Josef Rebenda

Stability of a Functional Differential System with a Finite Number of Delays

The paper is devoted to the study of asymptotic properties of a real two-dimensional differential system with unbounded nonconstant delays. The sufficient conditions for the stability and asymptotic stability of solutions are given. Used methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of Lyapunov-Krasovskii functional. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one or more constant delays or one nonconstant delay were studied.

Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability

The asymptotic behaviour of a real two-dimensional differential system ∑𝑥′(𝑡)=𝖠(𝑡)𝑥(𝑡)

A differential transformation approach for solving functional differential equations with multiple delays

In the paper an efficient semi-analytical approach based on the method of steps and differential transformation is proposed for numerical approximation of solutions of retarded logistic models of delayed and neutral type, including models with several constant delays. Algorithms for both commensurate and non-commensurate delays are described, applications are shown in examples. Validity and efficiency of the presented algorithms is compared with variational iteration method, Adomian decomposition method and polynomial least squares method numerically. Matlab package DDE23 is used to produce reference numerical values.

A numerical approach for solving of fractional Emden-Fowler type equations

In the paper, we utilize the fractional differential transformation (FDT) to solving singular initial value problem of fractional Emden-Fowler type differential equations. The solutions of our model equations are calculated in the form of convergent series with fast computable components. The numerical results show that the approach is correct, accurate and easy to implement when applied to fractional differential equations.

A semi-analytical approach for solving of nonlinear systems of functional differential equations with delay

In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented.In the paper, we propose a correct and efficient semi-analytical approach to solve initial value problem for systems of functional differential equations with delay. The idea is to combine the method of steps and differential transformation method (DTM). In the latter, formulas for proportional arguments and nonlinear terms are used. An example of using this technique for a system with constant and proportional delays is presented.

A new iterative method for linear and nonlinear partial differential equations

In this paper, a new iterative method for solving initial problem for linear and nonlinear partial differential equations is presented. Convergence analysis of iterative process is included. The proposed iterative scheme does not require any discretization, linearization or small perturbation and, compared to e.g. Adomian decomposition method or homotopy perturbation method, there is no need for calculation of generally n-dimensional integrals or derivatives, respectivelly. The approximate solution is calculated in the form of a sequence with easily computable terms. Depending on the form of the sequence it is possible to identify exact solution. Illustrative example is presented to demonstrate reliability and performance of the presented method.

Convergence analysis of an iterative scheme for solving initial value problem for multidimensional partial differential equations

Existence and uniqueness of solutions of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations are proved using Banach fixed-point theorem. An iterative scheme is derived and rigorous convergence analysis of this scheme and an error estimate are included as well. Several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme.

Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Wazewskis method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

Comparison of differential transformation method with adomian decomposition method for functional differential equations with proportional delays

In this paper, we will introduce two methods to obtain the numerical solutions for functional differential equations with proportional delays. The first method is the differential transformation method (DTM) and the second method is Adomian decomposition method (ADM). Moreover, we will make comparison between the solutions obtained by the two methods. Consequently, the results of our system tell us the two methods can be alternative ways for solution of the linear and nonlinear functional differential and integro-differential equations. New formulas for DTM were proven for these types of equations.