Sinc-Galerkin solution to eighth-order boundary value problems

Abstract

Many problems that arise in astrophysics, hydrodynamic and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory are modeled as eighth-order boundary-value problems. In this paper we show that the sinc-Galerkin method is an efficient and accurate numerical scheme for solving these problems. The inner product approximations to eighth and seventh order derivatives and the corresponding error estimates are derived. Compared to the non-polynomial spline technique and the reproducing kernel space method, we show that the sinc-Galerkin method provides more accurate results. In addition, for problems with singular solutions or singular source functions, the sinc-Galerkin method is shown to maintain the exponential convergence rate.

Appendix

The functions $\left\{ g_{8,i}\right\} _{i=0}^8$ in (20) are obtained by expanding the derivatives under the integral (18)

$$\begin{aligned}&g_{8,8}=w(\phi ')^{8},\qquad g_{8,7}=8 w'(\phi ')^{7}+28w (\phi ')^6 \phi '',\\&g_{8,6}=28 w'' (\phi ')^6+168 w'(\phi ')^5\phi ''+210 w (\phi ')^4 (\phi '')^2+56 w (\phi ')^5 \phi ''',\\&g_{8,5} = 56w'''(\phi ')^{5}+420 w''(\phi ')^{4}\phi '' + 840 w'(\phi ')^{3}(\phi '')^{2}+280 w'(\phi ')^{4}\phi '''\\&\quad +\, 420w(\phi ')^{2}(\phi '')^{3}+560 w(\phi ')^{3}\phi ''\phi '''+70 w(\phi ')^{4}\phi ^{(4)},\\&g_{8,4} = 70 w^{(4)} (\phi ')^5 + 560 w''' (\phi ')^3 \phi '' + 560 w'' (\phi ')^3 \phi ''' +1260 w'' (\phi ')^2 (\phi '')^2\\&\quad +\,280 w' (\phi ')^3 \phi ^{(4)}+1680 w' (\phi ')^2 \phi '' \phi '''+840 w' \phi ' (\phi '')^3+56 w (\phi ')^3 \phi ^{(5)}\\&\quad +\, 420 w (\phi ')^2 \phi '' \phi ^{(4)}+280 w (\phi ')^2 (\phi ''')^2 +840 w \phi ' (\phi '')^2 \phi '''+105 w (\phi '')^4,\\&g_{8,3}= 56 w^{(5)} (\phi ')^3 + 420 w^{(4)} (\phi ')^2 \phi '' + 840 w''' \phi ' (\phi '')^2 +560 w''' (\phi ')^2 \phi '''\\&\quad +\,420 w'' (\phi ')^2 \phi ^{(4)}+1680 w'' (\phi ')^2 \phi ' \phi '' \phi '''+420 w'' (\phi '')^3+168 w' (\phi ')^2 \phi ^{(5)}\\&\quad +\, 840 w' \phi ' \phi '' \phi ^{(4)}+560 w' \phi ' (\phi ''')^2 +840 w' (\phi '')^2 \phi ''' + 168 w \phi ' \phi '' \phi ^{(5)}\\&\quad +\, 28 w (\phi ')^2 \phi ^{(6)}+280 w \phi ' \phi ''' \phi ^{(4)}+210 w (\phi '')^2 \phi ^{(4)}+280 w \phi '' (\phi ''')^2,\\&g_{8,2} =28 w^{(6)} (\phi ')^2 + 168 w^{(5)} \phi ' \phi '' + 280 w^{(4)} \phi ' \phi ''' +210 w^{(4)} (\phi '')^2\\&\quad +\,280 w''' \phi ' \phi ^{(4)}+560 w''' \phi '' \phi '''+168 w'' \phi ' \phi ^{(5)} + 420 w'' \phi '' \phi ^{(4)}\\&\quad +\, 280 w'' (\phi ''')^2+ 56 w' \phi ' \phi ^{(6)} +168 w' \phi '' \phi ^{(5)}+ 280 w' \phi ''' \phi ^{(4)}\\&\quad +\, 8 w \phi ' \phi ^{(7)}+ 28 w \phi '' \phi ^{(6)}+56 w \phi ''' \phi ^{(5)}+35 w (\phi ^{(4)})^2,\\&g_{8,1} =8 w^{(7)} \phi ' + 28 w^{(6)}\phi '' + 56 w^{(5)} \phi ''' +70 w^{(4)} \phi ^{(4)}+56 w''' \phi ^{(5)}\\&\quad +\,28 w'' \phi ^{(6)}+ 8 w' \phi ^{(7)}+ w \phi ^{(8)}, \end{aligned}$$

