# Preface

Separation of variables is the basic mathod for solving linear partial differential equations (PDE for short). Not every linear PDE admits separation of variables and some classes of such equations are presented. Partial differential equations are usually suplemented by the initial and/or boundary conditions that reduces separation of variable further. This method could be extended to so called integrable evolution PDEs (linear or nonlinear) that can be written in the form of Lax pairs. For example, a classical diffusion equation $$u_t = \alpha\,u_{xx}$$ can be decomposed into associated Lax pair consists of the following two linear equations

$\phi_x + {\bf j}k \,\phi = u, \qquad \phi_t + k^2 \phi = \alpha\,u_x - {\bf j}k \,u\,\alpha ,$
where k is a complex parameter.

Introduction to Linear Algebra with Mathematica

# Separation of variables

Theorem: (Principle of Superposition) If ui is a solution of the second-order, linear partial diﬀerential equation
$A\, u_{tt} + B\, u_{xt} + C\, u_{xx} + D\, u_t + E\,u_x + F\,u = G_i$
for i = 1, 2, …, n on a domain Ω ⊂ ℝ², then for any constants c1, c2, …, cn, the function $$u(x,t) = c_1 u_1 + c_2 u_2 + \cdots + c_n u_n$$ is a solution to
$A\, u_{tt} + B\, u_{xt} + C\, u_{xx} + D\, u_t + E\,u_x + F\,u = \sum_{i=1}^n c_i G_i$
on Ω. Here $$u_{xx} = \partial u/\partial x^2 , u_{xt} = \partial u/\partial x\partial t$$ and so on are shortcuts for partial derivatives. In particular, if all Gi = 0 and every ui is a solution of the corresponding homogeneous equation
$A\, u_{tt} + B\, u_{xt} + C\, u_{xx} + D\, u_t + E\,u_x + F\,u = 0$
on a domain Ω ⊂ ℝ², then their linear combination $$u(x,t) = c_1 u_1 + c_2 u_2 + \cdots + c_n u_n$$ is also a solution of the given homogeneous partial differential equation on Ω.

1. Costa, G. and Levine, L.E., Families of separable partial differential equations, International Journal of Mathematical Education in Science and Technology, 1993, Vol. 24, No. 5, pp. 621--635; doi: 10.1080/0020739930240503
2. Fokas, A.S. and Spence, E.A., Synthesis, as opposed to separation of variables, SIAM Review, 2012, Vol. 54, No. 2, pp. 291--324.