Boundary condition problem for diffusion equation

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SUMMARY

The discussion centers on the boundary condition problem for the diffusion equation in a channel domain where a chemical is introduced. The original boundary conditions were c(0,t)=1 and c(a,t)=0, but the latter is invalid if the chemical disperses at the channel's outlet. Participants suggest exploring alternative boundary conditions, specifically proposing c(a,t) = f(t) for an unknown function f, to derive a solution that accommodates the dispersion of the chemical.

PREREQUISITES
  • Understanding of diffusion equations and their applications.
  • Familiarity with numerical methods for solving partial differential equations.
  • Knowledge of boundary conditions in mathematical modeling.
  • Basic concepts of chemical dispersion in fluid dynamics.
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  • Research alternative boundary conditions for diffusion equations.
  • Explore numerical methods for solving partial differential equations, focusing on finite difference methods.
  • Investigate the implications of chemical dispersion in channel flow dynamics.
  • Learn about the method of characteristics for solving boundary value problems.
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Mathematicians, engineers, and students involved in fluid dynamics, chemical engineering, or numerical analysis who are tackling boundary condition issues in diffusion problems.

brambram
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Homework Statement


BOUNDARY CONDITION PROBLEM
I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation:

δc/δt=D*((δ^2c)/(δx^2))-kc

assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is channel's length.

What if domain is not-infinite (as it is above) and we cannot assume that the chemical is not dispersed at the end of the channel- so the boundary condition c(a,t)=0 is no longer valid?


Homework Equations



What boundary condition can we use when we cannot assume that the chemical is not dispersed at the outlet of the channel?
 
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Perhaps it is asking you to set c(a, t) = f(t) for some unknown function f and obtain a solution in terms of f. But I'm only guessing.
 

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