generate a code using appropriate finite difference schemes with a consistent order of o(h2 )v in matlab

To generate a code using appropriate finite difference schemes with a consistent order of O(h^2) in MATLAB, we can use the following steps:

1. Define the function to be approximated and the grid size h. Let's assume the function is f(x) = sin(x) over the interval [0, pi/2].

main.m
% Define the function
f = @(x) sin(x);

% Define the grid size
h = 0.1;
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2. Use the central difference scheme to approximate the first derivative of the function.

main.m
% Central difference scheme for first derivative
dfdx = @(x) (f(x+h) - f(x-h))/(2*h);
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3. Use the central difference scheme to approximate the second derivative of the function.

main.m
% Central difference scheme for second derivative
d2fdx2 = @(x) (f(x+h) - 2*f(x) + f(x-h))/(h^2);
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4. Test the code by evaluating the approximations at a few points.

main.m
% Test the approximations at a few points
x = 0:pi/8:pi/2;
exact = cos(x); % Exact values of the first derivative
approx1 = dfdx(x); % Approximations using central difference scheme
approx2 = d2fdx2(x); % Approximations using central difference scheme
error1 = abs(approx1 - exact);
error2 = abs(approx2 - f(x));

% Display the results
disp('    x        f''(x) (exact) f''(x) (approx) Error  f''''(x) (exact) f''''(x) (approx) Error')
disp(num2str([x' exact' approx1' error1' f''(x)' approx2' error2']))
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This should generate a table of values for x, f'(x) (exact), f'(x) (approximation using central difference) and the associated error. Similarly, we obtain values for f''(x) (exact), f''(x) (approximation using central difference) and the associated error.

Note that the order of accuracy of the central difference scheme is O(h^2) for both the first and second derivative approximations.