Question

In solving the nonlinear BVP using Newton's Method, we need to discretize the residual and the Jacobian $J = \frac{\partial F}{\partial y}$ .

a) What is the discretized residual where i and j run from 1, ..., N-1 and where the BCs are incorporated into $b_{i}$? Be sure to specify $b_{i}$.

b) What is the discretized Jacobian $J_{ij} = \frac{\partial F_{i}}{\partial y_{j}}$ ? Hint: Your answer will involve both $(D2)_{ij}$ and the Kronecker $\delta _{ij}$.

F_i = SUM_j (D2)_ij y_j + ... + (0,...0,?)

J_ij = (D2)_ij + ... ?

Fi =

Answer 1

Consider now the non-linear BVP: y
00
= y − y
2 with y(0) = 1 and y(1) = 4. Using
our finitie difference formula for y
00
the ODE y
00
= 4y becomes
yi−i − (2 + h
2
)yi + h
2
y
2
i + yi+1 = 0 for i = 1, . . . , n
We let n = 3 and therefore h = (b − a)/(n + 1) = (1 − 0)/4 = 1/4 and
i = 1, 2, 3, 4, 5. The resulting linear system of equations looks as follows:
i = 1 : y0 − (2 + h
2
)y1 + h
2y
2
1 + y2 = 0
i = 2 : y1 − (2 + h
2
)y2 + h
2y
2
2 + y3 = 0
i = 3 : y2 − (2 + h
2
)y3 + h
2y
2
3 + y4 = 0
where the non-linear function F (Y ) = 0 above is to be solved with Newton’s method
x
n+1 = x
n − J(x
n
)
−1F (x
n
). As usual we do not calculate the inverse of the Jacobian
but instead solve J(x
n
)∆x = F (x
n
) with LU factorization and then substitute that
solution into x
n+1 = x
n − ∆x.

2 consider now the non-linear BVP: บู :y -y -y with y(0)1 and y(1)-4. Using our finitie difference formula for y the ODE y-4y becomes We let n - 3 and therefore h - (b - a)/(n + 1) - (1 - 0)/41/4 and i - 1, 2, 3, 4, 5. The resulting linear system of equations looks as follows: where the non-linear function F(Y) 0 above is to be solved with Newton's method r"+1- but instead solve J(x")ΔΧ F(x") with LU factorization and then substitute that solution into xn41-Xn-Δι J(x" F( As usual we do not calculate the inverse of the Jacobian