Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.     

Energy criterion of global existence for energy-critical Schrödinger equations with subcritical perturbations

We present a variational approach to study the energy-critical Schrödinger equations with subcritical perturbations. Through analysing the Hamiltonian property we establish two types of invariant evolution flows, and derive a new sharp energy criterion for blowup of solutions for the equation. Furthermore, we answer the question: how small are the initial data such that the solutions of this equation are bounded in H ¹(R ^N )?     

A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform and inverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.

Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.

Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.

Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.     

Combined sinh-cosh-Gordon equation: Symmetry reductions,exact solutions and conservation laws

Abstract

In this paper we study the combined sinh-cosh-Gordon equation, which arises in mathematical physics and has a wide range of scientific applications that range from chemical reactions to water surface gravity waves. We employ Lie symmetry analysis along with the simplest equation method to obtain exact solutions based on the optimal systems of one-dimensional subalgebras for the combined sinh-cosh-Gordon equation. Furthermore, conservation laws for the combined sinh-cosh-Gordon equation are derived by employing two different methods; the direct method and new conservation theorem.     

Logical equations in monadic logic

A logical formula F(X, P) can be treated as an equation to be satisfied by the solutions X ₀(P) for the predicates X with the expressions P as parameters (if there are such solutions). J. McCarthy considered the parametrization of the solutions, gave the general solution in the case of propositional logic, and stated the problem for other logics. We find the general solution for formulas in the first-order language with monadic predicates and equality. The solutions are obtained via quantifier elimination and parametrized by ϵ-terms. Bibliography: 10 titles. Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 358, 2008, pp. 251–270.     

THE KRALL POLYNOMIALS: A NEW CLASS OF ORTHOGONAL POLYNOMIALS

Abstract

A new set of orthogonal polynomials is found that are solutions to a sixth order formally self adjoint differential equation. These polynomials are shown to generalize the Legendre and Legendre type polynomials. We also show that these polynomials satisfy many properties shared by the classical orthogonal polynomials of Jacobi, Laguerre and Hermite.     

D'Alembert's equation and spherical functions

Summary The observation that the solutions to d'Alembert's functional equation are Z₂-spherical functions onR ² gives us a natural way of extending d'Alembert's functional equation to groups. We deduce in this setting that the general solutions are joint eigenfunctions for a system of partial differential operators, and we find a formula for the bounded solutions.     

The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation

Abstract In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. Keywords: Nonlinear Dirac equation, Dirac-Klein-Gordon equation, Pauli matrix Mathematics Subject Classification (2000): 35C05, 35L45     

New compacton-like and solitary patterns-like solutions to nonlinear wave equations with linear dispersion terms

It has been shown that many fully nonlinear wave equations with nonlinear dispersion terms possess compacton solutions and solitary patterns solutions. In this paper, with the aid of Maple, the mKdV equation, the equation with a source term, the five order KdV-like equation and the KdV–mKdV equation are investigated using some new, generalized transformations. As a consequence, it is shown that these equations with linear dispersion terms admit new compacton-like solutions and solitary patterns-like solutions. These transformations can be also extended to other nonlinear wave equations with nonlinear dispersion terms to seek new compacton-like solutions and solitary patterns-like solutions.     

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

Substitution Delone Sets

   Abstract. Substitution Delone set families are families of Delone sets X =(X ₁ , . . ., X _n ) which satisfy the inflation functional equation

Global attractor for the damped Benjamin–Bona–Mahony equations on R1

We utilize a new necessary and sufficient condition to verity the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood–Paley projection operators. We then use this condition to prove the existence of an attractor for the damped Benjamin–Bona–Mahony equation in the phase space H ¹(R ¹) by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to H ^3/2?? for every ?>0.     

Exact Solutions to a Finite-difference Model of a Nonlinear Reaction-advection Equation: Implications for Numerical Analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

Solving the linear difference equation with periodic coefficients via Fibonacci sequences

ABSTRACT

The aim of this paper is to establish explicit solutions of homogeneous linear difference equations with periodic coefficients. For this purpose, we get around the problem by converting each equation of this class to an equivalent linear difference equation with constant coefficients. Second, we provide some expressions of the solutions via the combinatorial and the Binet formulas of weighted generalized Fibonacci sequences. Finally, some numerical examples and applications are proposed.     

On the center conditions of certain cubic systems

This paper provides a new simple proof of a recent result by C. B. Collins (Differential and Integral Equations 10 (1997), 333-356) to derive the center conditions for a class of planar cubic systems. The idea is to consider periodic solutions of a related scalar non-autonomous equation.

     

SHARP UPPER BOUNDS FOR THE DEGREE OF REGULARITY OF THE SOLUTIONS TO AN ELLIPTIC EQUATION

ABSTRACT

We show that regularity of solutions to the well known Serrin equation is governed by that of particular ones, that we call principal solutions.     

Invariant Measures for a Stochastic Kuramoto–Sivashinsky Equation

Abstract

For the one-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are presented.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract

We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.