 3.3.1: Solve ut =kuxx; u(x,0)=ex; u(0,t)=0 on the halfline 0<x<.
 3.3.2: Solve ut =kuxx; u(x,0)=0; u(0,t)=1 on the halfline 0 < x <.
 3.3.3: Derive the solution formula for the halfline 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...
 3.3.2.1: Solve the Neumann problem for the wave equation on the halfline 0 ...
 3.3.2.2: The longitudinal vibrations of a semiinnite exible rod satisfy the...
 3.3.2.3: A wave f(x + ct) travels along a semiinnite string (0 < x < ) for ...
 3.3.2.4: Repeat Exercise 3 if the end is free.
 3.3.2.5: Solveutt =4uxx for0 < x < ,u(0,t)=0,u(x,0)1,ut(x,0)0 using the reec...
 3.3.2.6: Solveutt=c2uxx in 0 < x < ,0t < ,u(x,0)=0,ut(x,0)=V, ut(0,t)+aux(0,...
 3.3.2.7: a) Show that odd(x)=(sign x)(x). (b) Showthatext(x)=odd(x2l[x/2l]...
 3.3.2.8: Forthewaveequationinaniteinterval(0,l)withDirichletconditions, expl...
 3.3.2.9: (a) Find u(2 3,2) if utt =uxx in 0 < x < 1,u(x,0)= x2(1x),u t(x,0)=...
 3.3.2.10: Solve utt =9uxx in 0 < x < / 2,u(x,0)=cosx,ut(x,0)=0,u x(0,t)=0,u(/...
 3.3.2.11: Solve utt=c2uxx in 0 < x < l,u(x,0)=0,ut(x,0)=x,u(0,t)= u(l,t)=0.
 3.3.3.1: SolvetheinhomogeneousdiffusionequationonthehalflinewithDirichlet b...
 3.3.3.2: Solve the completely inhomogeneous diffusion problem on the halfli...
 3.3.3.3: Solve the inhomogeneous Neumann diffusion problem on the halfline ...
 3.3.4.1: Solve utt =c2uxx +xt, u(x,0)=0, ut(x,0)= 0.
 3.3.4.2: Solve utt =c2uxx +eax, u(x,0)=0, ut(x,0)= 0.
 3.3.4.3: Solve utt =c2uxx +cosx, u(x,0)=sinx, ut(x,0)=1+x.
 3.3.4.4: Show that the solution of the inhomogeneous wave equation utt =c2ux...
 3.3.4.5: Let f(x,t)beanyfunctionandletu(x,t)=(1/2c) f,where isthe triangle o...
 3.3.4.6: Derivetheformulafortheinhomogeneouswaveequationinyetanother way. (a...
 3.3.4.7: Let A be a positivedenite nn matrix. Let S(t)= m=0 (1)mA2mt2m+1 (2...
 3.3.4.8: Show that the source operator for the wave equation solves the prob...
 3.3.4.9: Letu(t)=t 0 s(t s)f(s)ds.UsingonlyExercise8,showthatusolves the inh...
 3.3.4.10: Use any method to show that u =1/(2c)D f solves the inhomogeneous w...
 3.3.4.11: Show by direct substitution that u(x, t)=h(t x/c) for x < ct and u(...
 3.3.4.12: Derive the solution of the fully inhomogeneous wave equation on the...
 3.3.4.13: Solve utt =c2uxx for 0 < x < , u(0,t)=t2, u(x,0)= x, ut(x,0)=0.
 3.3.4.14: Solvethehomogeneouswaveequationonthehalfline(0,)withzero initial d...
 3.3.4.15: Derive the solution of the wave equation in a nite interval with in...
 3.3.5.1: Prove that if is any piecewise continuous function, then 1 4 0 ep2/...
 3.3.5.2: Use Exercise 1 to prove Theorem 2.
Solutions for Chapter 3: REFLECTIONS AND SOURCES
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 3: REFLECTIONS AND SOURCES
Get Full SolutionsThis textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. Since 36 problems in chapter 3: REFLECTIONS AND SOURCES have been answered, more than 5514 students have viewed full stepbystep solutions from this chapter. Chapter 3: REFLECTIONS AND SOURCES includes 36 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Partial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567.

Cholesky factorization
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.

Cyclic shift
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.

Factorization
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.

GramSchmidt 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, offdiagonal 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).

Orthogonal subspaces.
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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , 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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness 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.