Using the RK4 method, give the numerical approximation for the IVP: y'y(0) = e; with a step size of hto Euler's Method (e.g. simply using the ki values to approximate this ODE)?.ty;20. How does this compare0.5 for the point at tThe exact solution to the IVP above is y(t) = et“/2+1. In a single figure, plot theresults of RK4, Euler methods and the exact function (thus, your plot should contain3 lines). To show the full Y-range, the Y-axis should be plotted using a log-scale,however the X-axis should remain linear. To do this, type "set(gca, 'YScale','log')"on the line below that used to create the plot. Make sure that your figure has alegend that describes what each line corresponds to. t_in=0;*S Matlab code for Euler, Improved Euler, RK4 and ode45 methodclear allclose allFunction for which solution have to dof=@ (t,y) t.*y;ext_val=@ (t) exp( (t. 2/2)+1);SEuler method* amount of intervals* initial t* initial y* at what point we have to evaluate* Number of stepsh=0.5;y_in-exp(1);t_eval=20;n= (t_eval-t_in)/h;t_elrl (1)=t_in;y_elrl(1)=y_in;for i-1:nSEular Stepsm=double (f(t_in,y_in));in=t_in+h;y_in=y_in+h*m;t_elrl (i+1)=t_in;y_elrl (i+l)=y_in;endfprintf('\nThe solution using Euler Method for h=8.2f at t(8.2f)is te\n',h,t_elrl(end),y_elrl(end))*RK4 method* amount of intervalsh=0.5;t_in=0;Y_in-exp(1);t_eval=20;n= (t_eval-t_in)/h;t_rk1 (1)=t_in;y_rk1(1) "y_in;tial t* initial yS at what point we have to evaluate* Number of stepsfor i-1:n&RK4 Stepskl=h*double (f (t_in,y_in));k2=h*double (f( (t_in+h/2), (y in+kl/2)));k3=h*double(f( (t in+h/2), (y in+k2/2)));k4=h*double (f( (t_in+h), (y_in+k3)));dy= (1/6)* (k1+2*k2+2 *k3+k4);t in=t in+h;y_in=y_in+dy;t rkl (i+1)=t_in;Y_rkl (i+1)=y_in;endfprintf('\nThe solution using Runge Kutta Method for h=t.2f att(8.1f) is Be\n',h,t_rk1 (end),y_rkl (end))ext_sol=ext_val(t_rk1);semilogy (t_elr1,y_elrl)1hold onsemilogy (t_rkl,y_rkl)semilogy (t_rk1,ext_sol)xlabel ('time')ylabel ('y(t)')title('y(t) vs. t plot')legend ('Euler','RK4','Exact Solution')The solution using Euler Method for h=0.50 at t(20.00) is 2.264072e+28The solution using Runge Kutta Method for h=0.50 at t(20.0) is2.283976e+67y(t) vs. t plot100FEuler-RK4Exact Solution100100100TO100100101010°4101214161820time

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sing the RK4 method, give the numerical approximation for the IVP: y' = ty; 0) = e; with a step size of h = 0.5 for the point at t = Euler's Method (e.g. simply using the k1 values to approximate this ODE)?. 20. How does this compare he exact solution to the IVP above is y(t) = et“/2+1. In a single figure, plot the sults of RK4, Euler methods and the exact function (thus, your plot should contain lines). To show the full Y-range, the Y-axis should be plotted using a log-scale, owever the X-axis should remain linear. To do this, type "set(gca, 'YScale','log')" n the line below that used to create the plot. Make sure that your figure has a gend that describes what each line corresponds to.

sing the RK4 method, give the numerical approximation for the IVP: y' = ty; 0) = e; with a step size of h = 0.5 for the point at t = Euler's Method (e.g. simply using the k1 values to approximate this ODE)?. 20. How does this compare he exact solution to the IVP above is y(t) = et“/2+1. In a single figure, plot the sults of RK4, Euler methods and the exact function (thus, your plot should contain lines). To show the full Y-range, the Y-axis should be plotted using a log-scale, owever the X-axis should remain linear. To do this, type "set(gca, 'YScale','log')" n the line below that used to create the plot. Make sure that your figure has a gend that describes what each line corresponds to.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Recurrence relation

A recurrence relation or recursive relation is an equation that represents a function in terms of the values of its smaller inputs. Every recurrence relation T(n) is a recursive function of integer n and consists of a base case and a recursive case.

Question

Using the RK4 method, give the numerical approximation for the IVP: y'
y(0) = e; with a step size of h
to Euler's Method (e.g. simply using the ki values to approximate this ODE)?.
ty;
20. How does this compare
0.5 for the point at t
The exact solution to the IVP above is y(t) = et“/2+1. In a single figure, plot the
results of RK4, Euler methods and the exact function (thus, your plot should contain
3 lines). To show the full Y-range, the Y-axis should be plotted using a log-scale,
however the X-axis should remain linear. To do this, type "set(gca, 'YScale','log')"
on the line below that used to create the plot. Make sure that your figure has a
legend that describes what each line corresponds to.

t_in=0;
*S Matlab code for Euler, Improved Euler, RK4 and ode45 method
clear all
close all
Function for which solution have to do
f=@ (t,y) t.*y;
ext_val=@ (t) exp( (t. 2/2)+1);
SEuler method
* amount of intervals
* initial t
* initial y
* at what point we have to evaluate
* Number of steps
h=0.5;
y_in-exp(1);
t_eval=20;
n= (t_eval-t_in)/h;
t_elrl (1)=t_in;
y_elrl(1)=y_in;
for i-1:n
SEular Steps
m=double (f(t_in,y_in));
in=t_in+h;
y_in=y_in+h*m;
t_elrl (i+1)=t_in;
y_elrl (i+l)=y_in;
end
fprintf('\nThe solution using Euler Method for h=8.2f at t(8.2f)
is te\n',h,t_elrl(end),y_elrl(end))
*RK4 method
* amount of intervals
h=0.5;
t_in=0;
Y_in-exp(1);
t_eval=20;
n= (t_eval-t_in)/h;
t_rk1 (1)=t_in;
y_rk1(1) "y_in;
tial t
* initial y
S at what point we have to evaluate
* Number of steps
for i-1:n
&RK4 Steps
kl=h*double (f (t_in,y_in));
k2=h*double (f( (t_in+h/2), (y in+kl/2)));
k3=h*double(f( (t in+h/2), (y in+k2/2)));
k4=h*double (f( (t_in+h), (y_in+k3)));
dy= (1/6)* (k1+2*k2+2 *k3+k4);
t in=t in+h;
y_in=y_in+dy;
t rkl (i+1)=t_in;
Y_rkl (i+1)=y_in;
end
fprintf('\nThe solution using Runge Kutta Method for h=t.2f at
t(8.1f) is Be\n',h,t_rk1 (end),y_rkl (end))
ext_sol=ext_val(t_rk1);
semilogy (t_elr1,y_elrl)
1
hold on
semilogy (t_rkl,y_rkl)
semilogy (t_rk1,ext_sol)
xlabel ('time')
ylabel ('y(t)')
title('y(t) vs. t plot')
legend ('Euler','RK4','Exact Solution')
The solution using Euler Method for h=0.50 at t(20.00) is 2.264072e+28
The solution using Runge Kutta Method for h=0.50 at t(20.0) is
2.283976e+67
y(t) vs. t plot
100
FEuler
-RK4
Exact Solution
100
100
100
TO
100
100
1010
10°
4
10
12
14
16
18
20
time

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