Question

obtain second order derivative y'' using Taylor series Expansion using 4 point

Answer 1

Five point central difference formula for the second order derivative can expressed as

$$f''(x_i)\approx{af(x_i)+bf(x_{i-1})+cf(x_{i+1})+df(x_{i-2})+ef(x_{i+2}) \over (\Delta x)^2}$$

Taylor series expansions for $f(x_{i-1}),\ f(x_{i+1}),\ f(x_{i-2})$ and $\ f(x_{i+2}),$ we get

$${af(x_i)+bf(x_{i-1})+cf(x_{i+1})+df(x_{i-2})+ef(x_{i+2}) \over (\Delta x)^2}=$$

$$={(a+b+c+d+e) \over (\Delta x)^2}f(x_i)+$$

$$+{(-b+c-2d+2e)f(x_i) \over \Delta x}f'(x_i)+$$

$$+{1 \over 2}(b+c+4d+4e)f''(x_i)+$$

$$+{\Delta x \over 6}(-b+c-8d+8e)f'''(x_i)+$$

$$+{(\Delta x)^2 \over 24}(b+c+16d+16e)f^{(IV)}(x_i)+$$

$$+{(\Delta x)^3 \over 120}(-b+c-32d+32e)f^{(V)}(x_i)+$$

$$+{(\Delta x)^4 \over 720}(b+c+64d+64e)f^{(VI)}(x_i)+H$$

The coefficients must satisfy the following conditions

$$a+b+c+d+e=0$$ $$-b+c-2d+2e=0$$ $${1 \over 2}(b+c+4d+4e)=1$$ $$-b+c-8d+8e=0$$ $$b+c+16d+16e=0$$

$$b=c$$ $$d=e$$ $$b=-16d$$ $$-32d+8d=2$$ $$a-32d+2d=0$$

$$a=-{5 \over 2}$$ $$b={4 \over 3}$$ $$c={4 \over 3}$$ $$d=-{1 \over 12}$$ $$e=-{1 \over 12}$$

$$f''(x_i)\approx{-{5 \over 2}f(x_i)+{4 \over 3}f(x_{i-1})+{4 \over 3}f(x_{i+1})-{1 \over 12}f(x_{i-2})-{1 \over 12}f(x_{i+2}) \over (\Delta x)^2}$$

$$f''(x_i)\approx{-30f(x_i)+16f(x_{i-1})+16f(x_{i+1})-f(x_{i-2})-f(x_{i+2}) \over 12(\Delta x)^2},$$ $$O((\Delta x)^4)$$