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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Write in normal form and solve by the simplex method, assuming all xj to be nonnegative.The problem in the example in the text with the constraints interchanged.
Solve the heat problem (1)–(3) by the explicit method with h = 0.2 and k = 0.01, 8 time steps, when f(x) = x if 0 ≤ x ≤ 1/2, f(x) = 1 - x if 1/2 ≤ x ≤ 1. Compare with the 3S-values 0.108,
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = (y - x - 1)2 +
Find the potential in Fig. 472, using the given grid and the boundary values:u = 70 on the upper and left sides, u = 0 on the lower and right sides P2 P P. 13 33 23 P22 P12 P. P21 P31 |P11 P., P. 30
Find the potential in Fig. 472, using the given grid and the boundary values:u(P01) = u(P03) = u(P41) = u(P43) = 200,u(P10) = u(P30) = -400, u(P20) = 1600,u(P02) = u(P42) = u(P14) = u(P24) = u(P34) =
A laterally insulated homogeneous bar with ends at x = 0 and x = 1 has initial temperature 0. Its left end is kept at 0, whereas the temperature at the right end varies sinusoidally according toFind
Apply Runge–Kutta for systems to y'1 = 6y1 + 9y2, y'2 = y1 + 6y2, y1(0) = -3, y2(0) = -3, h = 0.05, 3 steps.
Apply Euler’s method for systems to y'1 = y2, y'2 = -4y1, y1(0) = 2, y2(0) = 0, h = 0.2, 10 steps. Sketch the solution.
Apply the A–M method to y' = (x + y - 4)2, y(0) = 4, h = 0.2, x = 0, · · ·, 1, starting with 4.00271, 4.02279, 4.08413.
Solve y' = 2x-1√y - lnx + x-1, y(1) = 0 for 1≤ x ≤ 1.8(a) By the Euler method with h = 0.1,(b) By the improved Euler method with h = 0.2(c) By RK with h = 0.4. Verify that the exact
Solve y' + y = (x + 1)2, y(0) = 3 by the improved Euler method, 10 steps, with h = 0.1, Determine the errors.
Apply the program to Example 3 (10 steps, h = 0.1).
Kutta€™s third-order method is defined by yn+1= yn+ 1/6(k1+ 4k2+ k*3) with k1and k2as in RK (Table 21.3) and k*3= hf(xn+1, yn- k1+ 2k2). Apply this method to (4) in (6). Choose h = 0.2 and
Do Prob. 17 with h = 0.01, 10 steps. Compute the errors. Compare the error for x = 0.1 with that in Prob. 17.Data from Prob. 17Solve y' = y, y(0) = 1 by Euler’s method, 10 steps, h = 0.1.
Apply the program to the square grid in 0 ≤ x ≤ 5, 0 ≤ y ≤ 5 with h = 1 and u = 220 on the upper and lower edges, u = 110 on the left edge and u = -10 on the right edge. Solve the linear
Do 10 steps. Compare as indicated. Show details.Do Prob. 15 with h = 0.2, 5 steps, and compare the errors with those in Prob. 15.Data from Prob. 15y' + y tan x = sin 2x, y(0) = 1, h = 0.1
Apply the ADI method to the Dirichlet problem in Prob. 9, using the grid in Fig. 456, as before and starting values zero.Data from Prob. 9For the grid in Fig. 456 compute the potential at the four
Solve (1)–(3) by Crank–Nicolson with r = 1 (5 steps), where:f(x) = x(1 - x), h = 0.1. (Compare with Prob.15)Data from Prob. 15f(x) = x(1 - x), h = 0.2
Do 10 steps. Compare as indicated. Show details.y' = (1 - x-1)y, y(1) = 1, h = 0.1
The classical RK for a first-order ODE extends to second-order ODEs. If the ODE is y" = f(x, y), not containing y', thenApply this RKN (Runge€“Kutta€“NystrÃ¶m) method
How much can you reduce the error in Prob. 13 by halfing h (20 steps, h = 0.05)? First guess, then compute.Data from Prob. 13 Using Prob. 12, solve y' = 2xy, y(0) = 1 (10 steps, h = 0.1, RK
If, in Prob. 13, the axes are grounded (u = 0), what constant potential must the other portion of the boundary have in order to produce 220 V at P11?Data from Prob. 13Solve the mixed boundary value
Using the answer to Prob. 13, try to sketch some isotherms.Data from Prob.13For the square 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 let the boundary temperatures be on the horizontal and 50°C on the vertical
Solve the heat problem (1)–(3) by Crank–Nicolson for 0 ≤ t ≤ 0.20 with h = 0.2 and k = 0.04 when f(x) = x if 0 ≤ x ≤ 1/2, f(x) = 1 - x if 1/2 ≤ x ≤ 1. Compare with the exact
Do 10 steps. Compare as indicated. Show details.y' = y - y2, y(0) = 0.2, h = 0.1. Compare with Prob. 8.Data from Prob. 8y' = y - y2, y(0) = 0.2, h = 0.1
xy" + y' + xy = 0, y(1) = 0.765198, y'(1) = -0.440051, h = 0.5, 5 steps. (This gives the standard solution J0(x) in Fig. 110 0.5 - LO
Do Prob. 9 by Gauss€“Seidel, starting from 0. Compare and comment.Data from Prob. 9For the grid in Fig. 456 compute the potential at the four internal points by Gauss and by 5
Do 10 steps. Solve exactly. Compute the error. Show details.Do Prob. 7 using the improved Euler method, 20 steps with h = 0.05. Compare.Data from Prob. 7y' - xy2 = 0, y(0) = 1, h = 0.1
Solve by the classical RK.The system in Prob. 4Data from Prob. 4Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = -3y1 + y2, y'2 = y1 - 3y2, y1(0) = 2,
If the left end of a laterally insulated bar extending from x = 0 to x = 1 is insulated, the boundary condition at x = 0 is un(0, t) = ux(0, t) = 0. Show that, in the application of the explicit
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = x/y, y(1) = 3, h
For the grid in Fig. 456 compute the potential at the four internal points by Gauss and by 5 Gauss€“Seidel steps with starting values 100, 100, 100, 100 (showing the details of your work)
Do 10 steps. Solve exactly. Compute the error. Show details.y' = y - y2, y(0) = 0.2, h = 0.1
Solve by the classical RK.The system in Prob. 2Data from Prob. 2y'1 = -y1 + y2, y'2 = -y1 - y2, y1(0) = 0, y2(0) = 4, h = 0.2, 5 steps
In a laterally insulated bar of length 1 let the initial temperature be f(x) = x if 0 ≤ x < 0.5, f(x) = 1 -x if 0.5 ≤ x ≤ 1. Let (1) and (3) hold. Apply the explicit method with h = 0.2, k =
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = 1 - 4y2, y(0) =
For the grid in Fig. 456 compute the potential at the four internal points by Gauss and by 5 Gauss€“Seidel steps with starting values 100, 100, 100, 100 (showing the details of your work)
Compute approximate values in Prob. 7, using a finer grid (h = 0.1, k = 0.1), and notice the increase in accuracy.Data from Prob. 7If the string governed by the wave equation (1) starts from its
Do 10 steps. Solve exactly. Compute the error. Show details.y' = 2(1 + y2), y(0) = 0, h = 0.05
Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = y1, y'2 = -y2, y1(0) = 2, y2(0) = 2, h = 0.1, 10 steps
Solve ˆ‡2u = -Ï€2y sin 1/3Ï€x for the grid in Fig. 462 and uy(1, 3) = uy(2, 3) = 1/2ˆš243, u = 0 on the other three sides of the square. и%3D 9 3 P, P.
For the grid in Fig. 456 compute the potential at the four internal points by Gauss and by 5 Gauss€“Seidel steps with starting values 100, 100, 100, 100 (showing the details of your work)
Solve (1)–(3) (h = k = 0.2, 5 time steps) subject to f(x) = x2, g(x) = 2x, ux(0, t) = 2t, u(1, t) = (1 + t)2.
