(2) To solve a nonlinear BVP Fi(yj) = 0 using Newton's method, we make a guess...

Transcribed Image Text:

(2) To solve a nonlinear BVP Fi(yj) = 0 using Newton's method, we make a guess for y; and then iterate until the norm of the residual ||Fi|| e. Set (k+1) yj = (k) = y; " + Ayj, k = 0, 1, 2, ... where k labels the Newton iterations. Newton's equation for Ay, is derived in the following way: or (k) OF ; j 0 = F (y() + Ay;) F (y) Fi + where the Jacobian Jij = J(k) Ay = -F (y(k)) Fi/Oyj. (a) Discretize the residual for the nonlinear Poisson-Boltzmann BVP 2 F(y) = y" + y' - e = 0, y(1) = 1, y(2) = 0. X (2) Hint: F = ; Dyj+ BCs are incorporated into + b where i and j = 1, ..., N - 1 and where the = 1 2 0, h 2h' ,0,0]. (b) What is the discretized Jacobian Jij = Fi/y;? Hint: Your answer will involve D), D), and the Kronecker dij. ij ij (3) [cf. Moler 7.22 (d)] In solving the nonlinear BVP y = y 1, y(0) = 0, y(1) = 1 using Newton's method, we need to discretize the residual F = y y + 1 and the Jacobian J = F/y. (2) To solve a nonlinear BVP Fi(yj) = 0 using Newton's method, we make a guess for y; and then iterate until the norm of the residual ||Fi|| e. Set (k+1) yj = (k) = y; " + Ayj, k = 0, 1, 2, ... where k labels the Newton iterations. Newton's equation for Ay, is derived in the following way: or (k) OF ; j 0 = F (y() + Ay;) F (y) Fi + where the Jacobian Jij = J(k) Ay = -F (y(k)) Fi/Oyj. (a) Discretize the residual for the nonlinear Poisson-Boltzmann BVP 2 F(y) = y" + y' - e = 0, y(1) = 1, y(2) = 0. X (2) Hint: F = ; Dyj+ BCs are incorporated into + b where i and j = 1, ..., N - 1 and where the = 1 2 0, h 2h' ,0,0]. (b) What is the discretized Jacobian Jij = Fi/y;? Hint: Your answer will involve D), D), and the Kronecker dij. ij ij (3) [cf. Moler 7.22 (d)] In solving the nonlinear BVP y = y 1, y(0) = 0, y(1) = 1 using Newton's method, we need to discretize the residual F = y y + 1 and the Jacobian J = F/y.