Consider 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?
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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) = 0Given the data below: Xo = 1 X1= 2 x2 = 4 Axo) = 2 Ax1) = 3 Ax2 : = 8 (i) Calculate the second-order interpolating polynomial using the method of the Newton's interpolating polynomial. (ii) Use the interpolating polynomial in (i) to calculate the approximated/interpolated functional value at x = 3, i.e., (3). (iii)Calculate the percentage relative error if the true value of f(3) is 4.8.Q1 Use Matlab to find the roots of the following polynomials: (a) P₁ = 2s5 — 4s² + 3s³ − 2s² + 3s + 5 (b) P2 =S s6 + 5s³ s² + 10 -
- If you can please take a look, I struggled with figuring out problem 2, 4, , and 6. Please let me know if you have any solutions.Please solve the following by hand and without the use of AI. I am working to understand the step by step procedure of solving this problem so pleaase give a detailed step by step procedure, explaining each part as you go. Thank you!Only part d. please show all work.
- (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 episPlease don't provide handwritten solution ......1. Solve the following initial value problems by the Laplace Transform. Show all details including the partial fraction solutions. i. y' + 4y = 0, when y(0) = 2.8. ii. y' +y = 17 sin(2t),when y(0) = –1. iii.y" – y' – 6y = 0, when y(0) = 6 and y(0) = 13. iv.y" – 2y' – 3y = 0,when y(1) = -3 and y' (1) = -17. %3D 1 %3D - %3D
- The natural exponential function can be expressed by . Determine e2by calculating the sum of the series for:(a) n = 5, (b) n = 15, (c) n = 25For each part create a vector n in which the first element is 0, the incrementis 1, and the last term is 5, 15, or 25. Then use element-by-element calculations to create a vector in which the elements are . Finally, use the MATLAB built-in function sum to add the terms of the series. Compare thevalues obtained in parts (a), (b), and (c) with the value of e2calculated byMATLAB.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.Hello, 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.