Theory and Modern Applications
Table 2 Corresponding Green functions and their maximums
α,β | \(\mathcal{G}(t,s)\) | \(\max \mathcal{G}(t,s)\) |
|---|---|---|
β: = 1 | \(\frac{1}{\Gamma (\alpha )} \begin{cases} (\frac{t-a}{b-a} )(b-s)^{\alpha -1}-(t-s)^{\alpha -1};& a\leq s\leq t\leq b,\\ (\frac{t-a}{b-a} )(b-s)^{\alpha -1};& a\leq t\leq s\leq b. \end{cases}\) | \(\frac{{(b-a)^{\alpha -1}(\alpha -1)^{\alpha -1}}}{{\Gamma (\alpha )\alpha ^{\alpha }}}\) |
β: = 0 | \(\frac{1}{\Gamma (\alpha )}\begin{cases} (\frac{t-a}{b-a} )^{\alpha -1}(b-s)^{\alpha -1}-(t-s)^{\alpha -1}; & a\leq s\leq t\leq b,\\ (\frac{t-a}{b-a} )^{\alpha -1}(b-s)^{\alpha -1};& a\leq t\leq s\leq b. \end{cases}\) | \(\frac{(b-a)^{\alpha -1}}{\Gamma (\alpha )4^{\alpha -1}}\) |
α: = 2 | \(\begin{cases} (\frac{s-a}{b-a} )(b-t); & a\leq s\leq t\leq b,\\ (\frac{t-a}{b-a} )(b-s);& a\leq t\leq s\leq b. \end{cases}\) | \(\frac{b-a}{4}\) |