R. Grzymkowski

Stefan problem solved by Adomian decomposition method

The solution of the one-phase Stefan problem is presented. This problem consists of finding the distribution of temperature in the domain and the position of the moving interface (freezing front). The proposed solution is based on the Adomian decomposition method and optimalization. The validity of the approach is verified by comparing the results obtained with the analytical solution.

Comparing the Adomian decomposition method and the Runge–Kutta method for solutions of the Stefan problem

The solution of the one-phase Stefan problem is presented. A Stefans task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.

A Certain Analytical Method Used for Solving the Stefan Problem

The paper presents an analytic method applied for finding the approximate solution of Stefan problem reduced to the one-phase solidification problem of a plate with the a priori unknown, and varying in time, boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial extension of a sought function describing the temperature field into the power series, some coefficients of which can be determined with the aid of boundary conditions, and also based on the approximation of a function defining the freezing front location with the broken line, parameters of which can be obtained by solving the appropriate differential equations. Results received by applying the proposed procedure are compared with the results obtained using a classical numerical method for solving the Stefan problem.

An application of the adomian decomposition method for inverse stefan problem with neumann’s boundary condition

In this paper the solution of one-phase inverse Stefan problem with Neumann’s boundary condition is presented. This problem consists of the reconstruction of the function which describes the heat flux on the boundary, when the position of the moving interface is well-known. The proposed solution is based on the Adomian decomposition method and the least square method.

Multi-phase Inverse Stefan Problems Solved by Approximation Method

In this paper the approximation method is used to solve the multi-phase inverse Stefan problem. The solution is found in a linear combination form of the functions satisfying the equation of heat conduction. The coefficients of this combination are determined by the least square method for the boundary of a domain. The numerical example is presented.

Application of the Taylor Transformation to the Systems of Ordinary Differential Equations

In the paper the Taylor transformation is applied to systems of ordinary differential equations, including nonlinear differential equations. Apart from the description of the method, its computational effectiveness is demonstrated on example. Efficiency of the proposed method is confirmed it with the selected classical methods devoted to problems of considered kind. The present paper is an introduction to some further research in this area, which is very important for a wide range of problems described by means of the systems of ordinary differential equations.