Differential and Difference Equations with Applications

New results are proved on the asymptotic behavior of blow-up solutions to a higher-order equation with general potential are proved. Several author’s results are presented concerning both positive and oscillatory solutions to equations with regular and singular nonlinearities. Some applications of the results obtained are proposed.

In this paper, we investigate the generalized partial difference operator and propose a model of it in discrete heat equationHeat equation with several parameters and shift values. The diffusion of heat is studied by the application of Fourier’s law of heat conduction in dimensions up to three and several solutions are postulated for the same. Through numerical simulationsSimulation using MATLAB, solutions are validated and applicationsApplications are derived.

In this work, turbulent flows through porous media are considered. We begin by making a historical review of the equations governing laminar flows in porous media, from Darcy’s law to Darcy–Brinkman–Forchheimer’s more general model. Using the double averaging concept (in time and in space) we explain how to obtain the more general system of equations that governs turbulent flows through porous media. For the one-equation turbulent problem in the steady-state we show that the known existence results can be generalized to any space dimension $$d\ge 2$$ and for a more general function of turbulence production.

In this paper, using variational methods and critical point theory we discuss the existence of at least three solutions for nonlinear Kirchhoff-type difference equations with Dirichlet boundary conditions. We also provide examples in order to illustrate the main results.

We study a Cauchy problem associated to a second-order integro-differential inclusion. The general framework of evolution operators that define the problem that we consider has been developed by Kozak and, afterwards, improved by Henriquez. Our aim is to show the existence of mild solutions continuously depending on a parameter for the problem studied in the case when the set-valued map is Lipschitz in state variables. Moreover, as a consequence, we deduce the existence of a continuous selection of the set of all mild solutions of the problem considered. The proof our main result is based on a result of Bressan and Colombo concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values.

This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually adjoint first order difference operators. These classes encompass equations of hypergeometric type describing classical orthogonal polynomials of a discrete variable.

We consider nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic. We analyze the problem on the domain that is bounded by two coaxial cylinders which present solid thermo-insulated walls. Therefore we assume the cylindrical symmetry of the solution. In this work we present the existence and uniqueness results for corresponding problem with homogeneous boundary data for velocity, microrotation and heat flux, under the additional assumption that the initial density and initial temperature are strictly positive.

For full singular integro-differential equations with Gilbert kernel, the collocation method is justified. The approximate solution is sought in the form of Hermite–Fejer polynomial. The convergence of the method is proved and the rate of convergence is estimated.

We consider nonexistence of nontrivial solutions for several classes of nonlinear functional differential inequalities. In particular, we obtain sufficient conditions for nonexistence of such solutions for the following types of inequalities: semilinear elliptic inequalities with a transformed argument in the nonlinear term, including higher order ones; quasilinear elliptic inequalities with a transformed argument in the nonlinear term dependent on the absolute value of the gradient of the solution; elliptic inequalities with the principal part of the p-Laplacian type with similar transformations in the lower order terms; parabolic partial differential inequalities with a transformed temporal argument in the nonlinear term. In the case of the untransformed argument these results coincide with the well-known optimal results of Mitidieri and Pohozaev, but in the general case they depend on the character of the transformation of the argument. The results apply to different types of transformations of the argument, such as dilatations, rotations, contractions, and shifts.

Gielis transformations, with their origin in botany, are used to define square waves and trigonometric functions of higher order. They are rewritten in terms of Chebyshev polynomials. The origin of both, a uniform descriptor and the origin of orthogonal polynomials, can be traced back to a letter of Guido Grandi to Leibniz in 1713 on the mathematical description of the shape of flowers. In this way geometrical description and analytical tools are seamlessly combined.

In this work, Variational Iteration Method is employed to solve parabolic partial differential equations subject to initial and nonlocal inhomogeneous boundary conditions of integral type. Since nonlocal boundary conditions considerably complicate the application of standard functional and numerical techniques, equations having such conditions are first transformed to local (classical) boundary conditions Then they are solved by Variational Iteration Method.

