6. (17pts) Derive the stencil of the modified forward Euler method for the following heat equation with an extra "advection" (transport) term, thermal diffusivity of k=1, and length of L=2: Sou du +24 (0 0) at Ox? u(0.t) = u(2,1)=1 Use the centered difference approximation ...) . mi -U for the advection term. For the other two 2AX du au derivatives, and use the same finite difference approximations that we used in class and in the at ax? homework (forward difference for the time derivative, and centered difference for the spatial derivative). Finally, calculate the solution over one time step, using At -0.02 and N=5 spatial intervals, with initial condition Wes=(1, 10, 0, 1, 1). To simplify calculations, use quotients instead of decimals. 6. (17pts) Derive the stencil of the modified forward Euler method for the following heat equation with an extra "advection" (transport) term, thermal diffusivity of k=1, and length of L=2: Sou du +24 (0 0) at Ox? u(0.t) = u(2,1)=1 Use the centered difference approximation ...) . mi -U for the advection term. For the other two 2AX du au derivatives, and use the same finite difference approximations that we used in class and in the at ax? homework (forward difference for the time derivative, and centered difference for the spatial derivative). Finally, calculate the solution over one time step, using At -0.02 and N=5 spatial intervals, with initial condition Wes=(1, 10, 0, 1, 1). To simplify calculations, use quotients instead of decimals