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In this work, we present some useful results of discrete fractional calculus for interval-valued functions. The composition rules for interval fractional operators are introduced, which are used to construct the general form of the solutions to nonlinear interval fractional difference equations. An illustrative example is provided in which the method of recursive iterations is applied to obtain explicit formulas for the solutions of linear interval fractional difference equations.

The theory of interval-valued difference equations under $gH$-difference is an interesting topic, since it can be applied to study numerical solutions to interval-valued or fuzzy-valued differential equations. In this paper, we estimate the number of solutions to a class of first-order interval-valued difference equations under $gH$-difference, which reveals the complexity of the stability analysis in this area, as well as the difficulty for prediction and control problems. Then, based on the relative positions of initial values and equilibrium points, we provide sufficient conditions for the existence of convergent solutions. We also provide examples to illustrate the validity of our results.

In this article, we use a finite difference technique to solve variable-order fractional integro-differential equations (VOFIDEs, for short). In these equations, the variable-order fractional integration (VOFI) and variable-order fractional derivative (VOFD) are described in the Riemann-Liouville's and Caputo's sense, respectively. Numerical experiments, consisting of two examples, are studied. The obtained numerical results reveal that the proposed finite difference technique is very effective and convenient for solving VOFIDEs.

This paper aims to apply and investigate the compact finite difference methods for solving integer-order and fractional-order Riccati differential equations. The fractional derivative in the fractional case is described in the Caputo sense. In solving the Riccati equation, we first approximate first-order derivatives using the approach of compact finite difference. In this way, the system of nonlinear equations is obtained, which solves the Riccati equation. In addition, we examine the convergence analysis of the proposed approach for the fractional and nonfractional cases and prove that the methods are convergent under some suitable conditions. Examples are also given to illustrate the efficiency of our method compared to other methods.

In this paper an coupled Burgers' equation is considered and then a method entitled interval finite-difference method is introduced to find the approximate interval solution of interval model in level wise cases. Finally for more illustration, the convergence theorem is confirmed and a numerical example is solved.

In this paper, we study the existence and uniqueness of solutions for the boundary value problem -Dvv-my(t)=f (t, y (t+v-1)), Diy (v-N)=0, I ϵ {0, …, N-3}, DN-2y (v-N)=g (y), Dmv-Ny (b+M+v-m)=0, where v³2, 1£m<v, f: {0, …, b+M}xR®R is continuous, and nonnegative for y ³0, g: C ([v-N, …, b+M+v] ,R) is a given function. We give a representation for the solution to this problem, and we prove the existence and uniqueness of solution to this problem by contraction mapping theorem and Brouwer theorem.

‎In this paper‎, ‎the time-space distributed-order fractional diffusion equations with weakly singular solutions are considered‎. We provide the difference scheme using a nonuniform mesh to solve ‎equations‎‎. ‎The stability and convergence of the difference scheme are discussed‎, ‎We prove that the difference scheme is unconditionally stable‎. ‎We find that the difference scheme is convergent‎. ‎We also show that the temporal convergence order on the nonuniform mesh is higher than on the uniform mesh‎. Finally, some numerical examples are given to verify the theoretical analysis‏.