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We investigate the damping of inviscid surface gravity waves when they propagate over a permeable seabed. Traditionally, this problem is solved by considering the wave motion in the upper water layer interacting with the Darcy flow in the porous bottom layer through matching of the solutions at the permeable interface. The novel approach in this study is that we describe the interaction between the upper fluid layer and the permeable bottom layer by the application of a Robin boundary condition. By varying the magnitude of the Robin parameter R , we can model the bottom structure of the fluid layer from rigid to completely permeable. In the case when the bottom is a porous medium where Darcy's law applies, or composed by densely packed vertical Hele-Shaw cells, R is small, and can by determined by comparison with analytical results for a two-layer structure. In this case the wave damping is small. For larger values of R , the damping increases, and for very large R , the wave is almost critically damped. For the nonlinear transport in spatially damped shallow-water waves, increasing bottom permeability (larger R), reduces the magnitude of the horizontal Stokes drift velocity, while the drift profile tends to become parabolic with height. The vertical Stokes drift in the limit of large permeability is linear with height, with a magnitude that is larger than the horizontal drift. It is suggested that the implementation of a permeable bottom bed in some cases could prevent shorelines from damaging erosion by incoming surface waves. [ABSTRACT FROM AUTHOR]

We consider the problem of identifying an unknown portion $ \Gamma $ of the boundary of a $ d $-dimensional ($ d = 2, 3 $) body $ \Omega $ by a pair of Cauchy data $ (f, g) $ on the accessible part $ \Sigma $ of the boundary of a harmonic function $ u $. On the unknown boundary, a Robin homogeneous condition is assumed. For a fixed constant impedance $ \alpha $, it was shown by Cakoni and Kress in [1] through concrete examples that a single measurement of $ (f, g) $ on $ \Sigma $ can give rise to infinitely many different domains $ \Omega $. Nonetheless, shape optimization techniques can provide fair detections of the unknown boundary given a single pair of Cauchy data as we will showcase here. On this purpose, the inverse problem is recast into three different shape optimization formulations and the shape derivative of the cost function associated with each formulations are obtained. The shape gradient informations are then utilized in a Sobolev gradient-based scheme via finite element method to solve the optimization problems. Numerical results are provided to illustrate the feasibility of the proposed numerical methods in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in Rn,n≥2$$ {\mathbb{R}}^n,n\ge 2 $$. The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in Rn,n=2,3$$ {\mathbb{R}}^n,n=2,3 $$. Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem. [ABSTRACT FROM AUTHOR]

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type. [ABSTRACT FROM AUTHOR]

A new crack-tip finite element able to improve the accuracy of Finite Element Method (FEM) solutions for cracks growing along the Winkler-type spring interfaces between linear elastic adherents is proposed. The spring model for interface fracture, sometimes called Linear-Elastic (perfectly) Brittle Interface Model (LEBIM), can be used, e.g., to analyse fracture of adhesive joints with a thin adhesive layer. Recently an analytical expression for the asymptotic elastic solution with logarithmic stress-singularity at the interface crack tip considering spring-like interface behaviour under fracture Mode III was deduced by some of the authors. Based on this asymptotic solution, a special 5-node triangular crack-tip finite element is developed. The generated special singular shape functions reproduce the radial behaviour of the first main term and shadow terms of the asymptotic solution. This special element implemented in a FEM code written in Matlab has successfully passed various patch tests with spring boundary conditions. The new element allows to model cracks in spring interfaces without the need of using excessively refined FEM meshes, which is one of the current disadvantages in the use of LEBIM when stiff spring interfaces are considered. Numerical tests carried out by h-refinement of uniform meshes show that the new singular element consistently provides significantly more accurate results than the standard finite elements, especially for stiff interfaces, which could be relevant for practical applications minimizing computational costs. The new element can also be used to solve other problems with logarithmic stress-singularities. [ABSTRACT FROM AUTHOR]

