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This paper presents unified solutions for elastic–plastic expansion analysis of a cylindrical or spherical cavity in an infinite medium, adopting a flow theory of strain gradient plasticity. Previous cavity expansion analyses incorporating strain gradient effects have mostly focused on explaining the strain localization phenomenon and/or size effects during infinitesimal expansions. This paper is however concerned with the size-dependent behaviour of a cavity during finite quasi-static expansions. To account for the non-local influence of underlying microstructures to the macroscopic behaviour of granular materials, the conventional Mohr–Coulomb yield criterion is modified by including a second-order strain gradient. Thus the quasi-static cavity expansion problem is converted into a second-order ordinary differential equation system. In the continuous cavity expansion analysis, the resulting governing equations are solved numerically with Cauchy boundary conditions by simple iterations. Furthermore, a simplified method without iterations is proposed for calculating the size-dependent limit pressure of a cavity expanding to a given final radius. By neglecting the elastic strain increments in the plastic zone, approximate analytical size-dependent solutions are also derived. It is shown that the strain gradient effect mainly concentrates in a close vicinity of the inner cavity. Evident size-strengthening effects associated with the sand particle size and the cavity radius in the localized deformation zone is captured by the newly developed solutions presented in this paper. The strain gradient effect will vanish when the intrinsic material length is negligible compared to the instantaneous cavity size, and then the conventional elastic perfectly-plastic solutions can be recovered exactly. The present solutions can provide a theoretical method for modeling the size effect that is often observed in small-sized sand-structure interaction problems.

International audience; This paper investigates the boundary feedback control for a class of semi-linear hyperbolic partial differential equations with nonlinear relaxation, which is local Lipschitz continuous with a stable matrix structure. A sufficient condition in terms of linear inequalities is developed for the existence of global Cauchy solutions and the exponential stability by seeking a balance between the relaxation term and the boundary condition. These results are illustrated with an application to the boundary feedback control for a class of hyperbolic Lotka-Volterra models.

The diffusion equation for neutrons, or other neutral particles, is important in nuclear engineering and radiological sciences. In this chapter we present how to solve source-driven diffusion problems in one-dimensional geometries: slabs, cylinders, and spheres. We begin by presenting how to cast a time-dependent problem in terms of the solution of a series of steady-state problems using the backward Euler method. To solve steady-state problems we impose a mesh of cells on the problem domain and integrate the diffusion equation over each cell. Generic boundary conditions of the Dirichlet, Marshak, albedo, and reflecting type are allowed, and the harmonic mean diffusion coefficient is used for heterogeneous problems. Python is used to solve the resulting linear system of equations. Test problems and numerical demonstrations are included.

2-s2.0-85073881117, In this paper, we investigate the existence of positive solutions for the nonlinear m-point boundary value problems with integral boundary condition. By using fixed-point index theorem and Leggett- Williams fixed point theorem, the existence and multiplicity of positive solutions are obtained. As an application, two examples are given to demonstrate our results. © 2018 Hacettepe University. All rights reserved.

Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary int egral equation method. The ideal situation is when the right-hand side in the partial differential equation and th e initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficien ts, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the addition al time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the elliptici ty of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practi ce, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

We study the Cauchy problem for nonlinear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak a priori assumption on the exact solution, we propose a new regularization method for stabilising the ill-posed problem. These new results extend some earlier works on Cauchy problems for nonlinear elliptic equations. Numerical results are presented and discussed.

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.

This article deals with artificial boundaries which you encounter when a large spatial domain is confined to a smaller computational domain. Such an artificial boundary condition should not preferably interact with the fluid at all. Standard boundary conditions, e.g., a pressure or velocity condition, result in unphysical reflections. So far, existing artificial boundary conditions for the lattice Boltzmann method (LBM) are transferred from macroscopic formulations.In this work we propose novel discrete artificial boundary conditions (ABCs) which are tailored on the LBM's mesoscopic level. They are derived directly for the chosen LBM with the aim of higher accuracy. We describe the idea of discrete ABCs in a three velocity (D1Q3) model governing the Navier-Stokes equations in one dimension. Numerical results finally demonstrate the superiority of our new boundary condition in terms of accuracy compared to previously used ABCs.

