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UDC 517.947.42
GENERAL SOLUTION OF THE POLYHARMONIC EQUATION WITH AN ARBITRARY NUMBER OF INDEPENDENT VARIABLES
S. N. Perevezentsev
Let the operator
\[ \Delta = \sum_{l=1}^{n} \frac{\partial^2}{\partial x_l^2}, \tag{1} \]
where \(x_l\) is an independent variable; \(n\) is a nonnegative integer.
The equation
\[ \Delta^m W = 0 \tag{2} \]
is a homogeneous polyharmonic equation (P-equation) of the \(m\)-th harmonic order with respect to the function \(W\).
For various particular forms of the P-equation, general solutions have been constructed. The general solution of the Laplace equation \((m = 1, n = 3)\) [1] has the form of a linear combination of arbitrary functions, each of which depends on one variable.
I. N. Vekua [2] obtained the general solution of the P-equation for any \(m\) \((n = 2)\). This solution is a series of arbitrary functions of conjugate complex variables.
Important results belong to Almansi [3], who obtained the general solution of equation (2) in the following form:
\[ W = \sum_{k=0}^{m-1} |x|^{2k} W_k, \tag{3} \]
where
\[ x^2 = \sum_{l=1}^{n} x_l^2, \]
and \(W_k(x_1, x_2, \ldots, x_n)\) are arbitrary functions harmonic in the domain \(D\).
In the present paper the following expression is given for the general solution of the polyharmonic equation (2):
\[ W = \sum_{\alpha=1}^{m} W_\alpha, \tag{4} \]
where
\[ W_\alpha = \sum_{l=1}^{n} \sum_{k=0}^{\infty} \frac{(-1)^k (k+\alpha-1)!}{k!(2k+2\alpha-2)!} \left( \Delta^k P_{l\alpha} + \frac{x_l \Delta^k Q_{l\alpha}}{2k+2\alpha-1} \right) x_l^{2k+2\alpha-2}, \tag{5} \]
\(P_{l\alpha}\) and \(Q_{l\alpha}\) are arbitrary (not dependent on one another) analytic functions of \(n-1\) variables (all except \(x_l\)). To prove that expression (4) is a solution of equation (2), it is evidently sufficient to substitute into it only one general term \(W_\alpha\) (5).
The formula for the harmonic derivative of \(W_\alpha\) of order \(r\) (in view of the symmetry of all \(n\) terms in (5), we take only one) has the form
\[ \Delta^r(W_\alpha) = (\alpha - 1)(\alpha - 2)\ldots(\alpha-r) \times \]
\[ \times \sum_{k=0}^{\infty} \frac{(-1)^k (k+\alpha-1-r)!}{k!(2k+2\alpha-2-2r)!} \left( \Delta^k P_{l\alpha}+ \frac{x_l \Delta^k Q_{l\alpha}}{2k+2\alpha-1-r} \right) x_l^{2k+2\alpha-2-2r}, \tag{6} \]
where \(r\) is a positive integer. The validity of expression (6) for any \(r\) is easily proved by passing to the \(r+1\)-st harmonic derivative. For \(r=\alpha=m\), from (6) we obtain \(\Delta^m(W_m)=0\), which corresponds to equation (2), as was required to prove. In the case when \(m=1\), we obtain Laplace’s equation, which, according to (4) and (5), is satisfied by the solution
\[ W_1=\sum_{l=1}^{n}\sum_{k=0}^{\infty} \frac{(-1)^k x_l^{2k}}{(2k)!} \left( \Delta^k P_{l1}+ \frac{x_l \Delta^k Q_{l1}}{2k+1} \right). \tag{7} \]
If the functions \(P_{l1}\) and \(Q_{l1}\) are chosen in the form of polynomials of integral positive powers of their independent variables, then the sums in the solution (7) become finite. Consequently, expression (7) for \(W_1\) exhausts all solutions of Laplace’s equation in polynomials. Therefore the preceding summation condition should be regarded as conditional.
A. V. Bitsadze [4], studying the solution of Laplace’s equation expressed in the form (7), gave a formula by which one can compute the total number of linearly independent polynomials of a given degree.
