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In the paper, we solve two Stefan problems. The first problem recovers an unknown moving boundary by specifying the Cauchy boundary conditions on a fixed left-end. The second problem finds a time-d...

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrodinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.

In this study, the well-known Cauchy problem of elliptic type equation possibly with variable coefficient is contemplated while a part of right-hand side source is unknown as well whereas overspecified boundary data is imposed on boundary. It is a supposition that the right-hand side source can be observed as a sum of two-parts which are independent of each other and at the same time, each of them being in terms of own one-variable. It is proved that such an inverse problem possesses unique solution. To approximate this unique solution, a kind of domain type meshless collocation method is proposed so that the boundary data are imposed directly. This is not troublesome because the original problem, by variable transforming through homogenization function, is converted to an inverse problem with homogeneous Cauchy boundary conditions. This surprisingly diminishes the ill-posedness of the right-hand side construction of Cauchy problem. As a result, it does not require any regularization algorithms and therefore reduces the computational time. The convergence and error analysis of the proposed approximation method is fully discussed. It is worth-mentioning that the considered domain is of arbitrary shape and discussed in the polar coordinate for simplicity and, it does not matter how scattered points are chosen, therefore the method is truly meshless one. The accuracy and robustness of this homogenization meshless collocation method (HMCM) is tested on several numerical examples.

A stationary problem of radiative-conductive heat transfer in a three-dimensional domain is studied in the $${{P}_{1}}$$ -approximation of the radiative transfer equation. A formulation is considered in which the boundary conditions for the radiation intensity are not specified but an additional boundary condition for the temperature field is imposed. Nonlocal solvability of the problem is established, and it is shown that the solution set is homeomorphic to a finite-dimensional compact. A condition for the uniqueness of the solution is presented.

In this article, a novel meshless boundary function method (BFM) is proposed for solving the boundary identification problem of steady-state nonlinear heat conduction in arbitrary plane domain. Firstly, the original governing equation is transformed to a new one with homogeneous Cauchy boundary conditions by using a homogenization technique. Secondly, the domain type meshless collocation method is employed to solve the new partial different equation in a reduced domain, in which the numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the known boundary. After that, a nonlinear equation corresponding to each angle is formed and then is solved by the Newton iterative method in order to determine the missing boundary shape. Finally, the accuracy and robustness of the proposed BFM are examined through three numerical examples.

In this paper, we establish some Lyapunov-type inequalities for a class of linear and nonlinear fractional q-difference boundary value problems under Cauchy boundary conditions. As applications, we use the inequality to obtain an interval, where Mittag-Leffler function has no real zeros. In addition, we also derive nonexistence results for fractional q-difference boundary value problem.

We solved the problem of the Floquet-Bloch waves propagation in one-dimensional magnetophotonic crystal with hyperbolic and gyrotropic layers in the presence of a transverse magnetic field. New fundamental solutions of the Hill equation based on the third boundary-value problem with Cauchy boundary conditions are explicitly obtained in crystal layers. The dispersion equation is obtained in analytical form and its roots are found. The dispersion properties of hyperbolic media and magnetophotonic crystals with hyperbolic and gyrotropic layers are analyzed, and the main features of the propagation of TE and TM waves for two types of hyperbolicity in the presence of gyrotropy of the medium layers are elucidated.

In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the right-hand sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance.

We study a control problem where the state equation is a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise. We allow the control to act on the boundary and set boundary conditions which result in a stochastic differential equation for the trace of the solution on the boundary. This work provides necessary and sufficient conditions of optimality in the form of a maximum principle. We also provide a result of existence for the optimal control in the case where the control acts linearly.

