a). Solve the following two-point Boundary-Value problem by using finite element method:
d
(
(x + 1)
du
) + (2 + x2)u(x) = x2 4; x 2
(0; 1)
dx
dx
u(0) = u(1) = 0;
by using piecewise linear polynomials and using trapezoidal rule and Simpsons rule for the numerical quadra-ture.
b). Solve the following two-point Boundary-Value problem by using finite element method:
d
((x2
2)
du
) + (1 + 2x)u(x) = x2 + 4x 5; x 2
(0; 1)
dx
dx
u′(1) = 0;
u(0) = 2;
by using piecewise linear polynomials and using trapezoidal rule and Simpsons rule for the numerical quadra-ture.
c). Consider the following Black-Scholes PDE for European call:
@V
1
2S2
@2V
@V
rV = 0; (0; 1) (0; T ]; T 0
+
+ (r )S
@t
2
@S2
@S
V (S; t) = 0;
for S = 0;
V (S; t) = S
Ke
r(T
t)
;
for S
−
−
! 1
with suitable initial condition V (S; 0):
Solve the transformed PDE yt = yxx with suitable initial and boundary conditions by using finite elements mentioned in problem (a)and the Crank-Nicolson scheme.
Plot V (S; t) for T = 1; K = 10; r = 0:06; = 0:3, and the payoff.
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