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Questions and Answers of Numerical Analysis

The boundary-value problem Y" = 4(y − x), 0 ≤ x ≤ 1, y(0) = 0, y(1) = 2
Show that if y ∈ C6 [a, b] and if w0, w1, . . . ,wN+1 satisfy Eq. (11.18), then wi − y(xi) = Ah2 + O(h4), where A is independent of h, provided q(x) ≥ w > 0 on [a, b] for some w.
The boundary-value problem Y" = y' + 2y + cos x, 0 ≤ x ≤ π/2, y(0) = −0.3, y (π/2) = −0.1
Use the Linear Finite-Difference Algorithm to approximate the solution to the following boundary-value problems. a. y" = −3y' + 2y + 2x + 3, 0 ≤ x ≤ 1, y(0) = 2, y(1) = 1; use h = 0.1. b. y" =
Although q(x) < 0 in the following boundary-value problems, unique solutions exist and are given. Use the Linear Finite-Difference Algorithm to approximate the solutions, and compare the results to
Use the Linear Finite-Difference Algorithm to approximate the solution y = e−10x to the boundaryvalue problem Y" = 100y, 0 ≤ x ≤ 1, y(0) = 1, y(1) = e−10. Use h = 0.1 and 0.05. Can you
Repeat Exercise 3(a) and (b) using the extrapolation discussed in Example 2.In exercise 3(a) and (b)a. y" = ˆ’3y' + 2y + 2x + 3, 0 ‰¤ x ‰¤ 1, y(0) = 2, y(1) = 1; use
The lead example of this chapter concerned the deflection of a beam with supported ends subject to uniform loading. The boundary-value problem governing this physical situation iswith boundary
The deflection of a uniformly loaded, long rectangular plate under an axial tension force is governed by a second-order differential equation. Let S represent the axial force and q the intensity of
Prove Theorem 11.3.
Use the Nonlinear Finite-Difference method with h = 0.5 to approximate the solution to the boundary-value problem Y" = −(y')2 − y + ln x, 1≤ x ≤ 2, y(1) = 0, y(2) = ln 2. Compare your results
Use the Nonlinear Finite-Difference method with h = 0.25 to approximate the solution to the boundary-value problem Y" = 2y3, −1 ≤ x ≤ 0, y(−1) = 1/2 , y(0) = 1/3 Compare your results to the
Use the Nonlinear Finite-Difference Algorithm with TOL = 10−4 to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results. a.
Use the Nonlinear Finite-Difference Algorithm with TOL = 10−4 to approximate the solution to the following boundary-value problems. The actual solution is given for comparison to your results. a.
Repeat Exercise 4(a) and 4(b) using extrapolation. In exercise 4(a) and 4(b)
In Exercise 7 of Section 11.3, the deflection of a beam with supported ends subject to uniform loading was approximated. Using a more appropriate representation of curvature gives the differential
Showthat the hypotheses listed at the beginning of the section ensure the nonsingularity of the Jacobian matrix J for h < 2/L.
Use the Piecewise Linear Algorithm to approximate the solution to the boundary-value problem using x0 = 0, x1 = 0.3, x2 = 0.7, x3 = 1. Compare your results to the actual solution y(x) = −1/3 cos
Use Algorithm 12.1 to approximate the solution to the elliptic partial differential equation Use h = k = 1/2 , and compare the results to the actual solution u(x, y) = (x − y)2.
Use Algorithm 12.1 to approximate the solution to the elliptic partial differential equation Use h = k = 1/3, and compare the results to the actual solution u(x, y) = ln(x2 + y2).
Approximate the solutions to the following elliptic partial differential equations, using Algorithm 12.1: a. ∂2u / ∂x2 + ∂2u / ∂y2 = 0, 0 < x < 1, 0 < y < 1; u(x, 0) = 0, u(x, 1) =
Repeat Exercise 3(a) using extrapolation with h0 = 0.2, h1 = h0/2, and h2 = h0/4. Repeat exercise 3(a) a. ∂2u / ∂x2 + ∂2u / ∂y2 = 0, 0 < x < 1, 0 < y < 1; u(x, 0) = 0, u(x, 1) =
Construct an algorithm similar to Algorithm 12.1, except use the SOR method with optimal ω instead of the Gauss-Seidel method for solving the linear system.
Repeat Exercise 3 using the algorithm constructed in Exercise 5. Repeat exercise 3
A coaxial cable is made of a 0.1-in.-square inner conductor and a 0.5-in.-square outer conductor. The potential at a point in the cross section of the cable is described by Laplace's equation.
