The dynamic stability of confined, exothermically reacting fluids

The numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.

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Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

The aim of this work is to point out that within a similarity approach some classes of free boundary value problems governed by ordinary differential equations can be transformed to initial value problems. The interest in the numerical solution of free boundary problems arises because these are always nonlinear problems. Furthermore we show that free boundary problems arise also via a similarity analysis of moving boundary hyperbolic problems and they can be obtained as approximations of boundary value problems defined on infinite intervals. Most of the theoretical content of this survey is original: it generalizes and unifies results already available in literature. As far as applications of the proposed approach are concerned, three problems of interest are considered and numerical results for each of them are reported.

http://mat521.unime.it/~fazio/preprints/sr98.pdf

A Similarity Approach to the Numerical Solution of Free Boundary Problems

Heat transfer—A review of 1969 literature

Methods for transforming partial differential equations into forms more suitable for analysis and solution are investigated The idea of Lie's infinitesimal contact transformation group is introduced to develop a systematic method which involves mostly algebraic manipulations A thorough presentation of the application of this general method to the problem of similarity analysis in a broader sense - namely, the similarity between partial and ordinary differential equations, boundary value and initial value problems, and nonlinear and linear equations - is given with new and very general methods evolved for deriving the possible groups of transformations

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Similarity analysis of differential equations by Lie group.

The equation governing the one-dimensional heat conduction with non-linear heat generation is transformed from a boundary value to an initial value problem by using a method of transformation recently presented by Na [2], [3]. The resulting equation is then integrated numerically by the Runge-Kutta method. Critical values of a parameter are found which serve as a limit for the existence of steady-state solutions.

T. Y. Na

Papers

A Method for the Solution of Conduction Heat Transfer with Non‐linear Heat Generation

MHD Free Convection from a Vertical Semi‐Infinite Flat Plate with a Step Change in the Magnetic Field