This paper deals with the existence and energy estimates of positive solutions for a class of Kirchhoff-type boundary-value problems on the real line, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, applying a consequence of the local minimum theorem for differentiable functionals due to Bonanno the existence of a positive solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti–Rabinowitz. In what follows, employing two consequences of the local minimum theorem for differentiable functionals due to Bonanno by combining two algebraic conditions on the nonlinear term which guarantees the existence of two positive solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third positive solution for our problem. Moreover, concrete examples of applications are provided.

The aim of this paper is to present our recent existence and uniqueness results for a one-dimensional damped model of a suspension bridge and compare them to previous results for either damped or non-damped one-dimensional beam models.

Let $$R_k$$ be the class of functions with bounded radius rotation and let $$S_H$$ be the class of sense-preserving harmonic mappings. In the present paper we investigate a certain class of harmonic mappings related to the function of bounded radius rotation.

Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite naturally to discrete-time nonautonomous dynamical systems given in the process formulation. This paper provides an overview of the author’s work on this subject. Also an example is presented that has not appeared before in the literature.

We propose, for the sake of dialogue, that the nonautomomous reciprocal max-type difference equation, $$\begin{aligned} x_{n+1}=\max \left\{ \frac{A_{n}^{(0)}}{x_{n}}, \frac{A_{n}^{(1)}}{x_{n-1}}, \ldots , \frac{A_{n}^{(k)}}{x_{n-k}}\right\} , \ \ n=0, 1, \ldots , \end{aligned}$$where the parameters are positive periodic sequences and the initial conditions are positive, when $$k=1$$ may serve as a phenomenological model of seizure activity as occurs in mesial (or middle) temporal lobe epilepsy.

In this article we present the Pontryagin maximum principle of a time-optimal control problem for general form of functional-differential equations. The obtained results are the direct generalization of the case for ordinary differential equations: if the delay disappear then the results turn into the classic Pontryagin maximum principle for finite dimensional systems. In this work we apply the methodology and constructions of the i-Smooth analysis.

We believe that the difference between time scale systems and ordinary differential equations is not as big as people use to think. We consider linear operators that correspond to linear dynamic systems on time scales. We study solvability of these operators in $${\mathbb L}^\infty $$. For ordinary differential equations such solvability is equivalent to hyperbolicity of the considered linear system. Using this approach and transformations of the time variable, we spread the concept of hyperbolicity to time scale dynamics. We provide some analogs of well-known facts of Hyperbolic Systems Theory, e.g. the Lyapunov–Perron theorem on stable manifold.

In this paper, we study the oscillation and asymptotic properties of solutions of a certain nonlinear neutral third-order differential equation with either delay or advanced argument.

In this paper we study properties of the Neumann discrete problem. We investigate so called polar Pareto spectrum of a specific matrix which represents the Neumann discrete operator. There is a known relation between polar Pareto spectrum of any discrete operator and its Fučík spectrum. We also state a conjecture about asymptotes of Fučík curves with respect to the matrix and we illustrate a variety of polar Pareto eigenvectors corresponding to a fixed polar Pareto eigenvalue.

In the paper we resemble interval difference method of second order designed by us earlier and present new, fourth order interval difference methods for solving the Poisson equation with Dirichlet boundary conditions. Interval solutions obtained contain all possible numerical errorsError bounds. Numerical solutions presented confirm the fact that the exact solutions are within the resulting intervals.

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex coefficients and two unknown initial data. In order to prove our results, we first construct an explicit solution of a family of problems with initial data depending on a parameter $$\eta > 0$$ and then we obtain an asymptotic representation of a solution as $$\eta $$ tends to infinity.

The oscillationOscillation of all solutions of a linear autonomous difference system with two delays is studied. Explicit necessary and sufficient conditions in terms of the coefficient matrix and the delays are established, which are some extensions of the previous results. As an application, we can completely classify the oscillation and the asymptotic stabilityStability of a delay difference system.

