Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method

from the above question fourth order approimation in second order differenence

i find out the central difference by using the taylors equation

our numerical stencil will be

f(i-2),f(i-1),f(i),f(i+1),f(i+2) are the points from the center for the f(i)

and the distance between the two points of the equation is (deltax)

based on the above analysis equation we write is

diff(f,x,2)=a*f(i-2)+b*f(i-1)+c*f(i)+d*f(i+1)+e*f(i+2)

inorder to find the value of coefficents of a,b,c,d,e we right as the follows the from the taylors equation table

there fore from the above equation we find the

a+b+c+d+e=0

-2a-b+0+d+2e=0

4a/2+b/2+0+d/2+4e/2=1

-8a/6-b/6+0+d/6+8e/=0

16a/24+b/24+0+d/24+16e/24=0

we right this equation in matrix from as show below

a=[1 1 1 1 1;-2 -1 0 1 2;2 1/2 0 1/2 2;-8/6 -1/6 0 1/6 8/6;16/24 1/24 0 1/24 16/24]

b=[0 ;0 ;1; 0;0]

x=[a; b; c ;d ;e]

inverse (A)

a=-0.0833

b=1.33

c=-2.5000

d=1.333

e=0.0833

we subsitite this in the above equation as shown

similaary for the skewed right hand difference

defined for the five pont from the right hand side as showm

f(i),f(i+1),f(i+2),f(i+3),f(i+4),f(i+5)

we right as the

diff(f,x,2)=a*f(i)+b*f(i+1)+c*f(i+2)+d*f(i+3)+e*f(i+4)+g*f(i+5)

in the above equation a,b,c,d,e,g are the coefficient of the above equation

in the similary way, we find the

a=3.75

b=-12.88

c=17.888

d=-13.00

e=5.088

g=-0.833

using the skewed left hand difference

defined for the five pont from the right hand side as showm

f(i-5),f(i-4),f(i-3),f(i-2),f(i-1),f(i)

e right as the

diff(f,x,2)=a*f(i-5)+b*f(i-4)+c*f(i-3)+d*f(i-2)+e*f(i-1)+g*f(i)

in the above equation a,b,c,d,e,g are the coefficient of the above equation

in the similary way, we find the

a=-0.8333

b=5.0833

c=13.000

d=-17.833

e=-12.8333

g=3.75

program i divided the program in four parts as the

central difference

skewed right side difference

skewed left side difference

function [error_1]= numerical_function(x,dx)

%second order differenece

anatical_difference= -2*exp(x)*sin(x)

%fourth order central difference approixmation

%f"(x)=((a*f(i-2)+b*f(i-1)+c*f(i)+d*f(i+1)+e*f(i+2))/(dx)^2

%a,b,c,d,e are coefficient of the equation

a= -0.0833

b= 1.333

c= -2.5000

d= 1.3333

e= -0.08333

error_1= ((a*exp(x-2*dx)*cos(x-2*dx)+b*exp(x-dx)*cos(x-dx)+c*exp(x)*cos(x)+d*exp(x+dx)*cos(x+dx)+e*exp(x+2*dx)*cos(x+2*dx)))/(dx)^2;

%errors in fourth order central diference approixmation

error_fcda =abs(error_1-anatical_difference)

end

function [error_2]=numerical_function1(x,dx)

%second order differenece

anatical_difference= -2*exp(x)*sin(x)

%fourth order skewed right side approixmation

%f"(x)=(g*f(i))+(h*f(i+1)+l*f(i+2)+j*f(i+3)+k*f(i+4)+m*f(i+5))/(dx)^2

%g,h,k,l,m are coefficent of approximation in taylors equation

g=3.75

h=-12.3

j=17.83

k=-13.00

l=5.08

m=-0.800

error_2 =(g*(exp(x)*cos(x)))+(h*(exp(x+dx)*cos(x+dx)))+(j*(exp(x+2*dx)*cos(x+2*dx)))+(k*(exp(x+3*dx)*cos(x+3*dx)))+(l*(exp(x+4*dx)*cos(x+4*dx))+(m*(exp(x+5*dx)*cos(x+5*dx)))/(dx)^2);

%error in skewed_right_side_approixmation

error_fsra=abs(error_2-anatical_difference)

end

function [error_3]=numerical_function3(x,dx)

%second order differenece

anatical_difference =-2*exp(x)*sin(x)

%fourth order skewed right side approixmation

%f(x)=p*f(i-5)+q*f(i-4)+r*f(i-3)+s*f(i-2)+t*f(i-1)+u*f(i)/(dx)^2

%n,p,q,r,s,t are coefficent of approximation in taylors equation

n=-0.8333

p=5.088

q=-13.000

r=17.800

s=-12.800

t=3.7500

%by using talyors equation

error_3=((n*exp(x-5*dx)*cos(x-5*dx))+(p*exp(x-4*dx)*cos(x-4*dx))+(q*(exp(x-3*dx)*cos(x-3*dx)))+(r*(exp(x-2*dx)*cos(x-2*dx)))+(s*(exp(x-dx)*cos(x-dx)))+(t*(exp(x)*cos(x))))/(dx)^2

%error finding the equation

error_fsla = abs(error_3-anatical_difference)

end

%program_on_the_fourth_order_derivation

%inputs

%functionf(x)=exp(x)*cos(x)

x=pi/3;

%discretization of size x

dx=linspace(pi/4000,pi/40,20);

%error in numerical discretizatio for the first,second and fourth order

%approixmation

for i= 1:length(dx)

error_fcda=numerical_function(x,dx(i));

error_fsra=numerical_function1(x,dx(i));

error_fsla=numerical_function3(x,dx(i));

end

%%plotting for the graph dx vs errors

loglog(dx,error_fcda,'b-*')

hold on

loglog(dx,error_fsra,'g-*')

hold on

loglog(dx,error_fsla,'r-*')

hold off

xlabel('range of dx')

ylabel('error in approximation')

legend('first order','second order','fourth order')

title('error for differentization')