2. (1.5 marks) Write down the solution u(x, t) of the diffusion equation with D =...

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2. (1.5 marks) Write down the solution u(x, t) of the diffusion equation with D = 1 on an infinite domain -x < x < +∞ corresponding to the initial condition u(x,0)= = 0 x < -1 1-1 ≤ ≤ 1 0 1 <r in integral form, and then use a change of variables to express the solution with the error function, which is defined as er f(x) = 2 I #S dx. Plot the solution in the interval z € [-5, 5] for t = 0.01, 0.1 and 1 on the same graph. 3. (2 marks) Use the Matlab pde solver pdepe() to solve the diffusion problem in Q2, but on a finite domain of -5 ≤ x ≤ 5 using a) zero flux boundary conditions: Our--5= 0,0₂ur5 = 0 b) u(-5, t) = 0, u(5, t) = 0. Plot the solutions generated by pdepe() in a) and b) at times t = 1,2,3,4 together with the infinite domain solution obtained in Q2. Briefly explain the similarities and differences between these three solutions. [Matlab hint: the step-like initial condition can be specified using an if else command inside the function u0= pdexlic(x)] 2. (1.5 marks) Write down the solution u(x, t) of the diffusion equation with D = 1 on an infinite domain -x < x < +∞ corresponding to the initial condition u(x,0)= = 0 x < -1 1-1 ≤ ≤ 1 0 1 <r in integral form, and then use a change of variables to express the solution with the error function, which is defined as er f(x) = 2 I #S dx. Plot the solution in the interval z € [-5, 5] for t = 0.01, 0.1 and 1 on the same graph. 3. (2 marks) Use the Matlab pde solver pdepe() to solve the diffusion problem in Q2, but on a finite domain of -5 ≤ x ≤ 5 using a) zero flux boundary conditions: Our--5= 0,0₂ur5 = 0 b) u(-5, t) = 0, u(5, t) = 0. Plot the solutions generated by pdepe() in a) and b) at times t = 1,2,3,4 together with the infinite domain solution obtained in Q2. Briefly explain the similarities and differences between these three solutions. [Matlab hint: the step-like initial condition can be specified using an if else command inside the function u0= pdexlic(x)]