Approximate Asymptotic Solutions to the d-dimensional Fisher Equation - PowerPoint PPT Presentation

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Approximate Asymptotic Solutions to the d-dimensional Fisher Equation

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Approximate Asymptotic Solutions to the d-dimensional Fisher Equation S.PURI, K.R.ELDER, C.DESAI Nonlinear reaction-diffusion equation (1) We will confine ... – PowerPoint PPT presentation

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Title: 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

6
?Approximation 2 In (5), the dominant term
is the one with the largest
  • The largest is for
  • Under this approximation, we have

7
By simplifying and calculating, becomes
  • (6)

8
?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)

10
?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)

11
In (8), taking the inverse Fourier transformation
on both sides, we have
  • (9)

12
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

13
The solution corresponding to (9) for this
initial condition is
  • (10)

14
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.
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