Combined Method of Separation of Variables. 1. Critical Analysis of the Known Approach
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 4, July, 2020
COMBINED METHOD OF SEPARATION OF VARIABLES. 1. CRITICAL ANALYSIS OF THE KNOWN APPROACH V. A. Kot
UDC 517.518.8:519.633:536.2
For the purpose of development of a new high-accuracy method of separation of variables, allowing one to effectively use the method of differential transformations in combination with the method of weighted residuals, by the example of the problems on the nonstationary heat conduction of an unbounded plate, an unbounded cylinder, and a sphere with the first-kind boundary conditions, a critical analysis of the known solution technique based on the combined use of the Fourier method of separation of variables, the method of differential transformations, and the method of weighted residuals has been performed. It is shown that the computational scheme realized with the indicated technique has a very low approximation accuracy, especially in the case where it is used for bodies of cylindrical and spherical symmetry. The main reasons for the low accuracy of the solutions obtained by this scheme were revealed. Keywords: heat conduction equation, boundary-value problem, method of separation of variables, weighted residual method, eigenvalues, eigenfunctions, method of differential transformations. Introduction. In solving boundary-value problems for practical applications, along with classical methods, among which are the Fourier method of separation of variables and the methods of integral transformations with respect to the time and space coordinates [1–6], of great importance are approximate methods allowing one to find, within an admissible error, analytical solutions simple in form [7–11]. For approximate solution of the differential equation on the heat conduction of a body, approaches involving the combined application of the Fourier method and the Galerkin method, the Laplace integral transform method and the projection methods of weighted residuals, and variational methods are used [12–17]. The orthogonal projection methods of weighted residuals are based on the two main principles in accordance with which a system of characteristic functions, satisfying the conditions of an initial problem, is introduced and approximation parameters are determined by equating the integral of the residual of the initial equations, taken over a definite system of weight functions, to zero [8, 17, 18]. Integral transformations with respect to the time coordinate are performed using one of the classical methods: the Fourier method of separation of variables or the Laplace integral transform method, and such transformations with respect to the space coordinate are performed by the variational method, the weighted residuals method, the collocation method, and other methods [7–9]. For determining the coefficients of a differential equation, a residual is set up, and this residual should be orthogonal to the coordinate functions. As a result, a system of algebraic equations with a matrix, which is ill-posed as a rule, is obtained. As f
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