A 6-cm by 5-cm rectangular silver plate has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3·s. Let x represent the distance along the edge of the plate of length 6 cm and y
Approximate the solution to the following partial differential equation using the Backward-Difference method. ∂u / ∂t − ∂2u / ∂x2 = 0, 0 < x < 2, 0 < t; u(0, t) = u(2, t) = 0, 0 < t, u(x,
Repeat Exercise 6 using the Crank-Nicolson Algorithm. Repeat exercise 6
Repeat Exercise 5 using Richardson's method.Repeat exercise 5
Repeat Exercise 6 using Richardson's method. Repeat exercise 6
Show that the eigenvalues for the (m ˆ’ 1) by (m ˆ’ 1) tridiagonal method matrix A given byare with corresponding eigenvectors v(i), where v(i)j = sin(ijÏ€/m).
Show that the (m ˆ’ 1) by (m ˆ’ 1) tridiagonal method matrix A given bywhere Î» > 0, is positive definite and diagonally dominant and has eigenvalues with
Modify Algorithms 12.2 and 12.3 to include the parabolic partial differential equation∂u / ∂t − ∂2u / ∂x2 = F(x), 0< x < l, 0 < t;u(0, t) = u(l, t) = 0, 0 < t;u(x, 0) = f (x),
Use the results of Exercise 15 to approximate the solution to ∂u / ∂t − ∂2u / ∂x2 = 2, 0 < x < 1, 0 < t; u(0, t) = u(1, t) = 0, 0 < t; u(x, 0) = sin πx + x(1 − x), with h = 0.1 and k =
Change Algorithms 12.2 and 12.3 to accommodate the partial differential equation∂u / ∂t − α2 ∂2u / ∂x2 = 0, 0 < x < l, 0 < t;u(0, t) = φ(t), u(l, t) = (t), 0< t;u(x, 0) = f
The temperature u(x, t) of a long, thin rod of constant cross section and homogeneous conducting material is governed by the one-dimensional heat equation. If heat is generated in the material, for
Sagar and Payne [SP] analyze the stress-strain relationships and material properties of a cylinder alternately subjected to heating and cooling and consider the equationˆ‚2T /
Approximate the solution to the following partial differential equation using the Backward-Difference method. ∂u / ∂t - 1 / 16 ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t; u(0, t) = u(1, t) = 0, 0 < t,
Repeat Exercise 1 using the Crank-Nicolson Algorithm. Repeat exercise 1
Repeat Exercise 2 using the Crank-Nicolson Algorithm. Repeat exercise 2
Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations. a. ∂u / ∂t − ∂2u / ∂x2 = 0, 0 < x < 2, 0 < t; u(0, t) = u(2, t) = 0,
Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations. a. ∂u / ∂t - 4 / π2 ∂2u / ∂x2 = 0, 0 < x < 4, 0 < t; u(0, t) = u(4,
Repeat Exercise 5 using the Backward-Difference Algorithm. Repeat exercise 5
Repeat Exercise 6 using the Backward-Difference Algorithm. Repeat exercise 6
Repeat Exercise 5 using the Crank-Nicolson Algorithm. Repeat exercise 5
Approximate the solution to the wave equation ∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t; u(0, t) = u(1, t) = 0, 0 < t, u(x, 0) = sin πx, 0≤ x ≤ 1, ∂u / ∂t (x, 0) = 0, 0 ≤ x ≤
Approximate the solution to the wave equation ∂2u / ∂t2 - 1/16π2 ∂2u/∂x2 = 0, 0 < x < 0.5, 0 < t; u(0, t) = u(0.5, t) = 0, 0 < t, u(x, 0) = 0, 0 ≤ x ≤ 0.5, ∂u / ∂t (x, 0) = sin 4πx,
Approximate the solution to the wave equation ∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < π, 0 < t; u(0, t) = u(π, t) = 0, 0 < t, u(x, 0) = sin x, 0≤ x ≤ π, ∂u / ∂t (x, 0) = 0, 0 ≤ x
Repeat Exercise 3, using in Step 4 of Algorithm 12.4 the approximation wi,1 = wi,0 + kg(xi), for each i = 1, . . . ,m − 1. Repeat exercise 3
Approximate the solution to the wave equation ∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t; u(0, t) = u(1, t) = 0, 0 < t, u(x, 0) = sin 2πx, 0≤ x ≤ 1, ∂u / ∂t (x, 0) = 2π sin 2πx,
Approximate the solution to the wave equation ∂2u / ∂t2 − ∂2u / ∂x2 = 0, 0 < x < 1, 0 < t; u(0, t) = u(1, t) = 0, 0 < t, ∂u / ∂t (x, 0) = 0, 0 ≤ x ≤ 1. using Algorithm 12.4 with h
