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...

this paper is concerned with the study of the existence of positive solutions for a navier boundaryvalue problem involving the p-biharmonic operator; the right hand side of problem is a nonsmoothfunctional with variable parameters. the existence of at least three positive solutions is establishedby using nonsmooth version of a three critical points theorem for discontinuous functions. our resul...

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 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...

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary...

This paper presents an optimal solver for the Morley element problem for the boundaryvalue problem of the biharmonic equation by decomposing it into several subproblems and solving these subproblems optimally. The optimality of the proposed method is mathematically proved for general shape-regular grids. Mathematics subject classification: 65F08, 65N30, 65N99

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...

We determine necessary and sufficient conditions for the existence of solutions a nonlinear boundaryvalue problem system difference-algebraic equations in case parametric resonance. construct convergent iterative algorithm finding approximations to As an example application constructed scheme, we obtain two-point boundary-value Mathieu-type with perturbation. To check accuracy obtained perturba...

In this article, we obtain Lyapunov-type inequalities for certain odd order linear boundary-value problems. Our inequalities involve integrals of both q+(t) and q−(t) in addition to that of |q(t)|. The Green’s function for even order boundary-value problems plays a key role in our proofs. Also, using the Fredholm alternative theorem, we obtain a criterion for the existence and uniqueness of sol...

This article concerns the existence of solutions to periodic boundaryvalue problems for second-order nonlinear differential equation involving fractional derivatives. Under certain linear growth condition of the nonlinearity, we obtain solutions, by using coincidence degree theory. An example illustrates our results.