Theory and Modern Applications
Table 2 Multicolored rooted trees for (5.1a)–(5.1b) with order less than or equal to 2.0
No. | t | ρ(t) | ϕ(t) | φ(t) |
|---|---|---|---|---|
1 | \(\bullet _{i}\) | 0.5 | \(J_{i}\) | \(\int _{0}^{1}J_{i}B_{\tau}^{(1)}\,\mathrm{d}\tau +\int _{0}^{1}\frac{J_{i0}}{h}B_{\tau}^{(2)}\,\mathrm{d}\tau \) |
2 | •0 | 1 | \(J_{0}\) | \(\int _{0}^{1} h B_{\tau}^{(0)}\,\mathrm{d}\tau \) |
3 |
| 1.5 | \(\int _{0}^{h}s\circ \mathrm{d}W_{i}(s)\) | \(\int _{0}^{1}hB_{\tau}^{(0)} ( \int _{0}^{1}J_{i}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi + \int _{0}^{1} \frac{J_{i0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\xi )\,\mathrm{d}\tau \) |
4 |
| 2 | \(\int _{0}^{1}s\,\mathrm{d}s\) | \(\int _{0}^{1}hB_{\tau}^{(0)} \int _{0}^{1}hA_{\tau ,\xi}^{(0)}\,\mathrm{d}\xi \,\mathrm{d}\tau \) |
5 |
| 2 | \(\int _{0}^{h}W_{i}(s)W_{j}(s)\,\mathrm{d}s\) | \(\int _{0}^{1}hB_{\tau}^{(0)} ( \int _{0}^{1}J_{i}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi +\int _{0}^{1} \frac{J_{i0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\tau ) ( \int _{0}^{1}J_{j}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi + \int _{0}^{1} \frac{J_{j0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\tau )\,\mathrm{d}\tau \) |
6 |
| 2 | \(\int _{0}^{h}W_{i}(s)^{2}\,\mathrm{d}s\) | \(\int _{0}^{1}hB_{\tau}^{(0)} ( \int _{0}^{1}J_{i}A_{\tau ,\xi}^{(1)}\,\mathrm{d}\xi + \int _{0}^{1} \frac{J_{i0}}{h}A_{\tau ,\xi}^{(2)}\,\mathrm{d}\xi )^{2}\,\mathrm{d}\tau \) |
