Solution Of Boundary Value Problem Research Articles

Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Spatio-temporal physics-informed neural networks to solve boundary value problems for classical and gradient-enhanced continua

Recent advances have prominently highlighted physics informed neural networks (PINNs) as an efficient methodology for solving partial differential equations (PDEs). The present paper proposes a proof of concept exploring the use of PINNs as an alternative to finite element (FE) solvers in both classical and gradient-enhanced solid mechanics. To this end, spatio-temporal PINNs are designed to represent continuous solutions of boundary value problems within spatio-temporal space. These PINNs directly incorporate the equilibrium and constitutive equations in their differential and rate forms, bypassing the requirement for incremental implementation. This simplifies application of PINNs to solve complex mechanical problems, particularly those involved in the context of gradient-enhanced continua. Moreover, traditional meshing is no longer required as it is replaced by a point cloud, making it possible to overcome meshing drawbacks. The results of this investigation prove the effectiveness of the proposed methodology, especially with regards to non-monotonic loading conditions and irreversible plastic deformation. Compared to classical FE approaches, the proposed spatio-temporal PINNs are more readily applied to complex problems, which are tackled in their raw form. This is especially true for gradient-enhanced continuum problems, where there is no need to introduce additional degrees of freedom as in classical FE approaches. However, PINNs training generally requires more computation time, a challenge that can be mitigated by employing the concept of transfer learning as shown in this paper. This concept, which is very useful when performing parametric studies, involves applying knowledge grained from solving one problem to another different but related one. The use of PINNs as mechanical solvers is shown to be highly promising in the forthcoming era, where advancements in GPU technology can further enhance their performance in terms of computation time.

Natural Vibrations and Stability of Composite Cylindrical Shells Containing a Quiescent Fluid

This paper presents the results of investigation of the natural vibrations and stability of circular vertical multilayered cylindrical shells, fully or partially filled with a quiescent compressible fluid subjected to hydrostatic and external static loads. The behavior of the elastic structure and fluid medium is described based on the classical shell theory and Euler’s equations. The linearized equations of motion of the shell, together with the corresponding geometrical and physical relations are reduced to a system of ordinary differential equations with respect to new unknowns. The acoustic wave equation is transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is developed using Godunov’s orthogonal sweep method. The dependences of the lowest vibration frequencies and critical external pressure on the ply angle and the filling level of two-layer and three-layer cylindrical shells are analyzed in detail. It is demonstrated that, in contrast to the ply angle and a given combination of boundary conditions, the lay-up scheme of composite materials plays different parts in the problems of maximizing the fundamental frequency of vibrations and extending the stability boundaries.

Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions

Abstract In this paper, we study the existence of positive solutions for a fourth order boundary value problem coupled to functional perturbed clamped beam boundary conditions. Our main ingredient is the classical fixed point index. The problem investigated is an extension of other problems studied in previous papers by covering very general nonlocal perturbed conditions on the boundary.

Wave propagation in the Kolmogorov–Petrovskii–Piskunov–Fischer equation with time delay and spatial deviation

For the logistic equation with diffusion two generalizations are considered: one with time delay and the other with deviation in the spatial variable. For these equations, the problem of density wave propagation is considered. These equations, known as Kolmogorov–Petrovsky–Piskunov–Fisher equations with delay or deviation (KPPF with delay or deviation), are investigated using both asymptotic and numerical methods.To investigate the qualitative behavior of solutions of such equations, the wave profile equation is considered and the conditions for the emergence of its oscillatory modes are identified.The local properties of solutions with periodic boundary conditions are examines, revealing that an increase in the period leads to the emergence of stable solutions with a more complex spatial structures.Numerical analysis of the wave propagation process indicates that at sufficiently small delay values, this equation has solutions that are close to those of the standard KPPF equation. Increasing the delay parameter leads first to the appearance of a damped oscillatory component in the spatial distribution of the solution. With further increases of this parameter, a complex spatial profile of the wave appears.In the final part of the chapter, which focuses on numerical calculation, the numerical analysis of the wave propagation process from two initial perturbations is performed. This analysis tracks the process of wave interaction. The complex spatially inhomogeneous structure that arises during wave propagation and interaction can be explained by the properties of the corresponding solutions of the periodic boundary value problem, especially as the interval of change in the spatial variable increases.

The Existence and Uniqueness Results of Solutions for a Fractional Hybrid Integro-differential System

This paper discuss two important results for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the researches and the advance in this field and also the importance of this subject in the modeling of nonlinear real phenomena corresponding to a great variety of events gives the motivation to study this boundary value problem. The results are as follow, the first result consider the existence and uniqueness results of solutions for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential system this result based on Krasnoslskii fixed point theorem for a sum of two operators, the second result is the uniqueness of solution for fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the main result is based on Banach fixed point theorem, both results comes after transforming the system into Volterra integral system then transform again into operator system, then using fixed point theory to prove the results, this articule was ended buy an example to well illustrat the results and ideas of proof.

Solvability of a boundary value problem with integrable conditions for partial differential equations of fractional order

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works; Especially, The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, we only mention here the apriori estimate method (energy inequality method). The current piece of work concentrates on exploring the existence and uniqueness of a solution for a non-linear boundary value problem that has integrable conditions in the case of fractional partial differential equations. For this we split the proof into two sections: linear and non-linear problem; for the associated linear problem, we derive the a priori bound and demonstrate the density of the operator generated by the problem posed; we solve the non-linear problem by introducing a iterative process. The results show the efficiency of energy inequality method in the case of time fractional order differential equations with integrable conditions our results illustrate the existence and uniqueness of the continuous dependence of solution on fractional order. The work of the authors could be considered as a contribution to the development of the functional analysis method. And to conclude we argue that the research carried out in this article could serve as a contribution for possible studies in the field of applied mathematics.

