An exploration of combined dynamic derivatives on time scales and their applications

It is becoming evident that different dynamic derivatives play increasingly important roles in approximating functions and solutions of nonlinear differential equations for their great flexibility in grid designs. Different dynamic derivatives on time scales not only offer a convenient way in practical applications, but also show their distinctive features in approximations. It may be worthwhile to investigate if such useful features can be maintained or even improved in certain senses while different dynamic derivatives are used in the same application simultaneously. Under this consideration, we will introduce the combined delta (ΔΔ, or forward) and nabla (∇∇, or backward) dynamic derivatives, explore their basic properties, and investigate their applications for approximating classical derivative functions and for solving differential equation problems in this paper. Proper forward jump, backward jump and step functions will be introduced and utilized. It is found that while the combined dynamic derivatives possess similar properties as ΔΔ and ∇∇ derivatives, they offer more balanced approximations to the targeted functions and differential equations at satisfactory accuracy. The combined dynamic derivatives also reduce the unexpected computational spuriosity, and therefore lead to more reliable numerical algorithm designs. Computational examples are given to further illustrate our results.