Advection Diffusion problem

  • Thread starter jkthejetplane
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In summary, the speaker is having trouble following a similar approach to a problem and is unsure of the periodic boundary conditions. The other speaker suggests using separation of variables, but the first speaker does not think it is the best approach and suggests a different method involving an analytic solution and adding periodicity and translation.
  • #1
jkthejetplane
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4
Homework Statement
Solving PDE with ainitial conditions
Relevant Equations
I have no idea where to go wit this i started following something we did in class before but it doesnt quite make sense. I could really use help. I think i will be able to get the program if i can figure out these solutions
Thanks
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  • #2
I'm having trouble seeing what you did in your handwritten analysis. Can you first tell us your rationale for solving this? That periodic BC seems very peculiar. What does it even mean?
 
  • #3
Chestermiller said:
I'm having trouble seeing what you did in your handwritten analysis. Can you first tell us your rationale for solving this? That periodic BC seems very peculiar. What does it even mean?
Yeah I actually don't know what i was doing haha
I started following our notes on something similar and got completely lost. My attempt has basically no merit. The boundary conditions were given by the problem
 
  • #4
Your PDE is linear, you should therefore try to look for a solution by separation of variables; i.e. assume that the solution is of the form ##T(t,x) = u(t)v(x)## and derive a space-independent equation for ##u(t)## and a time-independent equation for ##v(x)##.

Remember that ##v(x)## inherit the periodic boundary condition from ##T##.
 
Last edited:
  • #5
I would definitely not use separation of variables in obtaining an analytic solution to this problem. Here is the approach I would recommend:

1. Start with the analytic solution for the case of C = 0, and on the infinite x interval from minus infinity to plus infinity. This is a closed-form single term expression (not a series of sines and cosines).

2. Add the periodicity to this solution by putting reflective boundary conditions at x = L/2 and x =-L/2 by taking a superposition of terms in the form of 1. , based on delta functions being initially located not only at x = 0, but also at x = + nL and x = - nL, with n = 1,2,3,...

3. Add the translation (advection) by substituting x - Ct for x in all the terms of 2.

This will satisfy the differential equation, the initial condition, and the boundary conditions
 

1. What is an advection diffusion problem?

An advection diffusion problem is a mathematical model used to describe the combined effects of advection (the movement of a substance due to a flow) and diffusion (the spread of a substance from areas of high concentration to areas of low concentration) on the distribution of a substance in a fluid or solid medium.

2. What are some real-world applications of advection diffusion problems?

Advection diffusion problems are commonly used in fields such as environmental science, engineering, and biology to model the transport of pollutants, nutrients, and other substances in air, water, and soil. They are also used in the study of heat transfer and chemical reactions in materials.

3. How are advection and diffusion related in an advection diffusion problem?

In an advection diffusion problem, advection and diffusion are two separate processes that work together to determine the overall distribution of a substance. Advection causes the substance to move with the flow, while diffusion causes it to spread out and equalize in concentration. The relative strengths of these two processes can greatly affect the behavior of the substance.

4. What are some numerical methods used to solve advection diffusion problems?

Some common numerical methods used to solve advection diffusion problems include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the problem into a grid or mesh and using iterative calculations to approximate the solution.

5. How do boundary conditions affect the solution of an advection diffusion problem?

Boundary conditions are important in advection diffusion problems because they define the behavior of the substance at the edges of the domain. Different types of boundary conditions can greatly affect the resulting solution, so careful consideration must be given to choosing appropriate boundary conditions for a given problem.

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