Question

4. (Matlal) attatimient) Consider the initial valle probleni 1<t< 2 y(1) 1 Caleulate the approximate solutions using forward Euler method, two stage and four stage Runge Kutta method with h 1/10, 1/20,1/40 and compute the maximum errors between the exact solution and the approximate solutions. Use this maximum error to verify the convergence order of each method (1, 2, and 1). Note: the exact solution is

Answer 1

Answer: Code: Define method function [ t, y 1 - euler f, tRange, yInitial, h ) %Find value nums teps = ( tRange (2) -tRange (1) ) / h; Create vector t-zeros (numSteps+1,1) set value t (1) = tRange (1) ; %set value y (1) - yInitial; %Loop for kk-1 : numStep:s %Find value t (kkt 1) = t (kk) + h; %Find value %End

%End end %Define function function [t, y TwostageRK ( f, tRange, yInitial, h) %Find value numSteps = ( tRange (2) -tRange ( 1 ) ) / h; %Create vector t=zeros (nums teps+1, 1) ; %set value t (1) = tRange (1) ; %set value y(1) y!nitial; = %Loop for kk-1 numStep:s %Find value a =h*f(t(kk), y(kk)); %Find value

bf (t (kk) +0.5*h, y(kk) +0.5*a); %Compute t (kk+1)t (kk) hi %compute y (kkt1)- y(kk) bi %End end %End end %Define function function [ t, y ] = FourStageRK ( f, tRange, yinitial, h ) %Find value numSteps stRange (2) - tRange(1) hi %Create vector t-zeros (numSteps+1,1) %set value t (1)tRange (1) %set value

y (1)yInitial; %Loop for kk-1: numSteps &Compute &Compute b f(t (kk) +0.5*h, y(kk) +0.5*a); = %Compute d = f(t(kk) +h, y(kk) +ch); %Find value t (kk+1)t(kk) hi %Find value %End end %End

end %Define function defun (t,y) -1*y*y*Y: %define function f r exact s luti n soln (t) sqrt (1/ (2*t +1)) %Define h value hval[0.1 0.05 0.025]; %Loop for kk-1: length (hval) %Get current h value h-hval (kk); Print fprintf ("In H value : %f", h); %Print n" Print fprintf ("\n Forward Euler Method:" Call function

[t,y] = euler (defun , [1,2], 1, h); %Create realVal -zeros (length (t), 1) %Loop kk-1: length (t) %Find value realVal (kk) - soln (t (kk)) %End end %Find value maxErr1 = max (abs (realva 1-y) ) ; %Print fprintf("\n Maximum Error: %f\n", maxErr1); %print fprintf ("In Two Stage Runge Kutta Method:") %Call function [t2,y2] - TwoStageRK (defun, [1,2], 1, h); %Create vector

realVal2zeros (length (t2), 1); %Create vect r maxErr-zeros (length (t2), 1); %Loop for kk= 1: length (t2) %Find value realva12 (kk) = soln (t (kk)); Compute error maxErr (kk) abs(realva12 (kk) - y2 (kk)) %End end %Find max error maxErr2 = max (maxErr) ; %Print fprintf("\n Maximum Error: %f\n", maxErr2); %Define function fprintf("In Four Stage Runge Kutta Method:") %call function

t3,y3 -FourStageRK (defun, [1,2], 1, h) %Create vector realVal3 -zeros (length (t3),1) %create vector max_Err- zeros (length (t3), 1) %Loop for kk-1: length (t2) %Find value realva13 (kk) soln (t(kk)); %Find error max_Err (kk) abs (realVal3 (kk) - y3 (kk)): %End end %Find max error maxErr3 = max (max. Err) ; % Print fprintf("\n Maximum Error: %f\n", maxErr3); %End

end Sample Output Η value : Θ . 100000 Forward Euler Method: Maximum Error: 0.422650 Two Stage Runge Kutta Method Maximum Error: 6.422650 Four Stage Runge Kutta Method: Maximum Error: 6.422650 H value : . O50000 Forward Euler Method: Maximum Error: 6.422650 Two Stage Runge Kutta Method Maximum Error: 0.494821 Four Stage Runge Kutta Method: Maximum Error: 6.422650 Η value : Θ . 025000 Forward Euler Method: Maximum Error: 0.422650 Two Stage Runge Kutta Method Maximum Error: 6.535565 Four Stage Runge Kutta Method: Maximum Error: 6.422650

