EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 9780100254145
Author: Chapra
Publisher: YUZU
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Chapter 26, Problem 14P
Given the first-order ODE
Solve this stiff
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sole using laplace transforms
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Consider the following ODE in time (from Homework 6). Integrate in time using 4th order Runge-Kutta
method. Compare this solution with the finite difference and analytical solutions from Homework 6.
4 25
u(0)=0
(a) Use At = 0.2 up to a final time t = 1.0.
(b) Use At=0.1 up to a final time t = 1.0.
0
(0)=2
(c) Discuss the difference in the two solutions of parts (a) and (b). Why are they so different?
1 - Explain Van-Neumann stability by finite difference schemes for this
equations
a- Laplace equation.
J²u
dyz (x, y) = 0
J²u
dx2(x, y) +
b- poison equation.
J²u
J²u
{ (x, y) + əy² (x, y) = f (x, y)
əx²
Chapter 26 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 26 - Given dydx=200,000y+200,000exex (a) Estimate the...Ch. 26 - Given dydx=30(costy)+3sint If y(0)=1, use the...Ch. 26 - 26.3 Given
If, obtain a solution from using a...Ch. 26 - Solve the following initial-value problem over the...Ch. 26 - Repeat Prob. 26.4, but use the fourth-order Adams...Ch. 26 - Solve the following initial-value problem from...Ch. 26 - Solve the following initial-value problem from...Ch. 26 - Solve the following initial-value problem from...Ch. 26 - Develop a program for the implicit Euler method...Ch. 26 - 26.10 Develop a program for the implicit Euler...
Ch. 26 - Develop a user-friendly program for the...Ch. 26 - 26.12 Use the program developed in Prob. 26.11 to...Ch. 26 - 26.13 Consider the thin rod of length l moving in...Ch. 26 - Given the first-order ODE dxdt=700x1000etx(t=0)=4...Ch. 26 - 26.15 The following second-order ODE is...Ch. 26 - 26.16 Solve the following differential equation...
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- dy -3t 1. The complementary equation to +2 + 4y = 5e is: dt² dt d²y -3t 2. The particular solution trial function of +2 + 4y = 5e (before substitution in t dt² dt differential equation) is: 3. The complementary function (YCF) when solving the differential equation d²y - n²y = 5cos(2t), where n is a constant, is: dt²arrow_forwardConsider the following linear equations,arrow_forward(3) For the given boundary value problem, the exact solution is given as = 3x - 7y. (a) Based on the exact solution, find the values on all sides, (b) discretize the domain into 16 elements and 15 evenly spaced nodes. Run poisson.m and check if the finite element approximation and exact solution matches, (c) plot the D values from step (b) using topo.m. y Side 3 Side 1 8.0 (4) The temperature distribution in a flat slab needs to be studied under the conditions shown i the table. The ? in table indicates insulated boundary and Q is the distributed heat source. I all cases assume the upper and lower boundaries are insulated. Assume that the units of length energy, and temperature for the values shown are consistent with a unit value for the coefficier of thermal conductivity. Boundary Temperatures 6 Case A C D. D. 00 LEGION Side 4 z episarrow_forward
- 2) Sketch the graph of the following step functions and describe this function in terms of unit step functions and find the Laplace transforms. 0 ≤t <2 2 ≤t<4 a) f1(t) = 4 t, = { ₁ - 4. 0, t≥ 4 (0, 0 ≤ t < 1 1arrow_forwardQ1 The frequency equation of a 3 Degree of Freedom spring mass system (Figure Q1) is given as: m³w6 – 4km²w4 + 3k?mw² = 0 where the value of the mass, m = 0.1 kg and spring stiffness coefficient, k = 10 N/m. By performing your calculation (final answer in 3 decimal points): (a) Determine the most positive roots of the system, w using Bisection Method. Use initial value of [5.7, 15.5]. Iterate until 5th iteration. (b) Determine the most positive roots of the system, w using Secant Method. Use initial value of [5.7, 15.5]. Iterate until 5th iteration c) The exact value of the most positive roots, w is given as 10 rad/s. Based from your answer in Q1(a) and Q1(b), identify its percentage relative error and justify which method is more accurate.arrow_forwardSolve the initial value problem below using the method of Laplace transforms. y" - 3y' – 18y = 0, y(0) = 1, y'(0) = 42arrow_forward1. A spring mass system serving as a shock absorber under a car's suspension, supports the M=1000kgmass of the car. For this shock absorber,k=1000N/m and c=2000N s/m. The car drives over a corrugated road with force F=2000sin(wt)N. Use your notes to model the second order differential equation suited to thisapplication. Simplify the equation with the coefficient of x'' as one. Solve x (the general solution) interms of using the complimentary and particular solution method. In determining the coefficients ofyour particular solution, it will be required that you assume w2 -1=w or . Do not 1-w2=-wuse Matlab as its solution will not be identifiable in the solution entry. Do not determine the value of w.You must indicate in your solution:1. The simplified differential equation in terms of the displacement x you will be solving2. The m equation and complimentary solution3. The choice for the particular solution and the actual particular solution xp4. Express the solution x as a piecewise…arrow_forwardAn object attached to a spring undergoes simple harmonic motion modeled by the differential equation d²x = 0 where x (t) is the displacement of the mass (relative to equilibrium) at time t, m is the mass of the object, and k is the spring constant. A mass of 3 kilograms stretches the spring 0.2 meters. dt² Use this information to find the spring constant. (Use g = 9.8 meters/second²) m k = + kx The previous mass is detached from the spring and a mass of 17 kilograms is attached. This mass is displaced 0.45 meters below equilibrium and then launched with an initial velocity of 2 meters/second. Write the equation of motion in the form x(t) = c₁ cos(wt) + c₂ sin(wt). Do not leave unknown constants in your equation. x(t) = Rewrite the equation of motion in the form ä(t) = A sin(wt + ), where 0 ≤ ¢ < 2π. Do not leave unknown constants in your equation. x(t) =arrow_forwardMy question and answer is in the image. Can you please check my work? A 2 kg mass is attached to a spring with spring constant 50 N/m. The mass is driven by an external force equal tof(t) = 2 sin(5t). The mass is initially released from rest from a point 1 m below the equilibrium position. (Use theconvention that displacements measured below the equilibrium position are positive.)(a) Write the initial-value problem which describes the position of the mass. 2y"+50y=2cos(5t) (b) Find the solution to your initial-value problem from part (a). (1+(1/2)tcos(t))cos(5t)-(1/2)tcos(t) (c) Circle the letter of the graph below that could correspond to the solution. 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