Partial Differential Equations with MATLAB

Introduction 13

and that, ify 1 andy 2 are any functions in the domain ofL,then

L[y 1 +y 2 ]=L[y 1 ]+L[y 2 ].

We use the idea of an operator to define linear PDEs. First, given a PDE
inu=u(x 1 ,x 2 ,...,xn), we write the equation in the form

L[u]=f(x 1 ,x 2 ,...,xn),

wherefis a given function.

Example 1The heat equation,ut=α^2 uxx, can be written as

L[u]=0,

whereLis the operator defined by

L[u]=ut−α^2 uxx.

Example 2The PDEux+yuy−xy^2 +siny= 0 can be written as

L[u]=xy^2 −siny,

whereLis defined by

L[u]=ux+yuy.

Then, we define a linear PDE as follows:

Definition 1.1The PDE

L[u]=f

is alinear PDEif

1) L[cu]=cL[u], (1.8)

for all constantscand all functionsuin the domain ofL,and

2) L[u 1 +u 2 ]=L[u 1 ]+L[u 2 ], (1.9)

for all functionsu 1 andu 2 in the domain ofL.

Also, if an operator satisfies both (1.8) and (1.9), we say that it is alinear
operator. If an operator or PDE fails to be linear, we call it anonlinear
operatoror PDE (and we do not call the operatorL—for “linear”—if we
know that it is nonlinear).