FINITE ENERGY WEAK SOLUTIONS TO SOME DIRICHLET PROBLEMS WITH VERY SINGULAR DRIFTS AND NONLINEAR ADVECTION TERMS

This paper studies the existence of nodal solutions for a class of boundary value problems that arise in population dynamics models. These problems are degenerate in the sense that the nonlinear term vanishes in a subinterval of the underlying region. In contrast to the non-degenerate case, in these degenerate situations the structure of the set of nodal solutions might consist of several components, making the analysis of this problem more complicated. This paper provides an initial step towards the solution of this general problem.

Non-linear finite element analyses of structures (such as beams) involve construction of weak solutions for the governing equations. While a weak approach weakens the differentiability requirements of the so-called shape functions, the governing equations are only satisfied in an integral sense and not point-wise, or, even path-wise. Moreover, use of a finite mesh leads to a stiffening of the numerical model. While strong solutions obtained through some of the existing mesh-free collocation methods overcomes some of these lacunae to an extent, the quality of the numerical solutions would be considerably improved if the computational algorithm were able to faithfully reproduce (or approximate or preserve) certain geometrical features of the response surfaces or manifolds. This paper takes the first step towards realizing this objective and proposes a multi-step transversal linearization (MTL) technique for a class of non-linear boundary value problems, which are treated as conditionally dynamical systems. Numerical explorations are performed, to a limited extent, through applications to large deflection analyses of planar beams with or without plastic deformations.

The numerical solution of a class of boundary value problems for nonlinear fractional differential equations is discussed. Using a suitable integral equation reformulation of the boundary value problem and spline collocation techniques on graded grids, high order methods for solving fractional boundary value problems are constructed.

For a class of boundary value problems where the spectral parameter appears in the boundary condition in the form of a locally definitizable function linearizations are constructed and their local spectral properties are investigated.

A generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

An extremum variational principle for a class of boundary value problems

This article deals with a class of discrete type boundary value problems. Sufficient conditions guaranteeing the existence of at least three positive solutions of this class of boundary value problems are established by using a fixed point theorem in cones in Banach spaces. An example is given to illustrate the main theorem.

For a class of nonlinear elliptic boundary value problems including the von Karman equations considered by D. M. Duc, N. L. Luc, L. Q. Nam, and T. T. Tuyen [Nonlinear Anal., 2003, 55: 951–968], we give a new proof of a corresponding theorem of three solutions via Morse theory instead of topological degree theory. Several bifurcation results for this class of boundary value problems are also obtained with Morse theory methods. In addition, for the von Karman equations studied by A. Borisovich and J. Janczewska [Abstr. Appl. Anal., 2005, 8: 889–899], we prove a few of bifurcation results under Dirichlet boundary conditions based on the second named author’s recent work about parameterized splitting theorems and bifurcations for potential operators.

The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids

In this paper, we consider two classes of boundary value problems in the frame of local proportional fractional derivatives. For both of these classes, we obtain the associated Green?s functions and discuss their properties. Using these properties, we go about the uniqueness of the solutions. In addition, we establish Lyapunov-type and Hartman-Wintner-type inequalities and build sharp estimated for the unique solutions of the considered equations.

Dual extremum principles and error bounds for a class of boundary value problems

A mixed decomposition-spline approach for the numerical solution of a class of singular boundary value problems

The asymptotic of the solutions of a class of non-linear boundary value problems with a free boundary

The electrostatic potential describing the fields interior to a magnetohydrodynamic (MHD) power generating channel obeys a linear uniformly elliptic partial differential equation. The physically natural boundary conditions, however, do not fall within a class for which the solution has previously been shown to be unique. We discuss the boundary conditions appropriate to this situation and then generalize the discussion in Courant and Hilbert to demonstrate the uniqueness of the solution to an extended class of boundary value problems which includes the MHD channel as a special case.

We consider a class of linear second order boundary value problems of the form subject to boundary conditions . A seventh order global numerical procedure is developed based on spline functions of degree seven and eight. Matrix formulation and the error analysis of the method are discussed. Finally numerical behaviour and the convergence order of the scheme are verified through examples.