differential Equations (ODEs) - using the Taylor series method with automatic computation of higher Taylor series terms. The explicit and implicit scheme of Taylor series method is compared with numerical solvers implemented in MATLAB software [1]. The computation time and accuracy of our approach are compared with that of MATLAB ode solvers on a set of ODEs test examples [2].

Advanced VLSI Circuits Simulation

The paper deals with very accurate and effective simulation of Complementary Metal-Oxide-Semiconductor (CMOS) transistors which are used to construct basic logic gates (inverter, NAND and NOR) and their composites (XOR, AND, OR). The transistors are substituted by a resistor-capacitor (RC) circuit and the circuit is described by a system of differential algebraic equations (DAEs). These equations are numerically solved by the variable-step, variable-order Modern Taylor Series Method (MTSM). The same approach can be used for VLSI simulation — it was implemented by the corresponding author in a general purpose programming language. This approach is faster than the state of the art (SPICE) and uses less memory.

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.