In this paper, we present necessary and sufficient conditions for the existence of positive C3[0, 1]∩C4(0, 1) solutions for the singular boundaryvalue problem x′′′′(t) = p(t)f(x(t)), t ∈ (0, 1); x(0) = x(1) = x′(0) = x′(1) = 0, where f(x) is either superlinear or sublinear, p : (0, 1) → [0,+∞) may be singular at both ends t = 0 and t = 1. For this goal, we use fixed-point index results.

Lie symmetry method is applied to find analytic solutions of initial-boundaryvalue problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and...

The modal formalism is developed to deal with index-modulated structures ended by anisotropic regions (perfectly matched layers, PML). The method is applicable to scattering problems from dielectric or metallic objects as well as to propagation in planar slab waveguides. The boundaryvalue problem is solved and the dispersion relation inside the PML is derived. The general expression of the eige...

The problem that is studied concerns a yawed wedge under a free surface, moving at a uniform speed. The model involves a trailing cavity whose boundary is a dividing streamline through the vertex of the wedge. The cavity closure mechanism is described according to the Tulin– Terent’ev single-spiral-vortex model. A closed-form solution to the governing nonlinear boundaryvalue problem is found by...

We consider a broad class of linear boundary-value problems for systems m ordinary differential equations order r known as general problems. Their solutions y : [a, b] → ℂm belong to the Sobolev space $$ {\left({W}_1^r\right)}^m and boundary conditions are given in form By = q, where B: (C(r−1))m ℂrm is an arbitrary continuous operator. For this problem, we prove that its solution can be approx...

The numerical code for calculating the shielding current density in the high-Tc superconductor is developed by using the element-free Galerkin method. The behavior of the shielding current density is expressed by the integro-differential equations. In order to solve the initial-boundaryvalue problem of the equations, the equivalent weak form is derived. Since the form satisfies the essential b...

We study the existence of positive solutions for the boundaryvalue problem of the singular higher-order functional differential equation (Ly(n−2))(t) + h(t)f(t, yt) = 0, for t ∈ [0, 1], y(0) = 0, 0 ≤ i ≤ n− 3, αy(n−2)(t)− βy(n−1)(t) = η(t), for t ∈ [−τ, 0], γy(n−2)(t) + δy(n−1)(t) = ξ(t), for t ∈ [1, 1 + a], where Ly := −(py′)′ + qy, p ∈ C([0, 1], (0,+∞)), and q ∈ C([0, 1], [0,+∞)). Our main to...

In this paper we analyze the global existence of classical solutions to the initial boundaryvalue 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 Fokker-Planck equation presents a nonlinearity in t...

In general the mathematical model for steady, one-dimensional conduction in fins appears as a nonlinear, two-point boundaryvalue problem. The nonlinearities arise due to radiative surface heat transfer, temperature dependent thermal conductivity or heat transfer coefficient etc. [i]. Often in such cases a numerical solution has to be obtained. Examining the literature reporting such solutions i...

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundaryvalue problem for a general Schrödinger-type equation posed on a two space dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain and we approximate its solut...