Wavelet-Based Solutions for Linear Boundary-Value Problems

Abstract

The Galerkin method is one of the most popular weighted residual methods, as whose performance shows a good balance among accuracy, computation, and stability [1].

Appendix 5.1 Calculation of Connection Coefficients of Modified Basis Function

According to the definition of the modified scaling basis given in Eq. (5.6), the generalized connection coefficients can be expressed as

$$\Upgamma_{k.l}^{n,m} = \int_{0}^{1} {\frac{{d^{m} \phi_{n,k} (x)}}{{dx^{m} }}\phi_{n,l} (x)dx} = \left\{ {\begin{array}{*{20}l} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Upgamma }_{k.l}^{n,m} } \hfill & {{\text{for}}\;k \in \left[ {0,r - 1} \right],} \hfill \\ {\tilde{\Upgamma }_{k.l}^{n,m} } \hfill & {{\text{for}}\;k \in \left[ {r,2^{n} - r} \right],} \hfill \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Upgamma } 3_{k.l}^{n,m} } \hfill & {{\text{for}}\;k \in \left[ {2^{n} - r + 1,2^{n} } \right]} \hfill \\ \end{array} } \right.,$$

(5.43)

where

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Upgamma }_{k.l}^{n,m} = \left\{ {\begin{array}{*{20}l} {\sum\limits_{{l_{1} = - 9}}^{ - 1} {\sum\limits_{{l_{2} = - 9}}^{ - 1} {u_{0,k} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)u_{0,l} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} } } \hfill \\ { + \sum\limits_{{l_{1} = - 9}}^{ - 1} {u_{0,k} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \sum\limits_{{l_{2} = - 9}}^{ - 1} {u_{0,l} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} ,\;\;{\text{for}}\;l \in \left[ {0,r - 1} \right],} \hfill \\ {\sum\limits_{{l_{1} = - 9}}^{ - 1} {u_{0,k} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} ,\quad \quad \quad \quad \quad \quad {\text{for}}\;l \in \left[ {r,2^{n} - r} \right] ,} \hfill \\ {\sum\limits_{{l_{1} = - 9}}^{ - 1} {\sum\limits_{{l_{2} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{0,k} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)u_{{1,2^{n} - l}} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} } } \hfill \\ { + \sum\limits_{{l_{1} = - 9}}^{ - 1} {u_{0,k} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \sum\limits_{{l_{2} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - l}} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} ,\;{\text{for}}\;l \in \left[ {2^{n} - r + 1,2^{n} } \right],} \hfill \\ \end{array} } \right.$$

(5.44)

$$\widetilde{\Upgamma }_{k.l}^{n,m} = \left\{ {\begin{array}{*{20}l} {\sum\limits_{{l_{2} = - 9}}^{ - 1} {u_{0,l} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} ,} \hfill & {{\text{for}}\;l \in \left[ {0,r - 1} \right],} \hfill \\ {\overline{\Upgamma }_{k - 7,l - 7}^{n,m} ,} \hfill & {{\text{for}}\;l \in \left[ {r,2^{n} - r} \right],} \hfill \\ {\sum\limits_{{l_{2} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - l}} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} ,} \hfill & {{\text{for}}\;l \in \left[ {2^{n} - r + 1,2^{n} } \right],} \hfill \\ \end{array} } \right.$$

(5.45)

$$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\Upgamma }_{k.l}^{n,m} = \left\{ {\begin{array}{*{20}l} {\sum\limits_{{l_{1} = 2^{n} + 1}}^{{2^{n} + 6}} {\sum\limits_{{l_{2} = - 9}}^{ - 1} {u_{{1,2^{n} - k}} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)u_{0,l} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} } } \hfill \\ { + \sum\limits_{{l_{1} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - k}} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \sum\limits_{{l_{2} = - 9}}^{ - 1} {u_{0,l} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} ,\;\;{\text{for}}\;l \in \left[ {0,r - 1} \right],} \hfill \\ {\sum\limits_{{l_{1} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - k}} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} ,\quad \quad \quad \quad \quad \quad \;{\text{for}}\;l \in \left[ {r,2^{n} - r} \right] ,} \hfill \\ {\sum\limits_{{l_{1} = 2^{n} + 1}}^{{2^{n} + 6}} {\sum\limits_{{l_{2} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - k}} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)u_{{1,2^{n} - l}} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l_{2} - 7}}^{n,m} + \overline{\Upgamma }_{k - 7,l - 7}^{n,m} } } \hfill \\ { + \sum\limits_{{l_{1} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - k}} \left( {\frac{{l_{1} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{l_{1} - 7,l - 7}}^{n,m} + \sum\limits_{{l_{2} = 2^{n} + 1}}^{{2^{n} + 6}} {u_{{1,2^{n} - l}} \left( {\frac{{l_{2} }}{{2^{n} }}} \right)} \overline{\Upgamma }_{{k - 7,l_{2} - 7}}^{n,m} ,\;{\text{for}}\;l \in \left[ {2^{n} - r + 1,2^{n} } \right].} \hfill \\ \end{array} } \right.$$

(5.46)

According to Eq. (5.43), the calculation of the generalized connection coefficients $\Upgamma_{k,l}^{n,m}$ can be reduced to the calculation of the integral

$$\overline{\Upgamma }_{k,l}^{n,m} = \int_{0}^{1} {\frac{{d^{m} \varphi (2^{n} x - k)}}{{dx^{m} }}\varphi (2^{n} x - l)dx} .$$

(5.47)

Let $y = 2^{n} x - l$, Eq. (5.47) can be transformed into

$$\overline{\Upgamma }_{k,l}^{n,m} = 2^{{n\left( {m - 1} \right)}} \int_{ - l}^{{2^{n} - l}} {\frac{{d^{m} \varphi (y - k + l)}}{{dy^{m} }}\varphi (y)dy} .$$

(5.48)

Then Eq. (5.47) can be further expressed as

$$\overline{\Upgamma }_{k,l}^{n,m} = 2^{{n\left( {m - 1} \right)}} [\Upgamma_{k - l}^{m} (2^{j} - l) - \Upgamma_{k - l}^{m} ( - l)].$$

(5.49)

Since the compact support of the scaling function is $\left[ {0,\widetilde{N}} \right]$, the relationship can be obtained according to Sect. 2.5.3

$$\Upgamma_{k}^{m} (x) = 0,\quad \left| k \right| \ge \widetilde{N}\left\| {x \le 0} \right\|\;x \le k,$$

(5.50)

$$\Upgamma_{k}^{m} (x) = \overline{\Upgamma }_{k}^{m} = \int_{ - \infty }^{ + \infty } {\frac{{d^{m} \varphi (x - k)}}{{dx^{m} }}\varphi (x)dx} ,\quad x \ge \widetilde{N},$$

(5.51)

$$\overline{\Upgamma }_{k}^{m} = 0,\quad \left| k \right| \ge \widetilde{N}.$$

(5.52)

It can be seen from Eqs. (5.50) to (5.52) that the integral $\Upgamma_{k}^{m} (x)$ needs to be calculated separately. The detailed calculation process of these integrals can be found in Sect. 2.5.3.