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)
- Consider 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_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_forwardExample: Sketch the graphs of the following functions. 1. y=sin2x So w nY oy shrin ILYarrow_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_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_forwardVerify if the following functions are Linear or not. Support your conclusion with appropriate reason. a) F(x) = b) f(x) =rcos wtarrow_forward
- Q1: 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_forwardCalculate the value of the first order derivative at the point x=0.2 with a single finite difference formula using all the following values. 0.1 0.2 0.3 0.4 f(x) 0.000 000 0.078 348 0.138 910 0.192 916 0.244981arrow_forwardCircle final answer Question: Which series helps us to derive the discretized form of the governing equations? Remember to circle final answerarrow_forward
- Do not actually solve the problem numerically or algebraically, just pick the one equation and define the relevant knowns and single unknown. Don’t forget to include direction when called for by a vector variable 12) The air conditioner removes 2.7 kJ of heat from inside a house with 450 m3 of air in it. At a typical air density of 1.3 kg/m3 that means 585 kg of air. If the specific heat of air is 1.01 kJ/(kg oC), by how much would this cool the house if no heat got in through the rest of the house during that time?arrow_forwardConsider the following linear equations,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