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About 3307 results. 1294 free access solutions
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Condition |
free/or 0.5$ |
| m53024 | 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 ≤ π,
using the Finite-Difference Algorithm with h = π/10 and k = 0.05, with h = π/20 and k = 0.1, and then with h = π/20 and k = 0.05. Compare your results at t = 0.5 to the actual solution u(x, t) = cos t sin x. |
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| m53025 | 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, 0≤ x ≤ 1,
using Algorithm 12.4 with h = 0.1 and k = 0.1. Compare your results at t = 0.3 to the actual solution u(x, t) = sin 2πx(cos 2πt + sin 2πt). |
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| m53026 | 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 = 0.1 and k = 0.1. |
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| m53027 | 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) = x,...............0≤ x ≤ 1;
u(0, y) = 0, u(1, y) = y, ...............0≤ y ≤ 1.
Use h = k = 0.2, and compare the results to the actual solution u(x, y) = xy.
b. ∂2u / ∂x2 + ∂2u / ∂y2 = −(cos (x + y) + cos (x − y)), 0< x < π,0< y < π/2;
u(0, y) = cos y, u(π, y) = −cos y, ...............0≤ y ≤ π/2,
u(x, 0) = cos x, u (x, π/2)= 0, .....................0 ≤ x ≤ π.
Use h = π/5 and k = π/10, and compare the results to the actual solution u(x, y) = cos x cos y.
c. ∂2u / ∂x2 + ∂2u / ∂y2 = (x2 + y2)exy, 0< x < 2, 0 < y < 1;
u(0, y) = 1, u(2, y) = e2y, ...............0≤ y ≤ 1;
u(x, 0) = 1, u(x, 1) = ex, .................0≤ x ≤ 2.
Use h = 0.2 and k = 0.1, and compare the results to the actual solution u(x, y) = exy.
d. ∂2u / ∂x2 + ∂2u / ∂y2 = x/y + y/x, 1< x < 2, 1 < y < 2;
u(x, 1) = x ln x, u(x, 2) = x ln4x2, ................1≤ x ≤ 2;
u(1, y) = y ln y, u(2, y) = 2y ln(2y), ...............1≤ y ≤ 2.
Use h = k = 0.1, and compare the results to the actual solution u(x, y) = xy ln xy. |
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| m53029 | Approximate
Using h = 0.25. Use
a. Composite Trapezoidal rule.
b. Composite Simpson s rule.
c. Composite Midpoint rule. |
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| m53030 | Approximate
Using h = 0.25. Use
a. Composite Trapezoidal rule.
b. Composite Simpson s rule.
c. Composite Midpoint rule. |
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| m53063 | Assume that
(1−2x)/(1−x+x2) + (2x−4x3)/(1−x2+x4) + (4x3−8x7)/(1 − x4+x8) +· · · = (1 + 2x)/(1+x+x2), for x < 1, and let x = 0.25. Write and execute an algorithm that determines the number of terms needed on the left side of the equation so that the left side differs from the right side by less than 10−6. |
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| m53072 | Assume that the largest eigenvalue λ1 in magnitude and an associated eigenvector v(1) have been obtained for the n × n symmetric matrix A. Show that the matrix
has the same eigenvalues λ2, . . . , λn as A, except that B has eigenvalue 0 with eigenvector v(1) instead of eigenvector λ1. Use this deflation method to find λ2 for each matrix in Exercise 5. Theoretically, this method can be continued to find more eigenvalues, but round-off error soon makes the effort worthless.
In exercise
a.
Use x(0) = (1,−1, 2)t.
b.
Use x(0) = (−1, 0, 1)t.
c.
Use x(0) = (0, 1, 0)t.
d.
Use x(0) = (0, 1, 0, 0)t. |
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| m53161 | Calculate R4,4 for the integrals in Exercise 1.
In Exercise 1 |
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| m53162 | Calculate R4,4 for the integrals in Exercise 2.
In Exercise 2 |
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| m53163 | Calculate R4,4 for the integrals in Exercise 2.
In Exercise 2 |
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| m53189 | Change Algorithm 5.4 so that the corrector can be iterated for a given number p iterations. Repeat Exercise 7 with p = 2, 3, and 4 iterations. Which choice of p gives the best answer for each initial-value problem?
In Exercise 7 |
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| m53190 | 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 (x), 0≤ x ≤ l,
where f (0) = φ(0) and f (l) = Ψ(0). |
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| m53191 | Change the Adams Fourth-Order Predictor-Corrector Algorithm to obtain approximate solutions to systems of first-order equations. |
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| m53225 | Compute the absolute error and relative error in approximations of p by p∗.
a. p = π, p∗ = 22/7
b. p = π, p∗ = 3.1416
c. p = e, p∗ = 2.718
d. p =√2, p∗ = 1.414
e. p = e10, p∗ = 22000
f. p = 10π , p∗ = 1400
g. p = 8!, p∗ = 39900
h. p = 9!, p∗ =√18π(9/e)9 |
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| m53226 | Compute the eigenvalues and associated eigenvectors of the following matrices.
a.
b.
c.
d.
e.
f. |
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| m53227 | Compute the eigenvalues and associated eigenvectors of the following matrices
a.
b.
c.
d.
e.
f. |
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| m53228 | Compute the error E for the approximations in Exercise 3.
In Exercise 3
a. f (x) = x2 + 3x + 2, [0, 1];
b. f (x) = x3, [0, 2];
c. f (x) = 1/x, [1, 3];
d. f (x) = ex , [0, 2];
e. f (x) = 1/2 cos x + 1/3 sin 2x, [0, 1];
f. f (x) = x ln x, [1, 3]. |
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| m53229 | Compute the error E for the approximations in Exercise 4.
In Exercise 4
a. f (x) = x2 + 3x + 2, [0, 1];
b. f (x) = x3, [0, 2];
c. f (x) = 1/x, [1, 3];
d. f (x) = ex , [0, 2];
e. f (x) = 1/2 cos x + 1/3 sin 2x, [0, 1];
f. f (x) = x ln x, [1, 3]. |
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| m53232 | Compute the Simpson s rule approximations S(a, b), S(a, (a + b)/2), and S((a + b)/2, b) for the following integrals, and verify the estimate given in the approximation formula. |
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