Inverse heat transport problems for coefficients in two‐layer domains and methods for their solution

Abstract

In various fields of science and technology it is often necessary to solve inverse problems, where from measurements of state of the system or process it is required to determine a certain typesetting of the causal characteristics. It is known that infringement of the natural causal relationships can entail incorrectness of the mathematical stating of inverse problems. Therefore the development of efficient methods for solving such problems allows one to considerably simplify experimental research and to increase the accuracy and reliability of the obtained results due to certain complication of algorithms for processing the experimental data. The problem of determination of thermal diffusivity coefficients considering other known characteristics of heat transport process is among incorrect inverse problems. These inverse problems for coefficients are quite difficult even in the case of homogeneous media. In this paper it is supposed that the heat transport equation is non‐homogeneous and an algorithm for determination of the thermal diffusivity coefficients for both the media is proposed. At the first step, the non‐homogeneous inverse problem with piecewise‐constant function of non‐homogeneity is solved. For this auxiliary inverse problem, the proposed method allows one to determine both the coefficients of thermal diffusivity and to restore the heat transport process without any additional information, i.e. the algorithm also solves the direct problem. Then the initial non‐homogeneous inverse problem with a piecewise‐continuous function of non‐homogeneity is solved. The proposed method reduces the non‐homogeneous inverse problem for coefficients to a set of two transcendent algebraic equations. Finally, the analytical solution to direct problem is obtained using Green's function. Atvirkštiniai šilumos laidumo uždaviniai dvisluoksnėse srityse ir jų sprendimo metodai Santrauka Ivairiose mokslo ir technologijos srityse dażnai tenka spresti atvirkštinius użdavinius, kada remiantis sistemos ar proceso būsenos parametru matavimais reikia nustatyti priežastines charakteristikas. Yra żinoma, kad natūraliu prieżastiniu priklausomybiu nepaisymas gali nulemti neteisinga atvirkštinio użdavinio matematine formuluote. Todel efektyvūs tokiu użdaviniu sprendimo metodai leis żymiai supaprastinti eksperimentinius tyrimus, padidinti gaunamu rezultatu tiksluma ir patikimuma, jeigu bus pritaikyti tam tikri sudetingesni eksperimentiniu rezultatu apdorojimo būdai. Difuzijos koeficientu nustatymas naudojant kitas żinomas šilumos laidumo proceso charakteristikas priklauso nekorektišku użdaviniu kategorijai. Atvirkštiniai użdaviniai koeficientams yra sudetingi net ir homogeninese terpese. Šiame darbe daroma prielaida, kad terpe nehomogeniška, ir pasiūlytas algoritmas difuzijos koeficientu nustatymui tokiu atveju. Pirmajame etape sumażinamas nehomogeninis atvirkštinis użdavinys, laikant, kad nehomogeniškumas aprašomas dalimis pastoviomis funkcijomis. Šiam pagalbiniam atvirkštiniam użdaviniui siūlomas metodas leidżia apibreżti abu šilumos difuzijos koeficientus ir atkurti šilumos laidumo proceso eiga be papildomos informacijos, t.y., algoritmas sprendżia taip pat ir tiesiogini użdavini. Po to yra sprendżiamas atvirkštinis pradinis użdavinys esant dalimis tolydżiai nehomogeniškuma aprašančiai funkcijai. Siūlomas metodas redukuoja nehomogenini atvirkštini użdavini i dvieju transcendentiniu lygčiu sprendima. Taip pat yra gautas tiesioginio użdavinio analizinis sprendinys, taikant Gryno formule. First Published Online: 14 Oct 2010

Keywords

• inverse problems
• heat transport
• multi‐layered domains