The mean first-passage time represents the average time for a migrating cell within its environment, starting from a certain position, to reach a specific location or target for the first time. In this feature article, we provide an overview of the characterization of the mean first-passage time of cells moving inside two- or three-dimensional domains, subject to various boundary conditions (Dirichlet, Neumann, Robin, or mixed), through the so-called adjoint diffusion equation. We concentrate on reducing the latter to inhomogeneous linear integral equations for certain density functions on the boundaries. The integral equations yield the mean first-passage time exactly for a very reduced set of boundaries. For various boundary surfaces, which include small deformations of the exactly solvable boundaries, the integral equations provide approximate solutions. Moreover, the method also allows to deal approximately with mixed boundary conditions, which constitute a genuine long-standing and open problem. New plots, figures, and discussions are presented, aimed at clarifying the analysis. [ABSTRACT FROM AUTHOR]

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball B (0 , r) . Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system. [ABSTRACT FROM AUTHOR]

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein. [ABSTRACT FROM AUTHOR]

In this paper, we propose new formulas for representing the solution of the first (Dirichlet), second (Neumann) and third (Robin) initial-boundary value problems for one-dimensional fractional diffusion equations with the time-fractional Caputo derivative. These formulas are obtained by the continuation method used in the theory of differential equations with integer derivatives. The Green's functions of the problems are constructed in terms of the Fox -function. As an example of the use of the constructed formulas, the inverse problem of determining the zero-coefficient of the fractional diffusion equation with the Neumann boundary condition and the overdetermination condition at the boundary of the spatial domain is studied. The existence and uniqueness of the solution of the considered inverse problem is obtained by a method based on the integral equations using Green's function. It is shown that this solution depends continuously on the input data. [ABSTRACT FROM AUTHOR]

We are interested in the thermal insulation of a bounded open set Ω surrounded by a set whose thickness is locally described by ε h, where h is a nonnegative function defined on the boundary ∂Ω. We study the problem in the limit for ε going to zero using a first-order asymptotic development Γ-convergence [ABSTRACT FROM AUTHOR]

In this paper, we establish a general weighted Hardy type inequality for the $ p- $Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls centred at the origin with radius $ R $ in $ \mathbb{R} ^n $. [ABSTRACT FROM AUTHOR]

This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.

In the present paper, the relatively new method of Radial Basis Function-Generated Finite Difference (RBF-FD) is used to solve a class of Partial Differential Equations (PDEs) with Dirichlet and Robin boundary conditions. For this approximation, Polyharmonic Splines (PHS) are used alongside Polynomials. This combination has many benefits. On the other hand, Polyharmonic Splines have no shape parameter and therefore relieve us of the hassle of calculating the optimal shape parameter. As the first problem, a two-dimensional Poisson equation with the Dirichlet boundary condition is investigated in various domains. Then, an elliptic PDE with the Robin boundary condition is solved by the proposed method. The results of numerical studies indicate the excellent efficiency, accuracy and high speed of the method, while for these studies, very fluctuating and special test functions have been used. [ABSTRACT FROM AUTHOR]

The heat exchange between a rigid body and a fluid is usually modelled by the Robin boundary condition saying that the heat flux through the interface is proportional to the difference between their temperatures. Such interface law describes only the unilateral heat exchange. The goal of this paper is to compare the Robin boundary condition starting with the transmission condition (the temperature and the flux continuity) using rigorous mathematical analysis. Our main results are the following. We first show that a generalized version of the Robin boundary condition can be justified. Second, we prove that replacing the generalized by the standard Robin condition can be justified for high convection velocity if the conductivity of the surrounding liquid is much lower than that of the body. On the other hand, if the fluid conducts much better than the body, then the effective boundary condition is shown not to be the Robin one, but it involves second-order derivatives. We strongly believe that those findings bring new insights to the physics of the heat exchange processes and, thus, could prove useful in engineering practice. [ABSTRACT FROM AUTHOR]

