Method of seperation of variables
A function has input scalars/vectors, and output scalars/vectors. Similarly operators input functions and output functions.
An operator is linear if and only if it is satisifed that
L(c_1u_1 + c_2u_2) = c_1L(u_1) + c_2L(u_2)
Where u_1,u_2 are arbitrary functions and c_1,c_2 are arbitrary constants.
The heat operator is
L(u) = \frac{\partial u}{\partial t} - k\frac{\partial^2 u}{\partial x^2}
A linear equation for the unknown u is of the format
L(u) = f(x,t)
Where L is a linear operator and f(x,t) does not depend on u.
Moreover if in the linear equation L(u) = f(x,t) we have f(x,t) = 0 we say the equation is linear and homogeneous. If f(x,t) \not = 0, we say the equation is nonhomogeneous.
Principle of Superposition
The fundemental property of linear operators allows solutions to be added together as such:
The concept of linearity and homogeneity also apply to boundary conditions.