Yongjin Li

On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations

This paper proves the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of nonlinear first-order ordinary differential equation with single constant delay and finite impulses on a compact interval. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions.

On Coupled p-Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions

This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs) with nonlinear -Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.

Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations

We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs) with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov’s fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.

Application of Topological Degree Method for Solutions of Coupled Systems of Multipoints Boundary Value Problems of Fractional Order Hybrid Differential Equations

We established the theory to coupled systems of multipoints boundary value problems of fractional order hybrid differential equations with nonlinear perturbations of second type involving Caputo fractional derivative. The proposed problem is as follows: , , , , , , , , , where , is linear, is Caputo fractional derivative of order , with , , and is fractional integral of order . The nonlinear functions , are continuous. For obtaining sufficient conditions on existence and uniqueness of positive solutions to the above system, we used the technique of topological degree theory. Finally, we illustrated the main results by a concrete example.

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

Computational Analysis of Complex Population Dynamical Model with Arbitrary Order

This paper considers the approximation of solution for a fractional order biological population model. The fractional derivative is considered in the Caputo sense. By using Laplace Adomian decomposition method (LADM), we construct a base function and provide deformation equation of higher order in a simple equation. The considered scheme gives us a solution in the form of rapidly convergent infinite series. Some examples are used to show the efficiency of the method. The results show that LADM is efficient and accurate for solving such types of nonlinear problems.