On Estimates of Solutions of Boundary-Value Problems for Implicit Differential Equations with Deviating Argument

On Estimates of Solutions of Boundary-Value Problems for Implicit Differential Equations with Deviating Argument

Exploring existence, uniqueness, and stability in nonlinear fractional boundary value problems with three-point boundary conditions

This study investigates the analysis of the existence, uniqueness, and stability of solutions for a Ψ-Caputo three-point nonlinear fractional boundary value problem using the Banach contraction principle and Sadovskii’s fixed point theorem. We demonstrate the practical implications of our analytical advancements for each situation, illustrating how the components of the fractional boundary value problem emerge in real-life occurrences. Our work significantly enhances the field of applied mathematics by offering analytical solutions and valuable insights.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Existence of multiple unbounded solutions for a three-point boundary value problems on an infinite time scales

Abstract In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Error Identities for Parabolic Equations with Monotone Spatial Operators

Abstract The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper.

A unified discontinuous Galerkin formulation for interfacial multiphysics modeling of thermo-chemically driven fracture

Many engineering and natural materials exhibit coupled thermo-chemo-mechanical phenomena, which can result in embrittlement and fracture. These fractures, in turn, can alter the subsequent thermal, chemical, and mechanical response. We present a theoretical formulation and computational framework for the analysis of thermo-chemically fractured solids, with emphasis on the post-fracture thermal and chemical interfacial behavior. The theoretical model is based on the thermodynamically-consistent formulation of Loeffel and Anand (IJP, 2011). The computational method extends the scalable discontinuous Galerkin/Cohesive Zone Model (DG/CZM) of Radovitzky et al. (CMAME, 2011) to thermo-chemo-mechanics, which facilitates coupled, large-scale simulations of materials and structures containing failed interfaces. In the proposed framework, all balance laws are enforced weakly via the DG formalism, resulting in a unified formulation for multiphysics problems in solids. This naturally enables the incorporation of general interface models, e.g. to account for effects such as the aeolotropic reduction in thermochemical transport due to the presence of fractures, or the acceleration of chemical reactions along crack flanks. The approach is verified against two analytical solutions of boundary value problems drawn from thermo-poro-elasticity and thermally-driven delamination. A scalable, three-dimensional simulation of thermochemically-driven concrete cracking illustrates the complete capabilities of the interfacial multiphysics modeling framework.

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients

Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients

Nonintrusive thermal contact conductance estimation in double-layered pipelines: A reciprocity functional method perspective

Thermal contact conductance (TCC) is a critical factor in various engineering applications involving heat transfer between solid surfaces. Accurate knowledge of TCC enables a more precise thermal system analysis, leading to improved efficiency, reduced energy consumption, and prevention of thermal damages. Moreover, TCC estimation can help to identify material discontinuities or failures. A recent tool for TCC estimation is the Reciprocity Functional Method (RFM) coupled with the Classical Integral Transform Technique (CITT). This method allows the solution of inverse boundary value problems without the need of iterative methods or intrusive measurements, making it valuable for nondestructive testing of interfacial flaws between solid materials. Furthermore, it enables the computation of TCCs using analytical expressions, resulting in a computationally fast procedure. The technique involves the solution of two auxiliary problems: one for the temperature discontinuity estimation, and the other for the interfacial heat flux estimation, both at the inaccessible interface between the two contacting materials. In this study, we applied the RFM and the CITT to estimate different types of TCCs, using a three-dimensional double-layer hollow cylinder geometry as a representative model, like double-layer pipelines used in the oil industry. Temperature measurements for solving the inverse problem were considered available on the outer surface of the pipe which, in the case of a real experiment, could be obtained via an infrared camera. The results demonstrated a good estimate for various analyzed test-cases, even when the measurements were contaminated with Gaussian noises.

Stability analysis and solutions of fractional boundary value problem on the cyclopentasilane graph

The study is being applied to a model involving silane and on cyclopentasilane graph. We consider a graph with labeled vertices by 0 or 1 inspired by the molecular structure of cyclopentasilane. In this paper, we first study the existence of solutions to fractional conformable boundary value problem on the cyclopentasilane graph by applying Scheafer and Krasnoselskii fixed point theorems. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stable, generalized Ulam-Hyers stable, Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable for the given problem. Finally, we give an example to support our important results.

Conservation Laws and Solutions of the First Boundary Value Problem for the Equations of Two- and Three-Dimensional Elasticity

Conservation Laws and Solutions of the First Boundary Value Problem for the Equations of Two- and Three-Dimensional Elasticity

Existence Results for the System of Fractional-Order Sequential Integrodifferential Equations via Liouville–Caputo Sense

We investigate the conditions for the existence and uniqueness of solutions in a nonlinear system of sequential fractional differential equations using the Liouville–Caputo type with varying orders. This system is enriched by nonlocal coupled integral boundary conditions. The desired outcomes are attained by employing traditional fixed-point theorems. It is essential to emphasize that the fixed-point approach proves to be an effective method for establishing the existence of solutions in boundary value problems. Furthermore, we provide constructed examples to illustrate the obtained results.