In recent years, proper orthogonal decomposition (POD) has become a popular model reduction method in the field of groundwater modeling. It is used to mitigate the problem of long run times that are often associated with physically-based modeling of natural systems, especially for parameter estimation and uncertainty analysis. POD-based techniques reproduce groundwater head fields sufficiently accurate for a variety of applications. However, no study has investigated how POD techniques affect the accuracy of different boundary conditions found in groundwater models. We show that the current treatment of boundary conditions in POD causes inaccuracies for these boundaries in the reduced models. We provide an improved method that splits the POD projection space into a subspace orthogonal to the boundary conditions and a separate subspace that enforces the boundary conditions. To test the method for Dirichlet, Neumann and Cauchy boundary conditions, four simple transient 1D-groundwater models, as well as a more complex 3D model, are set up and reduced both by standard POD and POD with the new extension. We show that, in contrast to standard POD, the new method satisfies both Dirichlet and Neumann boundary conditions. It can also be applied to Cauchy boundaries, where the flux error of standard POD is reduced by its head-independent contribution. The extension essentially shifts the focus of the projection towards the boundary conditions. Therefore, we see a slight trade-off between errors at model boundaries and overall accuracy of the reduced model. The proposed POD extension is recommended where exact treatment of boundary conditions is required.

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn-Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn-Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and rigorously discussed., Key words: Cahn-Hilliard system, convection, dynamic boundary condition, initial-boundary value problem, well-posedness, regularity of solutions

A wave equation on a time-dependent domain is considered. The shape of the domain changes according to a prescribed space/time-dependent velocity field. On the moving boundary the solution satisfies zero Dirichlet condition. It is known that if the domain keeps expanding at a “subsonic” speed, then the associated finite energy decays uniformly. Here, the scenario of interest is when the domain remains bounded and undergoes phases of expansion and contraction. Although the energy identity in this case is not necessarily conservative, it is shown that the L 2 space–time norm of the normal trace remains a priori bounded at small normal speeds of the boundary, analogously to the classical Dirichlet wave problem on a static domain. In addition, it is demonstrated that small normal velocity but very large acceleration of the boundary is compatible with the known existence theory, provided the magnitude of the deformations is relatively small. An “adaptive” boundary movement control is proposed and implemented numerically. The control action is dynamically computed from the normal trace data and dissipates the energy by means of small deformations of the domain only.

The bifurcation of non-trivial steady state solutions of a scalar reaction–diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.

The aim of this paper is to prove various ill-posedness and well-posedness results on the Cauchy problem associated to a class of fractional Kadomtsev-Petviashvili (KP) equations including the KP version of the Benjamin-Ono and Intermediate Long Wave equations., 42 pages

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical...

In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the Neumann boundary condition and the Dirichlet boundary condition on specific parts of the domain boundary, respectively. The ill-posedness arising from the Neumann boundary condition in the strongly anisotropic diffusion limit is handled by the decomposition of the solution into a mean part and a fluctuation part. The boundary layer analysis due to the Dirichlet boundary condition is conducted for each order in the expansion for the fluctuation part. Our results suggest that the leading order is the combination of the mean part and the composite approximation of the fluctuation part for the general Dirichlet boundary condition., 18 pages, 5 figures, 1 table

Purpose This paper proposes a novel approach to impose the Neumann boundary condition for isogeometric analysis (IGA) of Euler–Bernoulli beam with 1-D formulation. The proposed method is for only IGA in which it is difficult to handle the Neumann boundary conditions. The control points of B-spline are equivalent to nodes in finite element method. With 1-D formulation, it is not possible to accommodate multiple degrees of freedom in IGA. This case arises in the analysis of beams. The paper aims to propose a way to work around this issue in a simple way. Design/methodology/approach Neumann boundary conditions, which are even-order derivatives (example: double derivative) of the primary variable, are inherently satisfied in the weak form. Boundary conditions with an odd number of derivatives (example: slope) are imposed with the introduction of a new penalty matrix. Findings The proposed method can impose a slope boundary condition for IGA of a beam using 1-D formulation. Originality/value From the literature, it can be observed that the beam is formulated in 1-D by considering it as either a rotation-free element or a 2-D formulation by considering shear strain along with the normal strain. The work represents 1-D formulation of a beam while considering the slope boundary condition, which is easy and effective to formulate, compared with the slope boundary conditions reported in previous works.

