Analytical and numerical solutions of dissipative systems​

2018. 10. 11. 10:15
Kovács Róbert (BME)

Our world is not ideal, in reality the processes are dissipative. The framework of non-equilibrium thermodynamics offers lot of possibilities to derive models, constitutive equations that describe the 
behavior of a dissipative system. The level of modeling is arbitrary, depends on our choice which is 
reflected by these models. However, they should be remain as simple as possible to be applicable for practical problems in question.

First, the non-equilibrium thermodynamical background and the structure of equations are discussed. Here mostly the parabolic - hyperbolic properties of the resulted partial differential equations (PDE) are emphasized. The role of boundary conditions and their effect on solutions are also presented through different examples. Such example is related to a particular experimental arrangement called heat pulse (or laser flash) experiment that used to detect different dissipative wave propagation phenomena.

In this presentation the way from generating the equations to their solutions for experiments is presented. It covers analytical solution of a PDE for time dependent boundary condition and a particular numerical method that allows us to eliminate certain boundary conditions and related to the specific structure resulted by non-equilibrium thermodynamics.

A new successive approximation method of general non-local boundary value problems

2018. 10. 04. 10:15
Rontó Miklós (Miskolci Egyetem)

We propose a new successive approximation technique for the solvability analysis and approximate solution of general non-local boundary value  problems for non-linear systems of differential equations with locally Lipschitzian non-linearities. It will be studied the non-linear boundary value problem 
$\frac{dx(t)}{dt}=f(t,x(t)),\,t\in \left[ a,b\right] ,  \Phi (x)=d, $
where $\Phi :C(\left[ a,b\right] ,\mathbb{R}^{n})\rightarrow \mathbb{R}^{n}$ is a vector functional (possibly non-linear), $~f:\left[ a,b\right]
\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}~$is a function satisfying the Caratheodory conditions in a certain bounded set $D$, which will be concretized later, $d$ is a given vector and $f\in Lip(K,D),$ i.e. $f$ locally Lipschitzian 
$\left\vert f(t,u)-f(t,v)\right\vert \leq K\left\vert u-v\right\vert ,~\text{for all}\ \left\{ u,v\right\} \subset D~\text{ }a.e.t\in \left[ a,b\right] . $
By a solution of the problem, one means an absolutely continuous function satisfying the differential system almost everywhere on $\left[ a,b\right] .$ The analysis is constructive in the sense that it allows one to both study the solvability of the problem and approximately construct its solutions by operating with objects that are determined explicitly in finitely many steps of computation. The practical application of the technique is explained on a numerical example.