 5.5.1: In the expansion 1=%n odd (4/n) sin n, valid for 0 < x <, put x = /...
 5.5.2: Let (x) x2 for 0 x 1=l. (a) Calculate its Fourier sine series. (b) ...
 5.5.3: Consider the function (x) x on (0, l). On the same graph, sketch th...
 5.5.4: Find the Fourier cosine series of the function sin x in the inter...
 5.5.5: GiventheFouriersineseriesof(x) x on(0,l).Assumethattheseries can be...
 5.5.6: (a) By the same method, nd the sine series of x3. (b) Find the cosi...
 5.5.7: Put x=0 in Exercise 6(b) to deduce the sum of the series 1 (1)n n4 .
 5.5.8: A rod has length l = 1 and constant k = 1. Its temperature satises ...
 5.5.9: Solveutt =c2uxx for0<x<,withtheboundaryconditionsux(0,t)= ux(,t)=0 ...
 5.5.10: A string (of tension T and density ) with xed ends at x = 0 and x =...
 5.5.11: On a string with xed ends, show that if the center of a hammer blow...
 5.5.2.1: For each of the following functions, state whether it is even or od...
 5.5.2.2: For each of the following functions, state whether it is even or od...
 5.5.2.3: Prove property (5) concerning the integrals of even and odd functions.
 5.5.2.4: (a) Use(5)toprovethatif(x)isanoddfunction,itsfullFourierseries on (...
 5.5.2.5: Show that the Fourier sine series on (0, l) can be derived from the...
 5.5.2.6: Show that the cosine series on (0, l) can be derived from the full ...
 5.5.2.7: Show how the full Fourier series on (l,l) can be derived from the f...
 5.5.2.8: (a) Prove that differentiation switches even functions to odd ones,...
 5.5.2.9: Let (x) be a function of period . If(x)= n=1an sin nx for all x,nd ...
 5.5.2.10: (a) Let (x) be a continuous function on (0, l). Under what conditio...
 5.5.2.11: FindthefullFourierseriesofex on(l,l)initsrealandcomplexforms. (Hint...
 5.5.2.12: Repeat Exercise 11 for cosh x.(Hint: Use the preceding result.)
 5.5.2.13: Repeat Exercise 11 forx.
 5.5.2.14: Repeat Exercise 11 forx.
 5.5.2.15: Without any computation, predict which of the Fourier coefcients of...
 5.5.2.16: Use the De Moivre formulas (11) to derive the standard formulas for...
 5.5.2.17: Show that a complexvalued function f(x) is realvalued if and only...
 5.5.3.1: (a) Findtherealvectorsthatareorthogonaltothegivenvectors[1,1,1] and...
 5.5.3.2: (a) On the interval [1, 1], show that the function x is orthogonal ...
 5.5.3.3: Consider utt =c2uxx for 0 < x < l, with the boundary conditions u(0...
 5.5.3.4: Consider the problem ut =kuxx for 0 < x < l, with the boundary cond...
 5.5.3.5: (a) Showthattheboundaryconditionsu(0,t)=0,ux(l,t)=0leadto the eigen...
 5.5.3.6: Findthecomplexeigenvaluesoftherstderivativeoperatord/dxsubject to ...
 5.5.4.1: (a) Findtherealvectorsthatareorthogonaltothegivenvectors[1,1,1] and...
 5.5.4.2: (a) On the interval [1, 1], show that the function x is orthogonal ...
 5.5.4.3: Consider utt =c2uxx for 0 < x < l, with the boundary conditions u(0...
 5.5.4.4: Consider the problem ut =kuxx for 0 < x < l, with the boundary cond...
 5.5.4.5: (a) Showthattheboundaryconditionsu(0,t)=0,ux(l,t)=0leadto the eigen...
 5.5.4.6: Findthecomplexeigenvaluesoftherstderivativeoperatord/dxsubject to ...
 5.5.4.7: Show by direct integration that the eigenfunctions associated with ...
 5.5.4.8: Showdirectlythat(X 1X2 + X1X 2)b a =0ifbothX1 andX2 satisfythesame...
 5.5.4.9: Show that the boundary conditions X(b)= X(a)+X(a) and X(b)= X(a)+X(...
 5.5.4.10: (The GramSchmidt orthogonalization procedure) IfX1, X2,...is any se...
 5.5.4.11: (a) Show that the condition f(x)f(x)b a 0 is valid for any functio...
 5.5.4.12: Prove Greens rst identity: For every pair of functions f(x), g(x) o...
 5.5.4.13: Use Greens rst identity to prove Theorem 3. (Hint: Substitute f(x)=...
 5.14: What do the terms in the series 4 =sin1+ 1 3 sin3+1 5sin5+ looklike...
