About positivity of green's functions for nonlocal boundary value problems with impulsive delay equations.
Ontology highlight
ABSTRACT:
SUBMITTER: Domoshnitsky A
PROVIDER: S-EPMC3956640 | biostudies-other | 2014
REPOSITORIES: biostudies-other
Similar Datasets
Project description:This paper presents an analysis technique of high-order contact potential problems and its application to an elastic settlement analysis of a shallow foundation system subjected to a combined traction boundary condition. Closed-form solutions of potential functions are derived for an elastic half-space subjected to bilinear tangential traction boundary conditions over rectangular surface regions. Using the principle of superposition, the present solutions provide a means to form an approximate and continuous solution of elastic contact problems with higher-order tangential boundary conditions. As an application example, an elastic settlement analysis of a rigid footing founded on a dense granular soil is performed under a tangential traction boundary condition prescribed in an analogy with the stress equilibrium states of static sandpiles. A generalized solution approach to combined normal and tangential traction boundary value problems is discussed in the context of foundation engineering.
| S-EPMC6458418 | biostudies-literature
Project description:Boundary value problems (BVPs) play a central role in the mathematical analysis of constrained physical systems subjected to external forces. Consequently, BVPs frequently emerge in nearly every engineering discipline and span problem domains including fluid mechanics, electromagnetics, quantum mechanics, and elasticity. The fundamental solution, or Green's function, is a leading method for solving linear BVPs that enables facile computation of new solutions to systems under any external forcing. However, fundamental Green's function solutions for nonlinear BVPs are not feasible since linear superposition no longer holds. In this work, we propose a flexible deep learning approach to solve nonlinear BVPs using a dual-autoencoder architecture. The autoencoders discover an invertible coordinate transform that linearizes the nonlinear BVP and identifies both a linear operator L and Green's function G which can be used to solve new nonlinear BVPs. We find that the method succeeds on a variety of nonlinear systems including nonlinear Helmholtz and Sturm-Liouville problems, nonlinear elasticity, and a 2D nonlinear Poisson equation and can solve nonlinear BVPs at orders of magnitude faster than traditional methods without the need for an initial guess. The method merges the strengths of the universal approximation capabilities of deep learning with the physics knowledge of Green's functions to yield a flexible tool for identifying fundamental solutions to a variety of nonlinear systems.
| S-EPMC8566504 | biostudies-literature
Project description:By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by -D(0+)(ν1)y1(t) = λ1a1(t)f(y1(t), y2(t)), - D(0+)(ν2)y2(t) = λ2a2(t)g(y1(t), y2(t)), where D(0+)(ν) is the standard Riemann-Liouville fractional derivative, ν1, ν2 ∈ (n - 1, n] for n > 3 and n ∈ N, subject to the boundary conditions y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = 0 = [D(0+ (α)y2(t)] t=1, for 1 ≤ α ≤ n - 2, or y1((i))(0) = 0 = y ((i))(0), for 0 ≤ i ≤ n - 2, and [D(0+)(α)y1(t)] t=1 = ϕ1(y1), [D(0+)(α)y2(t)] t=1 = ϕ2(y2), for 1 ≤ α ≤ n - 2, ϕ1, ϕ2 ∈ C([0,1], R). Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.
| S-EPMC3921945 | biostudies-other
Project description:This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution (quasi-solution) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.
| S-EPMC7069490 | biostudies-literature
Project description:We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
| S-EPMC5582177 | biostudies-literature
Project description:We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order α (0 < α ≤ 1) involving the two-point and integral boundary conditions. Some new results on existence and uniqueness of a solution are established by using fixed point theorems. Some illustrative examples are also presented. We extend previous results even in the integer case α = 1.
| S-EPMC3980930 | biostudies-other
Project description:We unify the concepts of G-metric, metric-like, and b-metric to define new notion of generalized b-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes of G-α -admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.
| S-EPMC4033538 | biostudies-other
Project description:Gielis curves and surfaces can describe a wide range of natural shapes and they have been used in various studies in biology and physics as descriptive tool. This has stimulated the generalization of widely used computational methods. Here we show that proper normalization of the Levenberg-Marquardt algorithm allows for efficient and robust reconstruction of Gielis curves, including self-intersecting and asymmetric curves, without increasing the overall complexity of the algorithm. Then, we show how complex curves of k-type can be constructed and how solutions to the Dirichlet problem for the Laplace equation on these complex domains can be derived using a semi-Fourier method. In all three methods, descriptive and computational power and efficiency is obtained in a surprisingly simple way.
| S-EPMC3459917 | biostudies-literature
Project description:Efficient solution of physical boundary value problems (BVPs) remains a challenging task demanded in many applications. Conventional numerical methods require time-consuming domain discretization and solving techniques that have limited throughput capabilities. Here, we present an efficient data-driven DNN approach to non-iterative solving arbitrary 2D linear elastic BVPs. Our results show that a U-Net-based surrogate model trained on a representative set of reference FDM solutions can accurately emulate linear elastic material behavior with manifold applications in deformable modeling and simulation.
| S-EPMC11002675 | biostudies-literature
Project description:Impulsive delayed reward discounting (DRD) has been linked to nicotine dependence, but with some inconsistency. This may be related to the considerable variability in the literature with regard to the DRD assessments used, particularly in the case of the reward magnitudes assessed. In addition, previous studies have often not considered concurrent substance use when examining the relationship between DRD and nicotine dependence. The current study sought to further clarify the relationship between DRD and nicotine dependence by characterizing DRD across diverse reward magnitudes and incorporating other substance use. Daily smokers (N = 933) were assessed for DRD preferences across nine reward magnitudes (delayed reward range: $2.50-$850), comorbid substance use, and relevant demographic variables (age, education, income). A significant large effect size magnitude effect was found for DRD, reflecting steeper discounting for smaller delayed rewards, but significant correlations across magnitudes also suggested similar relative levels of discounting. Principal components analysis (PCA) was used to generate a single latent index of discounting across all magnitudes that accounted for 69% of the total variance. In correlation and regression analyses, steeper composite DRD was significantly associated with nicotine dependence severity. This relationship remained statistically significant after incorporating demographic variables and alcohol and illicit drug use. These findings provide evidence of a specific link between impulsive DRD and nicotine dependence and reveal that this association is robust across a broad range of monetary rewards. The study also demonstrates the utility of using PCA to generate latent indices of delay discounting across multiple magnitudes of delayed reward.
| S-EPMC4165767 | biostudies-literature