Do 10 steps. Solve exactly. Compute the error. Show details.y' = (y + x)2, y(0) = 0, h = 0.1
Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = -3y1 + y2, y'2 = y1 - 3y2, y1(0) = 2, y2(0) = 0, h = 0.1, 5 steps
(a) Write programs for the explicit and the Crank€”Nicolson methods.(b) Apply the programs to the heat problem of a laterally insulated bar of length 1 with u(x, 0) = sin Ï€x and
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are givenDo Prob. 2 by RK, 5
Solve the mixed boundary value problem for the Laplace equation ˆ‡2u = 0 in the rectangle in Fig. 458a (using the grid in Fig. 458b) and the boundary conditions ux= 0 on the left edge, ux=
Finer grid of 3 × 3 inner points. Solve Example 1, choosing h = 12/4 = 3 (instead of h = 12/3 = 4) and the same starting values.
Show that (12) gives the starting formula(where one can evaluate the integral numerically if necessary). In what case is this identical with (8)? (из+1,0 + и-1,0) + Иi,1 8(s) ds
Do 10 steps. Solve exactly. Compute the error. Show details.y' = 1/2π√1 - y2, y(0) = 0, h = 0.1
Solve by the Euler’s method. Graph the solution in the y1y2-plane. Calculate the errors.y'1 = -y1 + y2, y'2 = -y1 - y2, y1(0) = 0, y2(0) = 4, h = 0.2, 5 steps
Solve the initial value problem by Adams–Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are giveny' = 2xy, y(0) = 1, h
Using the present method, solve (1)–(4) with h = k = 0.2 for the given initial deflection and initial velocity 0 on the given t-interval.f(x) = x2 - x3, 0 ≤ t ≤ 2
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Fit a parabola (7) to the points (x, y). Check by sketching.t[hr] = Worker’s time on duty, y[sec] = His/her reaction time, (t, y) = (1, 2.0), (2, 1.78), (3, 1.90), (4, 2.35), (5, 2.70)
Compute the norms (5), (6), (7). Compute a corresponding unit vector (vector of norm 1) with respect to the l∞-norm.[0 0 0 1 0]
Do 4 steps with scaling for the matrix in Prob. 19, starting for [1 1 1] and computing the Rayliegh quotients and error bounds.Data from Prob. 19Compute the inverse of: 15
Find and graph three circular disks that must contain all the eigenvalues of the matrix:In Prob. 20Data from Prob. 20Compute the inverse of: 5 1 6.
Find and graph three circular disks that must contain all the eigenvalues of the matrix:In Prob. 18Data from Prob. 18Compute the inverse of: Г2.0 0.1 3.3 1.6 4.4 0.5 0.3 -4.3 2.8
Fit and graph:A straight line to (-1, 0), (0, 2), (1, 2), (2, 3), (3, 3)
Compute the l1-, l2-, and l∞-norms of the vectors.[0 0 0 -1 0]T
Compute the l1-, l2-, and l∞-norms of the vectors.[0.2 -8.1 0.4 0 0 -1.3 2]T
(a) Vector norms in our text are equivalent, that is, they are related by double inequalities; for instance,Hence if for some x, one norm is large (or small), the other norm must also be large (or
Show that κ(A) ≥ 1 for the matrix norms (10), (11), and κ(A) ≥ √n for the Frobenius norm (9).
Do 3 steps without scaling, starting from [1 1 1]T.0.2x1 + 4.0x2 - 0.4x3 = 32.00.5x1 - 0.2x2 + 2.5x3 = -5.17.5x1 + 0.1x2 - 1.5x3 = -12.7
Compute the norms (9), (10), (11) for the following (square) matrices. Comment on the reasons for greater or smaller differences among the three numbers. -k 2k -k k -2k -k -k 2k
Solve Ax = b1, Ax = b2. Compare the solutions and comment. Compute the condition number of A. 3.0 1.7 [4.7 4.7 A bị b2 |2.71 1.7 1.0 2.7 ||
Prove Theorem 2. Let A = B + C, B = diag(αjj), At = B + tC, and let t increase continuously from 0 to 1.
