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The convective-diffusion equation in one dimension will be expressed in terms of velocity and dispersion,
The independent variables can be transformed from to a spatial coordinate that translates with the velocity of the wave in the absence of dispersion, .
This transforms the equation to the diffusion equation in the transformed coordinates.
To see this, we will transform the differentials from to .
The total differentials expressed as a function of or are equal to each other.
The total differentials expressed either way are equal. The partial derivatives in and can be expressed in terms of partial derivatives in and by equating the total differentials with either or equal to zero and dividing by the non-zero differential.
Substitution into the original equation results in the transformed equation. This result could have been derived in fewer steps by using the chain rule but would not have been as enlightening.
The boundary condition at is now at changing values of . We will seek an approximate solution that has the boundary condition . A simple solution can be found for the following initial and boundary conditions.
This system is a step with no dispersion at . Dispersion occurs for as the wave propagates through the system. The solution can be found with a similarity transform, which we will discuss later. For now, the approximate solution is given as
The boundary condition at will be approximately satisfied after a small time unless the Peclet number is very small.
In some cases a partial differential equation and its boundary conditions (and initial condition) can be transformed to an ordinary differential equation with boundary conditions by combining two independent variables into a single independent variable. We will illustrate the approach here with the diffusion equation. It will be used later for hyperbolic PDEs and for the boundary layer problems.
The method will be illustrated for the solution of the one-dimensional diffusion equation with the following initial and boundary conditions. The approach will follow that of the Hellums-Churchill method.
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