The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensors technique

TL;DR

This paper introduces an advanced mathematical approach using irreducible Cartesian tensors within the generalized method of separation of variables to solve complex diffusion-influenced reaction problems involving multiple spherical sinks.

Contribution

It presents a novel application of ICT technique to the GMSV framework, enabling explicit solutions for multi-sink reaction-diffusion problems with arbitrary configurations.

Findings

Explicit series solution for diffusion equations using ICT.

Reduction of boundary value problems to linear algebraic systems.

Numerical results for reactions in sink arrays.

Abstract

Motivated through various applications of the trapping diffusion-influenced reactions theory in physics, chemistry and biology, this paper deals with irreducible Cartesian tensors (ICT) technique within the scope of the generalized method of separation of variables (GMSV). Presenting a survey from the basic concepts of the theory, we spotlight the distinctive features of the above approach against known in literature similar techniques. The classical solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in the ICT. By means of proved translation addition theorem we straightforwardly reduce the general boundary value diffusion problem for $N$ spherical sinks to the corresponding resolving infinite system of linear algebraic equations with respect to unknown tensor coefficients. These coefficients comprise explicit dependence on the arbitrary three-dimensional configurations of $N$ sinks with different radii and surface reactivities. Specific application of the ICT technique is illustrated by some numerical calculations for the reactions, occurring in the arrays of two and three spherical sinks. For the first time we treat the statement and solution to the above reaction-diffusion problem, applying the smooth manifold concept.