Application of the Method of Separation of Variables in the Computation of a Certain Type of Volume Integrals
Abstract Generated abstract
This paper presents a separation of variables approach for computing triple volume integrals of a form involving a time dependent function and the square of another spatial-temporal function. The method expands both functions in the same eigenfunction system, reducing the governing first order differential equation to equations for the expansion coefficients and rewriting the integral as a double series involving modal coupling terms. Citing prior analysis for Helmholtz eigenfunctions, the authors note that in practical computations only diagonal terms may need to be retained, yielding a simplified single summation formula for the integral.
Full Text
APPLICATION OF THE METHOD OF SEPARATION OF VARIABLES IN THE COMPUTATION OF A CERTAIN TYPE OF VOLUME INTEGRALS
V. N. Padalko, S. M. Neelov, V. K. Yasel’skii, E. M. Belov
(Presented by the scientific seminar of the Faculty of Physics and Technology)
UDC 518.517.392
When computing a triple integral of the form
\[ N(t)=\int\limits_V I(r,t)\varphi^2(r,t)\,dv, \tag{1} \]
where the behavior of the function \(I(r,t)\) in time is described by a first-order differential equation with separable variables
\[ \frac{d}{dt}I(r,t)=AI(r,t)+B\varphi(r,t), \tag{2} \]
where \(A\) and \(B\) are constant coefficients, and the function \(\varphi(r,t)\) can be represented as a series in eigenfunctions
\[ \varphi(r,t)=\sum_{n=1}^{\infty} a_n(t)\psi_n(r), \tag{3} \]
with
\[ a_n(t)= \frac{\int\limits_V \varphi(r,t)\psi_n(r)\,dv} {\int\limits_V \psi_n^2(r)\,dv}, \tag{4} \]
one may use the method of separation of variables. For this purpose it is necessary to expand the function \(I(r,t)\) in a series in the same system of eigenfunctions as the function \(\varphi(r,t)\), that is,
\[ I(r,t)=\sum_{n=1}^{\infty} C_n(t)\psi_n(r). \tag{5} \]
Substituting expressions (5) and (3) into equation (2), we obtain a system of equations for computing the expansion coefficients of the function \(I(r,t)\):
\[ \frac{d}{dt}C_n(t)=AC_n(t)+Ba_n(t). \tag{6} \]
The integral \(N(t)\) in this case will have the following form:
\[ N(t)=\sum_{n=1}^{\infty} C_n(t)\int\limits_V \psi_n(r)\varphi^2(r,t)\,dv. \tag{7} \]
Performing certain transformations of the integrand, we obtain
\[ \begin{aligned} N(t) &= \sum_{n=1}^{\infty} C_n(t) \int_V \psi_n(r)\,\varphi(r,t) \sum_{m=1}^{\infty} a_m(t)\psi_m(r)\,dv \\ &= \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} C_n(t)a_m(t)\int_V \psi_n(r)\psi_m(r)\varphi(r,t)\,dv \\ &= \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} C_n(t)a_m(t)F_{nm}(t), \end{aligned} \tag{8} \]
where
\[ F_{nm}(t)=\int_V \psi_n(r)\psi_m(r)\varphi(r,t)\,dv. \tag{9} \]
In [1], where the function \(F_{nm}(t)\) was investigated for the case in which the functions \(\Psi_n(r)\) are determined from the Helmholtz equations, it was shown that, with accuracy sufficient for practice, in such calculations one may take into account only the terms for which \(n=m\). Then the expression for computing the integral \(N(t)\) is substantially simplified and assumes the form
\[ N(t)\simeq \sum_{n=1}^{\infty} C_n(t)a_n(t)F_{nn}(t). \tag{10} \]
REFERENCES
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