I am an astronomer. I am a postdoc at Iowa State University. I use computer simulations to test models of planet formation.
[ 2 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 2 ] [ f 1 ′ f 2 ′ f 3 ′ f 4 ′ f 5 ′ f 6 ′ f 7 ′ ] = 1 h [ − 3 4 − 1 − 3 3 − 3 3 − 3 3 − 3 3 − 3 3 1 − 4 3 ] [ f 1 f 2 f 3 f 4 f 5 f 6 f 7 ] {\displaystyle {\begin{bmatrix}2&&&&&&\\1&4&1&&&&\\&1&4&1&&&\\&&1&4&1&&\\&&&1&4&1&\\&&&&1&4&1\\&&&&&&2\\\end{bmatrix}}{\begin{bmatrix}f_{1}'\\f_{2}'\\f_{3}'\\f_{4}'\\f_{5}'\\f_{6}'\\f_{7}'\\\end{bmatrix}}={\frac {1}{h}}{\begin{bmatrix}-3&4&-1&&&&\\-3&&3&&&&\\&-3&&3&&&\\&&-3&&3&&\\&&&-3&&3&\\&&&&-3&&3\\&&&&1&-4&3\\\end{bmatrix}}{\begin{bmatrix}f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}\\\end{bmatrix}}}
f i + 1 = f i + f i ′ h + f i ″ 2 ! h 2 + f i ( 3 ) 3 ! h 3 + f i ( 4 ) 4 ! h 4 + f i ( 5 ) 5 ! h 5 + f i ( 6 ) 6 ! h 6 + ⋯ {\displaystyle f_{i+1}=f_{i}+f'_{i}h{\color {red}+{\frac {f''_{i}}{2!}}h^{2}}{\color {blue}+{\frac {f_{i}^{(3)}}{3!}}h^{3}+{\frac {f_{i}^{(4)}}{4!}}h^{4}+{\frac {f_{i}^{(5)}}{5!}}h^{5}+{\frac {f_{i}^{(6)}}{6!}}h^{6}+\cdots }}
f i + 1 ′ = f i ′ + f i ″ h + f i ( 3 ) 2 ! h 2 + f i ( 4 ) 3 ! h 3 + f i ( 5 ) 4 ! h 4 + f i ( 6 ) 5 ! h 5 + ⋯ {\displaystyle f_{i+1}'=f'_{i}{\color {red}+f''_{i}h}{\color {blue}+{\frac {f_{i}^{(3)}}{2!}}h^{2}+{\frac {f_{i}^{(4)}}{3!}}h^{3}+{\frac {f_{i}^{(5)}}{4!}}h^{4}+{\frac {f_{i}^{(6)}}{5!}}h^{5}+\cdots }}
f 1 ′ + 3 f 2 ′ = − 17 f 1 + 9 f 2 + 9 f 3 − f 4 6 h + O ( h 4 ) {\displaystyle f_{1}'+3f_{2}'={\frac {-17f_{1}+9f_{2}+9f_{3}-f_{4}}{6h}}+{\color {blue}{\mathcal {O}}(h^{4})}}
f N ′ + 3 f N − 1 ′ = 17 f N − 9 f N − 1 − 9 f N − 2 + f N − 3 6 h + O ( h 4 ) {\displaystyle f_{N}'+3f_{N-1}'={\frac {17f_{N}-9f_{N-1}-9f_{N-2}+f_{N-3}}{6h}}+{\color {blue}{\mathcal {O}}(h^{4})}}
1 3 f i − 1 ′ + f i ′ + 1 3 f i + 1 ′ = 14 9 f i + 1 − f i − 1 2 h + 1 9 f i + 2 − f i − 2 4 h + O ( h 6 ) {\displaystyle {\frac {1}{3}}f'_{i-1}+f'_{i}+{\frac {1}{3}}f'_{i+1}={\frac {14}{9}}{\frac {f_{i+1}-f_{i-1}}{2h}}+{\frac {1}{9}}{\frac {f_{i+2}-f_{i-2}}{4h}}+{\color {blue}{\mathcal {O}}(h^{6})}}
f i ′ = − 49 f i 20 h + 6 f i + 1 h − 15 f i + 2 2 h + 20 f i + 3 3 h − 15 f i + 4 4 h + 6 f i + 5 5 h − f i + 6 6 h + O ( h 6 ) {\displaystyle f'_{i}=-{\frac {49f_{i}}{20h}}+{\frac {6f_{i+1}}{h}}-{\frac {15f_{i+2}}{2h}}+{\frac {20f_{i+3}}{3h}}-{\frac {15f_{i+4}}{4h}}+{\frac {6f_{i+5}}{5h}}-{\frac {f_{i+6}}{6h}}+{\color {blue}{\mathcal {O}}(h^{6})}}
f i ′ = + 49 f i 20 h − 6 f i − 1 h + 15 f i − 2 2 h − 20 f i − 3 3 h + 15 f i − 4 4 h − 6 f i − 5 5 h + f i − 6 6 h + O ( h 6 ) {\displaystyle f'_{i}=+{\frac {49f_{i}}{20h}}-{\frac {6f_{i-1}}{h}}+{\frac {15f_{i-2}}{2h}}-{\frac {20f_{i-3}}{3h}}+{\frac {15f_{i-4}}{4h}}-{\frac {6f_{i-5}}{5h}}+{\frac {f_{i-6}}{6h}}+{\color {blue}{\mathcal {O}}(h^{6})}}
