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Laplace Transform Method:

Diffusion Equation With Robin Condition

M.M. Yovanovich

LPTDEROB.MWS

Laplace transform of one-dimensional diffusion equation in half-space with homogeneous Robin boundary condition at . The PDE is and the initial condition is , with boundary conditions: and . The parameters are: is the fluid temperature, is the heat transfer coefficient, and is the thermal conductivity of the half-space.

> restart:

> with(inttrans):

> pde:= diff(u(x,t),x,x) = 1/alpha*diff(u(x,t),t);

> alias(U(x,s)=laplace(u(x,t),t,s)):

> ode:= laplace(pde,t,s);

> ic:= u(x,0) = 0;

> bc1:= subs(x=0, diff(u(x,t),x)) = -h/k*(u[f]-u(0,t));

> bc2:= u(infinity,t) = 0;

> ode1:= subs(ic,ode);

> ode2:= subs(U(x,s) = U(x), ode1);

> solode2:= dsolve(ode2, U(x));

To have a bounded solution at , we can eliminate the positive exponential by setting .

> _C1:= 0;

> solode3:= solode2;

Take the Laplace transform of the boundary condition at .

> bc1:= laplace(diff(u(x,t),x),t,s) = - laplace(h/k*(u[f]-u(x,t)), t,s);

Obtain the second constant of integration from the transformed Neumann boundary condition.

> rhs(solode3);
diff(rhs(solode3), x);
bc1:= subs(x=0, diff(rhs(solode3),x)) + laplace(h/k*u[f], t,s) - h/k*subs(x=0, rhs(solode3))= 0;

> _C2:= solve(bc1, _C2);

> solode4:= solode3;

> solve(solode4, U(x));

The solution in the transform domain can be written in the form:

> U(x,s):= ((h/k)*u[f]*exp(-sqrt(s/alpha)*x)/s)/(h/k + sqrt(s/alpha));

The solution of the problem is the inverse Laplace tranform of the above relation.

The Laplace transform tables cannot provide the inverse directly from the above expression. It will be necessary to let and before the tables can be used. We will try to use Maple to get the inverse of the following relation.