Consider the initial value problem (a) Verify by direct differentiation that the analytical solution of this problem is given by (b) An explicit scheme for the initial value problem (2.53) can be...

Consider the initial value problem

(a) Verify by direct differentiation that the analytical solution of this problem is given by

(b) An explicit scheme for the initial value problem (2.53) can be written in the form

Explain the derivation of this scheme

(c) Similarly, derive the following implicit20 scheme

(d) Set

and compute, by hand, y1; y2; y3 and z1;z2;z3: Compare these values with the analytical values given by r

(e) Write a computer program for the implicit and the explicit scheme. The program should accept N and T as input data and make a graph of the explicit,21 implicit, and exact solutions for t ranging from 0 to T . The numerical schemes should use t

(f) Test the program for T D 1: Evaluate the error for various values of t: Do the computations indicate that the error is