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Draft:Coimbra derivative

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  • Comment: Declined, then re-submitted with no edits. Please address the notability concerns. Particularly, show that there are secondary, not just primary references to this topic. Also, only 4 of the 15 references here are not by Coimbra, not a good indication of widespread coverage of the topic. Newystats (talk) 11:12, 12 December 2023 (UTC)
  • Comment: It would be best to start with an article on order derivatives in general. Newystats (talk) 06:01, 12 December 2023 (UTC)

The Coimbra derivative is the most versatile variable order derivative, and consequently it is the most often used for physical modeling:[1]

For 

where the lower limit can be taken as either or as long as is identically zero from or to . Note that this operator returns the correct fractional derivatives for all values of and can be applied to either the dependent function itself with a variable order of the form or to the independent variable with a variable order of the form .

The Coimbra derivative defined above returns the correct value of the zero-th order derivative when and returns the first derivative when . Also, the operator returns the -th derivative of when , a necessary property for dynamical modeling that is often overlooked when the focus is placed on mathematical properties that resemble fixed order differential operators (for example, whether the operators abide by the exponent rule, etc.). Also of great importance to dynamic modeling is the fact that this operator is dynamically consistent with causal behavior in the initial conditions. In other words, when is a true constant from to the initial time , the Coimbra operator returns zero for all values of . However, if is discontinuous between and , the operator returns the appropriate Heaviside contribution to the integral value of . In accordance with this causal definition, we take the value of the physical variable to be identically null from to as a representation of dynamic equilibrium. A nonzero initial condition is treated as a Heaviside function at , and, therefore, it is included in the second term of the definition of the operator above. This way, the Coimbra operator allows for discontinuities at the initial condition, as long as there is dynamic equilibrium for , i.e., when . Since one is free to place the initial counter of time at any given instant, this operator is both causal and allows for discontinuous initial conditions.:[2]

The Coimbra derivative can be generalized to any order[3], leading to the Coimbra Generalized Order Differintegration Operator (GODO)[4]

For 

where is an integer larger than the larger value of for all values of . Note that the second (summation) term on the right side of the definition above can be expressed as


so to keep the denominator on the positive branch of the Gamma () function and for ease of numerical calculation.

Applications and Numerical Implementation

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A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,[5][6][7][8][9][10][11] as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors[12][13][14][15]

References

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  1. ^ C. F. M. Coimbra (2003) “Mechanics with Variable Order Differential Equations,” Annalen der Physik (12), No. 11-12, pp. 692-703.
  2. ^ C. M. Soon, C. F. M. Coimbra, and M. H. Kobayashi (2005). “The Variable Viscoelasticity Oscillator,” Annalen der Physik (14), N.6, pp. 378-389.
  3. ^ C. F. M. Coimbra “Methods of using generalized order differentiation and integration of input variables to forecast trends," U.S. Patent Application 13,641,083 (2013).
  4. ^ J. Orosco and C. F. M. Coimbra (2018) “Variable-order Modeling of Nonlocal Emergence in Many-body Systems: Application to Radiative Dispersion,” Physical Review E (98), 032208.
  5. ^ L. E. S. Ramirez, and C. F. M. Coimbra (2007) “A Variable Order Constitutive Relation for Viscoelasticity”– Annalen der Physik (16) 7-8, pp. 543-552.
  6. ^ H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra (2008) “Variable Order Modeling of Diffusive-Convective Effects on the Oscillatory Flow Past a Sphere” – Journal of Vibration and Control, (14) 9-10, pp. 1569-1672.
  7. ^ G. Diaz, and C. F. M. Coimbra (2009) “Nonlinear Dynamics and Control of a Variable Order Oscillator with Application to the van der Pol Equation” – Nonlinear Dynamics, 56, pp. 145—157.
  8. ^ L. E. S. Ramirez, and C. F. M. Coimbra (2010) “On the Selection and Meaning of Variable Order Operators for Dynamic Modeling”– International Journal of Differential Equations Vol. 2010, Article ID 846107.
  9. ^ L. E. S. Ramirez and C. F. M. Coimbra (2011) “On the Variable Order Dynamics of the Nonlinear Wake Caused by a Sedimenting Particle,” Physica D (240) 13, pp. 1111-1118.
  10. ^ E. A. Lim, M. H. Kobayashi and C. F. M. Coimbra (2014) “Fractional Dynamics of Tethered Particles in Oscillatory Stokes Flows,” Journal of Fluid Mechanics (746) pp. 606-625.
  11. ^ J. Orosco and C. F. M. Coimbra (2016) “On the Control and Stability of Variable Order Mechanical Systems” Nonlinear Dynamics, (86:1), pp. 695–710.
  12. ^ E. C. de Oliveira, J. A. Tenreiro Machado (2014), "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459.
  13. ^ S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh (2012) "Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation Volume 218, Issue 22, pp. 10861-10870.
  14. ^ H. Zhang and S. Shen, "The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation," Numer. Math. Theor. Meth. Appl. Vol. 6, No. 4, pp. 571-585.
  15. ^ H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert (2013) "A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model," Computers & Mathematics with Applications, 66, issue 5, pp. 693–701.