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Meccanica

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17-12-2018

Functional solutions for problems of heat and mass transfer

Author: Giovanni Cimatti

Published in: Meccanica | Issue 1-2/2019

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Abstract

We prove the existence and, in certain cases, the uniqueness of functional solutions for boundary value problems of systems of P.D.E. in divergence form with constant boundary conditions. We are motivated by various problems of heat and mass transfer. After giving a suitable definition of functional solutions, we reformulate the boundary value problem as non-standard one-dimensional two point problem for a system of O.D.E coupled with a mixed problem for the laplacian. If ${{{\mathcal {C}}}}_F$ and ${{{\mathcal {C}}}}$ denote respectively the set of functional and classical solutions of the starting problem we settle, in simple cases, the question if ${{{\mathcal {C}}}}_F={{{\mathcal {C}}}}$.

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The case

$$\begin{aligned} {\nabla }\cdot \biggl [\sum _{j=1}^{n}a_{ij}(u_1,\ldots ,u_n){\nabla }u_j\biggl ]=0 \end{aligned}$$

can be treated with the same method used in this paper with only minor changes in the proofs.

Since the three components of the boundary of ${\Omega }$ have no points in common there is no problems at the interfaces.

The constant boundary conditions are the main limitation of the present method. However, the more general case $u_1=u_{11}^*{\ \hbox {on}\ {\Gamma }_1} $, $u_1=u_{12}^*{\ \hbox {on}\ {\Gamma }_2}$, $u_2=u_{21}^*{\ \hbox {on}\ {\Gamma }_1} $, $ u_2=u_{22}^*{\ \hbox {on}\ {\Gamma }_2}$, all the $u_{ij}$ constants, can be treated with minor changes.

$u_1|_{{\Gamma }1}$ denotes the restriction of $u_1$ to ${\Gamma }_1$.

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Title: Functional solutions for problems of heat and mass transfer
Author: Giovanni Cimatti
Publication date: 17-12-2018
Publisher: Springer Netherlands
Published in: Meccanica / Issue 1-2/2019
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-018-00924-x

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