For the biharmonic equation \((m=2)\), from (4) and (5) we obtain
\[ W=W_1+W_2, \tag{8} \]
where \(W_1\) is a solution of Laplace’s equation, and
\[ W_2=\sum_{l=1}^{n}\sum_{k=0}^{\infty} \frac{(-1)^k (k+1)x_l^{2k+2}}{(2k+2)!} \left( \Delta^k P_{l2}+ \frac{x_l \Delta^k Q_{l2}}{2k+3} \right). \tag{9} \]
Here \(P_{l2}\) and \(Q_{l2}\) are arbitrary analytic functions of \(n-1\) independent variables, except \(x_l\).
Let us return to the case \(m=1\). Let \(n=4\):
\[ x_1=x,\quad x_2=y,\quad x_3=z,\quad x_4=\tau; \]
\[ P_x=Q_x=P_y=Q_y=P_z=Q_z=0,\quad P_\tau\ne0;\quad Q_\tau\ne0. \tag{10} \]
Then the solution of Laplace’s equation and its derivative with respect to \(\tau\) will have the form
\[ W=\sum_{k=0}^{\infty} \frac{(-1)^k \tau^{2k}}{(2k)!} \left( \Delta^k P_\tau+ \frac{\tau \Delta^k Q_\tau}{2k+1} \right), \]
\[ \frac{\partial W}{\partial \tau} = \sum_{k=0}^{\infty} \frac{(-1)^k \tau^{2k}}{(2k)!} \left( \Delta^k Q_\tau- \frac{\tau}{2k+1}\Delta^{k+1}P_\tau \right). \tag{11} \]
Expressions (11) are the solution of the well-known problem for the wave equation [5]
\[ \frac{\partial^2 W}{\partial t^2} = a^2\left( \frac{\partial^2 W}{\partial x^2} + \frac{\partial^2 W}{\partial y^2} + \frac{\partial^2 W}{\partial z^2} \right), \tag{12} \]
\[ W\big|_{t=0}=f_1(x,y,z); \qquad \left.\frac{\partial W}{\partial t}\right|_{t=0} = f_2(x,y,z), \]
where it is sufficient merely to put
\[ \tau=iat;\quad P_\tau=f_1(x,y,z);\quad Q_\tau=if_2(x,y,z). \tag{13} \]
An equation of a more general form
\[ \frac{\partial^{2m}}{\partial t^{2m}}W=\delta^m W, \tag{14} \]
where \(\delta\) is an arbitrary linear differential operator in any number of independent variables, provided that the time function \(t\) does not enter into it; \(m\) is a positive integer, \(m \geqslant 1\); with the initial conditions
\[ W\big|_{t=0}=f_1;\quad \frac{\partial W}{\partial t}\bigg|_{t=0}=f_2\ldots \frac{\partial^{2m-1}}{\partial t^{2m-1}}\bigg|_{t=0}=f_{2m} \tag{15} \]
has the solution
\[ W=\sum_{\alpha=1}^{m} W_{\alpha}, \tag{16} \]
where
\[ W_{\alpha}=\sum_{k=0}^{\infty} \frac{(k+\alpha-1)!\,t^{2k+2\alpha-2}}{k!(2k+2\alpha-2)!} \left( \delta^{k}f_{2\alpha-1} + \frac{t\,\delta^{k}f_{2\alpha}}{2k+2\alpha-1} \right). \tag{17} \]
As everywhere above, here the upper limit of the sum \(\infty\) is set conditionally, for if one requires that for some term with number \(k=\beta\) the function \(f_i\) be a solution of the equation
\[ \delta^{\beta} f=0, \tag{18} \]
then the series terminates at this term, and the solution (17) assumes a finite form.
References
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Whittaker E. T., Watson G. A Course of Modern Analysis, vol. II. Fizmatgiz, 1963.
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Vekua I. N. New Methods for Solving Elliptic Equations. Gostekhizdat, 1948.
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Almansi E. Sull integrazione dell’ Equazione differenziale \(\Delta^{2n}=0\). Annali di matematica pura ed appli. S. III, t. II, 1898.
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Bitsadze A. V. On the theory of harmonic functions. Transactions of Tbilisi State University, Series of Mechanics and Mathematics, 34, 1962.
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Koshlyakov N. S., Gliner E. B., Smirnov M. N. Differential Equations of Mathematical Physics. Fizmatgiz, Moscow, 1962.
Received by the editors January 4, 1965.