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Insights into the fractional order initial value problem via semi-infinite systems
Tom T Hartley1, Carl F Lorenzo
1Department of Electrical and Computer Engineering, University of Akron, Akron, Ohio, USA. thartley@uakron.edu
Critical Reviews in Biomedical Engineering
|August 4, 2009
Summary
Fractional differential equations require time-varying initial conditions. This is due to past information storage in distributed systems, impacting fractional differintegral operators.
Area of Science:
- Mathematics
- Applied Mathematics
- Physics
Background:
- Fractional calculus extends traditional calculus to non-integer orders.
- Initial value problems (IVPs) are fundamental in modeling dynamic systems.
- Understanding the behavior of fractional differential equations (FDEs) is crucial for accurate system representation.
Purpose of the Study:
- To investigate the nature of initial conditions in fractional order differential equations.
- To demonstrate the necessity of time-varying initial conditions for FDEs.
- To connect the properties of fractional operators with information storage in distributed systems.
Main Methods:
- Analysis of initial value problems for fractional order differential equations.
- Utilizing solutions from known spatially distributed systems.
- Mathematical demonstration of the time-varying nature of initial conditions.
Main Results:
- Fractional differintegral operators necessitate an initial condition term.
- This initial condition term is shown to be time-varying.
- The time-varying nature arises from the distributed storage of past information.
Conclusions:
- The standard assumption of constant initial conditions is insufficient for FDEs.
- Fractional calculus inherently incorporates memory effects, requiring dynamic initial states.
- Accurate modeling with FDEs must account for the historical information embedded in initial conditions.
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