Combined Method of Separation of Variables. 2. Sequences of Differential Relations: Plate, Cylinder, Sphere
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
COMBINED METHOD OF SEPARATION OF VARIABLES. 2. SEQUENCES OF DIFFERENTIAL RELATIONS: PLATE, CYLINDER, SPHERE V. A. Kot
UDC 517.518.8:519.633:536.2
By the example of the Sturm–Liouville problem, solved for the function V(y) determined in the region Ωy ∈ [0, 1] and corresponding to problems on the nonstationary heat conduction of a lengthy plate, a lengthy cylinder, and a sphere with symmetric boundary conditions of the first, second, and third kind, the existence of infinite sequences of differential relations at the boundary points y = 0 and y = 1 has been established. It is shown that these sequences of differential relations can be used to advantage in approximately solving the Sturm–Liouville problem for the function V(y) defined by a power polynomial with coefficients determined from the solution of the corresponding systems of linear algebraic equations. Keywords: heat conduction equation, boundary-value problem, method of separation of variables, differential relations. Introduction. The combined method of separation of variables involves the solution of the boundary-value problem on nonstationary heat conduction for the temperature function T (y, t) by the Fourier method with the representation of this function as the product T (y, t) = V(y)ϕ(t), the subsequent approximate solution of the Sturm–Liouville problem for N the function V ( y ) = ∑ Ai ωi ( y ), and the integration of the weighted residual for determining the initial values of the i = 0 amplitudes Ai. In the present work, prominence is given to the solution of the Sturm–Liouville problem, forming, as known, the main part of the Fourier method of separation of variables. This problem is applied, in particular, to bodies canonical in shape, such as a lengthy plate, a lengthy cylinder, and a sphere, with the three main boundary conditions of the first, second, and third kind. Analysis of the results of works [1–5] and of our first investigation on the subject being considered [6] shows that, in solving the Sturm–Liouville problem for the function V(y) determined in the region Ωy ∈ [0, 1], this function is represented in the form of a power polynomial whose coefficients are determined from the solution of the system of linear algebraic equations constructed with the use of additional boundary conditions set at the points y = 0 and y = 1. As noted in [6], these conditions can be determined by a number of known approximate methods of solving the boundaryvalue problems defined by ordinary differential equations, among them the method of differential transformations [7–11] and the boundary method [12]. However, the process of obtaining the indicated additional boundary conditions (differential relations) is very routine and, one might say, tedious, which is especially true for problems with a cylindrical symmetry or a special one. By way of example we consider the procedure of obtaining one of the additional boundary conditions for the problem on the heat conduction of a cyl
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