Anzeige
Erschienen in:
2013 | OriginalPaper | Buchkapitel
Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equation
verfasst von : Ya-ling Li, Shi-you Lin
Erschienen in: Proceedings of The Eighth International Conference on Bio-Inspired Computing: Theories and Applications (BIC-TA), 2013
Verlag: Springer Berlin Heidelberg
Aktivieren 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 is concentrated on the following coupled system of the nonlinear fractional differential equationwhere \( f,\;K,\;g,\;H:\;\left[ {0,\,1} \right]\, \times \,\Re \, \to \,\left[ {0,\, + \infty } \right) \) are the positive continuous functions. \( D^{\alpha } \) and \( D^{\beta } \) are the standard Riemann–Liouville fractional derivatives with the order \( \alpha ,\, \beta, \) respectively. We give the existence and the uniqueness of the solution by using the Schauder fixed point theorem and the generalized Gronwall inequality.
$$ \left\{ \begin{aligned} &D^{\alpha } u\left( t \right) = f\left(
{t,v\left( t \right)} \right) + \int_{0}^{t} {K\left( {s,v\left( s
\right)} \right)ds,\quad 5 < \alpha ,\beta \le 6,\;0 < t < 1}
\hfill \\ &D^{\beta } v\left( t \right) = g\left( {t,u\left( t
\right)} \right) + \int_{0}^{t} {H\left( {s,u\left( s \right)}
\right)ds} \hfill \\ &u\left( 1 \right) = \mathop {\lim
}\limits_{{t \to 0}} u\left( t \right) \cdot t^{{2 - \alpha }} =
v\left( 1 \right) = \mathop {\lim }\limits_{{t \to 0}} v\left( t
\right) \cdot t^{{2 - \beta }} = 0. \hfill \\ \end{aligned}
\right. $$