Search

Abstract: 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 (, 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 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. [Copyright &y& Elsevier]

There has been a considerable number of recent discussions on applications of different types of dynamic derivatives. The dynamic derivatives not only play a central role in time scales theory and methods, but also provide a remarkable way of approximations in mathematical modelling and computational applications. In this paper, we will explore connections between existing dynamic and conventional derivatives over the time scales used. We will show that while first order dynamic derivatives provide acceptable accuracies in approximating conventional derivatives, second order dynamic derivatives are in general inconsistent with conventional derivatives. It is evident that, before incorporating structures of particular time scales used into the dynamic derivative formulae properly, one needs to be prudential when bridging discrepancies between the differential and difference equations modelling similar applications by higher order dynamic derivatives. Numerical experiments are given to further illustrate our conclusions. [ABSTRACT FROM AUTHOR]

Crossed symmetric solutions of nonlinear boundary value dynamic problems play an important role in many applications, in particular in adaptive algorithm designs. This article is devoted to the continuation of our investigation on second-order nonlinear companion dynamic boundary value problems on time scales. Monotonically convergent upper and lower solutions of the problems and their quasilinear approximations are investigated. It is shown that, under proper smoothness constraints, the iterative sequences constructed not only converge to the analytic solutions of the desired companion problems monotonically, but also preserve important crossed symmetry properties. The quasilinearization offers an efficient way in the solution approximation. Computational examples are given to illustrate our results. [Copyright &y& Elsevier]

This paper studies the numerical solution of two-dimensional nonlinear degenerate reaction-diffusion differential equations with singular forcing terms over rectangular domains. The equations considered may generate strong quenching singularities. This investigation focuses on a variable time step Peaceman­Rachford splitting method for the aforementioned problem. The time adaptation is implemented based on arc-length estimations of the first time derivative of the solution. The two-dimensional problem is split into several one-dimensional problems so that the computational cost is significantly reduced. The monotonicity and localized linear stability of the variable step scheme are investigated. We give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem. It is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem. [ABSTRACT FROM AUTHOR]