Answer to Question #89652 in Quantitative Methods for lana majeed

Question #89652
obtain second order derivative y'' using Taylor series Expansion using 4 point
1
Expert's answer
2019-05-16T14:11:20-0400

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)"


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