Modelling phenomena with interval differential equations (IDEs) is an effective way to consider the uncertainties that are unavoidable when collecting data. Similarly to the theory of ordinary differential equations, IDEs have been parallelly investigated with the interval difference equations from the beginning. These two branches can be regarded as one when unifying continuous and discrete solution domains. A conspicuous advantage when merging these areas is that the proof of several analogous properties in both theories need not be repeated. The paper provides a common and efficient tool for studying IDEs not only with continuous or discrete solution domains but also with more general ones. We propose the diamond-$\alpha$ derivative for interval-valued functions (IVFs) on time scales with respect to the generalized Hukuhara difference. Differently from most of the studies on the derivatives of functions on time scales, using the language of epsilon-delta, the novel concept is naturally studied according to the limit of IVFs on time scales as in classical mathematics. A particular class of IDEs on time scales is then considered with respect to the diamond-$\alpha$ derivative. Numerical problems are elaborated to illustrate the necessity and efficiency of the latter.

There are two key points in this work as the main objectives. The first is how to convert $ n^{th} $ order fuzzy differential equation into a first-order system of fuzzy differential equations using the notion of upper and lower bounds of the fuzzy solution to constitute the so-called interval fuzzy solution. The second is to solve the obtained system from the first step using a powerful method (the backstepping method) to provide an asymptotically stable solution by applying direct methods of stability (Lyapunov direct method).

In this paper, the continuse Legendre wavelets constructed on the interval [0, 1] are used to solve the nonlinear Fredholm integro-differential equation. The nonlinear part of integro-differential is ap- proximated by Legendre wavelets, and the nonlinear integro-differential is reduced to a system of nonlinear equations. We give some numerical examples to show applicability of the proposed method.

In this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.