Partial Differential Equations: Fourier Series MethodsDiffusion is the process of movement of molecules under a concentration gradient. It is an important process occurring in all living beings. Additionally, diffusion helps in the movement of substances in and out of the cells. The molecules naturally move from regions where they are highly concentrated to regions where they are not as concentrated. For instance, liquid and gases undergo diffusion process as the molecules are able to move randomly. This biological process can be investigated using a diffusion model (also known as a heat equation):∂????/∂???? = ∂^2????/∂????^2 ????(0,????) = 20, ????(30,????) = 50, ???? > 0,????(????, 0) = 60 − 2????, 0 < ???? < 30,with ???? = ????(????,????) corresponds to the concentration of molecules moving over time, t, along a one-dimensional spatial domain, ????.

a) Perform the calculation to obtain the steady-state concentration of molecules. 

b) Construct the initial-boundary value problem that depicts the transient concentration of molecules

c) Using the Fourier series techniques, perform the calculation to obtain the solution ????(????,????)of the PDE (1)