Zdeněk Svoboda

Solving certain classes of Lane-Emden type equations using the differential transformation method

In this paper, the differential transformation method (DTM) is applied to solve singular initial problems represented by certain classes of Lane-Emden type equations. Some new differential transformation formulas for certain exponential and logarithmic nonlinearities are derived. The approximate and exact solutions of these equations are calculated in the form of series with easily computable terms. The results obtained with the proposed methods are in good agreement with those obtained by other methods. The advantages of this technique are shown as well.

An existence criterion of positive solutions of p-type retarded functional differential equations

The conditions of existence of a positive solution (i.e., a solntion with positive coordinates on a considered interval) of systems of retarded functional equations in the case of unbounded delay with finite memory are discussed. A general criterion for nonlinear case is given as well as its application to a linear system. Illustrative special cases are considered too.

Simple uniform exponential stability conditions for a system of linear delay differential equations

Abstract Uniform exponential stability of linear systems with time varying coefficients x i ( t ) = - ∑ j = 1 m ∑ k = 1 r ij a ij k ( t ) x j ( h ij k ( t ) ) , i = 1 , … , m is studied, where t ⩾ 0 , m and r ij , i , j = 1 , … , m are natural numbers, a ij k : [ 0 , ∞ ) → R and h ij k : [ 0 , ∞ ) → R are measurable functions. New explicit result is derived with the proof based on Bohl–Perron theorem. The resulting criterion has advantages over some previous ones in that, e.g., it involves no M-matrix to establish stability. Several useful and easily verifiable corollaries are deduced and examples are provided to demonstrate the advantage of the stability result over known results.

Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.

Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

The present investigation deals with global instability of a general 𝑛-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaevs method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

Exponential stability of linear delayed differential systems

Abstract Linear delayed differential systems x ˙ i ( t ) = − ∑ j = 1 m ∑ k = 1 r i j a i j k ( t ) x j ( h i j k ( t ) ) , i = 1 , … , m are analyzed on a half-infinity interval t ≥ 0. It is assumed that m and rij, i , j = 1 , … , m are natural numbers and the coefficients a i j k : [ 0 , ∞ ) → R and delays h i j k : [ 0 , ∞ ) → R are measurable functions. New explicit results on uniform exponential stability are derived including, as partial cases, recently published results.

The solutions of second-order linear differential systems with constant delays

The representations of solutions to initial problems for non-homogenous n-dimensional second-order differential equations with delays x″(t)−2Ax′(t−τ)+(A2+B2)x(t−2τ)=f(t) by means of special matrix delayed functions are derived. Square matrices A and B are commuting and τ > 0. Derived representations use what is called a delayed exponential of a matrix and results generalize some of known results previously derived for homogenous systems.