1) Boundary Value Problem:
(a) Solve the differential equation, u_xx = - (pi^2/4)*(u +1 ) [Eq. 3.54 in the textbook], with the boundary conditions: u_x(0) = 1 and u_x(1) = 1. Use the shooting method. You may modify the program 'shooting.f' (or its analogue in C). Make a plot of u(x) and u_x(x). Obtain the analytical result for this set of boundary conditions using Mathematica.
(b) Modify the program 'shooting.f' (or its analogue in C) to use equations 3.59 - 3.63 instead of the shooting method to solve the same problem given in the book, u_xx = - (pi^2/4)*(u +1 ), with the boundary conditions: u(0) = 0 and u(1) = 1. Again, use 'shooting.f' (or its analogue in C) as a starting point. Hand in a printout of your program, and mark the changes you made to the original program.
2) Integral Equation:
Write a program to find f(x) in the interval from x= -1 to x=1 from the integral equation:
f(x) = x + 0.5 Integral[ dt (t - x) f(t) ], where the integral limits are -1 and 1. You may use and modify the program presented in class. Plot your solution f(x) in the interval from x= -1 to x=1, and give an analytical expression for it. Please hand in a printout of your program.