We establish some embedding results for weighted Sobolev spaces. As an application, we obtain one nonzero solution for the equation - div (| ∇ u | N - 2 ∇ u) + V (x) | u | N - 2 u = λ Q (x) f (u) , in R N , where V , Q are nonnegative potentials, λ > 0 is a large parameter and f has critical growth in the Trudinger–Moser sense. [ABSTRACT FROM AUTHOR]

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

The mean first-passage time represents the average time for a migrating cell within its environment, starting from a certain position, to reach a specific location or target for the first time. In this feature article, we provide an overview of the characterization of the mean first-passage time of cells moving inside two- or three-dimensional domains, subject to various boundary conditions (Dirichlet, Neumann, Robin, or mixed), through the so-called adjoint diffusion equation. We concentrate on reducing the latter to inhomogeneous linear integral equations for certain density functions on the boundaries. The integral equations yield the mean first-passage time exactly for a very reduced set of boundaries. For various boundary surfaces, which include small deformations of the exactly solvable boundaries, the integral equations provide approximate solutions. Moreover, the method also allows to deal approximately with mixed boundary conditions, which constitute a genuine long-standing and open problem. New plots, figures, and discussions are presented, aimed at clarifying the analysis.

In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition − div (| ∇ u | N − 2 ∇ u) + h (x) | u | q − 2 u = λ k (x) | u | p − 2 u in  Ω , | ∇ u | N − 2 (∇ u ⋅ ν) + | u | N − 2 u = 0 on  ∂ Ω , where Ω ⊂ ℝ N is an exterior domain such that 0 ∉ Ω ¯. [ABSTRACT FROM AUTHOR]

Levy et al. [1] reported the breakdown of electroneutrality in confined nanopores embedded in a dielectric medium. A Robin boundary condition was derived which eliminates the need to include the dielectric medium explicitly when solving for the electric field within the nanopore. In this comment, we point out issues related to the approximations made during the derivation of the boundary condition. The errors caused by the use of this boundary condition can be significant even for nanochannels of large aspect (length to radius) ratio, a condition on which the approximations in Levy et al. [1] are based. [ABSTRACT FROM AUTHOR]

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball B(0,r). Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system.

The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton's law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically. [ABSTRACT FROM AUTHOR]

In this article, we study the nonexistence of solution with finite Morse index for the following nonlinear elliptic equation −Δu = |u|p− 1u with ∂ u ∂ υ + c u = 0 o n ∂ R + N . [ABSTRACT FROM AUTHOR]

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number,  R a c , and the wave number,  a c , are obtained in closed form. An asymptotic analysis is performed, to obtain  R a c  for the case of adiabatic boundaries, for which  a c ≃ 0 . A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known. [ABSTRACT FROM AUTHOR]

A semi-analytical approach is presented to solve three-dimensional dynamic Green's function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green's function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green's functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green's functions. [ABSTRACT FROM AUTHOR]

This paper is devoted to solve the backward problem for radially symmetric time‐fractional diffusion‐wave equation under Robin boundary condition. This problem is ill‐posed and we apply an iterative regularization method to solve it. The error estimates are obtained under the a priori and a posteriori parameter choice rules. Numerical results show that the proposed method is efficient and stable. [ABSTRACT FROM AUTHOR]

In this paper, a new Cartesian grid finite difference scheme is introduced for solving parabolic initial-boundary value problems involving irregular domains and Robin boundary condition in two and three dimensions. In spatial discretization, a ray-casting matched interface and boundary (MIB) method is utilized to enforce different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mixed combinations, along the normal direction to generate necessary fictitious values outside the irregular domain. This allows accurate approximations of jumps in derivatives at various boundary locations so that the fourth-order central difference can be corrected at all Cartesian nodes. By treating such corrections as additional unknowns, the order of finite difference discretization of the Laplacian operator can be preserved. Moreover, by constructing corrections for different types of irregular and corner points, the proposed augmented MIB (AMIB) method can accommodate complicated geometries, while maintaining the fourth order of accuracy in space. In temporal discretization, the standard Crank–Nicolson scheme is employed, which is second-order in time and unconditionally stable. Furthermore, a Fast Sine Transform acceleration algorithm is employed to efficiently invert the discrete Laplacian, so that the augmented linear system in each time step can be solved with a complexity of O (N log N) , where N stands for the total spatial degree-of-freedom. The accuracy, stability and efficiency of the proposed AMIB method are numerically validated by considering various parabolic problems in two and three dimensions. [ABSTRACT FROM AUTHOR]

