 10.1: SolvethewaveequationinthesquareS ={ 0 < x < ,0 < y <},with homogene...
 10.2: Solve the wave equation in the rectangle R ={ 0 < x < a,0 < y < b},...
 10.3: In the cube (0,a)3, a substance is diffusing whose molecules multip...
 10.4: Consider the eigenvalue problem v = v in the unit square D = {0 < x...
 10.5: Find the dimension of each of the following vector spaces. (a) The ...
 10.6: Illustrate the GramSchmidt orthogonality method by sketching two li...
 10.7: Prove Theorem 2 in the Neumann and Robin cases.
 10.1: Show that with the initial conditions (26), all the cosct terms in ...
 10.2: Determinethevibrationsofacirculardrumhead(heldxedontheboundary) wit...
 10.3: Determinethevibrationsofacirculardrumhead(heldxedontheboundary) wit...
 10.4: Findallthesolutionsofthewaveequationoftheformu =eitf(r)that are nit...
 10.5: Solve the diffusion equation in the disk of radius a, with u = B on...
 10.6: Do the same for the annulus{a2< x2 +y2< b2}with u=B on the whole bo...
 10.7: Let D be the semidisk {x2 + y2 < b2, y > 0}. Consider the diffusion...
 10.1: Calculate the normalizing constants for the spherical harmonics usi...
 10.2: Verify the rst six entries in the table of spherical harmonics.
 10.3: Show that the spherical harmonics satisfy Ym l
 10.4: Solve the wave equation in the ball{r < a}of radius a, with the con...
 10.5: Solve the diffusion equation in the ball of radius a, with u = B on...
 10.6: (A Recipe for Eggs Fourier, by J. Goldstein) Consider an egg to be ...
 10.7: (a) Consider the diffusion equation in the ball of radius a, with u...
 10.8: (a) Let B be the ball {x2 + y2 +z2 < a2}. Find all the radial eigen...
 10.9: Solve the diffusion equation in the ball{x2 + y2 +z2 < a2}with u=0 ...
 10.10: Find the harmonic function in the exterior {r > a}of a sphere that ...
 10.11: Find the harmonic function in the halfball{x2 + y2 +z2 < a2,z > 0}...
 10.12: Asubstancediffusesininnitespacewithinitialconcentration(r)=1 forr <...
 10.13: Repeat Exercise 12 by computer using the methods of Section 8.2.
 10.1: For the Dirichlet problem in a square whose eigenfunctions are give...
 10.2: Sketch the nodal set of the eigenfunctionv(x, y)=sin3x sin y+sin x ...
 10.3: Small changes can alter the nature of the nodal set drastically. Us...
 10.4: Read about the nodal patterns of ancient Chinese bells in [Sh].
 10.1: Show that J0(z)=1z 22 + 1 (2!)2z 24 1 (3!)2z 26 + andJ1(z)=J 0(z)=z...
 10.2: Write simple formulas for J3/2 and J3/2.
 10.3: Derive the recursion relations (6) and (7).
 10.4: Show that the substitution u = z1/2v converts Bessels equation into...
 10.5: Show that if u satises Bessels equation, then v = zu(z) satises the...
 10.6: Use (11) and the recursion relations to compute J3/2 and J5/2. Veri...
 10.7: Find all the solutions u(x) of the ODE xuu+xu=0. (Hint: Substitute ...
 10.8: Show that H 1/2(z)=2/zei(z/2) exactly!
 10.9: (a) Show that u(r, t)=eitH s (r/c) solves the threedimensional wav...
 10.10: ProvethatthethreedenitionsoftheNeumannfunctionofintegerorder given ...
 10.11: Fill in the details in the derivation of (17) and (18).
 10.12: Show that cos(x sin)= J0(x)+2% k=1 J2k(x)cos2k.
 10.13: Substitute t =ei in (17) to get the famous identity e(1/2)z(t1/t) =...
 10.14: Solve the equation uxx uyy +k2u =0 in the disk {x2 + y2 < a2} with ...
 10.15: Solvetheequationuxx uyy +k2u =0intheexterior{x2 + y2 > a2} of the d...
 10.16: Solve the equation uxx uyy uzz +k2u =0 in the ball {x2 + y2+ z2 < a...
 10.17: Solve the equation uxx uyy uzz +k2u =0 in theexterior {x2 + y2 +z2 ...
 10.18: Find an equation for the eigenvalues and nd the eigenfunctions of i...
 10.19: Find an equation for the eigenvalues and nd the eigenfunctions of i...
 10.1: Show that the Legendre polynomials satisfy the recursion relation (4).
 10.2: (a) Prove Rodrigues formula (7). (b) Deduce that Pl(1)=1.
 10.3: Show that P2n(0)=(1)n(2n)!/22n(n!)2.
 10.4: Show that1 1x2Pl(x)dx=0forl 3.
 10.5: Let f(x)= x for0 x < 1,andf(x)=0for 1 < x 0. Find the coefcients al...
 10.6: Find the harmonic function in the ball {x2 + y2 +z2 < a2} with u = ...
 10.7: Find the harmonic function in the ball {x2 + y2 +z2 < a2} with the ...
 10.8: Solvethediffusionequationinthesolidcone{x2 + y2 +z2 < a2, < } with ...
 10.1: Show that in spherical coordinates the angular momentum operator L ...
 10.2: ProvetheidentityLxLyLyLx=iLz andthetwosimilaridentitiesformed by cy...
 10.3: (a) Write down explicitly the eigenfunction of the PDE (18) in the ...
 10.4: Show that if is not the reciprocal of an integer, the ODE (11) has ...
 10.5: (a) Write Schrodingers equation in two dimensions in polar coordina...
Solutions for Chapter 10: BOUNDARIES IN THE PLANE AND IN SPACE
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 10: BOUNDARIES IN THE PLANE AND IN SPACE
Get Full SolutionsPartial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Since 63 problems in chapter 10: BOUNDARIES IN THE PLANE AND IN SPACE have been answered, more than 5832 students have viewed full stepbystep solutions from this chapter. Chapter 10: BOUNDARIES IN THE PLANE AND IN SPACE includes 63 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.