Partial Differential Equations with MATLAB

Introduction 17

Theorem 1.1Ifu 1 ,u 2 ,...,unare solutions of the linear, homogeneous PDE
L[u]=0, then so is any linear combination ofu 1 ,u 2 ,...,un. (This is the
principle of superposition of solutionsfor linear PDEs.)

PROOF The fact thatu 1 ,u 2 ,...,unare solutions gives us

L[u 1 ]=L[u 2 ]=···=L[un]=0.

Then, for any linear combinationc 1 u 1 +c 2 u 2 +···+cnun,

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

=c 1 ·0+c 2 ·0+···+cn·0=0.

Now, in the theory of ODEs, for annth-order linear, homogeneous equation,
we need only findnlinearly independent solutions. Then, the general solution
consists of all possible (finite) linear combinations of these solutions. However,
life is much more complicated in the realm of PDEs. Often, we will need to
findinfinitelymany solutions,u 1 ,u 2 ,..., of a linear, homogeneous PDE before
we are in a position to construct a general solution

u=c 1 u 1 +c 2 u 2 +···=

∑∞

n=1

cnun. (1.11)

And since this infinite linear combination actually is an infinite series, ques-
tions of convergence come to the forefront. Indeed, for any given choice of
the coefficients, expression (1.10) may diverge for all values ofx,oritmay
converge for some values ofxbut not for others.
Suffice it to say that, throughout this book, we will assume that, whenever
(1.11) converges, it satisfies the linearity condition

L

[∞

∑

n=1

cnun

]

=

∑∞

n=1

cnL[un] (1.12)

and, therefore, that if eachunis a solution ofL[u] = 0, then so is the linear
combination, (1.11), of these solutions. When (1.12) holds, we say that we
maydifferentiate the series term-by-term.

Exercises 1.5

In Exercises 1–4, verify directly that the principle of superposition holds for
any two solutions,u 1 andu 2 ,ofthegivenPDE.

1.yux−x^2 uy+2u=0