Variety Of Boundary Value Problems Research Articles

Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems

<abstract><p>The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.</p></abstract>

Residual-based error corrector operator to enhance accuracy and reliability of neural operator surrogates of nonlinear variational boundary-value problems

This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating appropriate training data, cost-accuracy trade-offs, and nontrivial hyperparameter tuning. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control. A framework based on the linear variational problem that gives the correction to the prediction furnished by neural operators is considered based on earlier work in JCP 486 (2023) 112104. The operator, called Residual-based Error Corrector Operator or simply Corrector Operator, associated with the corrector problem is analyzed further. Numerical results involving a nonlinear reaction–diffusion model in two dimensions with PCANet-type neural operators show almost two orders of increase in the accuracy of approximations when neural operators are corrected using the correction scheme. Further, topology optimization involving a nonlinear reaction–diffusion model is considered to highlight the limitations of neural operators and the efficacy of the correction scheme. Optimizers with neural operator surrogates are seen to make significant errors (as high as 80 percent). However, the errors are much lower (below 7 percent) when neural operators are corrected.

Prediction of nonlocal elasticity parameters using high-throughput molecular dynamics simulations and machine learning

Higher-order elasticity theories are able to model the mechanics at nanoscale. However, these theories have length-scale parameters that need to be evaluated either through experiments or molecular dynamics (MD) simulations. In the current literature, these length-scale parameters are assumed to be constant for a chosen material. However, this is not true, as shown in this paper through a set of high-throughput MD simulations on a variety of boundary value problems (BVP). For carbon and boron nitride nanotubes, the length-scale parameter in the modified strain gradient theory is found to vary with the absolute dimensions, type of the BVP, and the deformation level. This complex dependence makes prediction of the length-scale parameter l0 challenging. To address this issue, a supervised machine learning (ML) based framework is developed here. In this new framework, MD, continuum formulation, and ML are used in tandem to predict the length-scale parameter for a given material, dimension, and boundary condition. It is a three-step process. First, a number of MD simulations are performed for varying chirality, length, and boundary conditions. Then, the values of l0 are found using these MD output in the continuum formulation. Finally, an ML-based regression model is used to capture the variability of l0 and subsequently predict its value for any chirality, length, and boundary condition. Here three regression models – Gaussian process regression, support vector machine, and neural network – are used to make the predictions.This predictive tool eliminates need for expensive MD simulations for further continuum scale analysis. In a broader context, this novel three-step framework opens an accurate yet inexpensive door for applying non-classical continuum theories to nanoscale mechanics problems. The accuracy is imposed by the MD simulations, while the computational cost reduction is achieved by machine learning.

A universal Birkhoff pseudospectral method for solving boundary value problems

Inspired by the well-conditioned spectral collocation method of Wang et al., a new Birkhoff pseudospectral method for solving boundary value problems is proposed. More specifically, we develop new Birkhoff interpolants that jointly satisfy boundary conditions on both the derivatives and values of the dependent variable. This key design criterion obviates the need to generate different Birkhoff matrices for different boundary conditions while maintaining the O(1) condition number with respect to N, where, N2 is the size of the Birkhoff matrix. In addressing the resulting theoretical and numerical problems, we generate computationally efficient and stable formulas for producing the new Birkhoff matrices. Furthermore, by transforming a generic mth order differential equation to its canonical form, we show that the same first-order Birkhoff matrix can be repeatedly used to solve a wide variety of boundary value problems. Illustrative numerical examples demonstrate the generality, simplicity and utility of the new approach.

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations.

On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

We investigate the possibility to determine the divergence-free displacement uindependently from the pressure reaction p for a class of boundary-value problems in incompressible linear elasticity. If not possible, we investigate if it is possible to determine it pressure robustly, i.e. pollution free from the pressure reaction.For convex domains there is but one variational boundary value problem among the investigated that allows the independent determination. It is the one with essential no-penetration conditions combined with homogeneous tangential traction conditions.Further, for the investigated cases, the weakly divergence-free displacement can be computed pressure robustly provided the total body force is decomposed into its direct sum of divergence- and rotation-free components using the Helmholtz decomposition. The elasticity problem is solved using these components as separate right-hand sides. The total solution is obtained using the superposition principle.We employ a (u,p) higher-order finite element formulation with discontinuous pressure elements. It is inf–sup stable for polynomial degree p≥2 but not pressure robust by itself. We propose a three step procedure to solve the elasticity problem which includes by the Helmholtz decomposition of the total body force. The extra cost for the three-step procedure is essentially the cost for the Helmholtz decomposition of the assembled total body force, and the small cost of solving the elasticity problem with one extra right-hand side. The results are corroborated by theoretical derivations as well as numerical results.

