Time left 1:53:1 Consider the heat equation = K at ax With initial conditions, u(x,0) = sin/x, Oxs1 And Dirichlet boundary conditions 1(0, t) =0 and u(1,t) =0 Complete the following steps to find the solution: Taking / =3 and using the above procedure of the method of lines For stability point of view we consider K = 0.5, L =1 Initial condition u(x,0) = f(x) = sina OSxs1 (3.5) and Dirichlet boundary conditions 1(0,1) = 81 (1) =0 (3.6) and u(1.?) = 82 (1) =0 (3.7) Now replace a? with finite difference approximation formula: dll, - lli - 21, + 1 1 1=1,2,...,N dt (1 mark) Thus, i = 1 dt du z i = 2 dt 1 =3 dt (6 marks)1 1= Here N+1 (1 mark) The boundary condition at x =0 and x =1 are transformed as follow: 10 = 81 (1) =0 1341 = 14 = 82(1) =0 The initial condition transformed to domain of interest as follow: 1, = f(ih) = sin x(ih) i =1,2,3 i =1 1 =2 2 i =3 (3 marks) Taking U = 1, #2 u;] Therefore 1 0 du 0.5 0.5 1 2 1 U 0 (3.20) dt Applying boundary condition and value of h, we get du (2 marks)Using Eigen value and Eigen vector method to solve this system of differential equations The Eigen values are 2 =-27.3137 1, =-16 % =-4.6863 and it's corresponding Eigen vectors are 0.5 - 0.7071 - 0.5 X = - 0.7071 0 X= - 0.7071 0.5 0.7071 - 0.5 Thus, using the general solution of ODEs system u = ci exp(At) x1 + c2 exp(12t) x2 + c3 exp(13t) x3 1 = (3 marks) By solving this system of equations, the solution is -0.5 1/ = -1.4142 74.6863f 0.7071 -0.5