In this paper, we will show that $$\log 0 = \log \infty =0$$ by the division by zero $$z/0=0$$ and its fundamental applicationsApplications. In particular, we will know that the division by zero is our elementary and fundamental mathematics.

In this paper, we present a collocation method for solving the following second-order Cauchy integro-differential equation $$\begin{aligned} x''(s)+\oint _{-1}^{1}\frac{\omega (t)x(t)}{s-t}dt=f(s),\quad -1< s < 1, \end{aligned}$$$$ x'(-1)=x(1)=0, $$in the space $${\mathscr {X}}:={\mathscr {C}}^0([-1,1],\mathbb {C})$$, with domain $$ {\mathscr {D}}:=\left\{ x \in {\mathscr {X}}~:~x''\in {\mathscr {X}},\quad x'(-1)=x(1)=0\right\} . $$The integral is a Cauchy principal value, and $$ \omega ( s ):=\sqrt{\frac{1+s}{1-s}} $$is the weight function. We come up with a modified collocation method to build an approximate solution $$x_{n}$$ using the airfoil polynomials of the first kind. Finally, we establish a numerical example to exhibit the theoretical results.

Asymptotic properties of solutions to difference equations of the form $$ \varDelta ^m(x_n-u_nx_{n-k})=a_nf(x_{\sigma (n)})+b_n $$Using a new version of the Krasnoselski fixed point theorem and the iterated remainder operator, we establish sufficient conditions under which a given solution of the equation $$ \varDelta ^m(x_n-u_nx_{n-k})=b_n $$is an approximative solution to the above equation. Our approach, based on the iterated remainder operator, allows us to control the degree of approximation. We use $$\mathrm {o}(n^s)$$, for a given nonpositive real s, as a measure of approximation.

We study a third-order delay trinomial difference equation. We transform this equation to a binomial third-order difference equation with quasidifferences. Using comparison theorems with a certain first order delay difference equation we establish results on asymptotic properties of nonoscillatory solutions of the studied equation. We give an easily verifiable criterium which ensures that all nonoscillatory solutions tend to zero.

E.T. Copson generalized the well-known result about the convergence of bounded and monotonic sequences of real numbers. Over the years, generalizations of this result have been made concerning linear and nonlinear inequalities that gave us a wide range of criteria for the convergence of sequences in relationship to the characteristic polynomial, monotonicity of the variables, etc. In this paper, we present a survey about these generalizations of Copson’s result, focusing in the state-of-art of the problem, and bring up some open questions that could lead us to future research.

A preliminary mathematical model of diabetes has been proposed in [4], in which the evolution of the size of a population of diabetes mellitus patients and the number of patients with complications, has been modeled by second order system of nonlinear differential equations. The model, has already been analyzed for the linear local stabilityStability of the equilibria of the system. However, the global behavior of the flow of the nonlinear system has not been studied. The present article analyzes the global behavior of the trajectories of the population growth using LyapunovLyapunov’s theory stability analysis. Toward this, we construct a suitable Lyapunov function corresponding to an interior equilibrium point and show that it is asymptotically stable within the entire open first quadrant of the planar state space which is the region of interest. Further, transient or incremental stabilityStability in the phase plane has been studied via LyapunovLyapunov’s theory exponent analysis. The stability analysis has also been verified through numerical simulationsSimulation, under various parameters. A physical interpretation of the parametric dependence of the flows of the nonlinear system is provided from the point of view of diabetic population dynamics.

A nonlocal boundary value problem for nonlinear functional equations is studied. New effective conditions are found for solvability a unique solvability of considered problem. Obtained results are concretized for differential equation with deviating argument.

In this work, we analyze the existence of solution to a fuzzy differential equation of first order in the fuzzy functional interval determined by an upper and a lower solution. The approach followed consists in the study of an auxiliary problem that is defined through a proper ‘truncation operator’ based on the choice of well ordered upper and lower solutions to the problem of interest. To our purpose, we justify that the truncation operator is well defined and satisfies some monotonicity properties. Finally, using the lattice structure of some subsets of the space of continuous fuzzy-valued functions and imposing some restrictions on the nonlinearity, we conclude the existence of solution to the equation on the interval $$[0,+\infty )$$ by the application of Tarski’s fixed point theorem.

