Question

PDEs...Just 1 and 2

Problem 1: By writing nm1 where (An are the associated eigenpairs, solve e kus +9(x, t) with k = 1, g(z, t)-cos x, u,0,t)-ur(2nd) = 0, u(z,0) = cos r + cos 21. 2. 3. k = 1, g(z, t)--_exp_2t)sinz, u(0,t) = u(z,t) = 0, u(z,0) = sinz.

Answer 1

n(x, t) = 〉|an(t)e-watPn(x) (1) n=1 and g(x, t) = 〉 gn (t)e-k t Pn (x) (2) n-1 where an, φ.J are the associated eigenpairs, we need to solve ut kuxx +g(x, t)... (3) We first use u(x, 0) f(x) By putting t 0 in (1), we get n=1

Thus an(0) bn are the Fourier sine coefficients off(x), where Similarly, we can find gn (t), 2 In(t) =ī | g(x,t)Pn(x)dx (5) Differentiating (1), partially w.r.t. "t", Differentiating (1), partially w.r.t. "x", twice n-1 n=1

Now we put all these values in (3), then we n-1 n-1 n-1 n-1 n-1 Taking all terms into left side, we get n-1 n-1

As we are given that {hn, pn) are the associated eigenpairs, using thie orthogonality, we have the following ODEs, for each n Which is a linear equation in an (t), With integrating factor Thus or, an (t)e

u(x,t) n-1 2 an (0) - 0 2 il g(x, t)Pn(x)dx 9n (t) - 0