Approximate Asymptotic Solutions to the d-dimensional Fisher Equation

S.PURI, K.R.ELDER, C.DESAI

Nonlinear reaction-diffusion equation

(1)

We will confine ourselves to the physically interesting

case.

Consider the Fourier transform of (1).

We can write the expansion for as

(3)

We will make 4 approximations.

☆Approximation 1

We can rewrite (3) as

(4)

where

where and

Calculate using the time integral and Laplace transform, we

get

(5)

where and

☆Approximation 2 In (5), the dominant term is the one with the largest

The largest is for

Under this approximation, we have

By simplifying and calculating, becomes

(6)

☆Approximation 3 In (6), we need the point where the exponential term is maximum.

This maxima arises for

Thus, we can further approximate as

Then, it reduces to

(7)

☆Approximation 4 In (7), we will consider only the modes. (It is necessary so as to put the solution into a summable form.)

Under this approximation, we have

and from (4)

(8)

In (8), taking the inverse Fourier transformation on both sides, we have

(9)

An interesting condition is one in which we have a populated site in a background of zero population:

: “seed amplitude”

: the location of the initial seed

The solution corresponding to (9) for this initial condition is

(10)

Let’s assume the midpoint of the interface is located at time t and at the distance r(t). (also let =0 and =1)

Substituting into (10), we obtain

The analytic solution corresponds to domain growth with an

asymptotic velocity in all dimensions.