EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 9780100254145
Author: Chapra
Publisher: YUZU
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Textbook Question
Chapter 6, Problem 22P
Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration and (b) the Newton-Raphson method:
Employ initial guesses of
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For the DE: dy/dx=2x-y y(0)=2 with h=0.2, solve for y using each method below in the range of 0 <= x <= 3:
Q1) Using Matlab to employ the Euler Method (Sect 2.4)
Q2) Using Matlab to employ the Improved Euler Method (Sect 2.5
close all
clear all
% Let's program exact soln
for i=1:5
x_exact(i)=0.5*i-0.5;
y_exact(i)=-x_exact(i)-1+exp(x_exact(i));
end
plot(x_exact,y_exact,'b')
% now for Euler's
h=0.5
x_EM(1)=0;
y_EM(1)=0;
for i=2:5
x_EM(i)=x_EM(i-1)+h;
y_EM(i)=y_EM(i-1)+(h*(x_EM(i-1)+y_EM(i-1)));
end
hold on
plot (x_EM,y_EM,'r')
% Improved Euler's Method
h=0.5
x_IE(1)=0;
y_IE(1)=0;
for i=2:1:5
kA=x_IE(i-1)+y_IE(i-1);
u=y_IE(i-1)+h*kA;
x_IE(i)=x_IE(i-1)+h;
kB=x_IE(i)+u;
k=(kA+kB)/2;
y_IE(i)=y_IE(i-1)+h*k;
end
hold on
plot(x_IE,y_IE,'k')
Find the three unknown on this problems using
Elimination Method and Cramer's Rule. Attach your
solutions and indicate your final answer.
Problem 1.
7z 5y
3z
16
%3D
3z
5y + 2z
-8
%3D
5z + 3y
7z
= 0
Problem 2.
4x-2y+3z 1
*+3y-4z -7
3x+ y+2z 5
Problem1: Solve the system of linear equations by each of the methods listed below.
(a) Gaussian elimination with back-substitution
(b) Gauss-Jordan elimination
(c) Cramer's Rule
3x, + 3x, + 5x, = 1
3x, + 5x, + 9x3 = 2
5x, + 9x, + 17x, = 4
Chapter 6 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
Ch. 6 - 6.1 Use simple fixed-point iteration to locate the...Ch. 6 - 6.2 Determine the highest real root of...Ch. 6 - Use (a) fixed-point iteration and (b) the...Ch. 6 - Determine the real roots of f(x)=1+5.5x4x2+0.5x3:...Ch. 6 - 6.5 Employ the Newton-Raphson method to determine...Ch. 6 - Determine the lowest real root of...Ch. 6 - 6.7 Locate the first positive root of
Where x...Ch. 6 - 6.8 Determine the real root of, with the modified...Ch. 6 - 6.9 Determine the highest real root of:...Ch. 6 - 6.10 Determine the lowest positive root...
Ch. 6 - 6.11 Use the Newton-Raphson method to find the...Ch. 6 - 6.12 Given
Use a root location technique to...Ch. 6 - You must determine the root of the following...Ch. 6 - Use (a) the Newton-Raphson method and (b) the...Ch. 6 - 6.15 The “divide and average” method, an old-time...Ch. 6 - (a) Apply the Newton-Raphson method to the...Ch. 6 - 6.17 The polynomial has a real root between 15...Ch. 6 - Use the secant method on the circle function...Ch. 6 - You are designing a spherical tank (Fig. P6.19) to...Ch. 6 - 6.20 The Manning equation can be written for a...Ch. 6 - 6.21 The function has a double root at. Use (a)...Ch. 6 - 6.22 Determine the roots of the following...Ch. 6 - 6.23 Determine the roots of the simultaneous...Ch. 6 - Repeat Prob. 6.23 except determine the positive...Ch. 6 - A mass balance for a pollutant in a well-mixed...Ch. 6 - Fir Prob. 6.25, the root can be located with...Ch. 6 - 6.27 Develop a user-friendly program for the...Ch. 6 - Develop a user-friendly program for the secant...Ch. 6 - 6.29 Develop a user-friendly program for the...Ch. 6 - 6.30 Develop a user-friendly program for Brent’s...Ch. 6 - 6.31 Develop a user-friendly program for the...Ch. 6 - 6.32 Use the program you developed in Prob. 6.31...
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- 3. Using the trial function u¹(x) = a sin(x) and weighting function w¹(x) = b sin(x) find an approximate solution to the following boundary value problems by determining the value of coefficient a. For each one, also find the exact solution using Matlab and plot the exact and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution, and (iii) plotting the solution) a. (U₁xx -2 = 0 u(0) = 0 u(1) = 0 b. Modify the trial function and find an approximation for the following boundary value problem. (Hint: you will need to add an extra term to the function to make it satisfy the boundary conditions.) (U₁xx-2 = 0 u(0) = 1 u(1) = 0arrow_forward3. Using the trial function uh(x) = a sin(x) and weighting function wh(x) = b sin(x) find an approximate solution to the following boundary value problems by determining the value of coefficient a. For each one, also find the exact solution using Matlab and plot the exact and approximate solutions. (One point each for: (i) finding a, (ii) finding the exact solution, and (iii) plotting the solution) a. (U₁xx - 2 = 0 u(0) = 0 u(1) = 0 b. Modify the trial function and find an approximation for the following boundary value problem. (Hint: you will need to add an extra term to the function to make it satisfy the boundary conditions.) (U₁xx - 2 = 0 u(0) = 1 u(1) = 0arrow_forwardQ3) Find the optimal solution by using graphical method:. Max Z = x1 + 2x2 Subject to : 2x1 + x2 < 100 X1 +x2 < 80 X1 < 40 X1, X2 2 0arrow_forward
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