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We study the problem of a diffusing particle confined in a large sphere in the n-dimensional space being absorbed into a small sphere at the center. We first non-dimensionalize the problem using the radius of large confining sphere as the spatial scale and the square of the spatial scale divided by the diffusion coefficient as the time scale. The non-dimensional normalized absorption rate is the product of the physical absorption rate and the time scale. We derive asymptotic expansions for the normalized absorption rate using the inverse iteration method. The small parameter in the asymptotic expansions is the ratio of the small sphere radius to the large sphere radius. In particular, we observe that, to the leading order, the normalized absorption rate is proportional to the (n － 2)-th power of the small parameter for .

Search theory represents the birth of operations analysis [

In this paper, we would like to extend our earlier work [

From the next section, the paper is outlined as follows. We first present the mathematical formulation of the problem in Section 2. Then we consider the special case of the three dimensions in Section 3 and derive the exact solution for this case in Section 4. Section 5 and Section 6 describe the solutions for dimension four and dimension five, respectively. These asymptotic solutions are validated against the accurate numerical solutions of a Sturm-Liouville problem in Section 7. Finally, Section 8 summarizes the paper.

We consider a particle in the n-dimensional space

Let

Let

where

normal vector

We first perform non-dimensionalization to make the problem dimensionless. Let

The function

where

The solution of initial boundary value problem (2) can be expressed in terms of exponentially decays of eigenfunctions.

Here

In (3), the slowest decaying term is

We consider the survival probability:

probability

Quantity

The normalized decay rate of survival probability is

In the two-dimensional case (

In this study, we derive asymptotic expansions for the smallest eigenvalue

We use the inverse iteration method to derive an asymptotic expansion for

Specifically, we solve the linear differential equation with boundary conditions below to update the approximation from

In the first iteration (

An approximation to the smallest eigenvalue

In the subsequent sections, we show that

For the three dimensional case (

We first solve for two independent solutions of (16) in the case of

Next we solve

For

For

For

For

With these results, we start the inverse iteration. For the first iteration (

combination of two independent solutions

The corresponding approxomation for

In the second iteration (

formed using

The corresponding approxomation for

In the third iteration (

The solution of (10) is constructed using

The corresponding approxomation for

Therefore, in the three dimensional case,

For the three dimensional case, the smallest eigenvalue

For the special case of

Substituting it into (9) for

The boundary condition for

The boundary condition for

Thus,

A general solution of the differential equation has the expression

Enforcing the boundary condition

Here we have set

The smallest eigenvalue

This exact solution specified by Equation (26) provides an alternative derivation for the asymptotic expansion of

Using the Taylor expansion of

and subtracting 1 from both sides of (27), we get

Based on (28), we construct an iterative formula for expanding

The iterative formula gives us

Going from

which is the same as the asymptotic expnsion derived using inverse iteration method.

For the four dimensional case (

We first solve for two independent solutions of (31) in the case of

Next we solve

For

For

With these results, we start the inverse iteration. For the first iteration (

combination of two independent solutions

The corresponding approxomation for

In the second iteration (

formed using

The corresponding approxomation for

Therefore, in the four dimensional case,

For the five dimensional case (

We first solve for two independent solutions of (36) in the case of

Next we solve

For

For

With these results, we start the inverse iteration. For the first iteration (

combination of two independent solutions

The corresponding approxomation for

In the second iteration (

formed using

The corresponding approxomation for

Therefore, in the four dimensional case,

To demonstrate the accuracy of asymptotic expansions we obtained above, we solve numerically Sturm- Liouville problem (9). Instead of using a uniform grid in variable r, we use a uniform grid in variable

In variable

We use the central difference with

accurate than that in lower dimensional space (smaller n). The two-term asymptotic solution in

In each case (

The focus of this paper was to calculate the absorption rate into a small sphere for a diffusing particle which was confined in a large sphere. Under the assumption that the ratio of the small sphere radius to the large sphere radius was small, we derived asymptotic expansions for the normalized absorption rate with the inverse iteration method.

Hong Zhou would like to thank Naval Postgraduate School Center for Multi-INT Studies for supporting this work. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

Hongyun Wang,Hong Zhou, (2016) Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere. Applied Mathematics,07,709-720. doi: 10.4236/am.2016.77065