34.5. EXERCISES 691

12. Find the solution to the initial boundary value problem

ut = 4uxx,u(0, t) = 0,u(2, t) = 0,

u(x,0) =x for x ∈

[0, 2

3

]1− 1

2 x if x ∈ ( 23 ,2]

13. Find the solution to the initial boundary value problem

ut = 5uxx,u(0, t) = 0,u(1, t) = 0,

u(x,0) =x for x ∈

[0, 1

2

]1− x if x ∈ ( 1

2 ,1]

14. Find the solution to the initial boundary value problem

ut = 3uxx,u(0, t) = 0,u(5, t) = 0,

u(x,0) =x for x ∈

[0, 5

2

]5− x if x ∈ ( 5

2 ,5]

15. Find the solution to the initial boundary value problem

ut = 2uxx,ux (0, t) = 0,u(2, t) = 0,

u(x,0) =x for x ∈

[0, 1

2

]23 −

13 x if x ∈ ( 1

2 ,2]

16. Find the solution to the initial boundary value problem

ut = 5uxx,ux (0, t) = 0,u(1, t) = 0,

u(x,0) =x for x ∈

[0, 1

2

]1− x if x ∈ ( 1

2 ,1]

17. Find the solution to the initial boundary value problem

ut = 5uxx,ux (0, t) = 0,u(2, t) = 0,

u(x,0) =x for x ∈

[0, 2

3

]1− 1

2 x if x ∈ ( 23 ,2]

18. Consider the following initial boundary value problem,

ut = uxx, u(0, t) = 0,u(2, t)+ux (2, t) = 0,u(x,0) = f (x)

Determine the appropriate equation for the eigenfunctions and show that there existsa sequence of strictly positive eigenvalues converging to ∞. Also explain why thesolution u if it exists, must have a limit limt→∞ u(x, t) = w(x) and that this limitsatisfies w(x) = 0.