# Discrepancy in estimating boundary stencil for finite difference method

I am trying to estimate the FD stencil for boundary as mentioned in this paper (section 4.1) using MATLAB. The stencil order (6th) is higher than the one mentioned in paper (4th). $$f_1' +\alpha f_2' = \frac{1}{h} (c_1f_1 + c_2f_2 + c_3f_3 +c_4f_4+ c_5f_5 + c_6 f_6 + c_7 f_7 )$$ Each term of both sides is expanded using Taylor series expansion and then derivatives are matched to estimate the coefficients [$c1,c2,c3,c4,c5,c6,c7$ and $\alpha$].

I solved the above system as following:

Above system of equation can be written as M*x=y

Note:

1) The alpha was taken to the left to form a part of unknown vector x

2) the factorial on both side were simplified and adjusted to right side.

I wrote following code for solving the above system of equation.

LM = zeros(8);
LM(1,1:7)=1;                LM(1,8)=0;
LM(2,2:7)=1:6;              LM(2,8)=-1;
LM(3,2:7)=(1:6).^2;         LM(3,8)=-2;
LM(4,2:7)=(1:6).^3;         LM(4,8)=-3;
LM(5,2:7)=(1:6).^4;         LM(5,8)=-4;
LM(6,2:7)=(1:6).^5;         LM(6,8)=-5;
LM(7,2:7)=(1:6).^6;         LM(7,8)=-6;
LM(8,2:7)=(1:6).^7;         LM(8,8)=-7;
y=[0;1;0;0;0;0;0;0];
x=LM\y;
rats(x)


which gives completely different answer than reported one. Both are given below:

Solution i.e. $x =[c1,c2,c3,c4,c5,c6,c7, \alpha$] by

MATLAB estimated solution: [-69/20, -17/10, 15/2, -10/3, 5/4, -3/10, 1/30, 6]

Solution reported in literature: [-197/60, -5/12, 5, -5/3, 5/12, -1/20, 0, 5]

Any help is appreciated.