Partial differential equations are equations which involve the partial derivatives of some function. The most famous partial differential equations involve the Laplacian, named after Laplace2 .
Definition 11.6.1 Let u be a function of n variables. Then Δu ≡ ∑ k=1nuxkxk. This is also written as ∇2u. The symbol Δ or ∇2 is called the Laplacian. When Δu = 0 the function u is called harmonic.Laplace’s equation is Δu = 0. The heat equation is ut − Δu = 0 and the wave equation is utt − Δu = 0.
Example 11.6.2 Find the Laplacian of u
uxx = 2 while uyy = −2. Therefore, Δu = uxx + uyy = 2 − 2 = 0. Thus this function is harmonic, Δu = 0.
Example 11.6.3 Find ut − Δu where u
In this case, ut = −e−t cosx while uyy = 0 and uxx = −e−t cosx therefore, ut − Δu = 0 and so u solves the heat equation ut − Δu = 0.
Example 11.6.4 Let u
In this case, utt = −sintcosx while Δu = −sintcosx. Therefore, u is a solution of the wave equation utt − Δu = 0.