Consider the ODE (a) Derive and implement an explicit Euler scheme for (4.184). (b) Derive an implicit Euler scheme for XXXXXXXXXXUse Newton’s method to solve the nonlinear equation arising at each...

Consider the ODE

(a) Derive and implement an explicit Euler scheme for (4.184).

(b) Derive an implicit Euler scheme for (4.184). Use Newton’s method to solve the nonlinear equation arising at each time level. Implement your scheme on a computer.

 (c) Set
=1/100 and compute numerical solutions from t = 0 to t = 1, using the programs developed in (a) and (b).

 (d) Show that

solves (4.184). Use this to check the accuracy of the schemes discussed above.

(e)   Consider the scheme

Derive Newton’s method for solving the nonlinear equation arising at each time level. Implement the scheme and discuss its accuracy and compare the results with those obtained by the implicit and explicit Euler schemes.

(f)     Redo (e) using the scheme