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advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown. 3.0 13.0016 T= 100C T=0C
Do the task in Prob. 12 if U1 = 220 V and the outer cylinder is grounded, U2 = 0.Data from Prob. 12Find the potential and complex potential between the coaxial cylinders of axis 0 (hence the vertical axis in space) and radii r1 = 1 cm, r2 = 10 cm, kept at potential U1 = 220 V and U2 = 2 kV,
At z = ±1 in Fig. 405 the tangents to the equipotential lines as shown make equal angles. Why?
By applying a suitable conformal mapping, obtain from fig. 406 the potential Φ in the sector -1/4π
Find the real and complex potentials in the sector -π/6 ≤ θ ≤ π/6 between the boundary θ = ±π/6, kept at 0 V, and the curve x3 - 3xy2 = 1, kept at 220 V.
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = sinh 2z
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch a figure of the equipotential lines.Φ(1, θ) = 1 if -1/2π < θ < 1/2π and 0 otherwise
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown. T= 60C 45 T = -20C -Insulated x
Make a list of important potential functions, with applications, from memory.
Find the linear fractional transformation z = g(Z) that maps |Z| ≤ 1 onto |z| ≤ 1 with Z = i/2 being mapped onto z = 0. Show that Z1 = 0.6 + 0.8i is mapped onto z = -1 and Z2 = -0.6 + 0.8i onto z = 1, so that the equipotential lines of Example 2 look in |Z| ≤ 1 as shown in Fig. 407.
Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.First quadrant of the z-plane with y-axis kept at 100°C, the segment 0 < x < 1 of
Find the location and size of the maximum of |F(z)| in the unit disk |z| ≤ 1.F(z) = 2z2 - 2
Using (7), find the potential Φ(r, θ) in the unit disk r < 1 having the given boundary values Φ(1, θ). Using the sum of the first few terms of the series, compute some values of Φ and sketch a figure of the equipotential lines.Φ(1, θ) = θ2/π2 if -π < θ < π
Find the potential in the first quadrant of the xy-plane if the x-axis has potential 2 kV and the y-axis is grounded.
Find the complex and real potentials in the upper half-plane with boundary values 5 kV if x < 2 and 0 if x > 2 on the x-axis.
Show that the streamlines of F(z) = 1/z and circles through the origin with centers on the y- axis.
Do Φ and a harmonic conjugate Ψ in a region R have their maximum at the same point of R?
Formulate Prob. 11 in terms of electrostatics. Data from Prob. 11 Find the temperature distribution T(x, y) and the complex potential F(z) in the given thin metal plate whose faces are insulated and whose edges are kept at the indicated temperatures or are insulated as shown.
Find the temperature T in the upper half-plane if, on the x-axis, T = 30°C for x > 1 and -30°C for x < 1.
Do the same task as in Prob. 19 if the boundary values on the x-axis are V0 when -α < x < α and 0 elsewhere.Data from Prob. 19Find the complex and real potentials in the upper half-plane with boundary values 5 kV if x < 2 and 0 if x > 2 on the x-axis.
Find the streamlines and the velocity for the complex potential F(z) = (1 + i)z. Describe the flow.
Show that the isotherms of F(z) = -iz2 + z are hyperbolas.
Find V in Prob. 22 and verify that it gives vectors tangent to the streamlines.Data from Prob. 22Describe the streamlines for F(z) = 1/2z2 + z.
In your own words, and using as few formulas as possible, write a short report on spline interpolation, its motivation, a comparison with polynomial interpolation, and its applications.
Write 84.175, -528.685, 0.000924138, and -362005 in floating-point form, rounded to 5S (5 significant digits).
What is a numeric method? How has the computer influenced numerics?
What is an error? A relative error? An error bound?
Small differences of large numbers may be particularly strongly affected by rounding errors. Illustrate this by computing 0.81534/ (35 · 724 - 35.596) as given with 5S, then rounding step wise to 4S, 3S, and 2S, where “step wise” means round the rounded numbers, not the given ones.
Solve by fixed-point iteration and answer related questions where indicated. Show details.f = x - 0.5 cos x = 0, x0 = 1. Sketch a figure.
Calculate the Lagrange polynomial P2(x) for the values Г(1.00) = 1.0000, Г(1.02) = 0.9888, Г(1.04) = 0.9784 of the gamma function [(24) in App. A3.1] and from it approximations of Г(1.01) and Г(1.03).
Why are roundoff errors important? State the rounding rules.
What is an algorithm? Which of its properties are important in software implementation?
What do you know about stability?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Sketch f(x) = x3 - 5.00x2 + 1.01x + 1.88, showing roots near ±1 and 5. Write x = g(x) = (5.00x2 - 1.01x + 1.88)/x2. Find a root by starting from x0 = 5, 4, 1, -1. Explain the (perhaps unexpected) results.