The air pressure p(x, t) in an organ pipe is governed by the wave equation ∂2p / ∂x2 = 1 / c2 ∂2p / ∂t2, 0 < x < l, 0 < t, where l is the length of the pipe, and c is a physical constant. If
In an electric transmission line of length l that carries alternating current of high frequency (called a "lossless" line), the voltage V and current i are described by ∂2V / ∂x2 = LC ∂2V /
Use Algorithm 12.5 to approximate the solution to the following partial differential equation (see the figure): Let M = 2; T1 have vertices (0, 0.5), (0.25, 0.75), (0, 1); and T2 have vertices
Repeat Exercise 1, using instead the triangles T1: (0, 0.75), (0, 1), (0.25, 0.75); T2: (0.25, 0.5), (0.25, 0.75), (0.5, 0.5); T3: (0, 0.5), (0, 0.75), (0.25, 0.75); T4: (0, 0.5), (0.25, 0.5), (0.25,
Approximate the solution to the partial differential equation subject to the Dirichlet boundary condition u(x, y) = 0, using the Finite-Element Algorithm 12.5 with the elements given in the
Repeat Exercise 3 with f (x, y) = −25π2 cos 5π/2 x cos 5π/2 y, using the Neumann boundary condition∂u / ∂n (x, y) = 0.
A silver plate in the shape of a trapezoid (see the accompanying figure) has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3 · s. The steady-state temperature u(x, y) of the
Let a, b, c, d ∈ R and consider each of the following statements. Decide which are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If a < b and c < d <
Suppose that a, b, c ∈ R and a < b. a) Prove that a + c < b + c. b) If c > 0, prove that a ∙ c < b ∙ c.
Prove that (ab + cd)2 < (a2 + c2)(b2 + d2) for all a, b, c, d ∈ R.
a) Let R+ represent the collection of positive real numbers. Prove that R+ satisfies the following two properties. i) For each x ∈ R, one and only one of the following holds: x ∈ R+, -x ∈ R+,
Prove (7), (8), and (9). Show that each of these statements is false if the hypothesis a > 0 or a > 0 is removed.
The positive part of an a ˆˆ R is defined byand the negative part bya) Prove that a = a+ - a- and |a| = a+ + a-.b) Prove that
Solve each of the following inequalities for x ∈ R. a) |2x + 1| < 7 b) |2 - x| < 2 c) |x3 - 3x + 1| < x3 d) x/x - 1 < 1. e) x2/4x2 - 1 < ¼
Let a, b ∈ R. a) Prove that if a > 2 and b = 1 + √a - 1, then 2 < b < a. b) Prove that if 2 < a < 3 and = 2 + √a - 2, then 0 < a < b. c) Prove that if 0 < a < 1 and b = 1 - √l - a, then 0 < b
The arithmetic mean of a. b ∈ R is A(a, b) = (a + b)/2, and the geometric mean of a, b ∈ [0, ∞) is G(a, b) = √ab. If 0 < a < b, prove that a < G{a, b) < A(a, b) < b. Prove that G(a, b) = A(a,
Let x ∈ R. a) Prove that |x| < 2 implies |x2 - 4| < 4|x - 2|. b) Prove that |x| < 1 implies |x2 + 2x - 3| < 4|x - 1|. c) Prove that -3 < x < 2 implies |x2 + x - 6| < 6|x - 2|. d) Prove that -1 < x
For each of the following, find all values of n ˆˆ N that satisfy the given inequality.a)b)c)
a) Interpreting a rational m/n as m • n–1 ∈ R, use Postulate 1 to prove that for m, n, p, q, ℓ ∈ Z and n, q, ℓ ≠ 0.b) Using Remark 1.1, Prove that Postulate 1 holds with Q in
Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If A and B are nonempty, bounded subsets of R, then sup(A ∩
Find the infimum and supremum of each of the following sets.a) E = {x ∈ R : x2 + 2x = 3}b) E = {x ∈ R : x2 - 2x + 3 > x2 and x > 0}c) E = {p/q ∈ Q : p2 < 5q2 and p, q > 0}d) E = {x
Let xn ∈ R and suppose that there is an M ∈ R such that |xn| < M for n ∈ N. Prove that sn = sup{xn, xn+1,...} defines a real number for each n ∈ N and that s1 > s2 > ∙ ∙ ∙. Prove an
If a, b ∈ R and b - a > 1, then there is at least one k ∈ Z such that a < k < b.