Приближенное представление дилогарифмами решения одной вариационной краевой задачи для круга при граничном условии Неймана

It is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of the Dirichlet integral in the unit disc centered at the origin on some admissible set of functions for given values of the normal derivative on the circle is equivalent to the Neimann boundary value problem for the Laplace equation in this domain. An effective approximate dilogarithm representation of the solution of the above equivalent variational boundary value problem is constructed on the basis of the known exact solution of the Neumann Boundary value problem for a circle using a special approximate formula for the Dini integral. The approximate formula is effective in the sense that it is quite simple in numerical implementation, stable, and the error estimation, which is uniform over a circle, allows calculations with the given accuracy. A special quadrature formula for the Dini integral has a remarkable property – its coefficients are non-negative. Quadrature formulas with non-negative coefficients occupy a special place in the theory of approximate calculations of definite integrals and its applications. Naturally, this property becomes even more significant when the coefficient are not number, but some functions. The performed numerical analysis of the approximate solution confirms its effectiveness.

Poroelastic model parameter identification using artificial neural networks: on the effects of heterogeneous porosity and solid matrix Poisson ratio

Predictive analysis of poroelastic materials typically require expensive and time-consuming multiscale and multiphysics approaches, which demand either several simplifications or costly experimental tests for model parameter identification.This problem motivates us to develop a more efficient approach to address complex problems with an acceptable computational cost. In particular, we employ artificial neural network (ANN) for reliable and fast computation of poroelastic model parameters. Based on the strong-form governing equations for the poroelastic problem derived from asymptotic homogenisation, the weighted residuals formulation of the cell problem is obtained. Approximate solution of the resulting linear variational boundary value problem is achieved by means of the finite element method. The advantages and downsides of macroscale properties identification via asymptotic homogenisation and the application of ANN to overcome parameter characterisation challenges caused by the costly solution of cell problems are presented. Numerical examples, in this study, include spatially dependent porosity and solid matrix Poisson ratio for a generic model problem, application in tumour modelling, and utilisation in soil mechanics context which demonstrate the feasibility of the presented framework.

Fatou-type theorems and boundary value problems for elliptic systems in the upper half-space

We survey recent progress in a program aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space ${\mathbb{R}}^n_{+}$ for second-order, homogeneous, constant complex coefficient, elliptic systems $L$, formulated in a manner that emphasizes pointwise nontangential boundary traces of the null-solutions of $L$ in ${\mathbb{R}}^n_{+}$.

A high-order wideband direct solver for electromagnetic scattering from bodies of revolution

The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as ω→0 in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nyström discretization of a one-dimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples.

Constructing solutions to two-way diffusion problems

A variety of boundary value problems in linear transport theory are expressed as a diffusion equation of the two-way, or forward–backward, type. In such problems boundary data are specified only on part of the boundary, which introduces several technical challenges. Existence and uniqueness theorems have been established in the literature under various assumptions; however, calculating solutions in practice has proven difficult. Here we present one possible means of practical calculation. By formulating the problem in terms of projection operators, we derive a formal sum for the solution whose terms are readily calculated. We demonstrate the validity of this approach for a variety of physical problems, with focus on a periodic problem from the field of active matter.

Three-dimensional lower bound finite element limit analysis of Hoek-Brown material using semidefinite programming

A reliable and accurate stability analysis in Hoek-Brown rock masses under fully three-dimensional geometries requires an efficient and powerful computational method. This paper presents a computational implementation of a three-dimensional lower bound finite element limit analysis using semidefinite programming for Hoek-Brown rock masses. Its computational performance is investigated extensively through a variety of boundary value problems in rock mechanics, namely triaxial compression, bearing capacity and cavity contraction. New lower bound solutions are numerically derived, including bearing capacity of square footings, uniaxial compression of cube and cylindrical rock specimens with fully rough platens and tension test on double notched rock specimens.

