|
|
 |
| |
|
|
|
Adaptive LADM–Neural Remainder Correction for Atangana–Baleanu Fractional Hirota–Satsuma Coupled KdV Systems |
|
|
|
PP: 609-633 |
|
|
doi:10.18576/pfda/120308
|
|
|
|
Author(s) |
|
|
|
Amin M. Elfeky,
Mohammad A. Tashtoush,
Essa Melfi Aldosari,
Mohamed Sayed Abdellatif,
Adel S. Hussain,
Firas K. A. Al-Hamarneh,
|
|
|
|
Abstract |
|
|
| The Atangana Baleanu fractional derivative is used to describe some memory effects by the GHS-Kdv system of nonlinear dispersive waves. However, it makes difficult to solve numerically this system. On the other hand, the Laplace-Adomian Decomposition Method (LADM), can be efficiently used to obtain a good approximation of the analytical solution of the GHS-Kdv system. But the precision obtained by this method depends of the number of decomposition terms that are considered. Moreover, even if we consider high numbers of terms, there is no guaranty that the precision will be better than the one obtained with lower numbers of terms. Therefore, to overcome these limitations, an adaptive Laplace-Adomian Neural Remainder Correction Framework has been developed. This new framework uses a truncated LADM approximation as an analytical base. It also uses a Multi-Output Neural Network trained only to predict the truncation remainder of the truncated LADM approximation. The proposed formulation allows to enforce exactly the initial conditions, and to keep the same nonlocal memory operator as in the problem statement, in the residual equation. It includes an adaptive LADM truncation criterion and an adaptive collocation based on residuals. Two test cases have been used to evaluate this new approach. They were compared with the analytical solution of the problem, and with numerical reference solutions. The results show that the efficiency of the neural corrections depend on the choice of the truncation order of LADM. For example, in a representative low-order case, the error was reduced from approximately 9.4x10−3 for pure LADM to 6.9x10−3 or the hybrid approximation, corresponding to an error-reduction factor of approximately 1.37. On the contrary, higher truncation orders can lead to greater correction errors, showing again how important is to select adaptively the number of decomposition terms.
|
|
|
|
|
|
|
|
