Consider Fisher's equation x+u(1u) - (x-ct) with (-00) = 1 and o(+0) = 0 is given...

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Consider Fisher's equation x+u(1u) - (x-ct) with (-00) = 1 and o(+0) = 0 is given by U = (i) The only traveling "front" wave u(x,t) = | u(x,t) = [1. + exp Verify this assertion by deriving the ODE that (; 5 - has to satisfy and then use a numerical method to solve it. [Of course you should also try to solve it analytically!] (ii) Since u(x,t) 0 (resp. 1) as x + (resp. -), we can approximate the traveling wave given in (i) in (-L, L), where L is large enough so that the wave front does not reach the boundary x=L, by imposing the boundary conditions u(-L,t) =1, u(L,t) = 0, and taking the initial value as u(x, 0). Design and implement a Chebyshev collocation method to approximate the solution of the PDE and study the convergence of the approximate solution as N is increasing. Consider Fisher's equation x+u(1u) - (x-ct) with (-00) = 1 and o(+0) = 0 is given by U = (i) The only traveling "front" wave u(x,t) = | u(x,t) = [1. + exp Verify this assertion by deriving the ODE that (; 5 - has to satisfy and then use a numerical method to solve it. [Of course you should also try to solve it analytically!] (ii) Since u(x,t) 0 (resp. 1) as x + (resp. -), we can approximate the traveling wave given in (i) in (-L, L), where L is large enough so that the wave front does not reach the boundary x=L, by imposing the boundary conditions u(-L,t) =1, u(L,t) = 0, and taking the initial value as u(x, 0). Design and implement a Chebyshev collocation method to approximate the solution of the PDE and study the convergence of the approximate solution as N is increasing.