Question

choosing C to make P(t) match P(0) at t = 0. Compute the maximum error

over 0 ≤ t ≤ 1 for each solution you obtain. How do the errors change with

h for the two methods?

dP aP (PM-P). With a 1 and PM 10, solve this equation with both Euler's method and Heun's method for step sizes h = 10-k for k = 1, 2, 3 for the interval 0 t1. Use the initial value P(O) 1. Given that the exact solution is

Answer 1

Matlab code for Euler and Heun method clear all close all all initial conditions t in-0 t end-1: P_in-l; a-1; Pm-10; Function for exact solution P_ext-e t) Pm*C exp (a Pm*t))/(1+C*exp (Pm*a*t)); fprintf( Displaying exact solution\n) disp (P_ext) fprint f ("Here value C-%f\n\n",c) of %loop for step size for k-1:3 h-10A-k; [P_euler,t_elrl euler (h, P_in,t_in,t_end); [ P-heun, t_hnn-heun ( h, P-in, t-in, tend): fprintf("For step size h=%f\n',h) fprintf( "\tUsing Euler method at t-% f value of P(%f) is %f. n',telr(end),telr (end), P_euler (end)) fprint f ( '\tUsing Heun method at t=%f value of P(&E) is %f. n',t_hnn(end),t_hnn (end), P_heun (end)) tprint f ( '\tExact tprint f ( '\tError fprint f ( '\tError solution at t=% f ǐs %f"\n', t-end,P-ext(t-end)) in Euler method is %e.\n' ,abs (P-ext (t-end)- in Heun method is %e.\n\n', abs(P-ext (t-end)- P euler (end) )) P heun (end))) end 8888888888 88888888888888888888888888888888888888888888훔 function for euler method function [PI,t]-euler(h, P İn,t in,t end) all parameter values a= 1 ; Pn« 10 ; %function (i) f-e(t,P) a*P(Pm-P); initial conditions t(1)-t-in:P1 ( 1 )-P-in;

t_in-t (1); t max-t_ end; n« ( t-max-t-in )/h; %Initial t Final t of %number steps Runge Kutta 4 iterations for i-1: n Pl (i+1)- double (P1(i)+h*(f(t(i),P1(i))) end end 용号号응용号号응용号号응용号号응용号응응용号융융용융융용용융용용융융용용융융용용융융욤욤융융욤욤융융욤욤융융욤욤융융 %function for heun method function [P1, t-heun (h,P_in,t in,t_end) all parameter values function (i) f-et,P) a*P(Pm-P); %initial conditions t (1)-t_in;P1 (1)-P_in; %Final t t_max-t_end; tn-t in:h:t max; %Runge Kutta I iterations for i-1: length(tn)-1 ml-double(f (t, P1 (i))); m2-double ( f ( ( tth ) , ( PI ( İ ) +h *m1 ) ) ) ; P1 (i+1)-P1 (i)+double (h m1+m2)/2) end Displaying exact solution e(t) (Pm*c*exp (a*Pm*t))/ (1+C*exp(Pm*a*t)) For step size h=0.100000 Using Euler method at t-1.000000 value of P 1.000000) is 10.000000. Using Heun method at t-1.000000 value of P(1.000000) is 9.972242 Exact solution at t-1.000000 is 9.995916. Error in Euler method is 4.084325e-03 Error in Heun method is 2.36734 2e-02 For step size h-0.010000 Using Euler method at t-1.000000 value of P(1.000000) is 9.996939

using Heun method at t-1.000000 value of P( 1.000000) is 9.995863. Exact solution at t-1.000000 is 9.995916 Error in Euler method is 1.023244e-03 Error in Heun method is 5.279111e-05. For step size h 0.001000 Using Euler method at t=1.000000 value of p( 1.000000) is 9.996025. Using Heun method at t-1.000000 value of P(1.000000) is 9.995915. Exact solution at t-1.000000 is 9.995916 Error in Euler method is 1.093647e-04. Error in Heun method is 4.995105e-07. Published with MATLAB® R2018a

%Matlab code for Euler and Heun method
clear all
close all

%all initial conditions
t_in=0; t_end=1;
P_in=1;
a=1; Pm=10;
%Function for exact solution

C=(1/(Pm-1));
P_ext=@(t) (Pm*C*exp(a*Pm*t))/(1+C*exp(Pm*a*t));
fprintf('Displaying exact solution\n')
disp(P_ext)

fprintf('Here value of C=%f\n\n',C)

%loop for step size
for k=1:3
    h=10^-k;
    [P_euler,t_elr]=euler(h,P_in,t_in,t_end);
    [P_heun,t_hnn]=heun(h,P_in,t_in,t_end);
  
    fprintf('For step size h=%f\n',h)
    fprintf('\tUsing Euler method at t=%f value of P(%f) is %f.\n',t_elr(end),t_elr(end),P_euler(end))
  
    fprintf('\tUsing Heun method at t=%f value of P(%f) is %f.\n',t_hnn(end),t_hnn(end),P_heun(end))
  
    fprintf('\tExact solution at t=%f is %f.\n',t_end,P_ext(t_end))
  
    fprintf('\tError in Euler method is %e.\n',abs(P_ext(t_end)-P_euler(end)))
    fprintf('\tError in Heun method is %e.\n\n',abs(P_ext(t_end)-P_heun(end)))
end
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%function for euler method

function [P1,t]=euler(h,P_in,t_in,t_end)
  
    %all parameter values
    a=1; Pm=10;
    %function (i)
    f=@(t,P) a*P*(Pm-P);
  
    %initial conditions
    t(1)=t_in;P1(1)=P_in;
              
    t_in=t(1);            %Initial t
    t_max=t_end;          %Final t
    n=(t_max-t_in)/h; %number of steps
    %Runge Kutta 4 iterations
    for i=1:n
        t(i+1)=t_in+i*h;
        P1(i+1)=double(P1(i)+h*(f(t(i),P1(i))));

    end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%function for heun method

function [P1,t]=heun(h,P_in,t_in,t_end)

    %all parameter values
    a=1; Pm=10;
    %function (i)
    f=@(t,P) a*P*(Pm-P);
    %initial conditions
    t(1)=t_in;P1(1)=P_in;
              
    t_max=t_end;          %Final t
    tn=t_in:h:t_max;
    %Runge Kutta 1 iterations
    for i=1:length(tn)-1
      
        t(i+1)= t(i)+h;
        m1=double(f(t,P1(i)));
        m2=double(f((t+h),(P1(i)+h*m1)));
        P1(i+1)=P1(i)+double(h*((m1+m2)/2));
      
     
    end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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