Discrete least-squares finite element methods

A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz map on the test space. The resulting linear system is overdetermined. Two different approaches for solving the system are suggested (although others are discussed): (1) solving the associated normal equation with linear solvers for symmetric positive-definite systems (e.g. Cholesky factorization); and (2) solving the overdetermined system with orthogonalization algorithms (e.g. QR factorization). The finite element assembly algorithm for each of these approaches is described in detail. The normal equation approach is usually faster for direct solvers and requires less storage. The second approach reduces the condition number of the system by a power of two and is less sensitive to round-off error. The rectangular stiffness matrix of second approach is demonstrated to have condition number O(h−1) for a variety of formulations of Poisson’s equation. The stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least-squares finite element methods and it is proved that the two are identical in some cases. An example with Poisson’s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable (if round-off error is not an issue) and rapidly converge to each other. The orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems. Experiments with Poisson’s equation in single-precision arithmetic as well as the linear acoustics problem near resonance in double-precision arithmetic verify this conclusion. The methodology described here is an outgrowth of the discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan (2015). The strength of DPG is highlighted throughout, however, the majority of theory presented is more general. Extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments.

Invertibility Properties of Singular Integral Operators Associated with the Lamé and Stokes Systems on Infinite Sectors in Two Dimensions

In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on \(L^p(\partial \Omega )\), \(1<p<\infty \), whenever \(\Omega \) is an infinite sector in \({\mathbb {R}}^2\). This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lame system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector \(\Omega \) and the differential operator in question.

Existence of solutions to a semilinear elliptic boundary value problem with augmented Morse index bigger than two

Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in \cite{ccn}, we prove the existence of a solution with {\it augmented Morse index} at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.

A new coercivity condition applied to semipositone integral equations with nonpositive, unbounded nonlinearities and applications to nonlocal BVPs

We consider the perturbed Hammerstein integral equation $$\begin{aligned} y(t)=\gamma _1(t)H_1(\varphi _1(y))+\gamma _2(t)H_2(\varphi _2(y))+\lambda \int _0^1G(t,s)f(s,y(s)) \mathrm{d}s \end{aligned}$$ and demonstrate the existence of at least one positive solution in the case where this problem is semipostione—i.e., f is allowed to be negative on its domain. As applications of this abstract result, we demonstrate existence of at least one positive solution to a variety of boundary value problems in the ordinary differential equations setting as well as the setting of elliptic partial differential equations on annuli. Finally, two novel aspects of this study are that, first, our results allow for f to be strictly negative on its entire domain, and second, it can actually be the case that \(\lim _{y\rightarrow +\infty }f(t,y)=-\infty \), uniformly for \(t\in [0,1]\). In addition, these results can hold even if \(H_1\) and \(H_2\) are piecewise linear on their domains.

Solutions of the Wheeler-Feynman equations with discontinuous velocities.

We generalize Wheeler-Feynman electrodynamics with a variational boundary value problem for continuous boundary segments that might include velocity discontinuity points. Critical-point orbits must satisfy the Euler-Lagrange equations of the action functional at most points, which are neutral differential delay equations (the Wheeler-Feynman equations of motion). At velocity discontinuity points, critical-point orbits must satisfy the Weierstrass-Erdmann continuity conditions for the partial momenta and the partial energies. We study a special setup having the shortest time-separation between the (infinite-dimensional) boundary segments, for which case the critical-point orbit can be found using a two-point boundary problem for an ordinary differential equation. For this simplest setup, we prove that orbits can have discontinuous velocities. We construct a numerical method to solve the Wheeler-Feynman equations together with the Weierstrass-Erdmann conditions and calculate some numerical orbits with discontinuous velocities. We also prove that the variational boundary value problem has a unique solution depending continuously on boundary data, if the continuous boundary segments have velocity discontinuities along a reduced local space.

Variational inequalities in Musielak-Orlicz-Sobolev spaces

In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces. Some results concerning the Trace mapping have also been provided, as well as existence results for some strongly nonlinear elliptic equations in Musielak-Orlicz spaces.

Normally solvable nonlinear boundary value problems

We investigate nonlinear problems which appear as Euler–Lagrange equations for a variational problem. They include in particular variational boundary value problems for nonlinear elliptic equations studied by F. Browder in the 1960s. We establish a solvability criterion of such problems and elaborate an efficient orthogonal projection method for constructing approximate solutions.

An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces

In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces $W^mL_\varphi(\Omega)$, under the assumption that the conjugate function of $\varphi$ satisfies the $\Delta_2$ condition. An imbedding theorem has also been provided without assuming this condition.