We show that with the multidimensional system of gas dynamics with a special forcing one can associate a quadratically nonlinear ODE system which describes a special class of motion. The system can be obtained by two different ways. In particular, we study the influence of Coriolis and frictional terms. We review the result about the non-frictional case and study the influence of constant dry friction.

In this paper, we will show and give applications of the division by zero $$z/0=1/0=0/0=0$$ in calculus and differential equations. In particular, we will know that the division by zero is our elementary and fundamental mathematics.

We study multidimensional variational problems, where the Lagrange function depends on the partial Caputo–Katugampola fractional derivatives, generalizing the Caputo and the Caputo–Hadamard fractional derivatives. We present sufficient and necessary conditions which determine the extremizers of a functional.

We consider nonlinear elliptic systems satisfying componentwise coercivity condition. The nonlinear terms have controlled growths with respect to the solution and its gradient, while the behaviour in the independent variable x is governed by functions in Morrey spaces. We obtain maximum principle for such kind of systems.

Consider the first-order linear differential equation $$\begin{aligned} x^{\prime }(t)+p(t)x(\tau (t))=0,\;\;\;t\ge t_{0}, \end{aligned}$$where the functions $$p,\tau \in C([t_{0,}\infty ),\mathbb {R}^{+})$$, (here $$ \mathbb {R}^{+}=[0,\infty )),\tau (t)\le t$$ for $$t\ge t_{0}$$ and $$ \lim _{t\rightarrow \infty }\tau (t)=\infty .$$ A survey on the oscillationOscillation of all solutions to this equation is presented in the case of monotone and non-monotone argument and especially in the critical case where $$ \liminf _{t\rightarrow \infty }p(t)=1/e\tau $$ and also when the known oscillation conditions $$\underset{t\rightarrow \infty }{\lim \sup } \int \nolimits _{\tau (t)}^{t}p(s)ds>1$$ and $$\liminf _{t\rightarrow \infty }\int _{\tau (t)}^{t}p(s)ds>\frac{1}{e}$$ are not satisfied. Examples illustrating the results are given.

A discrete version of the plane wave solution to some discrete Dirac type equations in the spacetime algebra is established. The conditions under which a discrete analogue of the plane wave solution satisfies the discrete Hestenes equation are briefly discussed.

For more than almost 200 years the MöbiusMöbius geometries strip and its “mysterious” property attracts the attention of mathematicians. After a “complete cut” of this surface, one object appears, but already with a fourfold twist. The generalization of this phenomenon to figures of a more complex configuration led to an “unexpected” result: after the cut of the generalized Möbius-Listing body, more than two geometric shapes may appear. In this paper, we consider all possible cases of a complete cut of the generalized Möbius-Listing body with a regular hexagon as radial section. In early works, together with different colleagues, on the basis of importance, they separately examined the case of Möbius-Listing’s bodies with a radial section of regular 3, 4 and 5 angular figures. Also, cases of similar bodies with a radial section of convex regular two and three angular figures were considered separately. One possible application of these results is assumed in the description of the properties of the middle surfaces in the theory of elastic shells [14] (Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston, p. 287, 1985).

This paper deals with the existence and multiplicity of periodic solutions for the fractional p-Laplacian equations. The minimization argument and extended Clark’s theorem are applied to prove our results.

In this paper, we obtain some sufficient conditions for the oscillation of all solutions of the third order nonlinear neutral difference equation with mixed arguments of the form $$\begin{aligned} \Delta ^3(x_n+b_n x_{n-\tau _1}+c_nx_{n+\tau _2})^{\alpha }+q_nx_{n-\sigma _1}^{\beta }+p_n x_{n+\sigma _2}^{\gamma }=0, n\ge n_0, \end{aligned}$$where $$\alpha $$, $$\beta $$ and $$\gamma $$ are the ratios of odd positive integers. Some examples are provided to substantiate our results.

