Question

Write a MATLAB Code to

8.5 (9) LAB. Implement the classical procedure (8.82), and apply it to equation (8.66). Solve it with stepsizes of h = 0.25 and 0.125 Compare with results in Table 8.13, the fourth- order Fehlberg example. Y'(x) = -Y(x) + 2 co$(x), Y(0) = 1 (8.66) Runge-Kutta methods of higher order can also be developed. A popular classical method is the following fourth-order procedure: Vi = f (xn, yn) h h v2 = f ( xx + + - 2 h h + Yn + 02 (8.82) V4 = f(xn+h, yn + hv3) h Yn+1 = yn + õlui +202 + 2v3 + va] Table 8.13. Example of Fourth-Order Fehlberg Formula (8.86) h yn(x) Y(x) - yn(x) ûn(x) – yn(x) 0.25 2.0 0.493156301 -5.71E-6 -9.49E-7 4.0 -1.410449823 3.71E-6 1.62E-6 6.0 0.680752304 2.48E-6 -3.97E-7 8.0 0.843864007 -5.79E-6 - 1.29E-6 10.0 -1.383094975 2.34E-6 1.47E-6 0.125 2.0 0.493150889 -2.99E-7 -2.35E-8 4.0 -1.410446334 2.17E-7 4.94E-8 6.0 0.680754675 1.14E-7 -1.76E-8 8.0 0.843858525 -3.12E-7 -3.47E-8 10.0 -1.383092786 1.46E-7 4.65E-8

Answer 1

`Hey,

Note: If you have any queries related to the answer please do comment. I would be very happy to resolve all your queries.

clc

clear all

close all

format long

f=@(x,y) -y+2*cos(x);

disp('Solution for h=0.25')

[X1,Y1]=runge4(f,[0,1],1,0.25)

[X2,Y2]=runge4(f,[0,1],1,0.125)

plot(X1,Y1,X2,Y2);

legend('h=0.25','h=0.125')

function [x,y]=runge4(f,tspan,y0,h)

x = tspan(1):h:tspan(2); % Calculates upto y(3)

y = zeros(length(x),1);

y(1,:) = y0; % initial condition

for i=1:(length(x)-1) % calculation loop

k_1 = f(x(i),y(i,:));

k_2 = f(x(i)+0.5*h,y(i,:)+0.5*h*k_1);

k_3 = f((x(i)+0.5*h),(y(i,:)+0.5*h*k_2));

k_4 = f((x(i)+h),(y(i,:)+k_3*h));

y(i+1,:) = y(i,:) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h; % main equation

end

end

Efx 2 E . A=80 1.45 EDIT h=0.25 h=0.125 1.4 CS MATL tor-/Use eee.m X 1.35 y(1,:) 1.3 for i= 125 k_1 = 1.2 k_2 = 1.15 k_3 = 1.1 k_4 = 1.05 y(i+1, end 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 end hand Window MATLAB? See resources for Getting Started. = 1.000000000000000 1.116872039079460 1.216315682883794 1.296779144337874 1.357006816031186 1.396058863520134 1.413325891102499 1.408538451209107 1.381771249021456

Kindly revert for any queries

Thanks.