The main objective of this paper is the derivation and validation of a free surface Neumann boundary condition for the advection-diffusion lattice Boltzmann method. Most literature boundary conditions are applied on straight walls and sometimes on curved geometries or fixed free surfaces, but dynamic free surfaces, especially with fluid motion in normal direction, are hardly addressed. A Chapman-Enskog Expansion is the basis for the derivation of the advection-diffusion equation using the advection-diffusion lattice Boltzmann method and the BGK collision operator. For this numerical scheme, a free surface Neumann boundary condition with no flux in normal direction to the free surface is derived. Finally, the boundary condition is validated in different static and dynamic test scenarios, including a detailed view on the conservation of the diffusive scalar, the normal and tangential flux components to the free surface and the accuracy. The validation scenarios reveal the superiority of the new approach to the compared literature schemes, especially for arbitrary fluid motion.

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$ and the lower blow-up estimates of the solutions.

In this paper we discuss the local solvability of Cauchy problem for quasi-linear partial differential equations of first order. By using the classical method of characteristics we describe the non-uniqueness or the degree of freedom for solutions and also decide the conditions for the existence and the uniqueness of solutions for overdetermined systems of quasi-linear PDEs of first order.

We propose a new numerical method for the solution of Bernoulli's free boundary value problem for harmonic functions in a doubly connected domain $D$ in $\real^2$ where an unknown free boundary $\Gamma_0$ is determined by prescribed Cauchy data on $\Gamma_0$ in addition to a Dirichlet condition on the known boundary $\Gamma_1$. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar and Kress~\cite{AkKr,HaKr05} for the solution of a related inverse boundary value problem. For this we interpret the free boundary $\Gamma_0$ as the unknown boundary in the inverse problem to construct $\Gamma_0$ from the Dirichlet condition on $\Gamma_0$ and Cauchy data on the known boundary $\Gamma_1$. Our method for the Bernoulli problem iterates on the missing normal derivative on $\Gamma_1$ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet--Neumann boundary value problem in $D$. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach.

In this paper analytic boundary value problems for some classical domains in ℂ n are developed by using the harmonic analysis due to L.K. Hua. First it is discussed for the version of one variable in order to induce the relation between the analytic boundary value problem and the decomposition of function space L2 on the boundary manifold. Then an easy example of several variables, the version of torus in ℂ 2 , is stated. For the noncommutative classical group LI, the characteristic boundary of a kind of bounded symmetric domain in ℂ 4 , the boundary behaviors of the Cauchy integral are obtained by using both the harmonic expansion and polar coordinate transformation. At last we obtain the conditions of solvability of Schwarz problem on LI, if so, the solution is given explicitly.

In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$. Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.

Summary This paper introduces a geometric solution strategy for Laplace problems. Our main interest and emphasis is on efficient solution of the inverse problem with a boundary with Cauchy condition and with a free boundary. This type of problem is known to be sensitive to small errors. We start from the standard Laplace problem and establish the geometric solution strategy on the idea of deforming equipotential layers continuously along the field lines from one layer to another. This results in exploiting ordinary differential equations to solve any boundary value problem that belongs to the class of Laplace's problem. Interpretation in terms of a geometric flow will provide us with stability considerations. The approach is demonstrated with several examples. Copyright © 2015 John Wiley & Sons, Ltd.

We study a boundary value problem for a fractional partial differential equation of order ≤ 1 in a domain with curvilinear boundary.

This work is devoted to the solvability of some non classical boundary value problems for the polyharmonic equation. On the boundary we consider operators of fractional order in the Riemann–Liouville and Hadamard sense. The considered problems generalize the Dirichlet and the Neumann problems with boundary operators of a fractional order.