and

$$\begin{aligned} g_{8,0}=w^{(8)}. \end{aligned}$$

Similarly, the functions $\left\{ g_{7,i}\right\} _{i=0}^7$ in (30) are given by

$$\begin{aligned}&g_{7,7}= p_7 w(\phi ')^{7},\qquad g_{7,6}=7 (p_7 w)'(\phi ')^{6}+21 p_7 w (\phi ')^5 \phi '',\\&g_{7,5}=21 (p_7 w)'' (\phi ')^5+ 105 (p_7 w)'(\phi ')^4\phi ''+105 p_7 w (\phi ')^3 (\phi '')^2+ 35 p_7 w(\phi ')^4 \phi ''',\\&g_{7,4} =35 (p_7 w)'''(\phi ')^4 +210 (p_7 w)''(\phi ')^3 \phi '' + 140 (p_7 w)'(\phi ')^{3} \phi ''' +315 (p_7 w)'(\phi ')^2 (\phi '')^2\\&\quad +\, 35 p_7 w (\phi ')^{3}\phi ^{(4)}+210 p_7 w (\phi ')^2 \phi ''\phi '''+ 105 p_7 w \phi ' (\phi '')^3,\\&g_{7,3} = 35 (p_7 w)^{(4)} (\phi ')^3 +210 (p_7 w)''' (\phi ')^2 \phi ''+315 (p_7 w)'' \phi '(\phi '')^2 +210 (p_7 w)'' (\phi ')^2 \phi '''\\&\quad +\,105 (p_7 w)' (\phi ')^2 \phi ^{(4)}+420 (p_7 w)' \phi ' \phi '' \phi '''+105 (p_7 w)' (\phi '')^3+ 21 p_7 w (\phi ')^2 \phi ^{(5)}\\&\quad +\, 105 p_7 w \phi ' \phi '' \phi ^{(4)}+70 p_7 w \phi ' (\phi ''')^2 +105 p_7 w (\phi '')^2 \phi ''',\\&g_{7,2}= 21 (p_7 w)^{(5)} (\phi ')^2 + 105 (p_7 w)^{(4)}\phi ' \phi '' + 140 (p_7 w)''' \phi ' \phi ''' +105 (p_7 w)''' (\phi '')^2\\&\quad +\,105 (p_7 w)'' \phi ' \phi ^{(4)}+210 (p_7 w)'' \phi '' \phi '''+42 (p_7 w)' \phi ' \phi ^{(5)}+105 (p_7 w)' \phi '' \phi ^{(4)}\\&\quad +\, 70 (p_7 w)' (\phi ''')^2 + 7 p_7 w \phi ' \phi ^{(6)}+ 21 p_7 w \phi '' \phi ^{(5)}+35 p_7 w \phi ''' \phi ^{(4)},\\&\quad g_{7,1} = 7 (p_7 w)^{(6)} \phi ' + 21 (p_7 w)^{(5)} \phi '' + 35 (p_7 w)^{(4)} \phi ''' +35 (p_7 w)''' \phi ^{(4)}\\&\quad +\,21 (p_7 w)'' \phi ^{(5)}+7 (p_7 w)' \phi ^{(6)}+p_7 w \phi ^{(7)}, \end{aligned}$$

and

$$\begin{aligned} g_{7,0}=(p_7 w)^{(7)}. \end{aligned}$$