Partial differential equations

3* Consider the following initial and boundary-value problem Ut - Uxx = e"+ 2cos(x), -TT 0 u(x, 0) = X, - IT 0 Ux (-7, t) = ux(1, t), t> 0. Suppose we know that the homogeneous solution of the above problem is given by un(x, t) = ent(An cos(nx) + Bn sin(nx)) n=0 with An = 0 and By = 2(-1)#41 n for any integer n 2 0. Use the Duhamel principle to find the inhomogeneous solution.The solution to the homogeneous version of the above ODE (when f(t) = 0) is un(t) = e-atg. Now let S(t) = eat. Using S(t) we can represent the inhomogeneous equation as u(t) = S(t)g + / S(t - s)f(s) ds. Duhamel's principle suggests a general way of constructing the solution of inhomogeneous problem from solution of the homogeneous problem, as described in the example above. Now consider the heat equation ut - KAu = f(x, t), IEU, t>0 u(I, 0) = g(I). We claim based on the Duhamel's principle that if the solution to the homogeneous version of the above problem is un(x, t) = S(t)g(x) then a solution of the above equation is u(x, t) = S(t)g(x)+ S(t - s) f(I, s) ds. =:eh (z,t) =:w0(x,t) We need to show that w(x, t) satisfies inhomogeneous heat equation with zero initial data