An analytical expression of the effectiveness factor of photocatalytic reactors that accounts for external mass transfer limitations is presented. The solution is based on Robin boundary conditions introducing the Sherwood number as the parameter that accounts for the external mass transfer limitations. Regions highlighting free diffusion, diffusion limitations, and external mass transfer limitations are identified based on the Thiele Modulus and the Sherwood number. At low values of the Sherwood number, the internal diffusion and reaction rate-limiting regions are surpassed by the external mass transfer limitations. A cut-off value of the Sherwood number below which external mass transfer limitations cannot be ignored is 0.55. The evaluation of the model showed that under Dirichlet boundary condition the model converges to literature models from similar studies. This 1-D model looked at the Sherwood number across a broad scope and accounted for a broader range of limitations associated with immobilized photocatalytic films.

The source term identification of the time‐fractional diffusion equation under Robin boundary condition is studied. This problem is ill‐posed. Therefore, we apply Landweber iterative regularization method, fractional Landweber iterative regularization method, TSVD method, combining TSVD method, and fractional Landweber iterative regularization method, respectively. The comparisons of these four methods are given, which can help us select the most effective method. The error estimates between the regularized approximate solutions and the exact solution are given under the a priori and a posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

This manuscript presents a novel formulation for a linear elastic model of a heterogeneous arterial section undergoing uniform pressure in a quasi-static regime. The novelties are twofold. First, an elastic bed support on the external boundary (elastic bed boundary condition) replaces the classical Dirichlet boundary condition (i.e., blocking displacements at arbitrarily selected nodes) for elastic solids to ensure a solvable problem. In addition, this modeling approach can be used to effectively account for the effect of the surrounding material on the vessel. Secondly, to study many geometrical configurations corresponding to different patients, we devise an unfitted strategy based on the Immersed Boundary (IB) framework. It allows using the same (background) mesh for all possible configurations both to describe the geometrical features of the cross-section (using level sets) and to compute the solution of the mechanical problem. Results on coronary arterial sections from realistic segmented images demonstrate that the proposed unfitted IB-based approach provides results equivalent to the standard finite elements (FE) for the same number of active degrees of freedom with an average difference in the displacement field of less than 0.5%. However, the proposed methodology does not require the use of a different mesh for every configuration. Thus, it is paving the way for dimensionality reduction. [ABSTRACT FROM AUTHOR]

In this article, we present a physics-informed neural network combined with fictitious domain method (FDM-PINN) to study linear elliptic and parabolic problems with Robin boundary condition. Our goal here to develop a deep learning framework where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction on small domain ω (ω ¯ ⊂ Ω) , which is geometrically simple shaped domain. We study the applicability and accuracy of FDM-PINN for the elliptic and parabolic problems with fixed ω or moving ω. This method is of the virtual control type and relies on a well-designed neural network. Numerical results obtained by FDM-PINN for two-dimensional elliptic and parabolic problems are given, which are more accurate than the results obtained by least-squares/fictitious domain method in [R. Glowinski and Q. He, A least-squares/fictitious domain method for linear elliptic problems with robin boundary conditions, Commun. Comput. Phys. 9 (2011), pp. 587–606.]. [ABSTRACT FROM AUTHOR]

We are concerned with an inverse problem arising in thermal imaging in a bounded domain Ω ⊂ R n , n = 2 , 3 . This inverse problem consists in the determination of the heat exchange coefficient q(x) appearing in the boundary of a hyperbolic heat equation with Robin boundary condition. A double-logarithmic stability estimate is developed. [ABSTRACT FROM AUTHOR]

We introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a planar bounded simply-connected C 2 -smooth open set. The considered lower-order perturbation corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius for this property to hold. [ABSTRACT FROM AUTHOR]

We prove a Lieb-Thirring inequality for Schrödinger operators - d²/dx² + V on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P. Exner, A. Laptev, and M. Usman [Commun. Math. Phys. 362 (2014), 531-541] albeit under the additional assumption V ∈ L¹(R+). The main difference in our proof is that we use the double commutation method in place of the single commutation method. We also establish an improved inequality in the case of a Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

Let ω ⊂ Rn be a bounded domain with Lipschitz boundary. For ε > 0 and n ∈ N, consider the infinite cone Ωε:= {(x1; x0)/∈ (0, ∞)× Rn: x' ∈ εx1ω}⊂ Rn+1 and the operator Qεα acting as the Laplacian u ↦ -āu on Ωε with the Robin boundary condition ∂νu = αu at ∂Ωε, where ∂νu is the outward normal derivative and α > 0. We look at the dependence of the eigenvalues of Qεα on the parameter β: this problem was previously addressed for n = 1 only (in that case, the only admissible ω are finite intervals). In the present work, we consider arbitrary dimensions n ≥ 2 and arbitrarily shaped "cross-sections" ω and look at the spectral asymptotics as β becomes small, i.e., as the cone becomes "sharp" and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity Nω:= Voln-1 ∂ω/Volnω. More precisely, for any fixed j ∈ N and α > 0, the j -th eigenvalue Ej(Qεα)of Qεα exists for all sufficiently small β > 0 and satisfies Ej(Qεα)=-Nω²α²/(2j+n-2)²ε² +O(1/ε)as ε→0+ The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest. [ABSTRACT FROM AUTHOR]

In this article, we have considered a non-classical Stefan problem that involves the space-dependent thermal conductivity, variable latent heat and Robin boundary condition at the fixed boundary. The solution of the problem is obtained by using the operational matrix method based on Genocchi polynomials and the collocation scheme. To show the accuracy, we have compared the results obtained by the Genocchi operational matrix method and a finite difference scheme with the exact solution for a particular case of the considered Stefan problem. It is found that the proposed Genocchi operational matrix method is more accurate than the finite difference scheme. Moreover, it is shown that the proposed algorithm rapidly converges as the degree of Genocchi polynomials increases. We have also analyzed the effects of the parameters α and δ on the temperature distribution and the evolution of moving phase front in the case of the considered non-classical Stefan problem. It is observed that the phase change processes become fast as the parameters α or/and δ increase. [ABSTRACT FROM AUTHOR]