In this paper we are concerned with the regularity of solutions to the Navier-Stokes equations with the condition on the pressure on parts of the boundary where there is flow. For the steady Stokes problem a result similar to L-q-theory for the one with Dirichlet boundary condition is obtained. Using the result, for the steady Navier-Stokes equations we obtain regularity as the case of Dirichlet boundary conditions. Furthermore, for the time-dependent 2-D Navier-Stokes equations we prove uniqueness and existence of regular solutions, which is similar to J.M.Bernard's results([6]) for the time-dependent 2-D Stokes equations.

The inverse Cauchy problem for a nonlinear elliptic equation defined in an arbitrary doubly-connected plane domain is solved numerically. When the overspecified boundary data are imposed on the outer boundary, we seek the unknown data on the inner boundary, by using a mixed group preserving scheme (MGPS) to integrate the nonlinear inverse Cauchy problem as an initial value problem. We also reverse the order of the above inverse Cauchy problem by giving overspecified boundary data on the inner boundary and seeking the unknown data on the outer boundary. Several numerical examples are examined to show that the MGPS can overcome the highly ill-posed behavior of nonlinear inverse Cauchy problem defined in arbitrary doubly-connected plane domain. The proposed algorithm is robust against large noise and very time saving without needing of any iteration.

A boundary singular integral equation of the plane problem was constructed using an approach based on the representation of the unknown Lekhnitskii complex potentials in the form of Cauchy type integrals with unknown densities on the boundary of the region occupied by the body. The contours of the holes and cuts and the shape of the outer boundary are exactly or approximately represented in the form of a sequence of straight and curved (in the form of elliptical arcs) boundary elements. The unknown densities on the boundary elements are approximated by a linear combination of some regular functions or complex functions that have a known singularity. In the numerical solution of the integral equation by the collocation method or by the least-squares method and in the subsequent calculations of the stress–strain state, the integrals of all types along the boundary elements are calculated analytically, which significantly increases the accuracy of the results.

The representation by the Mittag-Leffler function of the solution to the Cauchy problem for the evolution equation solved for the higher derivative is used in the study of degenerate linear and quasilinear evolution equations under some special constraints on the nonlinear part of the equation. The solvability conditions for the Cauchy problem are simplified in the situation when the generalized Showalter–Sidorov condition is used as the initial condition. These results are applied to studying an initial boundary value problem for the motion equation of the Kelvin–Voigt fluid.

The existence and uniqueness of solutions to linear parabolic equations with nonlinear flux on the boundary driven by Gaussian boundary noise is studied. Both the case of heat equation with boundary conditions of Wentzell type and the case of white-noise boundary conditions are considered.

Hybrid computational aero-acoustic (CAA) solution schemes rely on knowledge of a scattering function known as a Green’s function to propagate source fluctuations to the far-field. Presently, these schemes are restricted to relatively simple geometries. We present here a computational method for evaluating Green’s functions within more geometrically complex regions, as a means of extending the versatility of existing hybrid schemes. The direct collocation implementation of the Boundary Element Method used in truncated, semi-infinite domains, introduces additional unknowns on the boundary. In this paper we develop a modified boundary element formulation to efficiently incorporate approximate non-reflecting boundary conditions for an arbitrary number of truncation boundaries. The boundary condition is based on the Dirichlet-to-Neumann mapping operator. Results are compared to known analytical Green’s functions for an infinite pipe as a means of validating the new code. The method achieves relative errors of less than 1% compared with the analytical solution for the highest mesh density tested. Execution time, known to be large for acoustic problems, is minimised through the use of multi-threading. The scheme is also applied to a more realistic example of a 2D throttle.

This is the author's PDF version of an article published in ESAIM: Mathematical modelling and numerical analysis© 2015. The definitive version is available at http://www.esaim-m2an.org/

We consider Dirichlet boundary value problem for systems of two second-order differential equations with nonlinear continuous and bounded functions in right-hand sides. We prove the existence of a nontrivial solution to the problem comparing behaviors of solutions of auxiliary Cauchy problems at zero solution and at infinity.

The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived.

We obtain exact sufficient conditions on the lower order coefficient of a parabolic equation guaranteeing the power rate of the uniform stabilization of the solution to the Cauchy problem on every compact set K in RN and for any bounded initial function. Bibliography: 9 titles.

