Anzeige
Erschienen in:
28.07.2016 | Original Article
Solutions of Bagley–Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm with error estimates
Erschienen in: Neural Computing and Applications | Ausgabe 5/2018
EinloggenAktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
This paper presents iterative reproducing kernel algorithm for obtaining the numerical solutions of Bagley–Torvik and Painlevé equations of fractional order. The representation of the exact and the numerical solutions is given in the \( \hat{W}_{2}^{3} \left[ {0,1} \right] \), \( W_{2}^{3} \left[ {0,1} \right] \), and \( W_{2}^{1} \left[ {0,1} \right] \) inner product spaces. The computation of the required grid points is relying on the \( \hat{R}_{t}^{{\left\{ 3 \right\}}} \left( s \right) \), \( R_{t}^{{\left\{ 3 \right\}}} \left( s \right) \), and \( R_{t}^{{\left\{ 1 \right\}}} \left( s \right) \) reproducing kernel functions. An efficient construction is given to obtain the numerical solutions for the equations together with an existence proof of the exact solutions based upon the reproducing kernel theory. Numerical solutions of such fractional equations are acquired by interrupting the \( n \)-term of the exact solutions. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulate data, numerical comparisons, and graphical results. Finally, the utilized results show the significant improvement in the algorithm while saving the convergence accuracy and time.