Projection Method for Cauchy Problem for an Operator-Differential Equation

In the current paper, we study a projection method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in a Hilbert space. The projection subspaces are linear spans of eigenvectors of an operator similar to A(t). It is assumed that the operators A(t) and K(t) are sufficiently smooth. Error estimates for the approximate solutions and their derivatives are obtained. The application of the developed method for solving the initial boundary value problems is given.     

Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator

In this paper, we study the existence of positive solutions for the nonlinear four-point singular boundary value problem for higher-order with p-Laplacian operator. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem with p-Laplacian operator are obtained.     

On a resolvent estimate of the Stokes system in a half space arising from a free boundary problem for the Navier–Stokes equations

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L_q framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

The representation of meromorphic solutions for a class of odd order algebraic differential equations and its applications

In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.     

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

In this paper, the classical Ambarzumyan’s theorem for the regular SturmLiouville problem is extended to the case in which the boundary conditions are eigenparameter dependent. Specifically, we show that if the spectrum of the operator D 2 +q with eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

The Cauchy Problem for Second-Order Elliptic Systems on the Plane

Regular solutions to second-order elliptic systems on the plane are representable in terms of A-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects

This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

On a class of boundary value problems involving the p-biharmonic operator

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p>1. It describes various problems in the theory of elasticity, e.g., the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p−1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular boundary conditions which are frequently considered in the literature.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

带有p-Laplace算子的分数阶边值问题的特征区间

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the Lp-Lq estimates.     

Ground state periodic solutions for difference equations with discrete p-Laplacian

We use the critical point theory to establish existence results for periodic solutions of some nonlinear boundary value problems involving the discrete p-Laplacian operator. As an application we give an alternative proof to the upper and lower solutions theorem.     

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 437–454, June, 2005.     

Structure of Regular Solutions of the Three-Body Schrödinger Equation near the Pair Impact Point

We investigate the six-dimensional Schrödinger equation for a three-body system with central pair interactions of a more general form than Coulomb interactions. Regular general and special physical solutions of this equation are represented by infinite asymptotic series in integer powers of the distance between two particles and in the sought functions of the other three-body coordinates. Constructing such functions in angular bases composed of spherical and bispherical harmonics or symmetrized Wigner D-functions is reduced to solving simple recursive algebraic equations. For projections of physical solutions on the angular bases functions, we derive boundary conditions at the pair impact point.     

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with -accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with (A,η)-accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.     

A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo‐differential operator generating a Markov process

We use the method proposed by H. Kumano‐go in the classical case to construct a parametrix of the equation + q (x, D )u = 0 where q (x, D ) is a pseudo‐differential operator with symbol in the class introduced by W. Hoh. In case where –q (x, D ) extends to a generator of a Feller semigroup our construction yields an approximation for the transition densities of the corresponding Markov process. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)