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

Given function:

f = exp(x)*cos(x)

2nd order derivative of the function

f'' = -2*exp(x)*sin(x)

1) Developing Taylor Table for the Forward Differencing Scheme Approximation method

For FDS, f = (af(x) + bf(x+dx) + cf(x+2dx) + df(x+3dx) + ef(x+4dx))/(dx^2)

Table:

2) Developing Taylor Table for the Backward Differencing Scheme Approximation method

 For BDS, f = (af(x) + bf(x-dx) + cf(x-2dx) + df(x-3dx) + ef(x-4dx))/(dx^2)

Table:

3) Developing Taylor Table for the Central Differencing Scheme Approximation method

 For CDS, f = (af(x-2dx) + bf(x-dx) + cf(x) + df(x+dx) + ef(x+2dx))/(dx^2)

Table:

Coefficient for all three approximation methods will be calcualted in MATLAB only. 

Code:

Main Code:

clear all

 close all

 clc

 %inputs

 x=pi/3;

 %stepsize range

 dx=linspace(pi/4,pi/400,2000);

 %Analytical derivative

 %f(x) = exp(x)*cos(x)

 %f'(x)=(exp(x)*cos(x)-(exp(x)*sin(x)=exp(x)*(cosx-sinx)

 %Double derivative f" = -2*exp(x)*sin(x)

 analytical_derivative = -2*exp(x)*sin(x);

 %Skewed right-sided(4th order approimation)

 %Solving for the coefficients

 A=[1 1 1 1 1 1;0 1 2 3 4 5; 0 1/2 4/2 9/2 16/2 25/2; 0 1/6 8/6 27/6 64/6 125/6; 0 1/24 16/24 81/24 256/24 625/24; 0 1/120 32/120 243/120 1024/120 3125/120];

 B=[0;0;1;0;0;0];

 x_r=inv(A)*B;

 %Skewed left-sided(4th order approimation)

 %Solving for the coefficients

 C=[1 1 1 1 1 1;0 -1 -2 -3 -4 -5; 0 1/2 4/2 9/2 16/2 25/2; 0 -1/6 -8/6 -27/6 -64/6 -125/6; 0 1/24 16/24 81/24 256/24 625/24; 0 -1/120 -32/120 -243/120 -1024/120 -3125/120];

 D=[0;0;1;0;0;0];

 x_l=inv(C)*D;

 %central differencing(4th order approimation)

 %Solving for the coefficients

 E=[1, 1, 1, 1, 1; -2, -1, 0, 1, 2; 4/2, 1/2, 0, 1/2, 4/2; -8/6, -1/6, 0, 1/6, 8/6; 16/24, 1/24, 0, 1/24, 16/24];

 F=[0;0;1;0;0];

 x_c=E^-1*F;

 %Anonymous Function

 f=@(x) exp(x).*cos(x);

 %numerical derivative

right_difference=(x_r(1)*f(x)+x_r(2)*f(x+dx)+x_r(3)*f(x+2.*dx)+x_r(4)*f(x+3.*dx)+x_r(5)*f(x+4.*dx)+x_r(6)*f(x+5.*dx))./(dx.^2)

 left_difference=(x_r(1)*f(x)+x_r(2)*f(x-dx)+x_r(3)*f(x-2.*dx)+x_r(4)*f(x-3.*dx)+x_r(5)*f(x-4.*dx)+x_r(6)*f(x5.*dx))./(dx.^2)

cent_difference=(x_c(1)*f(x2.*dx)+x_c(2)*f(x1.*dx)+x_c(3)*f(x)+x_c(4)*f(x+dx)+x_c(5)*f(x+2.*dx))./(dx.^2)

 %absolute error

 error_r=abs(analytical_derivative-right_difference);

 error_l=abs(analytical_derivative-left_difference);

 error_c=abs(analytical_derivative-cent_difference);

 error=[error_r;error_l;error_c];

 figure(1)

 loglog(dx,error_r)

 grid on

 hold on

 loglog(dx,error_l)

 loglog(dx,error_c)

 ylabel('Absolute Error (Log Scale)')

 xlabel('Grid Spacing (dx in Log Scale)')

 legend('Right Differencing','Left Differencing','Central Differencing')

RESULTS: