Multi-order fractional differential equations and their numerical
solution
(with N.
J. Ford)
Appl. Math.
Comput. 154 (2004), 621-640.
2000 Mathematics Subject Classification: Primary 65L05;
Secondary 26A33, 34A12, 34A45, 34D30, 65L06, 65R20.
Key words: Fractional differential equation, multi-term equation,
Caputo derivative, existence, uniqueness, structural stability, Adams method
Abstract
We consider the numerical solution of (possibly nonlinear)
fractional differential equations of the
form $y^{(\alpha)}(t)= f(t,y(t),y^{(\beta_1)}(t),y^{(\beta_2)}(t),
\ldots,y^{(\beta_n)}(t))$ with $\alpha>\beta_1>\beta_2>\ldots>\beta_n$
and $\alpha-\beta_1\le1$,$\beta_j-\beta_{j+1}\le1$, $0\le\beta_n\le1$,
combined with suitable initial conditions.
The derivatives are understood in the Caputo sense.
We begin by discussing the analytical questions of existence and
uniqueness of solutions, and we investigate how the solutions depend on the
given data. Moreover we propose
convergent and stable numerical methods for such initial value
problems.
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