## DISCRETIZATION OF A FIRST ORDER DERIVATIVE USING A RANGE OF dx TERMS

DESCRIPTION:

Here in this code, we are using a range of dx terms for discretizing the first order derivative and studying the results of the dx vs error plot

CODE:

clear all
close all
clc

x = pi/3;
dx = linspace(pi/4,pi/4000,30);

%y = (sinx)/(x^3)
%actual dy by equation
dy = (x^3 *cos(x) - 3*x^2 *sin(x))/x^6;

count = 1;
for i = 1:length(dx)
%first order appx by frwd method
dy1(i) = ((sin(x+dx(i))/(x+dx(i))^3) - (sin(x)/(x)^3))/(dx(i));

%second order appx by central method
dy2(i) = ((sin(x+dx(i))/(x+dx(i))^3) - (sin(x-dx(i))/(x-dx(i))^3))/(2*dx(i));

%fourth order appx
dy3(i) = ((sin(x-(2*dx(i)))/(x-(2*dx(i)))^3)-(8*(sin(x-dx(i))/(x-dx(i))^3))+(8*(sin(x+dx(i))/(x+dx(i))^3))-(sin(x+(2*dx(i)))/(x+(2*dx(i)))^3))/(12*dx(i));

error1(i) = abs(dy1(i)- dy);
error2(i) = abs(dy2(i)- dy);
error3(i) = abs(dy3(i)- dy);
count = count+1;
end

loglog(dx,error1,'b','LineWidth',2)
hold on
loglog(dx,error2,'r','LineWidth',2)
loglog(dx,error3,'g','LineWidth',2)

title('ERROR STUDY FOR A RANGE OF dx TERMS')
legend('first order error','second order error','fourth order error','location','southeast')
xlabel('log(dx term)')
ylabel('log(error)')

%find slope of error lines

slope1 = (log(error1(2))-log(error1(1)))/(log(dx(2))-log(dx(1)))

slope2 = (log(error2(2))-log(error2(1)))/(log(dx(2))-log(dx(1)))

slope3 = (log(error3(2))-log(error3(1)))/(log(dx(2))-log(dx(1)))

PLOT:

RESULT:

We can observe that the value of error for Fourth order approximation scheme is of the order 10e(-10) and for second order approximation it was of the order 10e-5 and for first order approximation which has the maximum error and it is of the order 10e-3.

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