 8.1: he Taylor expansion written in Section 8.1 is valid if u is a C4 fu...
 8.2: a) If u(x) is merely a C3 function, what is the error in the second...
 8.3: Suppose that we wish to approximate the rst derivative u(x) of a ve...
 8.1: (a) Solvetheproblemut =uxx intheinterval[0,4]withu=0atboth ends and...
 8.2: Dothesamewithx =1andt =1.Calculatebyhandorbycomputer up to t=7.
 8.3: Solve ut =uxx intheinterval[0,5]withu(0,t)=0andu(5,t)=1for t 0, and...
 8.4: Solve by hand the problem ut =uxx in the interval [0, 1] with ux =0...
 8.5: Using the forward scheme (2), solve ut =uxx in [0, 5] with the mixe...
 8.6: Dothesamewiththeconditionsux(0,t)=u(5,t)=0 and u(x,0)= x.
 8.7: Show that the local truncation error in the CrankNicolson scheme i...
 8.8: a) Write down the CrankNicolson scheme ( = 1 2) for ut =uxx.(b) Con...
 8.9: For the scheme (15) for the diffusion equation, provide the details...
 8.10: Forthediffusionequationut =uxx,usecentereddifferencesforbothut and ...
 8.11: Consider the equation ut =auxx+bu, where a and b are constants and ...
 8.12: (a) Solve by hand the nonlinear PDE ut =uxx+(u)3 for all x using th...
 8.13: Consider the following scheme for the diffusion equation:un+1 j un1...
 8.14: (a) Formulate an explicit scheme for ut =uxx+uyy. (b) What is the s...
 8.15: Formulate the CrankNicolson scheme for ut =uxx+uyy.
 8.1: (a) Write the scheme (2) for the wave equation in the case s = 1 4 ...
 8.2: Solve by hand for a few time steps the numerical scheme (2) for utt...
 8.3: (a) Use the scheme (2) with x = t =0.2 to approximately solve utt =...
 8.4: (a) Use the scheme (2) with x = t =0.25 to solve utt =uxx approxima...
 8.5: Solve by hand for a few time steps the equation utt =uxx in the nit...
 8.6: Considerthewaveequationonthehalfline0 < x < ,withtheboundary condi...
 8.7: Solve by hand the nonlinear equation utt =uxx+u3 up to t=4, using t...
 8.8: RepeatExercise7bycomputerfortheequationutt =uxxu3 usingan implicit ...
 8.9: Consider the scheme (12) for the nonlinear wave equation (10). Let ...
 8.10: Consider the equation ut =ux. Use forward differences for both part...
 8.11: Consider the rstorder equation ut +aux =0. (a) Solve it exactly wi...
 8.1: SetupthelinearequationstondthefourunknownvaluesinFigure2(a), write ...
 8.2: Apply Jacobi iteration to the example of Figure 2(a) with zero init...
 8.3: Apply four GaussSeidel iterations to the example of Figure 2(a).
 8.4: Solve the example of Figure 2(a) but with the boundary conditions (...
 8.5: Consider the PDE uxx+uyy =0 in the unit square {0 x 1, 0 y 1}with t...
 8.6: (a) Writedowntheschemeusingcentereddifferencesfortheequation uxx+uy...
 8.7: Solve uxx +uyy =0 in the unit square{0 x 1,0 y 1}with the boundaryc...
 8.8: Formulate a nite difference scheme for uxx+uyy = f(x, y) in the uni...
 8.9: Apply Exercise 8 to approximately nd the harmonic function in the u...
 8.10: Try to do the same with the boundary conditions ux(0, y)=0, ux(1, y...
 8.11: Show that performing Jacobi iteration (3) is the same as solving th...
 8.12: Do the same (solving the diffusion equation) with t = (x)2 and comp...
 8.1: Considertheproblemuxx+uyy =4intheunitsquarewithu(0, y)=0, u(1, y)=0...
 8.2: (a) Find the area A of the triangle with three given vertices (x1, ...
 8.3: (Linear elements on intervals) In one dimension the geometric build...
 8.4: (Finite elements for the wave equation) Consider the problem utt =u...
 8.5: (Bilinear elements on rectangles) On the rectangle with vertices (0...
Solutions for Chapter 8: COMPUTATION OF SOLUTIONS
Full solutions for Partial Differential Equations: An Introduction  2nd Edition
ISBN: 9780470054567
Solutions for Chapter 8: COMPUTATION OF SOLUTIONS
Get Full SolutionsChapter 8: COMPUTATION OF SOLUTIONS includes 46 full stepbystep solutions. Partial Differential Equations: An Introduction was written by and is associated to the ISBN: 9780470054567. Since 46 problems in chapter 8: COMPUTATION OF SOLUTIONS have been answered, more than 5517 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Partial Differential Equations: An Introduction, edition: 2.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

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).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.