Consider the nonlinear differential equation

subject to the conditions x = 3 and dx/dt = 2 at the time t = 1. Let’s solve this problem using implicit Euler with a time step of h = 1/2. You may recall that a continuation method is often useful in this problem; you should use x⁽⁰⁾_i+1 = xi as the initial guess for the Newton–Raphson steps in the solution. Determine the numerical values for the residual and Jacobian corresponding to the first iteration of Newton–Raphson during the first time step of the implicit Euler integration of the differential equation.

Transcribed Image Text:

1 = r+ P | XP