Fig. 1
Biorthogonal spline scaling and wavelet functions of order 2 and dual order 4, and of order 3 and dual order 5.
Fig. 2 Graphs of the exact solution, numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 2 and M = 8.
Fig. 3 Graphs of the numerical solution at different time steps and the error bound for n = 2 and M = 8.
Fig. 4 Graphs of the numerical solution at different time steps, and comparison between the exact and numerical solutions for n = 3 and M = 16.
Fig. 5 Graphs of the numerical solution at different time steps, and the error bound for n = 3 and M = 16.
Table 1
Numerical values for xl, t and corresponding u[xl,t] for two methods.
| t=0 | t=0 | t=0 | t=0.3 | t=0.3 | t=0.6 | t=0.6 | t=0.9 | t=0.9 |
|---|---|---|---|---|---|---|---|---|
| xl | BSW-2 | Haar | BSW-2 | Haar | BSW-2 | Haar | BSW-2 | Haar |
| 0 | 0 | 0.1867561 | 0.1850020 | 0.2158150 | 0.2140760 | 0.0752277 | 0.0745150 | |
| 0 | 0 | 0.1451483 | 0.1440110 | 0.1705720 | 0.1695220 | 0.0548761 | 0.0542550 | |
| 0 | 0 | 0.1130518 | 0.1120270 | 0.1330290 | 0.1320680 | 0.0434034 | 0.0430950 | |
| 0 | 0 | 0.0883459 | 0.0875280 | 0.1037956 | 0.1030230 | 0.0336308 | 0.0332450 | |
| 0 | 0 | 0.0688853 | 0.0682610 | 0.0810188 | 0.0803560 | 0.0263730 | 0.0261660 | |
| 0 | 0 | 0.0537768 | 0.0532920 | 0.0632059 | 0.0627320 | 0.0205080 | 0.0203022 | |
| 0 | 0 | 0.0419592 | 0.0415060 | 0.0493254 | 0.0488210 | 0.0160445 | 0.0159001 | |
| 0 | 0 | 0.0327424 | 0.0324060 | 0.0384910 | 0.0381090 | 0.0124974 | 0.0124034 |
Fig. 6 Convergence rate graphs of the numerical solutions for the Haar and BSW-2 wavelet methods, respectively.
Fig. 7 Log-log plots of numerical solutions for Haar and BSW-2 wavelet methods, respectively.