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We present a novel Material Point Method (MPM) discretization of surface tension forces that arise from spatially varying surface energies. These variations typically arise from surface energy dependence on temperature and/or concentration. Furthermore, since the surface energy is an interfacial property depending on the types of materials on either side of an interface, spatial variation is required for modeling the contact angle at the triple junction between a liquid, solid and surrounding air. Our discretization is based on the surface energy itself, rather than on the associated traction condition most commonly used for discretization with particle methods. Our energy based approach automatically captures surface gradients without the explicit need to resolve them as in traction condition based approaches. We include an implicit discretization of thermomechanical material coupling with a novel particle-based enforcement of Robin boundary conditions associated with convective heating. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with Affine-Particle-In-Cell (APIC) [Jiang et al. 2015]. We show that our approach enables implicit time stepping for complex behaviors like the Marangoni effect and hydrophobicity/hydrophilicity. We demonstrate the robustness and utility of our method by simulating materials that exhibit highly diverse degrees of surface tension and thermomechanical effects, such as water, wine and wax.

A 2D numerical solution was developed to estimate the temperature and moisture distribution of a spherical moist product. The object used in the study is cranberry (Vaccinium macrocarpon). Robin boundary condition was incorporated as the heat and mass transfer takes place from the object by convection. The governing transient partial differential equations were discretized by the finite difference method. The transient terms were discretized using an implicit scheme. A computer program was generated to solve the heat and moisture transport expressions simultaneously. The diffusion coefficient was assumed as temperature dependent which allowed solving of the heat and moisture transport expressions simultaneously. Temperature and moisture profiles were calculated at various drying times and hot air temperatures ranged from 313 to 348 K. Mean and center temperature and moisture content were calculated. The average moisture content was reduced from 7.49 to 2.58, 0.65 and 0.24 kg (kg of db)−1 at drying time of 10 h and hot air temperatures of 313, 333 and 348 K, respectively. The product needed to be dried up to 2, 3 and 5.5 h at drying temperature of 348 K to maintain 50%, 30% and 10% of its initial moisture content. The results obtained were compared to existing work, and it was in good agreement.

In this paper, we consider the existence of multiple solutions for a quasilinear Schrodinger equation with Robin boundary condition involving critical Hardy–Sobolev exponent as follows: −Δu−Δ(u2)u+...

In this study, an attempt has been made to explore the influence of four inflow boundary conditions on solute plume migration and compared their concentrations. A numerical model was used to simulate the solute transport for different boundary conditions at the injection well with three different recharge and six hydraulic conductivity scenarios. Dirichlet concentration boundary, constant point source, and two Robin boundary conditions were applied to exemplify their effects on the outcomes. Significant variations were visible in the solute concentration profiles and their respective spreading patterns. Results show discrepancies between solutions obtained from the first-type and the third-type inflow boundary conditions for smaller Peclet numbers. The applicability of the M5′ model tree, a tree-based machine learning approach, was investigated and thereby exploited its capability to interpret the plausible functional dependency among the input parameters. The model tree also produced a dominancy structure within the parameter space with its combined classification and regression features. It was concluded that adopting a general boundary condition or generalization of solutions remained highly challenging, given the different input space parameters dictating a hydrogeological model.

In this paper, the analysis and state-feedback control synthesis for reaction–diffusion linear parameter-varying (LPV) systems with time delays and Robin boundary conditions are addressed. We explo...

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

We investigate the boundary layers of a singularly perturbed reaction-diffusion equation in a 3D channel domain. The equation is supplemented with a Robin boundary condition especially when the smooth function on the boundary, appearing in the Robin boundary condition, depends on the perturbation parameter. By constructing an explicit function, called corrector, which describes behavior of the perturbed solution near the boundary, we obtain an asymptotic expansion of the perturbed solution as the sum of the corresponding limit solution and the corrector, and show the convergence in L 2 of the perturbed solution to the limit solution as the perturbation parameter tends to zero.