The paper deals with the logarithmic fractional equations with variable exponents (−Δ)p1(·)s1(·)(u)+(−Δ)p2(·)s2(·)(u)+|u|p‾1(x)−2u+|u|p‾2(x)−2u=λb(x)|u|α(x)−2u+μa(x)|u|r(x)−2ulog|u|+μc(x)|u|η(x)−2u,x∈Ω,Np1(·)s1(·)(u)+Np2(·)s2(·)(u)+β(x)(|u|p‾1(x)−2u+|u|p‾2(x)−2u)=0,x∈ℝN\Ω‾,$$ \left\{\begin{array}{ll}& \kern-5pt {\left(-\Delta \right)}_{p_1\left(\cdotp \right)}^{s_1\left(\cdotp \right)}(u)+{\left(-\Delta \right)}_{p_2\left(\cdotp \right)}^{s_2\left(\cdotp \right)}(u)+{\left|u\right|}^{{\overline{p}}_1(x)-2}u+{\left|u\right|}^{{\overline{p}}_2(x)-2}u=\lambda b(x){\left|u\right|}^{\alpha (x)-2}u\\ {}& +\mu a(x){\left|u\right|}^{r(x)-2}u\log \mid u\mid +\mu c(x){\left|u\right|}^{\eta (x)-2}u,\\ {}& x\in \Omega, \\ {}& \kern-5pt {\mathcal{N}}_{p_1\left(\cdotp \right)}^{s_1\left(\cdotp \right)}(u)+{\mathcal{N}}_{p_2\left(\cdotp \right)}^{s_2\left(\cdotp \right)}(u)+\beta (x)\left({\left|u\right|}^{{\overline{p}}_1(x)-2}u+{\left|u\right|}^{{\overline{p}}_2(x)-2}u\right)=0,\\ {}& x\in {\mathbb{R}}^N\backslash \overline{\Omega},\end{array}\right. $$where (−Δ)pi(·)si(·)$$ {\left(-\Delta \right)}_{p_i\left(\cdotp \right)}^{s_i\left(\cdotp \right)} $$ and Npi(·)si(·)$$ {\mathcal{N}}_{p_i\left(\cdotp \right)}^{s_i\left(\cdotp \right)} $$ denote the variable si(·)$$ {s}_i\left(\cdotp \right) $$‐order pi(·)$$ {p}_i\left(\cdotp \right) $$‐fractional Laplace operator and the nonlocal normal pi(·)$$ {p}_i\left(\cdotp \right) $$‐derivative of si(·)$$ {s}_i\left(\cdotp \right) $$‐order, respectively, with si(·):ℝ2N→(0,1)$$ {s}_i\left(\cdotp \right):{\mathbb{R}}^{2N}\to \left(0,1\right) $$ and pi(·):ℝ2N→(1,∞)$$ {p}_i\left(\cdotp \right):{\mathbb{R}}^{2N}\to \left(1,\infty \right) $$ (i∈{1,2}$$ i\in \left\{1,2\right\} $$) being continuous. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}^N $$ is a bounded smooth domain with N>pi(x,y)si(x,y)$$ N>{p}_i\left(x,y\right){s}_i\left(x,y\right) $$ (i∈{1,2}$$ i\in \left\{1,2\right\} $$) for any (x,y)∈Ω‾×Ω‾,λ$$ \left(x,y\right)\in \overline{\Omega}\times \overline{\Omega},\lambda $$ and μ$$ \mu $$ are a positive parameters, r(·)$$ r\left(\cdotp \right) $$ and η(·)$$ \eta \left(\cdotp \right) $$ are two continuous functions, while variable exponent α(x)$$ \alpha (x) $$ can be close to the critical exponent p2s2∗(x)=Np‾2(x)/(N−s‾2(x)p‾2(x))$$ {p}_{2{s}_2}^{\ast }(x)=N{\overline{p}}_2(x)/\left(N-{\overline{s}}_2(x){\overline{p}}_2(x)\right) $$, given with p‾2(x)=p2(x,x)$$ {\overline{p}}_2(x)={p}_2\left(x,x\right) $$ and s‾2(x)=s2(x,x)$$ {\overline{s}}_2(x)={s}_2\left(x,x\right) $$ for x∈Ω‾$$ x\in \overline{\Omega} $$. Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, α(x)

In this article, we consider the nonlinear problem ∂u∂t=Au+V(x)um+p−2+λuqinΩ×(0,T),u(x,0)=u0(x)≥0inΩ,|∇u|p−2∂u∂ν=β(x)up−1on∂Ω×(0,T).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}=\mathbf{A}u+V(x){u}^{m+p-2}+\lambda {u}^q\kern0.30em & \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)={u}_0(x)\ge 0\kern0.30em & \mathrm{in}\kern0.30em \Omega, \\ {}{\left|\nabla u\right|}^{p-2}\frac{\partial u}{\partial \nu }=\beta (x){u}^{p-1}\kern0.30em & \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$where Au=div(mum−1|∇u|p−2∇u)$$ \mathbf{A}u=\operatorname{div}\left(m{u}^{m-1}{\left|\nabla u\right|}^{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}^N $$ is a bounded domain with smooth boundary, m>0,10,λ∈ℝ,q>0$$ m>0,1<p<N,V\in {L}_{loc}^1\left(\Omega \right),m+p-2>0,\lambda \in \mathbb{R},q>0 $$ and β∈Lloc1(∂Ω)$$ \beta \in {L}_{loc}^1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions V(x),β(x)$$ V(x),\beta (x) $$, and the exponents m+p$$ m+p $$ and q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials V(x)$$ V(x) $$ are taken into account to demonstrate applications of our main result. [ABSTRACT FROM AUTHOR]

The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton’s law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically.

