1. A note on paper 'Anomalous relaxation model based on the fractional derivative with a Prabhakarlike kernel' [Z. Angew. Math. Phys. (2019) 70:42]
- Author
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Katarzyna Górska, Tibor K. Pogány, and Andrzej Horzela
- Subjects
Applied Mathematics ,General Mathematics ,Anomalous relaxation ,Colo-Cole model ,Debye relaxation ,Prabhakar function ,Fractional derivative ,General Physics and Astronomy ,FOS: Physical sciences ,Function (mathematics) ,Mathematical Physics (math-ph) ,Lambda ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,Range (mathematics) ,Kernel (algebra) ,0103 physical sciences ,Relaxation (physics) ,Beta (velocity) ,010301 acoustics ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
Inspired by the article “Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel” (Z. Angew. Math. Phys. (2019) 70:42) whose authors Zhao and Sun studied the integro-differential equation with the kernel given by the Prabhakar function $$e^{-\gamma }_{\alpha , \beta }(t, \lambda )$$ , we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times whose admissible range extends the limits given in Zhao and Sun (Z Angew Math Phys 70:42, 2019, Theorem 3.1) to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel, the solution comprises to known phenomenological relaxation patterns, e.g., to the Cole–Cole model (if $$\gamma = 1, \beta =1-\alpha $$ ) or to the standard Debye relaxation.
- Published
- 2019