We investigate properties of the fuzzy Henstock–Kurzweil delta integral (shortly, FHK $$\varDelta $$-integral) on time scales, and obtain two necessary and sufficient conditions for FHK $$\varDelta $$-integrability. The concept of uniformly FHK $$\varDelta $$-integrability is introduced. Under this concept, we obtain a uniformly integrability convergence theorem. Finally, we prove monotone and dominated convergence theorems for the FHK $$\varDelta $$-integral.

In this work, we establish the necessary conditions for oscillation of the second order neutral delay differential equations of the form: $$\begin{aligned} (r(t)((x(t)+p(t)x(\tau (t)))')^\gamma )'+q(t) x^\gamma (\sigma (t))+v(t)x^\gamma (\eta (t))=0 \end{aligned}$$under the assumptions $$\begin{aligned} \int _{0}^{\infty }\left( \frac{1}{r(t)}\right) ^\frac{1}{\gamma }dt= & {} \infty \end{aligned}$$and $$\begin{aligned} \int _{0}^{\infty }\left( \frac{1}{r(t)}\right) ^\frac{1}{\gamma }dt< & {} \infty \end{aligned}$$for various ranges of p(t), where $$0<\gamma \le 1$$ is a quotient of odd positive integers.

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier–Stokes system (N). This description corresponds to the so-called Eulerian approach. We develop a new approximation method for (N) in both the steady and the nonsteady case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid. The method leads to a sequence of uniquely determined approximate solutions with a high degree of regularity, which contains a convergent subsequence with limit function v such that v is a weak solution on (N).

We study some discrete boundary value problems for discrete elliptic pseudo-differential equations in a half-space. These statements are related with a special periodic factorization of an elliptic symbol and a number of boundary conditions depends on an index of periodic factorization. This approach was earlier used by authors for studying special types of discrete convolution equations. Here we consider more general equations and functional spaces.

We consider a system described by the linear heat equation with adiabatic boundary conditions. We impose a nonlinear perturbation determined by a family of interval maps characterized by a certain set of parameters. The time instants of the perturbation are determined by an additional dynamical system, seen here as part of the external interacting system. We analyse the complex behaviour of the system, through the scope of symbolic dynamics, and the dependence of the behaviour on the time pattern of the perturbation, comparing it with previous results in the periodic case.

It is often the case that the equilibrium values are the only piece of information required for the solution of a practical problem (although, sometimes, time to achieve equilibrium or size of the device is the real issue) and in such situations it is important to identify the conditions under which a complex partial differential equationsPartial Differential Equations model can be replaced with a simpler one. In this paper, we study a flow injection analysis of a bi-enzyme electrode, with the aim of finding the ratio of the two enzymes involved which yields the highest current amplitude.

In this paper, we introduce higher order difference operator and its inverse by which we obtain x-Fibonacci sequence and its series with several theorems and results. Suitable examples verified by MATLAB are provided to illustrate our main results.

In this paper, we present comparison theorems for the oscillation of solutions of second-order damped nonlinear differential equations with p-Laplacian. Proof is given by means of phase plane analysis of systems. Moreover, combining the comparison theorem and (non)oscillation criteria for the generalized Euler differential equation, we give new (non)oscillation criteria for the damped equations.

We consider explicit two-level three-point in space finite-difference schemes for solving 1D barotropic gas dynamics equations. The schemes are based on special quasi-gasdynamic and quasi-hydrodynamic regularizations of the system. We linearize the schemes on a constant solution and derive the von Neumann type necessary condition and a CFL type criterion (necessary and sufficient condition) for weak conservativeness in $$L^2$$ for the corresponding initial-value problem on the whole line. The criterion is essentially narrower than the necessary condition and wider than a sufficient one obtained recently in a particular case; moreover, it corresponds most well to numerical results for the original gas dynamics system.

We propose geometric and automorphic sides of a general correspondence for vertex operator algebras and review supporting examples.