Question 5.66: (a) Determine the constants in the following relations : h^–......
(a) Determine the constants in the following relations :
\begin{aligned} h^{-4} \delta^{4} & =D^{4}\left(1+a \delta^{2}+b \delta^{4}\right)+O\left(h^{6}\right), \\ h D & =\mu \delta+a_{1} \Delta^{3} E^{-1}+(h D)^{4}\left(a_{2}+a_{3} \mu \delta+a_{4} \delta^{2}\right)+O\left(h^{7}\right). \end{aligned}
(b) Use the relations in (a) to construct a difference method for the boundary value problem
y^{i v}(x)=p(x) y(x)+q(x)
y(0), y(1), y^{\prime}(0) and y^{\prime}(1) are given.
The step size is h=1 / N, where N is a natural number. The boundary conditions should not be approximated with substantially lower accuracy than the difference equation. Show that the number of equations and the number of unknowns agree.
(a) Applying the difference operators on y\left(x_{n}\right), we obtain the truncation error at x=x_{n} as
\begin{aligned} T_{n}^{(2)} & =\delta^{4} y\left(x_{n}\right)-h^{4}\left(1+a \delta^{2}+b \delta^{4}\right) y^{(4)}\left(x_{n}\right)+O\left(h^{10}\right) \\ & =C_{6} h^{6} y^{(6)}\left(x_{n}\right)+C_{8} h^{8} y^{(8)}\left(x_{n}\right)+O\left(h^{10}\right) \end{aligned}
where C_{6}=\frac{1}{6}-a, C_{8}=\frac{1}{80}-\frac{a}{12}-b.
Setting C_{6}=0, C_{8}=0, we obtain a=1 / 6, b=-1 / 720.
Next, we apply the first derivative operator h D on y\left(x_{n}\right) and write as
\begin{aligned} T_{n}^{(1)}= & h y^{\prime}\left(x_{n}\right)-\mu \delta y\left(x_{n}\right)-a_{1} \Delta^{3} y\left(x_{n}-h\right) \\ & -h^{4}\left(a_{2}+a_{3} \mu \delta+a_{4} \delta^{2}\right) y^{(4)}\left(x_{n}\right)+O\left(h^{7}\right) \\ = & h y^{\prime}\left(x_{n}\right)-\frac{1}{2}\left[y\left(x_{n+1}\right)-y\left(x_{n-1}\right)\right]-a_{1}\left[y\left(x_{n+2}\right)\right. \\ & \left.-3 y\left(x_{n+1}\right)+3 y\left(x_{n}\right)-y\left(x_{n-1}\right)\right]-h^{4}\left(a_{2}+a_{4} \delta^{2}\right) y^{(4)}\left(x_{n}\right) \\ & -\frac{1}{2} h^{4} a_{3}\left[y^{(4)}\left(x_{n+1}\right)-y^{(4)}\left(x_{n-1}\right)\right] \\ = & C_{3} h^{3} y^{(3)}\left(x_{n}\right)+C_{4} h^{4} y^{(4)}\left(x_{n}\right)+C_{5} h^{5} y^{(5)}\left(x_{n}\right)+C_{6} h^{6} y^{(6)}\left(x_{n}\right)+O\left(h^{7}\right) \end{aligned}
\begin{aligned} \text{where}\qquad & C_{3}=-\frac{1}{6}-a_{1}, C_{4}=-\frac{a_{1}}{2}-a_{2}, \\ & C_{5}=-\frac{1}{120}-\frac{a_{1}}{4}-a_{3}, C_{6}=-\frac{a_{1}}{12}-a_{4} . \end{aligned}Setting C_{3}=C_{4}=C_{5}=C_{6}=0, we obtain
a_{1}=-1 / 6, a_{2}=1 / 12, a_{3}=1 / 30, a_{4}=1 / 72.
(b) The difference scheme at x=x_{n}, can be written as
\begin{aligned} \delta^{4} y_{n} & =h^{4}\left(1+\frac{1}{6} \delta^{2}-\frac{1}{720} \delta^{4}\right)\left[p\left(x_{n}\right) y_{n}+q\left(x_{n}\right)\right], \\ n & =1(1) N-1, \\ y^{\prime}\left(x_{n}\right) & =h^{-1}\left[\mu \delta y_{n}-\frac{1}{6} \Delta^{3} E^{-1} y_{n}\right]+h^{3}\left(\frac{1}{12}+\frac{1}{30} \mu \delta+\frac{1}{72} \delta^{2}\right)\left[p\left(x_{n}\right) y_{n}+q\left(x_{n}\right)\right], \\ n & =0, N . \end{aligned}
When n=1, the first equation contains the unknown y_{-1} outside [0,1]. This unknown can be eliminated using the second equation at n=0. Similarly, when n=N-1, the first equation contains, y_{N+1} outside [0,1] which can be eliminated using the second equation at n=N. Further, y(0), y(1) are prescribed. Hence, we finally have (N-1) equations in N-1 unknowns.