Use (a) the Newton-Raphson method and (b) the modified secant method
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Chapter 6 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
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Advanced Engineering Mathematics
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Fundamentals of Differential Equations (9th Edition)
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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')arrow_forwardIntegrate the function X^2+2*x on the interval 0 to 3 with a step size of .5 using the trapezoid rule.arrow_forward3. 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_forward
- x^2-5x^(1/3)+1=0 Has a root between 2 and 2.5 use bisection method to three iterations by hand.arrow_forward2. Solve the following ODE in space using finite difference method based on central differences with error O(h). Use a five node grid. 4u" - 25u0 (0)=0 (1)=2 Solve analytically and compare the solution values at the nodes.arrow_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_forward
- 4. Solve the 2D Laplace's equation on a square domain using finite difference method based on central differences with error O(h2). Use four nodes in each direction. 0 (1.y)--5 (2.1) 0arrow_forwardConsider the function p(x) = x² - 4x³+3x²+x-1. If bisection is used with the function with a starting interval [2 3], which of the following is the absolute value of relative error at the end of the second iteration? 0.0213 0.091 0.0435 0.2000arrow_forwardSolve the following differential equation for axial deformation of a bar of length 12mm using Galerkin Weighted Residual method 2022/09/ d² dx² = -0.75(4- x)² One end of the bar is fixed whereas the displacement is 12 mm at the other end of the bar. You may use Matlab for computations. Use the trial function û(x) = Co + ₂x + ₂x²arrow_forward
- Which procedure provides a method that may be used to apply Cstigliano's second theorem?arrow_forwardUsing trapezoidal rule integration with 2 equal sized sub-intervals, find the area under the curve defined by the polynomial y = 0.10x5 + 0.1x³ + 24 between limits of 4.2 and 11.1. Give your answer to a precision of at least 3 significant figures. Your Answer:arrow_forward8. Use the Lagrange multiplier method to find the point on the line 3x + 8y = 146 that is closest to the origin.arrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
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