We consider a kind of scattering problem by a crack Γ that is buried in a bounded domain D , and we put a point source inside the domain D . This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Γ. Both sides of the crack Γ are given Dirichlet-impedance boundary conditions, and different boundary condition (Dirichlet, Neumann or Impedance boundary condition) is set on the boundary of D . Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.

The present study introduces a novel numerical approach for solving dispersion dominated problems with Cauchy boundary condition in an Eulerian-Lagrangian scheme. The study reveals the incapability of traditional Neuman approach to address the dispersion dominated problems with Cauchy boundary condition, even though it can produce reliable solution in the advection dominated regime. Also, the proposed numerical approach is applied to a real field problem of radioactive contaminant migration from radioactive waste repository which is a major current waste management issue. The performance of the proposed numerical approach is evaluated by comparing the results with numerical solutions of traditional FDM (Finite Difference Method), Neuman approach, and the analytical solution. The results show that the proposed numerical approach yields better and reliable solution for dispersion dominated regime, specifically for Peclet Numbers of less than 0.1. The proposed numerical approach is validated by applying to a real field problem of radioactive contaminant migration from radioactive waste repository of varying Peclet Number from 0.003 to 34.5. The numerical results of Neuman approach overestimates the concentration value with an order of 100 than the proposed approach during the assessment of radioactive contaminant transport from nuclear waste repository. The overestimation of concentration value could be due to the assumption that dispersion is negligible. Also our application problem confirms the existence of real field situation with advection dominated condition and dispersion dominated condition simultaneously as well as the significance or advantage of the proposed approach in the real field problem.

In this paper, we consider a class of boundary value problems of fractional differential equations with integral and anti-periodic boundary conditions, which is a new type of mixed boundary condition. Using the contraction mapping principle, Krasnosel’skii fixed point theorem, and Leray-Schauder degree theory, we obtain some results of existence and uniqueness. Finally, several examples are provided for illustrating the applications of our theoretical analysis.

We prove the existence of unique solutions to the Dirichlet boundary value problems for linear second-order uniformly parabolic operators in either divergence or non-divergence form with boundary blowup low-order coefficients. The domain is possibly time varying, non-smooth, and satisfies the exterior measure condition., Comment: 27 pages, 1 figure

Based on the analysis of the boundary value problem of Laguerre equation, this paper studies the similar structure of its solution expression. This paper is found that its solution can be obtained by combining similar kernel function with coefficients of left boundary condition. While the similar kernel function is constructed by both the function of guide solution and coefficients of right boundary condition. Hence, we proposed a method for solving this class of boundary value problems: the similar constructing method.

We propose two approaches to the study of the boundary value problem for the stationary Stokes equations with impermeability boundary condition. The first approach is classical and is based on a Friedrichs type inequality and a variant of the de Rham theorem. The second approach is based on solving the boundary value problem with the impermeability condition for the system of Poisson equations and decomposition of a Sobolev space into the sum of solenoidal and potential subspaces. We also study the gradient-divergence boundary value problem with impermeability boundary condition and establish the corresponding Ladyzhenskaya–Babushka–Brezzi inequality.

This paper deals with Cauchy problem for a class of impulsive partial hyperbolic differential equations involving the Caputo derivative. Our first purpose is to show that the formula of solutions in cited papers are incorrect. Next, we reconsider a class of impulsive fractional partial hyperbolic differential equations and introduce a correct formula of solutions for Cauchy problem in R n . Further, some sufficient conditions for existence of the solutions are established by applying fixed point method. At last, we consider the Cauchy problem in a Banach space via the technique of measures of noncompactness and Monch's fixed point theorem. Some examples are given to illustrate our results.

