Word Count : 921

---

We are asked to solve the heat equation [ u_t = u_{xx}, \quad 0 \le x \le 1, , t>0 ] with [ u(x,0) = 0, \quad u(0,t) = 0, \quad u(1,t) = t ] using the Crank–Nicolson method with (h = 0.2) and (\lambda = 1), for two time steps. --- Step 1: Discretization * Spatial step: (h = 0.2 \implies x_0 = 0, x_1 = 0.2, x_2 = 0.4, x_3 = 0.6, x_4 = 0.8, x_5 = 1) * Time step: let (k) satisfy (\lambda = \frac{k}{h^2} = 1 \implies k = h^2 = 0.04) * Interior nodes: (i = 1,2,3,4) (since (u_0) and (u_5) are boundary values) Crank–Nicolson formula for interior nodes: [ -\frac{\lambda}{2} u_{i-1}^{n+1} + (1+\lambda) u_i^{n+1} - \frac{\lambda}{2} u_{i+1}^{n+1} = \frac{\lambda}{2} u_{i-1}^n + (1-\lambda) u_i^n + \frac{\lambda}{2} u_{i+1}^n ] Given (\lambda = 1), this simplifies to: [ -0.5, u_{i-1}^{n+1} + 2, u_i^{n+1} - 0.5, u_{i+1}^{n+1} = 0.5, u_{i-1}^{n} + 0, u_i^n + 0.5, u_{i+1}^n ] --- Step 2: Initial and boundary conditions * (u_i^0 = 0) for all interior nodes * (u_0^n = 0), (u_5^n = t_n = n ___ ___ ________ _____ __________ ________ ____ _________ ____.
_______ ______ ______ _____ _______ ___ _____ ________ ___ ___.
___ ______ _____ ___ ____ ____ _________ ______ ______ _______.
_______ ____ ________ __________ ______ __________ ________ _____ _________ ________ __________.
__________ ______ _____ _____ _____ ___ _____.
___ ________ _______ __________ ___ ________ _________ ____ _______ ___ __________ _____.
___ ___ ___ ____ _______ _____.
_____ ________ ___ __________ __________ ____.
____ __________ ______ _________ ___ ________ __________ _____ ________.
_______ _______ ____ _________ ________ ________ __________ ________.
____ _______ _________ _______ __________ _____ ___ _____ _________ ____.
____ ___ _______ ____ _______ _____ _________ _______ _________ _____ _________ ____.
_______ ______ _______ _______ _____.
__________ ___ _____ __________ _____ _____ ________.
____ ________ ____ ___ ________ ______ ___ ____ ____.
____ ____ ________ _____ ______ _____ ______ _____.
____ ______ _______ ________ __________.
_____ _______ ________ __________ ___ ________ ___ ______ _______.
________ ___ ________ _________ ________ ___ _________ _______ __________ ________ _____.
___ _________ _________ ________ ____ ____ _________ ________ ______.
___ ______ ___ ________ _______ _________.
_______ ______ __________ _________ __________ _____ __________ _______ ________ ________.
__________ _________ __________ ________ ___ ______ __________ ______ ________.
_________ _____ ___ _______ ________ _________ ______ ___ _____ _____ ________.
___ ___ ______ _____ ____ _________ _______ _________ _______ __________.
_____ ____ ___ ______ ___ ___ _______ __________.
_________ ____ _________ ______ _______ __________.
____ _____ _____ __________ ________ __________ ________ ___ _______.
_________ ________ _____ ____ _________.
________ ______ ______ _____ ____ _________ ______ _________ ___ __________ __________.
_________ ______ _______ ______ ______ __________ ________ ________ _____ _____ _______ _______.
___ _________ _____ _____ _______ ________ __________ ______.
________ ________ _______ __________ _________ __________ ________ ____ ________ _________.
_______ __________ _____ ______ ______ _________.
_______ _______ _________ __________ ____ ___ _____ _________ ______ _______ _____.
_______ ________ ____ ____ _____ ___ ______ __________.
_____ _________ _________ _________ _________ __________ __________ __________ _______ ____ ______ _____.
________ _______ __________ _______ ______ _______ _____ __________ _______ ____ _________ _________.
____ ________ _____ ____ ___ __________ __________ _______ _____ ____.
________ ______ _________ ___ ___ ______ ____ ____.
_____ ______ _____ _________ __________ ______ ____ ____ _____ ________ _______ _____.
_____ ______ _____ ___ ________.
________ ___ ______ ___ ______.
_____ ______ ______ ________ ___ ___ ______ ____ ________ ______.
______ __________ ______ ___ _______ ___ _________ ________ ______ __________ _______.
________ __________ __________ _________ _________ _________ ___.
_____ _____ _______ ___ ___ ______ _____ _________.
_________ _________ ________ _______ _________ _______ ________ ______ ___ ________ _____ ____.
_______ __________ _______ ______ ___ ____ __________ _______ ____.
___ _____ _____ ____ ______ _______ __________ ____ ________ ________.
_________ ___ ____ __________ ______ ________ _______ ____ __________ ____ _____.
___ _________ ________ _________ _____ ______ ____ ____.
___ _____ ___ _________ _______ __________ _________ _________ _________ _______.
_______ ____ ________ ________ _____ _______ _______ _______.
_______ ____ _____ _______ _________ ___ _________ ______ ____.
___ __________ _________ _________ ________ ____ _______ _____ _____ ________ ________.
____ ______ _____ _________ __________ _______ ______ _________ ________.
_________ _______ __________ ____ _______ _________ ____.
____ _________ _______ __________ ___ ____ ____.
_____ _______ __________ _________ ______ _________ ____ ____ _________ ___ _______ ________.
______ ________ ____ _________ _____ ____.
_____ __________ _______ ____ _________ _________.
___ ___ ________ ____ ____ ________ _________ ________.
_______ _______ _______ ____ ____ _____ ________ __________.
________ _____ _____ _________ _______ _____ __________ __________ _________ ______ ____.
___ ____ ______ ________ ___.
______ _______ __________ _____ ________.
______ ______ __________ ______ _______ ______ ___ ________ _____ _______ ____.
_________ _________ _________ _________ _____ __________ _______.
___ __________ ___ _______ ____ ___ _________.
_________ ________ _____ ___ ______ ____ ____ ____ ________.
________ ____ _______ _____ _____ _____ _____ __________ ____ ___.
_______ _____ __________ _________ ___ ______ ______ _____ __________ _________ ______ ________.
__________ ___ _________ __________ __________ ___ ____ _________ ________ ____ _________.
___ _______ ____ ________ _______ __________ ____ ______ _________ __________ _______ ______.
_________ _____ ______ __________ ______ _________ ________ _______ ________ ______.
____ ____ ________ ______ _________ ______ ______.
________ _________ ____ ______ _____ _______ _________ __________ ________ ______ __________ _____.
______ _______ ________ ___ _________ _______ __________ _______ ____ _________ ___.
_________ ________ __________ ____ ______ ____ _______ ____.
_____ ________ ________ ___ __________ _____ ___ _____ ____ ______ ___ __________.
______ __________ _______ _____ __________ ____ _____ ___ _________.
__________ ____ _____ ______.
Get Full Answer on WhatsApp