Partial Differential Equations with MATLAB

16 An Introduction to Partial Differential Equations with MATLAB©R

8. Prove that the operatorL[u] is linear if and only if it satisfies property
   (1.10), that is, prove thatL[u] satisfies properties (1.8) and (1.9) if and
   only if it satisfies property (1.10).


9. We know from calculus (and from Exercise 8) that∂x∂(c 1 u 1 +c 2 u 2 )=

c 1 u (^1) x+c 2 u (^2) x, for all constantsc 1 andc 2 and all differentiable functions
u(and that the same is true not only forxbut, of course, for any
independent variable).
a) Use this fact to prove that the following higher-order derivatives
are linear operators, as well.
i)L[u]=uyy
ii)L[u]=uxxy
b) Use mathematical induction to prove that the operatorL[u]=
∂nu
∂xn=uxx︸︷︷···x︸
ntimes
is linear.

10. Prove that, ifu 1 andu 2 are solutions of the homogeneous PDEL[u]=0,
    then so is the functionc 1 u 1 +c 2 u 2 , for any choice of the constantsc 1
    andc 2. Is this true for nonhomogeneous PDEs, as well?


11. Ifu 1 andu 2 are solutions of the nonhomogeneous equationL[u]=f,
    what can we say about the functionu 1 −u 2?


12. Use mathematical induction to prove that, ifLis linear,

L[c 1 u 1 +c 2 u 2 +···+cnun]=c 1 L[u 1 ]+c 2 L[u 2 ]+···+cnL[un]

for all constantsc 1 ,c 2 ,...,cnand all functionsu 1 ,u 2 ,...,unin the do-

main ofL.

1.5 LinearPDEs—ThePrincipleofSuperposition

Here, again, we take our cue from the theory of linear ODEs.

Definition 1.3Given functionsu 1 ,u 2 ,...,un, any function of the form

c 1 u 1 +c 2 u 2 +···+cnun,

wherec 1 ,c 2 ,...,cnare constants, is called alinear combinationofu 1 ,u 2 ,
...,un.

The following theorem follows immediately from the result of Exercise 12
of the previous section.