The state-based peridynamics is considered a nonlocal method in which the equations of motion utilize integral form as opposed to the partial differential equations in the classical continuum mechanics. As a result, the enforcement of boundary conditions in solid mechanics analyses cannot follow the standard way as in a classical continuum theory. In this paper, a new approach for the boundary condition enforcement in the state-based peridynamic formulation is presented. The new method is first formulated based on a convex kernel approximation to restore the Kronecker-delta property on the boundary in 1-D case. The convex kernel approximation is further localized near the boundary to meet the condition that recovers the correct boundary particle forces. The new formulation is extended to the two-dimensional problem and is shown to reserve the conservation of linear momentum and angular momentum. Three numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed approach.

In this paper, a new method to impose 2-D solid wall boundary conditions in smoothed particle hydrodynamics is presented. The wall is discretised by means of a set of virtual particles and is simulated by a local point symmetry approach. The extension of a previously published modified virtual boundary particle (MVBP) method guarantees that arbitrarily complex domains can be readily discretised guaranteeing approximate zeroth and first-order consistency. To achieve this, three important new modifications are introduced: (i) the complete support is ensured not only for particles within one smoothing length distance, h, from the boundary but also for particles located at a distance greater than h but still within the support of the kernel; (ii) for a non-uniform fluid particle distribution, the fictitious particles are generated with a uniform stencil (unlike the previous algorithms) that can maintain a uniform shear stress on a particle-moving parallel to the wall in a steady flow; and (iii) the particle properties (density, mass and velocity) are defined using a local point of symmetry to satisfy the hydrostatic conditions and the Cauchy boundary condition for pressure. The extended MVBP model is demonstrated for cases including hydrostatic conditions for still water in a tank with a wedge and for curved boundaries, where significant improved behaviour is obtained in comparison with the conventional boundary techniques. Finally, the capability of the numerical scheme to simulate a dam break simulation is also shown. Copyright © 2015 John Wiley & Sons, Ltd.

We consider a second-order differential equation that describes the behavior of the "boundary function" -- the zeroth-order boundary function of the asymptotic boundary function method. The dual operator formalism is applied to this equation. The result is an approximate solution of the problem that satisfies the boundary conditions on both boundaries. The boundary function obtained in this way is compared with the results of numerical solution of the exact differential equation.

In this paper, we prove the existence and uniqueness of a solution for a second-order set-valued differential equation with three-point boundary conditions, where the perturbation is measurable with respect to the time variable and Lipschitzian with respect to the second and third variables.

A novel scheme for implementation of the no-slip boundary conditions in the lattice Boltzmann method is presented. In detail, we have substituted the classical bounce-back idea by the direct velocity boundary condition specification employing geometric-based manipulation of the equilibrium distribution functions. In this way we have constructed the equilibrium density function in such a way that it imposes the desired Dirichlet boundary conditions at numerical boundary points. Therefore, in fact a kind of equilibrium boundary condition is made. This specification for general curved solid surfaces is made by means of immersed boundary concepts, but without any need to interpolating density distribution values. On the other hand, the results show that the method presents a faster solution procedure in comparison to the bounce-back scheme.

Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrodinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schrodinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.

The hyperbolic partial differential equation, which contains integral condition in place of classical boundary condition arises in many application of modern physics and technologies. In this article, we propose a numerical method to solve the hyperbolic equation with nonlocal boundary condition using radial basis function based finite difference method. Several numerical experiments are presented and compared with some existing method to demonstrate the efficiency of the proposed method.

A nonlinear parabolic integro-differential equation ? t g ( u ) - Δ u = F + ? 0 t f ( s , u ( s ) ) d s with a known Neumann boundary condition on a part of the boundary and an unknown Dirichlet boundary condition α ( t ) on the other part of the boundary is studied. The inverse problem of identifying the unknown time-dependent function α ( t ) from an additional integral measurement E ( t ) = ? ? g ( u ( t , x ) ) d x is investigated. The well-posedness of the problem in suitable function spaces is shown and a numerical time-discrete scheme for approximations is designed. Convergence of the proposed scheme is supported by a numerical experiment.

We suggest a method for solving a boundary value problem for ordinary differential equations with boundary conditions in the presence of state and integral constraints. The method is based on the embedding principle, which permits one to reduce the original boundary value problem to a special optimal control problem with the use of the general solution of a Fredholm integral equation of the first kind.