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## Analytical solution to diffusion-convection PDE in 1D

July 7, 2017 compiled on — Friday July 07, 2017 at 08:35 PM

This is a diffusion-convection PDE.

Where is the diffusion constant and is the convection speed. Boundary conditions are

Initial conditions are

The first step is to convert the PDE to pure diffusion PDE using the transformation

Substituting this back in (1) gives

Dividing by and simplifying

To make (2) pure diffusion PDE, we want

From (4) or or which has the solution

 (5)

From (3) we want . Substituting the result just obtained for in (3) gives

Hence

For some constant . The constant ends up canceling out at the very end. Hence we set it to now instead of carrying along in all the derivation in order to simplify notations. Therefore . Substituting this into (5) gives the transformation function

Using this in (2) gives the pure diffusion PDE to solve

 (6)

Converting the original boundary conditions from to gives

And

And for the initial conditions

Therefore the new PDE to solve is

With time varying boundary conditions

And initial conditions

To solve this using separation of variables, the boundary conditions has to be homogenous. Therefore we use standard method to handle this as follows. Let

 (7)

Where is the steady state solution which needs to only satisfy the boundary conditions and satisfies the PDE but with homogenous boundary conditions. Therefore

And (7) becomes

Substituting the above in (6) gives

Or

 (8)

This is diffusion PDE with homogenous B.C. with source term

Now we find . Since this solution needs to satisfy homogenous boundary conditions, we know the solution to pure diffusion on bounded domain with source present is by given by the following eigenfunction expansion

 (8A)

Where eigenvalues are for and are the eigenfunction. Substituting the above in (8) in order to obtain an ODE to solve for gives

 (9)

Expanding in terms of eigenfunctions

Applying orthogonality

 (9A)

But

Hence from (9A) we find

Using the above in (9) gives

To solve the above ODE, the integrating factor is , therefore

Using the above in (7) gives

 (10)

is now found from initial conditions. At the above becomes

Applying orthogonality

But since is zero everywhere except at the end point. And

Therefore

And the solution (10) becomes

 (11)

But

Hence (11) becomes

 (12)

We now convert back to .  Since and then the final solution is

The following is animation of the solution for 30 seconds, side-by-side with numerical solution.

The following is the code used

References

• Paper ”Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients”. Dilip Kumar Jaiswal, Atul Kumar, Raja Ram Yadav. Journal of Water Resource and Protection, 2011, 3, 76-84
• Paper ”An analytical solution of the diffusion convection equation over a finite domain”. Mohammad Farrukh N. Mohsen and Mohammed H. Baluch, Appl. Math. Modelling, 1983, Vol. 7, August 285.
• Lecture 20: Heat conduction with time dependent boundary conditions using Eigenfunction Expansions. Introductory lecture notes on Partial Differential Equations By Anthony Peirce.