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The paper deals with extremely exact, stable, and fast numerical solutions of systems of differential equations. It also involves solutions of problems that can be reduced to solving a system of differential equations. The approach is based on an original mathematical method, which uses the Taylor series method for solving differential equations in a non-traditional way. Even though this method is...
In this paper an outline is given of historical and current developments in the application of recurrent Taylor series to the integration of systems of ordinary differential equations. Then an extremely accurate and fast method for the numerical solution of ordinary differential equations is presented. In general Taylor series method is not included or even mentioned in surveys on numerical integration...
In this paper an outline is presented of historical and current developments in the application of recurrent Taylor series to the integration of systems of ordinary differential equations. The idea of an extremely accurate and fast method for numerical solutions of differential equations is presented in the paper. In general Taylor series are not included or not even mentioned in surveys on numerical...
The application of Taylor series has become a standard concept in numerical analysis. Their ability of approximating functions arbitrarily close under certain conditions makes them an ideal tool for the integration of differential equations. Before the appearance of digital computers the analytical determination of Taylor series coefficients, i.e. the calculation of higher derivatives, was regarded...
Methods of numerical solutions of differential equations have been studied since the end of the last century. A large number of integration formulas have been published especially for solving special systems of differential equations. In general, it was not possible to choose the best method but for a subclass of tasks defined by similar properties the most suitable method could always be found. The...
It is the aim of the paper to give some of the basic philosophy of analogue computation before entering into a detailed discussion of the theory of analogue principles. The analogue technique has, in general, three control states; they are the reset, operate, and hold states. The idea of analogue principles is used in our Taylor series integrators. Again, three control states are applied (reset, operate,...
Main objective of this paper is to outline possible ways how to achieve a substantial acceleration in case of advection-diffusion equation (A-DE) calculation, which is commonly used for a description of the pollutant behavior in atmosphere. A-DE is a kind of partial differential equation (PDE) and in general case it is usually solved by numerical integration due to its high complexity. These types...
Homogeneous differential equations with constant coefficients are analysed in the paper to show how extremely high accuracy and speed of computations can be obtained using Taylor series for numerical solutions of differential equations. Two examples of homogeneous differential equations and their solutions by the modern Taylor series method will be analysed in this paper. The main idea behind the...
The paper deals with semi-analytical computations and gives the examples of absolutely exact solutions that can be obtained using numerical solutions of differential equations. Numerical solutions of differential equations based on the Taylor series are implemented in a simulation language TKSL. Polynomials functions, finite integrals, Fourier series and exponential functions are but a few examples...
The simulation language TKSL and modern Taylor series method has proved to be very powerful computing tools for extremely exact, stable and fast numerical solutions of systems of differential equations. In a natural way, TKSL also involves solutions of problems that can be reduced to solving a system of differential equations. It is known that the Taylor series method is a parallel one and therefore...
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