Compute the norms (9), (10), (11) for the following (square) matrices. Comment on the reasons for greater or smaller differences among the three numbers.The matrix in Prob. 10Data from Prob.
(a) Find Î± and b such that Î±x1+ x2= b, x1+ x2= 3 has(i) A unique solution(ii) Infinitely many solutions(iii) No solutions.(b) Apply the Gauss elimination to the following
Verify (12) for the matrices in Probs. 9 and 10.Data from Prob. 9Compute the matrix norm and the condition number corresponding to the l1-vector norm. 2 4
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 9Data from Prob. 9Show the factorization and solve.0.01x1 +
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.The system in
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 5.25 21 10.5 10.5 5.25 4.2 5.25 4.2 3.5 5.25 4.2 3.5 3.
Use (4) to obtain an upper bound for the spectral radius:In Prob. 5Data from Prob. 5Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find the spectrum and compare.
Find the inverse by the Gauss–Jordan method, showing the details.In Prob. 4Data from Prob. 4Show the factorization and solve by Doolittle’s method. 2x1 + x2 + 2x3 =
The least squares approximation of a function f(x) on an interval Î± ‰¤ x ‰¤ b by a functionwhere y0(x), · · ·, ym(x) are given functions,
Do 5 steps, starting from x0 = [1 1 1]. Compare with the Gauss–Seidel iteration. Which of the two seems to converge faster? Show the details of your work.The system in
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 0.01 0.01 1 0.01 0.01 1
Solve5x1 + x2 - 3x3 = 17 - 5x2 + 15x3 = -102x1 - 3x2 + 9x3 = 0
Solve 3x2 - 6x3 = 0 4x1 - x2 + 2x3 = 16-5x1 + 2x2 - 4x3 = -20
In Prob. 5, compute C(a) If you solve the first equation for x1, the second for x2, the third for x3, proving convergence(b) If you nonsensically solve the third equation for x1, the first for x2,
(a) Write a program for n Ã— n matrices that prints every step. Apply it to the (nonsymmetric!) matrix (20 steps), starting from [1 1 1]T.(b) Experiment in (a)
Compute the matrix norm and the condition number corresponding to the l1-vector norm. 6 5
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?).
Compute the matrix norm and the condition number corresponding to the l1-vector norm. [ 2.1 4.5 0.5 1.8
Do three QR-steps to find approximations of the eigenvalues of: 14.2 -0.1 -0.1 -6.3 0.2 0.2 2.1
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Fit a parabola (7) to the points (x, y). Check by sketching.(-1, 5), (1, 3), (2, 4), (3, 8)
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?).
Apply the power method (3 steps) with scaling, using x0= [1 1 1]Tor [1 1 1 1]T, as applicable. Give Rayleigh quotients and error
By what integer factor can you at most reduce the Gerschgorin circle with center 3 in Prob. 6?Data from Prob. 6Find and sketch disks or intervals that contain the eigenvalues. If you have a CAS, find
Do three QR-steps to find approximations of the eigenvalues of:The matrix in the answer to Prob. 1Data from Prob. 1Tridiagonalize. Show the details. 0.98 0.04 0.44 0.04 0.56 0.40 0.44 0.80 0.40
Solve the following linear systems by Gauss elimination, with partial pivoting if necessary (but without scaling). Show the intermediate steps. Check the result by substitution. If no solution or
Fit a straight line to the given points (x, y) by least squares. Show the details. Check your result by sketching the points and the line. Judge the goodness of fit.Estimate R from (i, U) = (2, 104),
Do 5 steps, starting from x0 = [1 1 1]T and using 6S in the computation. Make sure that you solve each equation for the variable that has the largest coefficient (why?).
Apply the power method (3 steps) with scaling, using x0= [1 1 1]Tor [1 1 1 1]T, as applicable. Give Rayleigh quotients and error

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