Magnetism

Analytical Heat Diffusion Theory by A. V. Luikov

By A. V. Luikov

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4) is integrated over the whole surface and the time interval Ax = x2 — Τχ. If a surface temperature is constant and the coefficient a is independent of the temperature, then we shall have AQ = β 2 - ô i = <*S Γ* [ta - φ)} dx. 5) c. The Third Calculation Method. The volume element dv = dx dy dz is heated for the time Ax = x2 — xx in the temperature range from tx to t2; it receives the heat amount equal to <*(/,-*)*. 6) The total heat amount AQ which was supplied for heating for the time Ax will be found after integration over the whole volume AQ = Q2-Qi =

3) is a particular solution to this equation, viz: -fi- = *Ce*+", θξ d2t θξ2 -1^- = lCe**+l"9 -J^jr- = kICe*+l*9 3η οξθη dH ' θη2 Substituting these relationships into our equation gives, upon cancellation by Ce***1*, the so-called equation of coefficients ak2 + bkl + cl2 + dk + el + / = 0. 4) Hence, Eq. 4). Thus, we may take an arbitrary value of one of these two coefficients ; however, the second one should be found from Eq. , we can obtain an infinite number of particular solutions. , with respect to k (we consider k to be variable and /to be constant) and, depending on the value of the discriminant, we may obtain for A: (1) two unequal real roots, (2) two equal real roots, and (3) two complex conjugate roots.

This method turned out to be so efficient, that many hitherto unsolvable problems have now been solved. Moreover, the method permits solution of problems in a simpler form. Subsequently, the operational methods found an application in thermal physics and chemical engineering for the solution of various problems of transient heat conduction and diffusion. In recent years these methods were extended to hydrodynamics, neutron transfer in absorbing media, etc. Strict mathematical justification of the Heaviside operational method has been made in the works of Bromwich [5], Jeffreys [51], Efros and Danilevsky [28], Doetsch [25], Van der Pol [122], Ditkin [22], etc.

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