Solution of a separable smoluchowski equation in one spatial dimension

Area of Science:

• Physics
• Physical Chemistry
• Applied Mathematics

Background:

• The Smoluchowski equation describes aggregation processes.
• Solving this equation, especially in one spatial dimension, presents computational challenges.
• Approximation methods are needed for practical applications.

Purpose of the Study:

• To develop an approximate solution for a separable Smoluchowski equation in one spatial dimension.
• To utilize a finite eigenfunction expansion for this approximation.
• To demonstrate the method's applicability and accuracy.

Main Methods:

• Constructing an approximate solution via finite eigenfunction expansion.
• Computing the Smoluchowski operator's spectrum and eigenfunctions using the shooting method of adjoints.
• Presenting explicit numerical solutions for static and fluctuating potentials.

Main Results:

• The finite eigenfunction expansion provides an accurate approximate solution.
• The method successfully computes the spectrum and eigenfunctions.
• The expansion is valid for smooth initial probability distributions across all time scales.
• The approach is generalizable to other linear eigenproblems on finite one-dimensional intervals.

Conclusions:

• The finite eigenfunction expansion is an effective method for approximating solutions to the Smoluchowski equation.
• The shooting method of adjoints is a viable technique for spectral computations.
• The developed method offers a robust approach for analyzing aggregation dynamics and related phenomena.