Efficient Solutions for the Poisson-Boltzmann Equation in a Rectangular Domain

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In summary, the person is looking for a method to solve a P-B equation with constant boundary conditions. Another person suggests using a Fourier transform, but the first person is unsure because the exponential term contains the function itself rather than independent variables. The only other method suggested is to write the right hand side as a power series and truncate it to a low order to solve the resulting PDE.
  • #1
Assaf Peled
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Hello,

I'm trying to find either an analytical or a semi-analytical method for solving the following P-B Eq.
upload_2017-8-19_20-15-16.png

with C and A being two constants. The equation is to be solved within a rectangle with constant boundary conditions.

If anyone has a clue, I'll be grateful.

Have a good evening.
 

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  • #2
Have you tried using a Fourier transform? The exponential in your equation makes that method seem promising.
 
  • #3
Not sure that you're right because the exponential term contains the function itself rather than the independent variables x or y.
 
  • #4
In that case, the only other method I know would be to write the right hand side as a power series centered at the origin of the domain and truncate the series to some low order, like first or second, and solve the resulting PDE.
 

What is a Poisson-Boltzmann case?

A Poisson-Boltzmann case is a type of mathematical model used to describe the interaction between charged particles in a solution. It takes into account both the electrostatic forces between the particles and the thermal motion of the particles.

Why is the Poisson-Boltzmann case important in scientific research?

The Poisson-Boltzmann case is important because it allows scientists to understand and predict the behavior of charged particles in a solution. This is crucial in various fields such as biochemistry, materials science, and electrochemistry, where the interactions between charged particles are essential to the overall behavior of the system.

What are the assumptions made in the Poisson-Boltzmann case?

The Poisson-Boltzmann case assumes that the solution is dilute, the particles are small compared to the distance between them, and the particles are in thermal equilibrium. It also assumes that the particles are spherical and the solvent is a continuous medium with a dielectric constant.

How is the Poisson-Boltzmann equation solved?

The Poisson-Boltzmann equation is a non-linear partial differential equation and can be solved using numerical methods. The most common approach is to discretize the equation and use iterative methods to find a numerical solution. There are also analytical solutions for simplified cases.

What are some real-world applications of the Poisson-Boltzmann case?

The Poisson-Boltzmann case has many real-world applications, such as predicting protein-ligand binding interactions, studying the behavior of electrolytes in batteries and fuel cells, and understanding the stability and behavior of colloidal suspensions. It is also used in various computational methods for biomolecular simulations and drug design.

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