Partial Differential Equations with MATLAB

12 An Introduction to Partial Differential Equations with MATLAB©R

however, for the ODE problems in Exercises 1–4. Each of these problems is
called aboundary-value problem, and we will study these problems in
detail in Section 1.7. For now, decide whether each of these problems is well-
posed, in terms of existence and uniqueness of solutions.

1.y′′+y=0,y(0) =y(2) = 0, 0 ≤x≤ 2

2.y′′+y=0,y(0) =y(π)=0, 0 ≤x≤π

3.y′′+y′− 2 y=0,y(0) = 0,y′(1) = 0, 0 ≤x≤ 1

4.y′′+25y=0,y(0) = 1,y(π)=− 1 , 0 ≤x≤π

5. For which values of the constantLis the following boundary-value prob-
   lem well-posed?

y′′+4y=8x, y(0) =A, y(L)=B, 0 ≤x≤L

Explain why each of the following problems is not well-posed.

6.uxx=0,u(0,y)=y^2 ,u(1,y)=3y,u(x,0) =x+2,x≥ 0 , 0 ≤y≤ 1

7.uxx+uyy=0,ux(0,y)=ux(1,y)=uy(x,0) =uy(x,2) = 0, 0 ≤x≤

1 , 0 ≤y≤ 2

1.4 LinearPDEs—Definitions.....................

Almost every PDE which we have met so far is what is called alinearPDE,
whichisdefinedinexactlythesamemanner as a linear ODE. Remember that
the latter was any ODE which could be written in the form

a 0 (x)y(n)+a 1 (x)y(n−1)+···+an− 1 (x)y′+an(x)y=f(x),

wherey=y(x)andy(k)=d

ky
dxk. However, a more fruitful way of looking at it
is to define the so-calledoperator,‡L,by

L[y]=a 0 (x)y(n)+a 1 (x)y(n−1)+···+an− 1 (x)y′+an(x)y.

It is then easy to show that, ifcis any constant andyany function in the
domain ofL,then
L[cy]=cL[y],

‡An operator is a “function of functions,” as it were. That is, it is a function which has the
property that its domain and range each consists of a certain class of functions.