A condition for solvability of an integral equation which is connected with the first boundary value problem for the heat equation is investigated. It is shown that if this condition is fulfilled then the boundary considered is 1⁄2-Hölder. Further, some simple concrete examples are examined.
In this paper we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem εy ′′ + ky = f(t, y), t ∈ ha,bi, k < 0, 0 < ε ≪ 1 satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions and delicate estimations.
The article deals with the numerical modelling of heat and mass transfer in the counterflow wet-cooling tower fill. Due to the complexity of this phenomenon the simplified model based on the set of four ODEs [1] was chosen. The used approach is generally applicable to the simulation of the distribution of moist air temperature. water temperature, specific humidity of air and water mass flow rate. Evaluation of the distribution of heat and mass sources is also done. Boundary condition for outlet water temperature are based on experimentally obtained Merkel number correlation. Numerical solution of chosen model was performed using Dormand-Prince method combined with shooting method. Results are compared with data available in the literature. and Obsahuje seznam literatury
We consider the boundary value problem involving the one dimensional pLaplacian, and establish the precise intervals of the parameter for the existence and nonexistence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.
In this paper, using a fixed point theorem on a convex cone, we consider the existence of positive solutions to the multipoint one-dimensional $p$-Laplacian boundary value problem with impulsive effects, and obtain multiplicity results for positive solutions.
Kolmogorov N-widths are an approximation theory concept that, for a given problem, yields information about the optimal rate of convergence attainable by any numerical method applied to that problem. We survey sharp bounds recently obtained for the N-widths of certain singularly perturbed convection-diffusion and reaction-diffusion boundary value problems.
In this article, we deal with the Boundary Value Problem (BVP) for linear ordinary differential equations, the coefficients and the boundary values of which are constant intervals. To solve this kind of interval BVP, we implement an approach that differs from commonly used ones. With this approach, the interval BVP is interpreted as a family of classical (real) BVPs. The set (bunch) of solutions of all these real BVPs we define to be the solution of the interval BVP. Therefore, the novelty of the proposed approach is that the solution is treated as a set of real functions, not as an interval-valued function, as usual. It is well-known that the existence and uniqueness of the solution is a critical issue, especially in studying BVPs. We provide an existence and uniqueness result for interval BVPs under consideration. We also present a numerical method to compute the lower and upper bounds of the solution bunch. Moreover, we express the solution by an analytical formula under certain conditions. We provide numerical examples to illustrate the effectiveness of the introduced approach and the proposed method. We also demonstrate that the approach is applicable to non-linear interval BVPs.
Consider boundary value problems for a functional differential equation ( x (n) (t) = (T +x)(t) − (T −x)(t) + f(t), t ∈ [a, b], lx = c, where T +, T − : C[a, b] → L[a, b] are positive linear operators; l: ACn−1 [a, b] → R n is a linear bounded vector-functional, f ∈ L[a, b], c ∈ ℝ n , n ≥ 2. Let the solvability set be the set of all points (T +, T −) ∈ ℝ + 2 such that for all operators T +, T − with kT ±kC→L = T ± the problems have a unique solution for every f and c. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl, A. Lomtatidze, J. Šremr: Some boundary value problems for first order scalar functional differential equations. Folia Mathematica 10, Brno, 2002.
We investigate two boundary value problems for the second order differential equation with p-Laplacian (a(t)Φp(x ′ ))′ = b(t)F(x), t ∈ I = [0, ∞), where a, b are continuous positive functions on I. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: i) x(0) = c > 0, lim t→∞ x(t) = 0; ii) x ′ (0) = d < 0, lim t→∞ x(t) = 0.