 5.15: Use the same idea as in Exercises 12 and 13 to show that none of th...
 5.1: n=0(1)nx2n is a geometric series. (a) Does it converge pointwise in...
 5.2: Consider any series of functions on any nite interval. Show that if...
 5.3: Letn beasequenceofconstantstendingto.Let fn(x)bethesequence of func...
 5.4: Letgn(x)= 1intheinterval1 4 1 n2,1 4 +1 n2 for odd n1intheinterval3...
 5.5: Let (x)=0 for 0 < x < 1 and (x)=1 for 1 < x < 3. (a) Findthe rstfou...
 5.6: Find the sine series of the function cos x on the interval (0,). Fo...
 5.7: Let(x)=#1x for1 < x < 0 +1x for 0 < x < 1.(a) Find the full Fourier...
 5.8: ConsidertheFouriersineseriesofeachofthefollowingfunctions.Inthis ex...
 5.9: Let f(x) be a function on (l, l) that has a continuous derivative a...
 5.10: Deduce from Exercise 9 that there is a constant k so that an+bn...
 5.11: (Term by term integration) (a) If f(x) is a piecewise continuous fu...
 5.12: Start with the Fourier sine series of f(x)=x on the interval (0, l)...
 5.13: Start with the Fourier cosine series of f(x)= x2 on the interval (0...
 5.14: Find the sum n=11/n6.
 5.15: Find the sum n=11/n6. (a) Find Bn. (b) Let 2<x < 2. For which such ...
 5.16: Let (x)= xin(,). If we approximate it by the function f(x)= 1 2a0...
 5.17: Modify the proofs of Theorems 5 and 6 for the case of complexvalue...
 5.18: Consider a solution of the wave equation with c = 1 on [0,l] with h...
 5.19: Here is a general method to calculate the normalizing constants. Le...
 5.20: Use the method of Exercise 19 to compute the normalizing constants ...
 5.1: Sketch the graph of the Dirichlet kernelKN()=sinN + 1 2sin 1 2 in c...
 5.2: Prove the Schwarz inequality (for any pair of functions): (f,g)fg...
 5.3: Prove the inequality ll 0 (f(x))2dx[f(l) f(0)]2 for any real funct...
 5.4: (a) Solve the problem ut =kuxx for 0 < x < l, u(x, 0)= (x), with th...
 5.5: Prove the Schwarz inequality for innite series: anbn a2 n1/2b2 n1/2...
 5.6: Consider the diffusion equation on [0, l] with Dirichlet boundary c...
 5.7: Let [f(x)2 +g(x)2]dx be nite, where g(x)= f(x)/(eix1).Let cn be...
 5.8: Prove that both integrals in (12) tend to zero.
 5.9: Fill in the missing steps in the proof of uniform convergence.
 5.10: Prove the theorem on uniform convergence for the case of the Fourie...
 5.11: Prove that the classical full Fourier series of f(x) converges unif...
 5.12: Show that if f(x) is aC1 function in [, ] that satises the periodic...
 5.13: A very slick proof of the pointwise convergence of Fourier series, ...
 5.14: ProvethevalidityoftheFourierseriessolutionofthediffusionequation on...
 5.15: Carry out the step going from (24) to (25).
 5.1: (a) Solve as a series the equation ut =uxx in (0,1) with ux(0,t)=0,...
 5.2: For problem (1), complete the calculation of the series in case j(t...
 5.3: Repeat problem (1) for the case of Neumann BCs.
 5.4: Solve utt =c2uxx +k for 0 < x < l, with the boundary conditions u(0...
 5.5: Solve utt=c2uxx +etsin5x for 0< x <, with u(0,t)=u(,t)=0 and the in...
 5.6: Solve utt =c2uxx +g(x)sint for 0 < x < l, with u =0 at both ends an...
 5.7: Repeat Exercise 6 for the damped wave equation utt =c2uxx rut+ g(x)...
 5.8: Solve ut =kuxx in (0,l), with u(0,t)=0, u(l,t)= At, u(x,0)=0, where...
 5.9: Use the method of subtraction to solve utt =9uxx for 0 x 1=l, with ...
 5.10: Findthetemperatureofametalrodthatisintheshapeofasolidcircular conew...
 5.11: Write out the solution of problem (11) explicitly, starting from th...
 5.12: Carry out the solution of (11) in the case that f(x,t)= F(x)costh (...
 5.13: If friction is present, the wave equation takes the form utt c2uxx ...
Solutions for Chapter 5: FOURIER SERIES
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 5: FOURIER SERIES
Get Full SolutionsThis textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2. Chapter 5: FOURIER SERIES includes 97 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. Since 97 problems in chapter 5: FOURIER SERIES have been answered, more than 5865 students have viewed full stepbystep solutions from this chapter.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·