Approximate Asymptotic Solutions to the d-dimensional Fisher Equation

1
Approximate Asymptotic Solutions to the
d-dimensional Fisher Equation

• S.PURI, K.R.ELDER, C.DESAI

2
Nonlinear reaction-diffusion equation (1)

• We will confine ourselves to the physically
  interesting
• case.
• Consider the Fourier transform of (1).

3
We can write the expansion for as
(3)

• We will make 4 approximations.
• ?Approximation 1
• We can rewrite (3) as
• (4)
  
• 
  
• where

4

• where and

5

• Calculate using the time integral and Laplace
  transform, we
• get
• (5)
• where and

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

• The largest is for
• Under this approximation, we have

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By simplifying and calculating, becomes

• (6)

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

• This maxima arises for
• Thus, we can further approximate as

9
Then, it reduces to

• (7)

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?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)

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

• (9)

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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

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

• (10)

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Lets 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.