4. Solve the 2D Laplace's equation on a square domain using finite difference method based on central differences with error O(h). Use four nodes in each direction. (2.0)=0 0= (2.1) 0
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- 2. 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.(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 episIntegrate the function X^2+2*x on the interval 0 to 3 with a step size of .5 using the trapezoid rule.
- Use 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.The mass and stiffness matrix of the system is given by: M = and m2 k, +k, -k, K = -k, k, +k, Take m1 =10 kg, m2-5 kg, k1=k3=100 N/m, k2=84 N/m Use modal analysis and determi ne: a) The normalized stiffness ki b) Its eigenvalues and eigenvectorsQ-12)-for the model below Max z=10X1+5X2 st 4X1+2X2540 3X1+3X25 30 X12 0, X2 20 M Q-13)For question-12,the solution kind is: O infeasible O unbounded O optimal multiple
- Verify if the following functions are Linear or not. Support your conclusion with appropriate reason. a) F(x) = b) f(x) =rcos wt3. 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) = 03. 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) = 0
- 1. Consider the following square matrix A = 9 6 (a) Determine the eigenvalues and eigenvector(s) of A. (b) Find the modal matrix M and diagonalize A through similarity transformation M 'AM. do we get a Jordan form or not? Explain why?For the following system, perform only the first elimination using Gaussian Elimination with partial pivoting.7. Consider an element that conducts heat as shown below with length L, cross sectional area A, and heat conductance k. Nodes 1 and 2 have temperatures of T, and T2. The heat flux q due to conduction is given by: dT ΔΤ q = - k dx Ax This relationship is analogous to Hooke's Law from the prior problem. Heat transfer by conduction Qc is given by: Oc = qA Use equilibrium requirements to solve for the heat transfer by conduction Qci and Qcz at the nodes and use these equations to derive a "conductance matrix" (or the stiffness matrix due to conduction which is the analog of the stiffness matrix) for this heat conducting element. For the sign convention, consider heat flux positive when heat flows into the element and negative when it flows out of the element. Show your full matrix equation and the conductance matrix. Oci T T2 Oc2 2