## Derive the computational scheme and show that (contrary to the Forward Euler scheme) you have to solve a nonlinear algebraic equation for each time step when using this scheme.

1.Let y be a scalar function of time t and consider the nonlinear ODE y + y = ty3, t ∈ (0, 4), y(0) = 1 2 . a) Assume you want to solve this ODE numerically by the Backward Euler method. Derive the computational scheme and show that (contrary to the Forward Euler scheme) you have to solve a nonlinear algebraic equation for each time step when using this scheme. b) Implement the scheme in a program that also solves the ODE by a Forward Euler method. With Backward Euler, use Newton’s method to solve the algebraic equation. As your initial guess, you have one good alternative, which one? Let your program plot the two numerical solutions together with the exact solution, which is known (e.g., from Wolfram Alpha) to be y(t) = √ 2 √ 7e2t + 2t + 1

2.Assume that the initial condition on u is nonzero in the finite difference method from Sect. 8.4.12: u
(0) = V0. Derive the special formula for u1 in this case. Filename: ic_with_V_0.pdf.

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