Separation of variables

Separation of variables seeks a product of one-variable factors, reducing a PDE to ordinary differential equations.

A solution to a partial differential equation that factors into functions of separate variables, for example $u(x,t)=X(x)T(t)$. For the heat equation $u_t=D{u}_{xx}$, substitution yields $T^{\prime }/(DT)=X^{\prime \prime }/X$, so both sides are a common constant wherever division is valid. The resulting ordinary differential equations must be solved together with the original boundary conditions. Individual separated solutions generate more general solutions by superposition when the equation and boundary data are linear.

A solution of a partial differential equation expressed as a product of functions of separate groups of variables. Substitution reduces many linear partial differential equations to ordinary differential equations; superposition then combines the resulting normal modes.

A separated Fourier mode with initial value $e^{iky}$ at $x=0$ obeys $X^{\prime }=-k^{3}X/2$ and $X(0)=1$. Its amplitude decays for $x\to +\infty$ precisely when $k>0$. The zero mode stays constant and negative modes grow. Thus a decay requirement on a half-space imposes a sign condition on the real wave number; a formal separated expression alone does not establish compatibility of both boundary conditions.

A separation constant arises when an equation is a sum of functions of independent variables. If $F(r)+G(\theta )=0$ on a product region, each function is constant, with opposite signs. Boundary or regularity conditions can then select discrete values, as for scalar spheroidal harmonics.

The harmonic function on $a<r<b$ with boundary values $u(a,\theta )=0$ and $u(b,\theta )=\cos (n\theta )$ is