Theory and Modern Applications
Table 3 Colored rooted tree for (5.8) with order equal to 2.5
No. | t | ρ(t) | ϕ(t) | φ(t) |
|---|---|---|---|---|
7 | \(\bullet _{i}\) | 2.5 | \(\int _{0}^{h}sW_{i}(s)\,\mathrm{d}s\) | \(\int _{0}^{1}hB_{\tau}^{(0)}\int _{0}^{1}hA_{\tau ,\xi}^{(0)}\,\mathrm{d}\xi \int _{0}^{1}J_{i}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi \,\mathrm{d}\tau +\int _{0}^{1}hB_{\tau}^{(0)}\int _{0}^{1}hA_{\tau ,\xi}^{(0)}\,\mathrm{d}\xi \int _{0}^{1}\frac{J_{i0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\xi \,\mathrm{d}\tau \) |
8 | •0 | 2.5 | \(\int _{0}^{1}\int _{0}^{1}s\,\mathrm{d}s\circ \mathrm{d}W_{i}(s)\) | \(\int _{0}^{1}hB_{\tau}^{(0)}\int _{0}^{1}hA_{\tau ,\xi}^{(0)}\int _{0}^{1} J_{i}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi \,\mathrm{d}\xi \,\mathrm{d}\tau +\int _{0}^{1}hB_{\tau}^{(0)}\int _{0}^{1}hA_{\tau ,\xi}^{(0)}\int _{0}^{1} \frac{J_{i0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\xi \,\mathrm{d}\xi \,\mathrm{d}\tau \) |