Establishing linearity in laplace's equation with no closed-form solution

[jdmo]

I'm writing a paper, and as a motivation to the forthcoming finite element modeling, I want to state, with some sort of "proof" that Laplace's equation in a heterogeneous volume:

\del (sigma \del V) = 0

exhibits linearity.

By "linearity", I mean that if a set of initial conditions (call them I_1) lead to solution V_1, and another set of initial conditions (call them I_2) lead to solution V_2, then the solution to the equation with initial conditions I_1+I_2 is V_1+V_2.

But I cannot do so in a straightforward manner because there is no closed-form solution which relates V_1 to I_1, for example. Is it enough just to say that Laplace's equation is linear and thus obeys superposition to initial conditions? I would like to make my claim more convincing.

Thanks in advance.

 

[Valerie Park]

Yes, you can make your claim more convincing by demonstrating the linearity of Laplace's equation. This can be done by showing that the equation satisfies the principle of superposition, which states that the total response of a system is the sum of its individual responses to separate stimuli. To do this, consider two sets of initial conditions I1 and I2, and the corresponding solutions V1 and V2. Now take I3 = I1 + I2 and show that V3 = V1 + V2. This demonstrates that the equation is linear and thus obeys the principle of superposition.