The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

We consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.      

Theory of bifurcations of the Schrödinger equation

We obtain solvability conditions and a representation of solutions for a boundary value problem for a linear nonstationary Schrödinger equation in a Hilbert space as well as sufficient conditions for the bifurcation of solutions of this equation.     

Boundary-value problem for a class of second-order parameter-dependent dynamic equations on a time scale

In this study, we consider a boundary value problem generated by a second-order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem.     

Factorized equation of vibrations of a finite string with nonstationary second directional derivatives at the endpoints

We obtain closed-form recursion formulas for the classical solutions of a mixed problem for the general inhomogeneous factorized equation of vibrations of a bounded string with second directional derivatives in the boundary conditions, in which the coefficients multi-plying the first of the two directional derivatives are independent of time. We study the case of boundary conditions in which all first directional derivatives are not directed along the characteristics of the equation. We obtain necessary and sufficient conditions on the right-hand side and the initial and boundary data of the problem for its well-posed global solvability in the set of classical solutions.     

具有多重解的非线性奇摄动问题

利用边界层法,研究了一类具有多重解的非线性奇摄动问题．在适当的假设下,通过给出外部解展开式系数及其对应边界条件的一般表达式,根据退化问题的边值作为某方程的根的重数,得到了此问题不同形式的渐近解．特别地,当这种根的重数为偶数时,问题具有二重解．另外,将相关结果应用于化学反应器理论,并通过对具有多重解的例子的渐近解和精确解的数值模拟说明如此构造的渐近解具有较高的精度．     

A Black-Scholes option pricing model with transaction costs

We consider a boundary value problem for a nonlinear differential equation which arises in an option pricing model with transaction costs. We apply the method of upper and lower solutions in order to obtain solutions for the stationary problem. Moreover, we give conditions for the existence of solutions of the general evolution equation.     

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Green's function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.     

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence of three positive solutions for a class of Riemann-Liouville fractional q-difference equation

In this paper, we confirm the existence of three positive solutions for a class of Riemann-Liouville fractional $q$-difference equation which satisfies the boundary conditions. We gain several sufficient conditions for the existence of three positive solutions of this boundary value problem by applying the Leggett-Williams fixed point theorem.     

A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.     

Smooth dependence on data of solutions and contact regions for a Signorini problem

We prove that the solutions to a 2D Poisson equation with unilateral boundary conditions of Signorini type as well as their contact intervals depend smoothly on the data. The result is based on a certain local equivalence of the unilateral boundary value problem to a smooth abstract equation in a Hilbert space and on an application of the Implicit Function Theorem to that equation.     

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A FOUR-POINT BOUNDARY VALUE PROBLEM FOR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we investigate the nonexistence of positive solutions for a class of four-point boundary value problem of nonlinear differential equation with fractional order derivative. We give sufficient conditions on nonlinear term and the parameter such that the boundary value problem has no positive solutions. Some examples are presented to illustrate the main results.     

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.     

分数阶微分方程组边值问题解的存在性

研究分数阶微分方程组边值问题在一类新型的边界条件——分数阶分离边界条件下解的存在性.通过将微分方程组边值问题转化为与之等价的积分方程组,利用Banach不动点定理和Leray-Schauder非线性更替得到边值问题解存在的充分条件,并给出两个例子说明了主要结论.     

On a stochastic wave equation with unilateral boundary conditions

We prove the existence and uniqueness of solutions to the initial boundary value problem for a one-dimensional wave equation with unilateral boundary conditions and random noise. We also establish the existence of an invariant measure.

     

Boundary integral equations for mixed Dirichlet,Neumann and transmission problems

We consider a Helmholtz equation in a number of Lipschitz domains in n ≥ 2 dimensions, on the boundaries of which Dirichlet, Neumann and transmission conditions are imposed. For this problem an equivalent system of boundary integral equations is derived which directly yields the Cauchy data of the solutions. The operator of this system is proved to be injective and strongly elliptic, hence it is also bijective and the original problem has a unique solution. For two examples (a mixed Dirichlet and transmission problem and the transmission problem for four quadrants in the plane) the boundary integral operators and the treatment of the compatibility conditions are described.     

The exact number of solutions for the second order nonlinear boundary value problem

A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well. The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava.     

Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.