Answer to Question #131397 in Differential Equations for Meenu Rani

Question

Does principle of superposition also hold for non-linear PDEs? If not, then give an example of non-linear PDE where principle of superposition does not hold.

Answer 1

Unlike linear PDEs, with non-linear PDEs the principle of superposition (which states that linear combination of solutions of equation is again the solution of this equation) does not hold in general case. Example:

$u_{x} +u^2 u_y=0$ (1)

One solutions of this PDE is $u_1 = \frac{-1+\sqrt{1+4xy}}{2x}$ . However, the function $u =cu_1$ does not solve the same PDE unless $c = 0, \pm1$.