Proof of the solvability of a system of partial differential equations in the nonlinear theory of shallow shells of Timoshenko type

We study the solvability of a system of second-order partial differential equations under given boundary conditions. To prove the existence of a solution of the system, we reduce it to a single nonlinear partial differential equation whose solvability is proved with the use of the contraction mapping principle.     

On the Solvability of Nonlinear Boundary Value Problems for the System of Differential Equations of Equilibrium of Shallow Anisotropic Timoshenko-Type Shells with Free Edges

Differential Equations - We study the solvability of a nonlinear boundary value problem for a system of five second-order partial differential equations under given boundary conditions. The system...     

Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.     

On the solvability of two-point,second-order boundary value problems

We gain solvability of a system of nonlinear, second-order ordinary differential equations subject to a range of boundary conditions. The ideas involve differential inequalities and fixed point methods. In particular, maximum principles are not employed.     

A priori estimates and solvability of the third two-point boundary value problem

We study a priori estimates and solvability of a nonlinear two-point boundary value problem for systems of second-order ordinary differential equations with leading positively homogeneous nonlinearity of order > 1 vanishing on a single surface. Assuming that an a priori estimate holds, we prove the invariance of the solvability of the problem under a continuous change of the leading nonlinear homogeneous terms and under arbitrary perturbations that do not affect the behavior of the leading nonlinear homogeneous terms at infinity.     

On an approach to the study of the solvability of boundary value problems for the system of differential equations of the 3D theory of elasticity

We study the solvability of a boundary value problem for a system of second-order linear partial differential equations. A theorem on the existence of a solution of the problem is proved. The method used in the study is to reduce the original system of equations to a system of 3D singular integral equations, whose solvability can be proved with the use of the notion of symbol of a singular operator.     

Analog of the first Fredholm theorem for higher-order nonlinear differential equations

We study the existence of solutions continuously depending on a parameter for higher-order nonlinear ordinary differential equations with linear boundary conditions. In particular, we prove a theorem of Fredholm type providing tests for the unique solvability of this problem.     

On the use of asymptotics in nonlinear boundary value problems

Summary We consider the solvability of some nonlinear boundary value problems for differential equations where the nonlinearity is bounded. This involves the study of the asymptotic behaviour of certain multivalued functionals.     

On some boundary value problems for a system of two second-order equations

For a two-point homogeneous boundary value problem for a system of two nonlinear second-order differential equations, we suggest sufficient solvability conditions (in particular, stated, like Bernstein conditions, in terms of the growth of the absolute values of the right-hand sides of the system with respect to the derivatives of the unknown functions). We obtain a priori estimates for solutions.     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

Construction and Study of Exact Solutions to A Nonlinear Heat Equation

We construct and study exact solutions to a nonlinear second order parabolic equation which is usually called the “nonlinear heat equation” or “nonlinear filtration equation” in the Russian literature and the “porous medium equation” in other countries. Under examination is the special class of solutions having the form of a heat wave that propagates through cold (zero) background with finite velocity. The equation degenerates on the boundary of a heat wave (called the heat front) and its order decreases. The construction of these solutions by passing to an overdetermined system and analyzing its solvability reduces to integration of nonlinear ordinary differential equations of the second order with an initial condition such that the equations are not solvable with respect to the higher derivative. Some admissible families of heat fronts and the corresponding exact solutions to the problems in question are obtained. A detailed study of the global properties of solutions is carried out by the methods of the qualitative theory of differential equations and power geometry which are adapted for degenerate equations. The results are interpreted from the point of view of the behavior and properties of heat waves with a logarithmic front.     

A Boundary Integral Formulation for Poroelastic Materials

Wave propagation in porous media is an important topic, e.g. in geomechanics or the oil-industry. We formulate a linear system of coupled partial differential equations based on Biot's theory with the solid displacements and the pore pressure as the primary unknowns. To solve this system of coupled partial differential equations in a semi-infinite homogeneous domain the BEM (Boundary element method) is especially suitable. Starting from a representation formula a system of two boundary integral equations is derived. These boundary integral equations are used to solve related boundary value problems via a direct approach. Coercivity of the resulting bilinear form is shown, from which unique solvability of the variational formulation follows from injectivity. Using these results we derive the unique solvability of the related boundary integral equations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

Covering mappings in a product of metric spaces and boundary value problems for differential equations unsolved for the derivative

We prove some assertions about solvability, solution estimates, and well-posed solvability of equations with covering mappings in a product of metric spaces. The results are applied to the analysis of boundary value problems for differential equations unsolved for the derivative.     

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.     

Solvability of problems with degeneration: Imbibition of fluid in a porous medium

For a degenerate system of equations such as the equations of motion of immiscible fluids in porous media, we study the solvability of an initial–boundary value problem. Using the process of capillary imbibition of a wetting fluid as an example, we study a class of self-similar solutions with degeneration on the movable boundary and on the entry into the porous layer. The considered problem can be reduced to the analysis of properties of a nonlinear operator equation. For the classical solution of the original problem, we prove existence and uniqueness theorems.     

Justification of an Approach to Application of Orthogonal Expansions for Approximate Integration of Canonical Systems of Second Order Ordinary Differential Equations

A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier—Cliebysliev coefficients is proved. This theorem is a theoretical substantiation for the numerical solution of second order ordinary differential equations using Chebyshev series and Markov quadrature formulas.     

Constructive methods for obtaining the solution of the periodic boundary value problem for a system of matrix differential equations of Riccati type

In the nonsingular case, we obtain sufficient coefficient conditions for the unique solvability of the periodic boundary value problem for a system of matrix differential equations of Riccati type. We develop efficient algorithms for constructing the solution.     

On the Cauchy type problem for systems of functional differential equations

Using theorems on functional differential inequalities, we establish new efficient conditions for the solvability as well as unique solvability of the Cauchy type problem for systems of functional differential equations in both linear and nonlinear cases.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.