Ps3 | Physical chemistry homework help

1. A 3-point stencil for derivatives by MATLAB. Let’s say we wanted to develop a 3-point stencil for calculating a derivative:

𝑓(𝑥)′ = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑏 ∙ 𝑓(𝑥) + 𝑐 ∙ 𝑓(𝑥 + ℎ) and a double derivative: 𝑓(𝑥)′′ = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑏 ∙ 𝑓(𝑥) + 𝑐 ∙ 𝑓(𝑥 + ℎ)

where the various points are defined as:

𝑎 ∙ 𝑓(𝑥 − ℎ) = 𝑎 ∙ 𝑓(𝑥) − 𝑎 ∙ ℎ 2 ∙ 𝑓(𝑥)′ + 𝑎 ∙ ℎ2 2 ∙ 𝑓(𝑥)′′ 𝑏 ∙ 𝑓(𝑥) = 𝑏 ∙ 𝑓(𝑥) 𝑐 ∙ 𝑓(𝑥 + ℎ) = 𝑐 ∙ 𝑓(𝑥) + 𝑐 ∙ ℎ 2 ∙ 𝑓(𝑥)′ + 𝑐 ∙ ℎ2 2 ∙ 𝑓(𝑥)′′ Consequently, one would find that, for the derivative 𝑓(𝑥)′ it must be true that: 𝑎 + 𝑏 + 𝑐 = 0 −𝑎 ∙ ℎ 2 + 𝑐 ∙ ℎ 2 = 1 𝑎 ∙ ℎ2 2 + 𝑐 ∙ ℎ2 2 = 0 You can solve this using the following Matlab code: syms a b c h; eqn1 = a + b + c == 0; eqn2 = -a*h/2+c*h/2 == 1; eqn3 = a*h^2/2 + c*h^2/2 == 0; [A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [a, b, c]); der = linsolve(A,B) der = -1/h 1/h Thus: 𝑓(𝑥)′ = − 1 ℎ 𝑓(𝑥 − ℎ) + ℎ 𝑓(𝑥 + ℎ) Please determine the coefficients a,b, and c for the double derivative 𝑓(𝑥)′′ using Matlab by modifying the code above. (10 pts)