Fourth-order boundary value problems with discontinuous boundary conditions are studied. We prove that the eigenvalues depend not only continuously but smoothly on the coefficients and on the boundary conditions and find formulas for the derivatives with respect to each of these parameters.

In this paper, using the Nehari manifold and fibering maps, we study the existence of at least two positive solutions for the following p-Kirchhoff equation where is a bounded domain, with a, b, , is the p-Laplacian operator, is a parameter.

In this paper, we study strong solutions to the steady compressible heat-conductive fluid near a non-zero constant flow with the Dirichlet boundary condition for the velocity on the inflow and outflow part of the boundary. We also consider the Dirichlet boundary condition for the temperature, and we do not need the thermal conductivity coefficient κ to be large. The existence of strong solutions is established for any Reynolds number and Mach number in the framework of perturbation.

We investigate the Markov process on the boundary of a bounded Lipschitz domain associated to the Neumann and Robin boundary value problems. We first construct L p -semigroups of sub-Markovian contractions on the boundary, generated by the boundary conditions, and we show that they are induced by the transition functions of the forthcoming processes. As in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected Brownian motion. The Robin problem is treated with a Kato type L p -perturbation method, using the Revuz correspondence. An exceptional (polar) set occurs on the boundary. We make the link with the Dirichlet forms approach.

Results of a numerical experiment of solving an ill-posed Cauchy problem for the wave equation are discussed. An instrumental function for the regularizing algorithm applied here is given, and an analysis of stability is carried out.

A piecewise interpolation approximation of the solution to the Cauchy problem for ordinary differential equations (ODEs) is constructed on a set of nonoverlapping subintervals that cover the interval on which the solution is sought. On each interval, the function on the right-hand side is approximated by a Newton interpolation polynomial represented by an algebraic polynomial with numerical coefficients. The antiderivative of this polynomial is used to approximate the solution, which is then refined by analogy with the Picard successive approximations. Variations of the degree of the polynomials, the number of intervals in the covering set, and the number of iteration steps provide a relatively high accuracy of solving nonstiff and stiff problems. The resulting approximation is continuous, continuously differentiable, and uniformly converges to the solution as the number of intervals in the covering set increases. The derivative of the solution is also uniformly approximated. The convergence rate and the computational complexity are estimated, and numerical experiments are described. The proposed method is extended for the two-point Cauchy problem with given exact values at the endpoints of the interval.

A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated.

We prove a theorem on the existence and uniqueness of a mild solution to the Cauchy problem for a stochastic differential equation of neutral type in the weighted Hilbert space.

In this paper, we study the non-reflecting boundary condition for the time-harmonic Maxwell's equations in homogeneous waveguides with an inhomogeneous inclusion. We analyze a series representation of solutions to the Maxwell's equations satisfying the radiating condition at infinity, from which we develop the so-called electric-to-magnetic operator for the non-reflecting boundary condition. Infinite waveguides are truncated to a finite domain with a fictitious boundary on which the non-reflecting boundary condition based on the electric-to-magnetic operator is imposed. As the main goal, the well-posedness of the reduced problem will be proved. This study is important to develop numerical techniques of accurate absorbing boundary conditions for electromagnetic wave propagation in waveguides.

The aim of the paper is to investigate the boundary value problem of the evolution equation Lu = K (x,t) ut - Δu + a (x,t) u = f (x,t). The characteristic property of this type of equations is the failure of the Petrovski’s “A” condition when coefficients are constant [1]. In this case, Cauchy problem is incorrect in the sense of Hadamard. Hence in this paper, the space, guaranteeing the correctness of the boundary value problem in the sense of Hadamard, is selected by adding some additional conditions to the coefficients of the equation.

In this paper, we study the Cauchy problem of the generalized Fokas–Olver–Resenau–Qiao equation. Firstly, by means of transport equation and Littlewood–Paley theory, we obtain the local well-posedn...

