The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional differential equations and study the effects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.
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Table 1.
Relationship between global and truncation errors with respect to Theorem 4.2 for
Table 2.
Relationship between global and truncation errors with respect to Theorem 4.2 for
Table 3.
Relationship between global and truncation errors with respect to Theorem 4.2 for
Table 4.
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for
| | ||
| 0.838 | 9.8267(-2) | 2.5169(-1) |
| 0.712 | 9.4799(-2) | 1.6371(-1) |
| 0.617 | 9.3258(-2) | 1.4043(-1) |
| 0.544 | 9.2595(-2) | 1.3162(-1) |
| 0.489 | 9.2311(-2) | 1.2770(-1) |
| 0.448 | 9.2198(-2) | 1.2578(-1) |
| 0.416 | 9.2154(-2) | 1.2477(-1) |
Table 5.
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for
| | | |
| 0.753 | 6.1846(-2) | 1.1423(-1) |
| 0.578 | 6.3482(-2) | 9.3146(-2) |
| 0.462 | 6.3552(-2) | 8.6966(-2) |
| 0.386 | 6.3352(-2) | 8.4792(-2) |
| 0.336 | 6.3249(-2) | 8.3971(-2) |
| 0.303 | 6.3206(-2) | 8.3645(-2) |
| 0.281 | 6.3189(-2) | 8.3511(-2) |
| 0.266 | 6.3183(-2) | 8.3455(-2) |
Table 6.
Perturbed problem: relationship between global error and bounds from Theorem 4.5 for
| | | |
| 0.839 | 2.7785(-2) | 1.5982(-1) |
| 0.568 | 4.4303(-2) | 7.4435(-2) |
| 0.412 | 4.8468(-2) | 6.7814(-2) |
| 0.322 | 4.9372(-2) | 6.6793(-2) |
| 0.271 | 4.9568(-2) | 6.6638(-2) |
| 0.241 | 4.9612(-2) | 6.6577(-2) |
| 0.224 | 4.9620(-2) | 6.6556(-2) |
| 0.215 | 4.9622(-2) | 6.6549(-2) |
| 0.209 | 4.9622(-2) | 6.6547(-2) |
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Plot of
Values of
Solution of the problem test and bound (6) for
Comparison of the difference