Question

In each of problems 1-7, solve Ut = kuxx with the given boundary and initial conditions. 1. u(O, t) = u(1, t) = 0 fort> 0; u(x, 0) = sin(1rx) for 0 < x < 1?

Answer 1

To solve the given partial differential equation Ut = kuxx, we can use separation of variables and solve for X(x) and T(t) separately. The general solution is u(x, t) = ∑Bnsin(nπx)exp(-n²kπ²t), where Bn are constants satisfying the given initial conditions.

Step-by-step explanation:

First, let's rewrite the given partial differential equation Ut = kuxx as a separable equation: Ut - kuxx = 0. We can use the method of separation of variables to find a solution.

Next, we assume that u(x, t) can be written as a product of two functions: u(x, t) = X(x)T(t).

By substituting this into the equation, we obtain (1/T)dT = k(1/X)d²X.

After simplifying, we can separate the variables and solve for X(x) and T(t) separately. The general solution is given by u(x, t) = ∑Bnsin(nπx)exp(-n²kπ²t), where Bn are constants satisfying the given initial conditions.