Notes on the diamond-\(\alpha\) dynamic derivative on time scales
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.
scientific article; zbMATH DE number 5083561