- 3.3.1: Solve ut =kuxx; u(x,0)=ex; u(0,t)=0 on the half-line 0<x<.
- 3.3.2: Solve ut =kuxx; u(x,0)=0; u(0,t)=1 on the half-line 0 < x <.
- 3.3.3: Derive the solution formula for the half-line Neumann problemwt kwx...
- 3.3.4: Consider the following problem with a Robin boundary condition: DE:...
- 3.3.5: (a) Use the method of Exercise 4 to solve the Robin problem: DE: ut...
- 188.8.131.52: Solve the Neumann problem for the wave equation on the half-line 0 ...
- 184.108.40.206: The longitudinal vibrations of a semi-innite exible rod satisfy the...
- 220.127.116.11: A wave f(x + ct) travels along a semi-innite string (0 < x < ) for ...
- 18.104.22.168: Repeat Exercise 3 if the end is free.
- 22.214.171.124: Solveutt =4uxx for0 < x < ,u(0,t)=0,u(x,0)1,ut(x,0)0 using the reec...
- 126.96.36.199: Solveutt=c2uxx in 0 < x < ,0t < ,u(x,0)=0,ut(x,0)=V, ut(0,t)+aux(0,...
- 188.8.131.52: a) Show that odd(x)=(sign x)(|x|). (b) Showthatext(x)=odd(x2l[x/2l]...
- 184.108.40.206: Forthewaveequationinaniteinterval(0,l)withDirichletconditions, expl...
- 220.127.116.11: (a) Find u(2 3,2) if utt =uxx in 0 < x < 1,u(x,0)= x2(1x),u t(x,0)=...
- 18.104.22.168: Solve utt =9uxx in 0 < x < / 2,u(x,0)=cosx,ut(x,0)=0,u x(0,t)=0,u(/...
- 22.214.171.124: Solve utt=c2uxx in 0 < x < l,u(x,0)=0,ut(x,0)=x,u(0,t)= u(l,t)=0.
- 126.96.36.199: Solvetheinhomogeneousdiffusionequationonthehalf-linewithDirichlet b...
- 188.8.131.52: Solve the completely inhomogeneous diffusion problem on the half-li...
- 184.108.40.206: Solve the inhomogeneous Neumann diffusion problem on the half-line ...
- 220.127.116.11: Solve utt =c2uxx +xt, u(x,0)=0, ut(x,0)= 0.
- 18.104.22.168: Solve utt =c2uxx +eax, u(x,0)=0, ut(x,0)= 0.
- 22.214.171.124: Solve utt =c2uxx +cosx, u(x,0)=sinx, ut(x,0)=1+x.
- 126.96.36.199: Show that the solution of the inhomogeneous wave equation utt =c2ux...
- 188.8.131.52: Let f(x,t)beanyfunctionandletu(x,t)=(1/2c) f,where isthe triangle o...
- 184.108.40.206: Derivetheformulafortheinhomogeneouswaveequationinyetanother way. (a...
- 220.127.116.11: Let A be a positive-denite nn matrix. Let S(t)= m=0 (1)mA2mt2m+1 (2...
- 18.104.22.168: Show that the source operator for the wave equation solves the prob...
- 22.214.171.124: Letu(t)=t 0 s(t s)f(s)ds.UsingonlyExercise8,showthatusolves the inh...
- 126.96.36.199: Use any method to show that u =1/(2c)D f solves the inhomogeneous w...
- 188.8.131.52: Show by direct substitution that u(x, t)=h(t x/c) for x < ct and u(...
- 184.108.40.206: Derive the solution of the fully inhomogeneous wave equation on the...
- 220.127.116.11: Solve utt =c2uxx for 0 < x < , u(0,t)=t2, u(x,0)= x, ut(x,0)=0.
- 18.104.22.168: Solvethehomogeneouswaveequationonthehalf-line(0,)withzero initial d...
- 22.214.171.124: Derive the solution of the wave equation in a nite interval with in...
- 126.96.36.199: Prove that if is any piecewise continuous function, then 1 4 0 ep2/...
- 188.8.131.52: Use Exercise 1 to prove Theorem 2.
Solutions for Chapter 3: REFLECTIONS AND SOURCES
Full solutions for Partial Differential Equations: An Introduction | 2nd Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.