Numerical Methods

3. Buckling of Column: Under a uniform load, small deflections y of a simply supported beam are given by where L = 10 feet is the length of the beam, EI = 1900 is modulus of elasticity times moment of inertia, and q = -0.6 is load distribution. The beam extends from z = 0 to z = L. The goal is to find y at every 0.02 foot using the Direct Method and plot y(z) versus I. (a) Using centered-differencing, formulate the finite difference approximation. To illustrate your work, use 5 grid points (2 of them are on the boundaries, y = 0 and y = L). Write down finite-difference approximations for the interior points, for a general uniform grid spacing of Ar. Then write your algebraic equations in the matrix-vector form with unknown vector on the left hand side and all known quantities on the right hand side. (b) Using a computer program for N + 1 total points, solve the above equation using the direct method with the grid spacing of 0.02 foot. Plot y(z) versus z. Compare your solution with the Exact Solution (clearly label your graphs).

The graph off over the interval [1, 9] is shown inthe figure. Using the data in the figure, find a midpointapproximation with 4 equal subdivisions forn \int_{i}^{9} f(x) d x

66. You purchase fresh strawberries in Mexico for 28 pesos per kilogram. 'What is the price in U.S. dollars per pound? (1 kg=2.205 lb)

29. A square meter is_____times as large as a square millimetre.

6) Determine whether the point (7, 8) is in the feasible set of the system of inequalities: \left\{\begin{array}{l} 2 x+6 y \leq 66 \\ 4 x+2 y \leq 48 \\ x+y \leq 14 \\ x \geq 0, y \geq 0 \end{array}\right. A) Yes B) No

5.17. The equation of state for a gas is given by the van der Waals equation \left(P+\frac{a}{v^{2}}\right)(v-b)=R T where P is the pressure, v is the specific volume, T'is the temperature,R is the gas constant, and a, b are constants that depend on the gas. For P= 70 atm, T= 200 K, R =0.08205 liter atm/mole K, a = 3.59, and b =0.0427, the specific volume is given in liters/mole. Find this value using the Newton-Raphson method, after obtaining the approximate value by the search method. Also, use the roots function in MATLAB to obtain the solution and compare the result with that obtained earlier.

Consider the linear equation Y^{\prime}(x)=\lambda Y(x)+(1-\lambda) \cos (x)-(1+\lambda) \sin (x), q u a d Y(0)=1 The true solution is Y (x) = sin(x)+cos(x). Solve this problem using Euler's method with several values of A and h, for 0 < x < 10. Comment on the results. a) = -1: h = 0.5.0. 25. 0.125. )\= 1; h = 0.5, 0.25, 0.125 \ = -5; h = 0.5, 0.25, 0.125, 0.0625II 1) A = 5; h = 0.0625

In Probs. 33-36 find the transform. In Probs. 37-45 find the inverse transform. Show the details of your work. \text { 35. } 0.5 e^{-4.5 t} \sin 2 \pi t

5) Which one of the following points is in the feasible set of the system of inequalities? \left\{\begin{array}{l} x \geq 0, y \geq 0 \\ x+y \leq 12 \\ 2 x+5 y \leq 50 \end{array}\right. A) (-2, 0) B) (1, 10) C) (1, -2) D) (3, 2) E) none of these

1. Using linear stability analysis, identify any restrictions on the step size for a stable solution.