Prove that for each a ∈ R and each n ∈ N there exists a rational rn such that |a - rn| < 1/n.
Prove that if a < b are real numbers, then there is an irrational such that
Prove that a lower bound of a set need not be unique but the infimum of a given set E is unique.
Show that if E is a nonempty bounded subset of Z, then inf E exists and belongs to E.
Use the Reflection Principle and analogous results about suprema to prove the following results. a) [APPROXIMATION PROPERTY FOR INFIMA] Prove that if a set E ⊂ R has a finite infimum and ε > 0 is
a) Prove that if x is an upper bound of a set E ⊂ R and x ∈ E, then x is the supremum of E. b) Make and prove an analogous statement for the infimum of E. c) Show by example that the converse of
Suppose that E, A, B ⊂ R and E = A U B. Prove that if E has a supremum and both A and B are nonempty, then sup A and sup B both exist, and sup E is one of the numbers sup A or sup B.
A dyadic rational is a number of the form k/2n for some k, n ∈ Z. Prove that if a and b are real numbers and a < b, then there exists a dyadic rational q such that a < q < b.
Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) If a > 0 and b ≠ 0, then (a + b)n > bn for all n ∈
a) Prove that if x1 > 2 and xn+1 = 1 + √xn - 1 for n ∈ N, then 2 < xn+1 < xn holds for all n ∈ N. b) Prove that if 2 < x1 < 3 and xn+1 = 2 + √xn - 2 for n ∈ N, then 0 < xn < holds for all n
Let a0 = 3, b0 = 4, and c0 = 5. a) Let ak = ak-1 + 2, bk = 2ak-1 + bk-1 + 2, and ck = 2ak-1 + ck-1 + 2 for ∈ N. Prove that ck - bk is constant for all k ∈ N. b) Prove that the numbers defined
Use the Binomial Formula or the Principle of Induction to prove each of the following.a) ∑nk=0(-1)k (n/k) = 0 for all n ∈ N.b) (a + b)n > an + bn for all n ∈ N and a, b > 0.c) (1 + \/n)n
Prove each of the following statements. a) 2n + 1 < 2n for n = 3, 4,.... b) n < 2n for n = 1, 2,.... c) n2 < 2n + 1 for n = 1, 2,.... d) n3 < 3n for n = 1, 2,....
Prove that the following formulas hold for all n ˆˆ N.a)b)c)d)
Prove that 0 < a < b implies 0 < an < bn and 0 < n√a < n√b for all n ∈ N.
Prove that 2n + 3" is a multiple of 5 for all odd neN.
Prove that 2n < n! + 2 for n ∈ N.
Prove thatfor n ˆˆ N.
a) Using Remark 1.28, prove that the square root of an integer m is rational if and only if m = k2 for some k ∈ N. b) Prove that √n + 3 + √n is rational for some n ∈ N if and only if n =
Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) Let f(x) = sin x. Then the functionis a bijection, and its
For each of the following, prove that f is 1-1 on E and find f(E). a) f(x) = 3x - 7, E = R β) f(x) = el/x, E = (0, ∞) γ) f(x) = tan x, E = (π/2, 3π/2) δ) f(x) = x2 + 2x - 5, E = (-∞, -6] e)
Find f(x) and f-1(E) for each of the following. a) f(x) = 2 - 3x, E = (-1, 2) b) f(x) = x2 + l, E = (-l, 2] c) f(x) = 2x - x2, E = [-2, 2) d) f(x) = log(x2 - 2x + 2), E = (0, 3] e) f(x) = cos x, E =
Give a simple description of each of the following sets.a)b)c)d)e)f)
Prove Theorem 1.37iii, iv, and v.
Let f(x) = x2. a) Find subsets B and C of R such that f(C\B) ≠ f(C)\f(B). b) Find a subset E of R such that f-1(f(E)) ≠ E.
Let X, Y be sets and f : X → Y. Prove that the following are equivalent. a) f is 1-1 on X. b) f(A\B) = f(A)\f(B) for all subsets A and B of X. c) f-1(f(E)) = E for all subsets £ of X. d) F(A ∩

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