Use the Galerkin method to solve the following boundary-value problem using a: (a) one-term approximation and (b) two-term approximation. Compare your results with the exact solution by plotting them on the same graph.
Hint: Use the following one- and two-term approximations
One-term approximation:
Two-term approximation:
The exact solution is
The approximate solution is split into two parts. The first term satisfies the given essential boundary conditions exactly, i.e.,
Want to see the full answer?
Check out a sample textbook solutionChapter 2 Solutions
Introduction To Finite Element Analysis And Design
Additional Engineering Textbook Solutions
Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
Fox and McDonald's Introduction to Fluid Mechanics
Engineering Mechanics: Statics
Statics and Mechanics of Materials (5th Edition)
Fundamentals Of Thermodynamics
Engineering Mechanics: Statics & Dynamics (14th Edition)
- (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_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_forwardUse the Lax method to solve the inviscid Burgers' equation using a mesh with 51 points in the x direction. Solve this equation for a right propagating discontinu- ity with initial data u = 1 on the first 11 mesh points and u = 0 at all other points. Repeat your calculations for Courant numbers of 1.0, 0.6, and 0.3 and compare your numerical solutions with the analytical solution at the same time.arrow_forward
- i need the answer quicklyarrow_forwardUse the graphical method to find the optimal solution for the following LP equations: Min Z=10 X1 + 25 X2 Subject to X1220, X2 ≤40 ,XI +X2 ≥ 50 X1, X2 ≥ 0.arrow_forwardVerify if the following functions are Linear or not. Support your conclusion with appropriate reason. a) F(x) = b) f(x) =rcos wtarrow_forward
- I have the answers but I want to know the steps with calculation to reach these answers. blank1 = 1.764 , blank2 = -2.18 , blank3= 43.307 , blank4 = 1.764arrow_forwardConsider 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?arrow_forwardQ1: The number of bacterial cells (P) in a given reactor is related to time in days (t) as described by the following mathematical model: dp dt 0.0000007 P², If at initial time (P = 106). Determine the number of cells when (t 2days) using the fourth order Runge-Kutta method and at time increment of (1 day). = = 0.3 P 1arrow_forward
- 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_forwardHello, could I get some help with a Differential Equations problem that involves Eigenvalues and Eigenvectors? The set up is: There are two toy rail cars, Car 1, and Car 2. Car 1 has a mass of 2 kg, and is traveling 3 m/s towards Car 2, which has a mass of 1 kg, and is traveling towards Car 1 at 2 m/s. There is a bumper on the second rail car that engages at the moment the cars hit (connecting Car 1 and Car 2), and does not let go. The bumper acts like a spring with spring constant K = 2 N/m. Car 2 is 7 m from the wall at the time of collision (Car 2 is between Car 1 and the wall). I have attached the work I have done so far, but I'm not understanding how to find x1(t) and x2(t), how we know Car 2 hits the wall (or moves away from it), and at what speed Car 1 travels to stay in place after link-up (given: 1 m/s, but not sure why that is). Thank you in advance.arrow_forwardConsider the function p(x) = x² - 4x³+3x²+x-1. Use Newton-Raphson's method with initial guess of 3. What's the updated value of the root at the end of the second iteration? Type your answer...arrow_forward
- Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning