Numerical Method question solutions

1. Consider the following second order differential equation: d2 u (N — 1) du

— + —u = O in (0, 1), (0.1) dr r dr r2 where 1 + I (N — and N 1, 2, 3.

• Find (erplicitly) all the solutions to (0.1) having the form

Hint: make the change of variables t log(r) for r e (0, 1). Find the ODE that v(log(r)) v(t) solves. Observe that this change of variable maps (0, 1) (—00, 0).

• Find the exact solution for the ODE (0.1) with the Final Values

u(l) 0, du (l) —— 1. dr

Hint: Use the explicit solutions found in i.

• Write one MATLAB script to solve numerically the ODE (0.1) + Final Value problem in ii.. Hint: rlYansforrn the Final Value Problem into an IVP reversing the interval (0, 1) using

The script must contain at least one implementation based on the Crank-Nicholson method and another based on the Runge-Kutta method of order 2.

Include plots to compare the implementatioms against the true sohltion. Provide also analysis of your findings.

[James Tursa]

What have you done so far? What specific problems are you having with your code?

Can you add an image showing the differential equation and instructions? I can't understand what you have posted.

[Mehmet Çuha]

ı am trying to use this code but I cant replaces variables .

function [t,u]=feuler(odefun,tspan,y0,Nh,varargin)
%FEULER Solves differential equations using the forward
%  Euler method.
%  [T,Y]=FEULER(ODEFUN,TSPAN,Y0,NH) with TSPAN=[T0,TF]
%  integrates the system of differential equations
%  y'=f(t,y) from time T0 to TF with initial condition
%  Y0 using the forward Euler method on an equispaced
%  grid of NH intervals.
%  Function ODEFUN(T,Y) must return a vector, whose
%  elements hold the evaluation of f(t,y), of the
%  same dimension of Y.
%  Each row in the solution array Y corresponds to a
%  time returned in the  column vector T.
%  [T,Y] = FEULER(ODEFUN,TSPAN,Y0,NH,P1,P2,...) passes
%  the additional parameters P1,P2,... to the function
%  ODEFUN as ODEFUN(T,Y,P1,P2...).

h=(tspan(2)-tspan(1))/Nh;
y=y0(:); % always creates a column vector
w=y; u=y.';
tt=linspace(tspan(1),tspan(2),Nh+1);
for t = tt(1:end-1)
 w=w+h*odefun(t,w,varargin{:});
 u = [u; w.'];
end
t=tt';
return