Let α1, · · ·, αn be numbers with αj correctly rounded to Sj digits. In calculating the sum α1 + · · · + αn retaining S = min Sj significant digits, is it essential that we first add and then round the result or that we first round each number to S significant digits and
Find e-0.25 and e-0.75 by linear interpolation of e-x with x0 = 0, x1 = 0.5 and x0 = 0.5, x1 = 1, respectively. Then find p2(x) by quadratic interpolation of e-x with x0 = 0, x1 = 0.5, x2 = 1 and from it e-0.25 and e-0.75. Compare the errors. Use 4S-values of e-x.
Do the tasks in Prob. 4 for f(x) = sin 1/2πx.Data from Prob. 4Integrate f(x) = x4 from 0 to 1 by (2) with h =1, h = 0.5, h = 0.25 and estimate the error for h = 0.5 and h = 0.25 by (5).
Why is the selection of a good method at least as important on a large computer as it is on a small one?
Can the Newton (–Raphson) method diverge? Is it fast? Same questions for the bisection method.
Solve by fixed-point iteration and answer related questions where indicated. Show details.Find the smallest positive solution of sin x = e-x.
Solve x2 - 30x + 1 = 0 by (4) and by (5), using 6S in the computation. Compare and comment.
Find the quadratic polynomial that agrees with sin x at x = 0, π/4, π/2 and use it for the interpolation and extrapolation of sin x at x = -π/8, π/8, 3π/8, 5π/8. Compute the errors.
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. A, 2m = 4
What is fixed-point iteration?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Solve x4 - x - 0.12 = 0 by starting from x0 = 1.
What is the advantage of Newton’s interpolation formulas over Lagrange’s?
Solve by fixed-point iteration and answer related questions where indicated. Show details.Find the negative solution of x4 - x - 0.12 = 0.
Do the computations in Prob. 7 with 4S and 2S.Data from Prob. 7Solve x2 - 30x + 1 = 0 by (4) and by (5), using 6S in the computation. Compare and comment.
Calculate the Lagrange polynomial p2(x) for the 5S-values f(0.25) = 0.27633, f(0.5) = 0.52050, f(1.0) = 0.84270 and from p2(x) an approximation of f (0.75) (= 0.71116).
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. B, 2m = 4
What is spline interpolation? Its advantage over polynomial interpolation?
List and compare the integration methods we have discussed.
Find the cubic spline g(x) for the given data with k0 and kn as given. If we started from the piece wise linear function in Fig. 438, we would obtain g(x) in Prob. 10 as the spline satisfying g'(-2) = f'(-2) = 0, g'(2) = f'(2) = 0. Find and sketch or graph the corresponding interpolation polynomial
Solve by fixed-point iteration and answer related questions where indicated. Show details.A partial sum of the Maclaurin series of J0(x) (Sec. 5.5) is f(x) = 1 - 1/4x2 + 1/64x4 - 1/2304x6. Conclude from a sketch that f(x) = 0 near x = 2. Write f(x) = 0 as x = g(x) (by dividing f(x) by 1/4x and
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. J, 2m = 4
Calculate and graph L0, L1, L2, L3 with x0 = 0, x1 = 1, x2 = 2, x3 = 3 on common axes. Find p3(x) for the data (0, 1), (1, 0.765198), (2, 0.223891), (3, -0.260052) [values of the Bessel function J0(x)]. Find p3 for x = 0.5, 1.5, 2.5 and compare with the 6S exact values 0.938470, 0.511828, -0.048384.
Overflow and underflow can sometimes be avoided by simple changes in a formula. Explain this in terms of √x2 + y2 = x√1 + (y/x)2 with x2 ≥ y2 and x so large that x2 would cause overflow. Invent examples of your own.
How did we use an interpolation polynomial in deriving Simpson’s rule?
What is adaptive integration? Why is it useful?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f (0) = 1, f1 = f(1) = 0, f2 = f(2) = 1, f3 = f(3) = 0, k0 = 0, k3 = 6
Solve by fixed-point iteration and answer related questions where indicated. Show details.Prove that if g is continuous in a closed interval I and its range lies in I, then the equation x = g(x) has at least one solution in I. Illustrate that it may have more than one solution in I.
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. Compute the integral J by Simpson’s rule with 2m = 8 and use the value and that in Prob. 11 to estimate the error by (10). Data from Prob. 11 Evaluate the integrals by
Find the degree of the interpolation polynomial for the data (-4, 50), (-2, 18), (0, 2), (2, 2), (4, 18), using a difference table. Find the polynomial.
In what sense is Gauss integration optimal?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 2, f1 = f(1) = 3, f2 = f(2) = 8, f3 = f(3) = 12, k0 = k3 = 0
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Design a Newton iteration. Compute 3√7, x0 = 2.
Set up (14) for the data in Prob. 3 and compute Г(1.01), Г(1.03), Г(1.05) Data from Prob. 3Calculate the Lagrange polynomial P2(x) for the values Г(1.00) = 1.0000, Г(1.02) = 0.9888, Г(1.04) = 0.9784, of the gamma function [(24) in App. A3.1] and from it approximations of Г(1.01) and
How did we obtain formulas for numeric differentiation?
Find the cubic spline g(x) for the given data with k0 and kn as given.f0 = f(0) = 4, f1 = f(2) = 0, f2 = f(4) = 4, f3 = f(6) = 80, k0 = k3 = 0
Compute ln α - ln b with 6S arithmetic for α = 4.00000 and b = 3.99900(a) As given(b) From ln (α/b)
Evaluate the integrals by Simpson’s rule with 2m as indicated, and compare with the exact value known from calculus. Find the smallest n in computing A (see Probs. 7 and 8) such that 5S-accuracy is guaranteed (a) by (4) in the use of (2) (b) by (9) in the use of (7). Data from Prob. 7 Evaluate
Write 46.9028104, 0.000317399, 54/7, 890/3 in floating-point form with 5S (5 significant digits, properly rounded).
Compute (5.346 - 3.644)/(3.444 - 3.055) as given and then rounded step wise to 3S, 2S, 1S. Comment. (“Step wise” means rounding the rounded numbers, not the given ones.)
If a cubic spline is three times continuously differentiable (that is, it has continuous first, second, and third derivatives), show that it must be a single polynomial.
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Solve Prob. 5 by Newton’s method with x0 = 5, 4, 1, -3. Explain the result.Data from Prob. 5Solve by fixed-point iteration and answer related questions where indicated. Show details.Sketch f(x) = x3 -
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. See App. A3.1. They occur in optics. Si(1) by (7), 2m = 2, 2m = 4
Use p2(x) in (18) and the values of erf x, x = 0.2, 0.4, 0.6 in Table A4 of App. 5, compute erf 0.3 and the error. (4S-exact erf 0.3 = 0.3286).
Let 19.1 and 25.84 be correctly rounded. Find the shortest interval in which the sum s of the true (un rounded) numbers must lie.
Calculate 1/e = 0.367879 (6S) from the partial sums of 5–10 terms of the Maclaurin series(a) Of e-x with w = 1(b) Of ex with x = 1 and then taking the reciprocal. Which is more accurate?
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Find the smallest positive zero of P24 = (1 - x2)P"4 = 15/2 (-7x4 + 8x2 - 1)(a) By Newton’s method(b) Exactly, by solving a quadratic equation.
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. See App. A3.1. They occur in optics. Si(1) by (7), 2m = 10
Write a program for the forward formula (14). Experiment on the increase of accuracy by successively adding terms. As data use values of some function of your choice for which your CAS gives the values needed in determining errors.
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.x + ln x = 2, x0 = 2
What is the relative error of nα∼ in terms of that of α∼?
Show that can be obtained by the division algorithm
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.f = x3 - 5x + 3 = 0, x0 = 2, 0, -2
The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. Compare your value with that possibly given by your CAS. Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. See App. A3.1. They occur in optics. C(1.25) by (7), 2m = 10
List 4–5 ideas that you feel are most important in this section. Arrange them in best logical order. Discuss them in a 2–3 page report.
Show that the relative error of α~2is about twice that of α~.
Solve x2 - 40x + 2 = 0 in two ways. Use 4S-arithmetic.
Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.Find the solution of cos x cosh x = 1 near x = 3/2π.
This simple but slowly convergent method for finding a solution of f(x) = 0 with continuous f is based on the intermediate value theorem, which states that if a continuous function f has opposite signs at some x = α and x = b (> α), that is, either f(α) < 0, f(b) > 0 or f(α) > 0,
Compute the solution of x4 = x + 0.1 near x = 0 by transforming the equation algebraically to the form x = g(x) and starting from x0 = 0.
Solve, using x0 and x1 as indicated:Prob 21, x0 = 1.0, x1 = 2.0Data from Prob. 21Apply Newton’s method (6S-accuracy). First sketch the function(s) to see what is going on.f = x3 - 5x + 3 = 0, x0 = 2, 0, -2
Solve Prob. 25 by bisection (3S-accuracy).Data from Prob. 25Compute the solution of x4 = x + 0.1 near x = 0 by transforming the equation algebraically to the form x = g(x) and starting from x0 = 0.
Consider f(x) = x4 for x0 = 0, x1 = 0.2, x2 = 0.4, x3 = 0.6, x4 = 0.8. Calculate f'2 from (14a), (14b), (14c), (15). Determine the errors. Compare and comment.
In Prob. 26 Using ex < e (0 < x < 1), conclude that |In| ≤ e/(n + 1) → 0 as n → ∞. Solve the iteration formula for In-1 = (e - In)/n, start from I15 ≈ 0 and compute 4S values of I14, I13, · · ·, I1.Data from Prob. 26Integrating by parts, show that In = ∫10 exxn dx = e
Solve, using x0 and x1 as indicated:sin x = cot x, x0 = 1, x1 = 0.5
Find the cubic spline for the data f(0) = 0, f(1) = 0, f(2) = 4, k0 = 1, k2 = 5.
The derivative f'(x) can also be approximated in terms of first-order and higher order differences. Compute f'(0.4) in Prob. 27 from this formula, using differences up to and including first order, second order, third order, fourth order. Data from Prob. 27 Consider f(x) = x4 for x0 = 0, x1 = 0.2,

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