You are given the following initial value problem: dy dt = 2y + t 2 y(1) = 5 (a)

Question

You are given the following initial value problem: dy dt = 2y + t 2 y(1) = 5 (a) Write down the time step update, yi+1, if we are using the Backward Euler method. Is this an explicit or implicit method? Justify. (b) Use the Backward Euler method to estimate y(2), using a time step size of t = 0.5. (c) Classify the ODE in terms of linearity, order, and homogeneity.

2. Consider the following boundary value problem. x 2 d 2u dx2 + x = 0, 1 < x < 5 u(1) = 0 u(5) = 1 (a) Use a second order difference formula to discretize the equation. Use an equal spacing of x = 1 in the domain. (b) Justify the accuracy of the difference formula. (c) Write the resulting linear system of equations in matrix form. Do not solve them.

3. Which one of the following is a boundary value problem?

Justify. (a) d 2y dx2 x 2 = y, y(0) = 1, dy dx (1) = 2 (b) d 2y dx2 x 2 = y, y(0) = 1, dy dx (0) = 2

4. The concentration of two chemical species A and B can be modeled by the following system of differential equations: A = A 2 + B B = 2B + A 2 with A(0) = 1 and B(0) = 0. (a) Use two steps of the forward Euler method, using a timestep of t = 0.5, to integrate this system in time.

5. You are given a matrix A and two vectors b1 and b2. You are tasked with computing the solution to Ax1 = b1 and Ax2 = b2. Would you use Gaussian elimination or LU decomposition? Justify.

8.Consider the function f(x) = x 2 sin(x). (a) Expand f(x) in a Taylor series up to second order about the point x = 1. (b) Compute f 0 (x). (c) Compute the forward, backward and central difference approximations to f 0 (x) at x = 1 using a spacing x = 0.5.

9. Consider the differential equation d 2y dt2 (1 y 2 ) dy dt + y = 0 Let y = z. Rewrite this equation as a system of first order differential equations for [y, z] T . 10.

10 Please write down the result of the following computation assuming it is performed using double precision floating point representation: 1 + 1.34 × 1020 =?

Explanation / Answer

3) Equation 'a' is a boundary value problem as the conditions for the variable are mentioned at two extremities. Equation 'b' is an initial value problem because the conditions mentioned are for the same point y(0) which is considered as the initial point.

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