Constraint counting

To answer this in the important special case of a linear partial differential equation, Einstein asked: how many of the partial derivatives of a solution can be linearly independent? It is convenient to record his answer using an ordinary generating function

${\displaystyle s(\xi )=\sum _{k=0}^{\infty }s_{k}\xi ^{k}}$

where ${\displaystyle s_{k}}$ is a natural number counting the number of linearly independent partial derivatives (of order k) of an arbitrary function in the solution space of the equation in question.

Whenever a function satisfies some partial differential equation, we can use the corresponding rewrite rule to eliminate some of them, because further mixed partials have necessarily become linearly dependent. Specifically, the power series counting the variety of arbitrary functions of three variables (no constraints) is

${\displaystyle f(\xi )={\frac {1}{(1-\xi )^{3}}}=1+3\xi +6\xi ^{2}+10\xi ^{3}+\dots }$

but the power series counting those in the solution space of some second order p.d.e. is

${\displaystyle g(\xi )={\frac {1-\xi ^{2}}{(1-\xi )^{3}}}=1+2\xi +5\xi ^{2}+7\xi ^{3}+\dots }$

which records that we can eliminate one second order partial ${\displaystyle u_{tt}}$, three third order partials ${\displaystyle u_{ttt},\,u_{ttx},\,u_{tty}}$, and so forth.

More generally, the o.g.f. for an arbitrary function of n variables is

${\displaystyle s[n](\xi )=1/(1-\xi )^{n}=1+n\,\xi +\left({\begin{matrix}n\\2\end{matrix}}\right)\,\xi ^{2}+\left({\begin{matrix}n+1\\3\end{matrix}}\right)\,\xi ^{3}+\dots }$

where the coefficients of the infinite power series of the generating function are constructed using an appropriate infinite sequence of binomial coefficients, and the power series for a function required to satisfy a linear m-th order equation is

${\displaystyle g(\xi )={\frac {1-\xi ^{m}}{(1-\xi )^{n}}}}$

Next,

${\displaystyle {\frac {1-\xi ^{2}}{(1-\xi )^{3}}}={\frac {1+\xi }{(1-\xi )^{2}}}}$

which can be interpreted to predict that a solution to a second order linear p.d.e. in three variables is expressible by two freely chosen functions of two variables, one of which is used immediately, and the second, only after taking a first derivative, in order to express the solution.

To verify this prediction, recall the solution of the initial value problem

${\displaystyle u_{tt}=u_{xx}+u_{yy},\;u(0,x,y)=p(x,y),\;u_{t}(0,x,y)=q(x,y)}$

Applying the Laplace transform ${\displaystyle u(t,x,y)\mapsto [Lu](\omega ,x,y)}$ gives

${\displaystyle -\omega ^{2}\,[Lu]+\omega \,p(x,y)+q(x,y)+[Lu]_{x}+[Lu]_{y}}$

Applying the Fourier transform ${\displaystyle [Lu](\omega ,x,y)\mapsto [FLU](\omega ,m,n)}$ to the two spatial variables gives

${\displaystyle -\omega ^{2}\,[FLu]+\omega \,[Fp]+[Fq]-(m^{2}+n^{2})\,[FLu]}$

or

${\displaystyle [FLu](\omega ,m,n)={\frac {\omega \,[Fp](m,n)+[Fq](m,n)}{\omega ^{2}+m^{2}+n^{2}}}}$

Applying the inverse Laplace transform gives

${\displaystyle [Fu](t,m,n)=[Fp](m,n)\,\cos({\sqrt {m^{2}+n^{2}}}\,t)+{\frac {[Fq](m,n)\,\sin({\sqrt {m^{2}+n^{2}}}\,t)}{\sqrt {m^{2}+n^{2}}}}}$

Applying the inverse Fourier transform gives

${\displaystyle u(t,x,y)=Q(t,x,y)+P_{t}(t,x,y)}$

where

${\displaystyle P(t,x,y)={\frac {1}{2\pi }}\,\int _{(x-x')^{2}+(y-y')^{2}<t^{2}}{\frac {p(x',y')\,dx'dy'}{\left[t^{2}-(x-x')^{2}-(y-y')^{2}\right]^{1/2}}}}$

${\displaystyle Q(t,x,y)={\frac {1}{2\pi }}\,\int _{(x-x')^{2}+(y-y')^{2}<t^{2}}{\frac {q(x',y')\,dx'dy'}{\left[t^{2}-(x-x')^{2}-(y-y')^{2}\right]^{1/2}}}}$

Here, p,q are arbitrary (sufficiently smooth) functions of two variables, so (due their modest time dependence) the integrals P,Q also count as "freely chosen" functions of two variables; as promised, one of them is differentiated once before adding to the other to express the general solution of the initial value problem for the two dimensional wave equation.

In the case of a nonlinear equation, it will only rarely be possible to obtain the general solution in closed form. However, if the equation is quasilinear (linear in the highest order derivatives), then we can still obtain approximate information similar to the above: specifying a member of the solution space will be "modulo nonlinear quibbles" equivalent to specifying a certain number of functions in a smaller number of variables. The number of these functions is the Einstein strength of the p.d.e. In the simple example above, the strength is two, although in this case we were able to obtain more precise information.