Partial Differential Equations: An Introduction 2nd Edition solutions

Partial Differential Equations: An Introduction 2
Solve uxx + uyy - 0 in the quarter disk {x2 + y2 < a2, x > 0, y > 0} with the following BCs:

Partial Differential Equations: An Introduction 2
Which of the following operators are linear? (a) lu =ux +xuy (b) lu =ux +uuy (c) lu =ux +u2 y (d) lu =ux +uy +1 (e) lu =1+x2 (cos y)ux +uyxy[arctan(x/y)]u

Partial Differential Equations: An Introduction 2
Find the general solution of 3utt +10uxt +3uxx =sin(x +t).

Partial Differential Equations: An Introduction 2
Solve the Neumann problem in the half-space{z > 0}.

Partial Differential Equations: An Introduction 2
Solve the equation ux +uy =1.

Partial Differential Equations: An Introduction 2
Solve the equation 3uy +uxy =0. (Hint:Letv =uy.)

Partial Differential Equations: An Introduction 2
Solve the Schrodinger equation ut =ikuxx for real k in the interval 0 < x < l with the boundary conditions ux(0,t)=0,u(l,t)=0.

Partial Differential Equations: An Introduction 2
Solve aux +buy +cu =0.

Partial Differential Equations: An Introduction 2
Solve the heat equation with convection: ut kuxx +Vux =0 for < x < with u(x,0)= (x), where V is a constant. (Hint: Go to a moving frame of reference by substituting y=xVt.)

Partial Differential Equations: An Introduction 2
Find the solutions that depend only on r of the equation uxx +uyy = k2u, where k is a positive constant. (Hint: Look up Bessels differential equation in [MF] or in Section 10.5.)

Partial Differential Equations: An Introduction 2
(a) Solve the equation yux +xuy =0 with u(0, y)=ey2. (b) Inwhichregionofthexyplaneisthesolutionuniquelydetermined?

Partial Differential Equations: An Introduction 2
Let a sequence of L2 functions fn(x) converge to a function f(x) in the mean-square sense. Show that it also converges weakly in the sense of distributions.

Partial Differential Equations: An Introduction 2
VerifythelinearityandnonlinearityoftheeightexamplesofPDEsgiven in the text, by checking whether or not equations (3) are valid.

Partial Differential Equations: An Introduction 2
Solve utt =c2uxx +eax, u(x,0)=0, ut(x,0)= 0.

Partial Differential Equations: An Introduction 2
Show that the functions (c1 +c2 sin2x +c3 cos2x) form a vector space. Find a basis of it. What is its dimension?

Partial Differential Equations: An Introduction 2
Solve utt =c2uxx, u(x,0)=ex,ut(x,0)=sinx.

Partial Differential Equations: An Introduction 2
Solve ux +uy +u =ex+2y with u(x,0)=0.

Partial Differential Equations: An Introduction 2
Consider waves in a resistant medium that satisfy the problem utt =c2uxx rut for 0 < x < l u =0 at both ends u(x,0)= (x) ut(x,0)= (x), where r is a constant, 0 < r < 2c/l. Write down the series expansion of the solution.

Partial Differential Equations: An Introduction 2
Separate the variables for the equation tut =uxx +2u with the boundary conditions u(0, t)=u(, t)=0. Show that there are an innite number of solutions that satisfy the initial condition u(x, 0)=0. So uniqueness is false for this equation!

Partial Differential Equations: An Introduction 2
Constructadirect(butunmotivated)proofofExample1asfollows,without using any knowledge of eigenvalues. Let w(x) be aC2 function such that w(0)=w(1)=0. (a) Expand 1 0 [w(x)w(x)cot(x)]2 dx and integrate the cross-term by parts. (b) Show that w2(x)cotx 0 asx 0 or 1. (c) Deduce that  1 0 [w(x)]2 dx2 1