UDC 518.517.392

USING HÖLDER’S INEQUALITY IN THE EVALUATION OF A CERTAIN TYPE OF INTEGRALS

S. M. Neyelov, V. K. Yasel’skii

(Submitted by the scientific seminar of the Faculty of Physics and Engineering)

The determination of integrals of the form

\[ I(t)=\int_V X(r,t)\varphi^2(r,t)\,dv \tag{1} \]

is considered in a number of works [1—3] for certain special cases, when the functions \(X(r,t)\) and \(\varphi(r,t)\) are constant at different points of the space \(r\) and do not change with time \(t\).

In the general case the functions \(X(r,t)\) and \(\varphi(r,t)\) are not constant in time. \(X(r,t)\) is described by a first-order differential equation with nonseparable variables [1, 4] and is positive in the integration volume \(V\), while the function \(\varphi(r,t)\) changes its value with time from zero to unity \((0\leq \varphi(r,t)\leq 1)\). In this case, to determine the integral (1), one may use a method consisting in finding the total derivative with respect to time of \(I(t)\)

\[ \frac{d}{dt}\int_V X(r,t)\varphi^2(r,t)\,dv. \tag{2} \]

As is evident from expression (2), the given problem reduces to the evaluation of integrals that are terms of the total derivative with respect to time of \(I(t)\), that is, integrals of the form

\[ \int_V X'(r,t)\varphi^2(r,t)\,dv \tag{3} \]

and

\[ \int_V X(r,t)(\varphi^2(r,t))'\,dv, \tag{4} \]

where \(X'(r,t)\) and \((\varphi^2(r,t))'\) are the time derivatives of the functions

\[ X(r,t)\ \text{and}\ \varphi^2(r,t). \]

It follows from works [1, 2] that, upon substituting into integral (3) specific expressions for \(X'(r,t)\), it can be expressed in terms of \(I(t)\)

\[ \int_V X'(r,t)\varphi^2(r,t)\,dv=f[I(t)]. \tag{5} \]

In turn, in order also to express integral (4) through integral (1), it is necessary to carry out some transformations of the integrand:

\[ \int_V X(r,t)\frac{d}{dt}\varphi^2(r,t)\,dv = \int_V X(r,t)\varphi^2(r,t)\frac{d}{dt}\ln\varphi^2(r,t)\,dv. \tag{6} \]

Expand \(\varphi(r,t)\) in a series in eigenfunctions [4]

\[ \varphi(r,t)=\sum_{n=1}^{\infty} a_n(t)\psi_n(r), \tag{7} \]

where \(\psi_n(r)\) are determined from the Helmholtz equations

\[ \nabla^2\psi_n(r)+B_n^2\psi_n(r)=0,\quad (n=0,1,2,\ldots), \tag{8} \]

and \(a_n\) are the expansion coefficients.

In the work of Geza [4], where this expansion was investigated, it is shown that series (7) is convergent, and for estimative calculations with an error of \(3\text{–}5\%\) the first four or five terms of the expansion are sufficient.

To separate the variables in the right-hand side of expression (7), we use Hölder’s inequality for series

\[ \left|\varphi(r,t)=\sum_{n=1}^{\infty} a_n(t)\psi_n(r)\right| \le \left[\sum_{n=1}^{\infty} a_n^2(t)\right]^{1/2} \left[\sum_{n=1}^{\infty} \psi_n^2(r)\right]^{1/2}. \tag{9} \]

From the obtained relation (9), the logarithmic derivative with respect to time of the function \(\varphi^2(r,t)\) can be represented in the form

\[ \frac{d}{dt}\ln\varphi^2(r,t) = \frac{d}{dt}\ln\left\{\sum_{n=1}^{\infty}a_n(t)\psi_n(r)\right\}^2 \le \]

\[ \le \frac{d}{dt}\ln\left\{ \left[\sum_{n=1}^{\infty}a_n^2(t)\right]^{1/2} \left[\sum_{n=1}^{\infty}\psi_n^2(r)\right]^{1/2} \right\}^2 = \]

\[ = \frac{d}{dt}\ln\sum_{n=1}^{\infty}a_n^2(t). \tag{10} \]

As a result of substituting (10) into (6), one can obtain an expression for estimating integral (4):

\[ \int_V X(r,t)\bigl(\varphi^2(r,t)\bigr)'\,dv \le \int_V X(r,t)\varphi^2(r,t)\,dv\, \frac{d}{dt}\ln\sum_{n=1}^{\infty}a_n^2(t). \tag{11} \]

The inequality sign, upon substitution of (10) into (6), is preserved by virtue of the properties of Stieltjes integrals.

The final formula for estimating expression (2) has the form

\[ \frac{d}{dt}I(t)\approx f[I(t)]+I(t)\frac{d}{dt}\ln\sum_{n=1}^{\infty}a_n^2(t) \tag{12} \]

and can be used as an algorithm for computing the integral \(I(t)\).

REFERENCES

1. P. Wenzel. Effektive stationäre Xenonvergiftung. Kernenergie, № 10/11, 1961.
2. P. Wenzel. Effektive Xenonvergiftung nach Abschalten. Kernenergie, № 12, 1961.
3. W. Hälg u. a. Theoretische und experimente Untersuchungen über Reaktivitätsänderungen, verursacht durch Xenonvergiftung. Zeitschrift für Angewandte Mathematik und Physik, Vol. 14, 1963.
4. G. L. Geza. The Effect of Modal Interaction in the Xenon Instability. Nuclear Science and Engineering, 13, № 4, 1962.