Copyable code:

%Define method
function [ t, y ] = euler ( f, tRange, yInitial, h )

   %Find value
   numSteps = ( tRange(2) - tRange(1) ) / h;

   %Create vector
   t=zeros(numSteps+1,1);

   %Set value
   t(1) = tRange(1);

   %Set value
   y(1) = yInitial;

   %Loop
   for kk = 1 : numSteps

       %Find value
       t(kk+1) = t(kk) + h;

       %Find value
       y(kk+1,:) = y(kk,:) + h * f(t(kk), y(kk));

   %End
   end

%End
end

%Define function
function [ t, y ] = TwoStageRK ( f, tRange, yInitial, h )

   %Find value
   numSteps = ( tRange(2) - tRange(1) ) / h;

   %Create vector
   t=zeros(numSteps+1,1);

   %Set value
   t(1) = tRange(1);

   %Set value
   y(1) = yInitial;

   %Loop
   for kk = 1 : numSteps

       %Find value
       a = h*f(t(kk), y(kk));

       %Find value
       b = f(t(kk)+0.5*h, y(kk)+0.5*a);

       %Compute
       t(kk+1) = t(kk) + h;

       %compute
       y(kk+1) = y(kk) + b;

   %End
   end

%End
end

%Define function
function [ t, y ] = FourStageRK ( f, tRange, yInitial, h )

   %Find value
   numSteps = ( tRange(2) - tRange(1) ) / h;

   %Create vector
   t=zeros(numSteps+1,1);

   %Set value
   t(1) = tRange(1);

   %Set value
   y(1) = yInitial;

   %Loop
   for kk = 1 : numSteps

       %Compute
       a = f(t(kk), y(kk));

       %Compute
       b = f(t(kk)+0.5*h, y(kk)+0.5*a);

       %Compute
       c = f(t(kk)+0.5*h, y(kk)+0.5*h*b);

       %Compute
       d = f(t(kk) +h, y(kk)+c*h);

       %Find value
       t(kk+1) = t(kk) + h;

       %Find value
       y(kk+1) = y(kk) + (1/6) *(a+2*b+2*c + d)*h;

   %End
   end

%End
end

%Define function
defun = @(t,y) -1*y*y*y;

%define function for exact solution
soln = @(t) sqrt(1/(2*t +1));

%Define h value
hval = [0.1 0.05 0.025];

%Loop
for kk=1: length(hval)

   %Get current h value
   h = hval(kk);

   %Print
   fprintf("\n H value: %f", h);

   %Print
   fprintf("\n-----------------------\n");

   %Print
   fprintf("\n Forward Euler Method:");

   %Call function
   [t,y] = euler(defun, [1,2], 1, h);

   %Create
   realVal = zeros(length(t), 1);

   %Loop
   for kk=1: length(t)

       %Find value
       realVal (kk) = soln(t(kk));

   %End
   end

   %Find value
   maxErr1 = max(abs(realVal-y));

   %Print
   fprintf("\n Maximum Error: %f\n", maxErr1);

   %Print
   fprintf("\n Two Stage Runge Kutta Method:");

   %Call function
   [t2,y2] = TwoStageRK(defun, [1,2], 1, h);

   %Create vector
   realVal2 = zeros(length(t2), 1);

   %Create vector
   maxErr = zeros(length(t2), 1);

   %Loop
   for kk=1: length(t2)

       %Find value
       realVal2 (kk) = soln(t(kk));

       %Compute error
       maxErr(kk) = abs(realVal2(kk) - y2(kk));

   %End
   end

   %Find max error
   maxErr2 = max(maxErr);

   %Print
   fprintf("\n Maximum Error: %f\n", maxErr2);

   %Define function
   fprintf("\n Four Stage Runge Kutta Method:");

   %Call function
   [t3,y3] = FourStageRK(defun, [1,2], 1, h);

   %Create vector
   realVal3 = zeros(length(t3), 1);

   %Create vector
   max_Err = zeros(length(t3), 1);

   %Loop
   for kk=1: length(t2)

       %Find value
       realVal3 (kk) = soln(t(kk));

       %Find error
       max_Err(kk) = abs(realVal3(kk) - y3(kk));

   %End
   end

   %Find max error
   maxErr3 = max(max_Err);

   %Print
   fprintf("\n Maximum Error: %f\n", maxErr3);

%End
end