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite strip. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data (or Cauchy data), a small perturbation in the data can cause a dramatically large error in the solution for 0

We consider the stability of a system of two strongly coupled wave equations by means of only one boundary feedback. We show that the stability of the system depends in a very complex way on all of the involved factors such as the type of coupling, the hinged regularity and the accordance of boundary conditions. We first show that the system is uniformly exponentially stable if the undamped equation has Dirichlet boundary condition, while it is only polynomially stable if the undamped equation is subject to Neumann boundary condition.Next, by a spectral approach, we show that this sensibility of stability with respect to the boundary conditions on the undamped equation is intrinsically linked with the transmission of the vibration as well as the dissipation between the equations.

A nonlinear second-order differential equation with a small parameter at the derivatives is considered in the case where the limit algebraic equation has a multiple root. The matching method is applied to construct an asymptotic expansion of the solution of the boundary value problem. Two boundary layer variables with different scales are used to describe the asymptotic solution near the boundary.

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).

A polynomial solution of the classical one-phase Stefan problem with a Neumann boundary condition is presented. As a result of the multiple integration of the heat-conduction equation, a sequence of identical equalities has been obtained. On the basis of these equalities, solutions were constructed in the form of the second-, third-, fourth-, and fifth-degree polynomials. It is shown by test examples that the approach proposed is highly efficient and that the approximation errors of the solutions in the form of the fourth- and fifth-degree polynomials are negligible small, which allows them to be considered in fact as exact. The polynomial solutions obtained substantially surpass the analogous numerical solutions in the accuracy of determining the position of the moving interphase boundary in a body and are in approximate parity with them in the accuracy of determining the temperature profile in it.

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.

Sedimentation processes of solid particles in a fluid with heat transfer are simulated using a coupled Lattice Boltzmann Method, Immersed Boundary Method and Discrete Element Method (LBM-IBM-DEM) scheme. In the numerical simulations, solid particles are specified either by a given temperature which is termed Dirichlet boundary condition or by a temperature gradient which is termed Neumann boundary condition. Several cases are examined containing one, two and 504 solid particles settling in a fluid, respectively. All the considered cases could be divided into two groups: Group Dirichlet and Group Neumann according to different styles of boundary conditions employed but under exactly the same initial states. The effects of these two boundary conditions on the particle behavior are quantized.

This article studies the solutions in H 1 of a steady transport equation with a divergence-free driving velocity that is W 1 , ∞ , in a two-dimensional bounded polygon. Since the velocity is assumed fully non-homogeneous on the boundary, existence and uniqueness of the solution require a boundary condition on the open part Γ − , where the normal component of u is strictly negative. In a previous article, we studied the solutions in L 2 of this steady transport equation. The methods, developed in this article, can be extended to prove existence and uniqueness of a solution in H 1 with Dirichlet boundary condition on Γ − only in the case where the normal component of u does not vanish at the boundary of Γ − . In the case where the normal component of u vanishes at the boundary of Γ − , under appropriate assumptions, we construct local H 1 solutions in the neighborhood of the end-points of Γ − , which allow us to establish existence and uniqueness of the solution in H 1 for the transport equation with a Dirichlet boundary condition on Γ − .

We obtain a solution of the Black–Scholes equation with a non-smooth boundary condition using symmetry methods. The Black–Scholes equation along with its boundary condition are first transformed into the one dimensional heat equation and an initial condition respectively. We then find an appropriate general symmetry generator of the heat equation using symmetries and the fundamental solution of the heat equation. The symmetry generator is chosen such that the boundary condition is left invariant; the symmetry can be used to solve the heat equation and hence the Black–Scholes equation.

This paper discusses the existence of singular solutions of a superlinear parabolic partial differential equation on bounded domains with homogeneous Neumann boundary conditions. It is shown that in some parameter range, there exists a solution with a spatial singularity whose position and strength depend on time. Both interior and boundary singularities are studied.