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Notes on the diamond-$\alpha$ dynamic derivative on time scales. (English) Zbl 1114.26003
The paper redefines the $\diamond_\alpha$ derivative independently of the standard $\Delta$ and $\nabla$ dynamic derivatives, and further examines its properties and relationship with the $\Delta$ and the $\nabla$ formulae. First, basic definitions and theorems of the time scales theory and of the $\Delta$ and $\nabla$ dynamic derivatives are presented. Then it is shown that the new definition of the $\diamond_\alpha$ derivative is correct and equivalent to a linear combination of the $\Delta$ and $\nabla$ derivatives at points where those derivatives exist. Several theorems concerning the properties of the $\diamond_\alpha$ derivative are proved. The authors consider two counterexamples that demonstrate that a $\diamond_\alpha$ antiderivative does not exist for some continuous functions on a time scale in the case of a fixed $\alpha$ value strictly between 0 and 1. Finally they discuss computational experiments where nonuniform time scales resulting from adaptive computations of the numerical solution of a solitary wave equation are employed. Numerical errors are computed and compared between different first order dynamic derivative aproximates over an interval which includes a singularity in the conventional derivative.

MSC:
 26A24 Differentiation of functions of one real variable 39A10 Additive difference equations
Full Text:
References:
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