In this work, a numerical method for the singularly perturbed parabolic convection–diffusion turning point problem with Robin boundary condition was developed. The solution to the considered problem has a boundary layer on the left side of the domain. The present method comprises an implicit trapezoidal method for time discretization on a uniform mesh and second-order central difference schemes for space discretization on a Shishkin mesh. The resultant scheme has been shown that the numerical approximation converges uniformly to the solution of the continuous problem regardless of the diffusion parameter. Finally, to validate the resultant scheme, numerical experiments were performed.

The gluing formula for the zeta-determinants of Laplacians with respect to the Robin boundary condition was proved in [15]. This formula contains a constant which is expressed by some curvature tensors on the cutting hypersurface including the scalar and principal curvatures. In this paper we compute this constant explicitly when the cutting hypersurface is a 2-dimensional closed submanifold in a closed Riemannian manifold, and discuss some related topics. We next use the conformal rescaling of the Riemannian metric to compute the value of the zeta function at zero associated to the generalized Dirichlet-to-Neumann operator defined by the Robin boundary condition on this cutting hypersurface. [ABSTRACT FROM AUTHOR]

We propose a numerical method tailored to perform interface-resolved simulations of evaporating multicomponent two-phase flows. The novelty of the method lies in the use of Robin boundary conditions to couple the transport equations for the vaporized species in the gas phase and the transport equations of the same species in the liquid phase. The Robin boundary condition is implemented with the cost-effective procedure proposed by Chai et al. [1] and consists of two steps: (1) calculating the normal derivative of the mass fraction fields in cells adjacent to the interface through the reconstruction of a linear polynomial system, and (2) extrapolating the normal derivative and the ghost value in the normal direction using a linear partial differential equation. This methodology yields a second-order accurate solution for the Poisson equation with a Robin boundary condition and a first-order accurate solution for the Stefan problem. The overall methodology is implemented in an efficient two-fluid solver, which includes a Volume-of-Fluid (VoF) approach for the interface representation, a divergence-free extension of the liquid velocity field onto the entire domain to transport the VoF, and the temperature equation to include thermal effects. We demonstrate the convergence of the numerical method to the analytical solution for multicomponent isothermal evaporation and observe good overall computational performance for simulating non-isothermal evaporating two-fluid flows in two and three dimensions. • A method is proposed to perform interface-resolved simulations of evaporating multicomponent two-phase flows. • Robin boundary conditions are used to couple the vaporized species and the same species in the liquid phase. • A VoF approach represents the interface, and a divergence-free extension of the liquid velocity field transports the VoF. • Convergence of the numerical method to the analytical solution for multicomponent isothermal evaporation is shown. [ABSTRACT FROM AUTHOR]

In this paper, a non-polynomial-based trigonometric cubic B-spline collocation method is developed to solve the reaction-diffusion singularly perturbed problems with Robin boundary conditions. These problems are more tedious to solve than those with Dirichlet and Neumann boundary conditions. The parameter ε in the differential equation results in a rapid change in the solution over a small region. A piecewise uniform mesh is constructed to handle this difficulty. Also, a modification of the proposed mesh is suggested to improve the accuracy of the numerical results by introducing a change in the transition parameter. Through rigorous analysis, it has been shown that the method is almost second-order uniformly convergent. The performance and theoretical findings of the proposed scheme are validated through numerical experiments presented for two test problems. The accuracy of the method is measured in the discrete maximum norm. The tabular results demonstrate that the newly added mesh produces better results. [ABSTRACT FROM AUTHOR]