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NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION
F Liu1, M M Meerschaert, R J McGough
1School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia.
Fractional Calculus & Applied Analysis
|June 18, 2013
Summary
This study introduces effective numerical methods for multi-term time-fractional wave-diffusion equations. The proposed techniques accurately simulate these complex fractional differential equations, showing their practical applicability.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Computational Physics
Background:
- Fractional calculus extends classical calculus to non-integer orders, enabling modeling of complex phenomena.
- Time-fractional wave-diffusion equations capture anomalous diffusion and wave propagation with memory effects.
- Multi-term fractional derivatives introduce more intricate dynamics compared to single-term models.
Purpose of the Study:
- To analyze and simulate multi-term time-fractional wave-diffusion equations.
- To develop computationally effective numerical methods for these equations.
- To validate the proposed methods through numerical simulations.
Main Methods:
- Definition of multi-term time fractional derivatives in the Caputo sense.
- Development of numerical schemes for simulating equations with fractional orders in [0,4).
- Implementation of techniques applicable to fractional Laplacian models.
Main Results:
- Effective numerical methods were proposed and demonstrated for multi-term time-fractional wave-diffusion equations.
- Numerical results confirmed the accuracy and effectiveness of the developed simulation techniques.
- The study provides a foundation for analyzing more complex fractional models.
Conclusions:
- The proposed numerical methods are effective for simulating multi-term time-fractional wave-diffusion equations.
- These methods offer a valuable tool for researchers working with fractional differential equations.
- The techniques can be extended to a broader class of fractional time-space models.
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