QuestionQuestion

Please see Questions.pdf

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

(This equation is called the Nagumo equation, or at least it has that form).

Let u(x,t)=v(x-sig t), (here sig = sigma)
Then ut=-sig v', uxx=v''
and we obtain the equation:
v''+sig v'+f(v)=0, where f(v)=-2v3+3v2-v

we want v(s) to go to 1 as s tends to infinity, to go to zero as s tends to -infinity,
and we want that v' tends to zero as s tends to + or – infinity.

If we write the ODE as a system:
v'=w
w'=-f(v) – sig w.

Then we want an equilibrium (also known as stationary) point at P=(1,0) which is the case since f(1)=0 and an equilibrium points at O=(0,0), which is the case since f(0) =0 and for the solution (v(s),w(s)) to go to 0 as s tends to -infinity and to go to P as s tends to +infinity.
Linear stability analysis shows that both P and O are saddle points for the linearization, which is the same for both and of the form [0 1; 1 -sig]: the determinant is negative, and this means one positive eigenvalue and one negative eigenvalue for each....
$15.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Differential Equations Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Decision:
Upload a file
Continue without uploading

SUBMIT YOUR HOMEWORK
We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats