Mathematical Formulation

MATLAB specifies such parabolic PDE in the form

c(x, t, u, ux)ut = x-m 𝜕𝜕𝑥

(xmb(x,t.u,ux))+s(x,t,u,ux)….............(1) with boundary conditions p(xl, t, u) + q(xl, t) · b(xl, t, u, ux) =0……………. (2) p(xr, t, u) + q(xr, t) · b(xr, t, u, ux) =0,……………..(3) where xl represents the left endpoint of the boundary and xr represents the right endpoint of the boundary, and initial condition u(0, x) = f(x).

For given problem the equation can be written as :

1𝐷𝑑𝐶𝑑𝑡

=𝑟−2 𝑑𝑑𝑟

(𝑟2 𝑑𝑐𝑑𝑟

)+o

On comparing with main equation 1:

c=1/D, m=2,s=0,b=du/dx,u=c,x=r

Boundary conditions given in the problem:-

0+1.(du/dx)=0 at xl on comparing with equation 2 pl=0 and ql=1

ur-u∞ +0.(du/dx)=0 at xr on comparing with equation 3 pr=( ur-u∞) and qr=0

initial condition at time t=0 all concentrations are equal to 0.

Algorithm

1.Functions parabolic and parabolic1 are defined to plot the concn vs position and surface contours respectively.

2. The constants required to be inputted by the user are fed into required variables using input function.

3. D stores diffusivity coefficient, k stores the bulk concentration stores the radius of cylinder,T stores the end time.

4. Functions bcl, initial1 and eqn1 are defined to pass the values of boundary conditions, initial conditions and the equation to the function function.

5. The values of concentration gets stored in a 2 dimensional array u which evaluated by calling the functions bcl,initial1 and eqn1 using function handles.

6. The function plot and surf produces the graphs and xlabel and ylabel is for naming the axis.

7 Potting concentration vs position at a particular time is done by varying the dimension associated with time through a loop the variable of the loop being z.

8. For a particular z u values at all x is ploted with respect to x.

9. hold on function used to retrieve all the plots on the same graph each corresponding to a particular time.

Code for generating concn vs position plot function parabolic global D global k D=input('Enter diffusivity constant'); k=input('Enter bulk concentation'); R=input('Enter radius of cylinder'); T=input('Enter the end time in minutes'); x = linspace(0,R,100); t = linspace(0,T*60,100); m=2; %Solving the PDE u = pdepe(m,@eqn1,@initial1,@bc1,x,t); %Plotting the solution for z=1:100 plot(x,u(z,:)) hold on; xlabel('Distance'); ylabel('concn'); end function [pl,ql,pr,qr] = bc1(xl,ul,xr,ur,t) %BC1: MATLAB function M-file that specifies boundary conditions %for a PDE in time and one space dimension. global k pl = 0; ql = 1; pr = ur-k; qr = 0; function[c,b,s] = eqn1(x,t,u,DuDx) %EQN1: MATLAB function M-file that specifies %a PDE in time and one space dimension. global D c = 1/D; b = DuDx; s = 0;

function u0 = initial1(x) %INITIAL1: MATLAB function M-file that specifies the initial condition %for a PDE in time and one space dimension. u0=0;

Code for generating surface contour function parabolic1 global D global k D=input('Enter diffusivity constant'); k=input('Enter bulk concentation'); R=input('Enter radius of cylinder'); T=input('Enter the maximum number of minutes'); x = linspace(0,R,100); t = linspace(0,T*60,100); m=2; %Solve the PDE u = pdepe(m,@eqn1,@initial1,@bc1,x,t); %Plot solution surf(x,t,u); title('Surface plot of solution.'); xlabel('Distance r'); ylabel('Time t'); zlabel('conc u'); function [pl,ql,pr,qr] = bc1(xl,ul,xr,ur,t) %BC1: MATLAB function M-file that specifies boundary conditions %for a PDE in time and one space dimension. global k pl = 0; ql = 1; pr = ur-k; qr = 0; function[c,b,s] = eqn1(x,t,u,DuDx) %EQN1: MATLAB function M-file that specifies %a PDE in time and one space dimension. global D c = 1/D; b = DuDx; s = 0; function u0 = initial1(x) %INITIAL1: MATLAB function M-file that specifies the initial condition %for a PDE in time and one space dimension. u0=0;

SAMPLE OUTPUT

Enter diffusivity constant0.00002

Enter bulk concentation0.05

Enter radius of cylinder0.1

Enter the maximum number of minutes1.0

Graph of concn vs position at a fixed time.

Plot for surface contour such that concentration on z axis time on y axis and distance on x axis.

With the progress of time the concentration at a given point increases.