This numerical study exclusively focused on the direct two point diagonally multistep block method of order four (2DDM4) in the form of Adams-type formulas. The proposed predictor–corrector scheme was applied in this study to compute two equally spaced numerical solutions for the third-order two point and multipoint boundary value problems (BVPs) subject to Robin boundary conditions concurrently at each step. The optimization of the computational costs was taken into consideration by not resolving the equation into a set of first-order differential equations. Instead, its implementation involved the use of shooting technique, which included the Newton divided difference formula employed for the iterative part, for the estimation of the initial guess. Apart from studying the local truncation error, the study also included the method analysis, including the order, stability, and convergence. The results of eight numerical problems demonstrated and highlighted competitive computational cost attained by the scheme, as compared to the existing method.

FreeHyTE Direct Boundary Method Toolbox is a new computational framework for the solution of interior and exterior boundary value problems in two dimensions using three classes of direct methods: the Boundary Element Method, the Method of Fundamental Solutions and the Trefftz-Herrera Method. The toolbox, currently including solvers for Laplace and Helmholtz boundary value problems, is straightforward to use, featuring a simple graphical user interface and automatic mesh generators, and amenable to extension, as it provides modular computational procedures, directly applicable to other types of boundary elements and differential equations. The toolbox supports the definition of simply or multiply-connected domains, boundary elements of any order, complex wavenumbers, and Dirichlet, Neumann and Robin boundary conditions. FreeHyTE Direct Boundary Method Toolbox is freely distributed under the GNU General Public License and supported by manuals to quickly get new users started.

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, $\mathcal{O}(h^{2})$ O ( h 2 ) and $\mathcal{O}(h^{3})$ O ( h 3 ) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

Solutions of similarity-type for a nonlinear non-classical Stefan problem with temperature-dependent thermal conductivity and a Robin boundary condition are obtained. The analysis of several particular cases are given when the thermal conductivity $ L(f) $ and specific heat $ N(f) $ are linear in temperature such that $ L(f) = \alpha +\delta f $ with $ N(f) = \beta+\gamma f. $ Existence of a similarity type solution also obtained for the general problem by proving the lower and upper bounds of the solution.

In this article, we present a highly-accurate wavelet-based approximation to study and analyze the physical and numerical aspects of two-parameter singularly perturbed problems with Robin boundary conditions. To explore the swiftly changing behavior of such problems, we have used a special type of non-uniform mesh known as Shishkin mesh. Using Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. We comprehensively explain an approach to solve the Robin boundary conditions involving the proposed Haar wavelet scheme. Through rigorous analysis, the order of convergence of the present scheme is shown quadratic and linear in the spatial and temporal directions, respectively. The robustness and proficiency of the contributed scheme are conclusively demonstrated with three test examples. Irrespective of the problem’s geometry, the proposed method is highly accurate and very economical.

This work aims to provide the asymptotic behavior of some class of quasilinear problems posed in a domain with small holes in RN for N>2, as ϵ→0. A nonlinear Robin boundary condition is prescribed ...

This paper describes the longitudinal dispersion of passive tracer molecules injected in a steady, fully developed, viscous, incompressible, laminar flow through an annular pipe with a first order heterogeneous boundary absorption at the outer wall, numerically using layer-adapted meshes. The model is based on steady advection-diffusion equation with Dirichlet and Robin boundary conditions. The solutions are discussed in the form of iso-concentration contours of the tracer molecules in the vertical plane. An artanh transformation is used to convert the infinite domain into a finite one. A combination of central finite difference and 2-point upwind scheme is adopted to solve the governing advection-diffusion equation. It is shown that how the mixing of tracers is affected by the shear flow, aspect ratio and the first-order boundary absorption. When the flow becomes convection dominated, the monotone finite difference on a uniform mesh does not work properly, so a layer-adapted mesh, namely a “Shishkin” mesh, is used to capture the layer phenomena at the different downstream stations. The present results are compared with existing experimental and numerical data and we have earned an excellent agreement with them. It is observed that, due to the use of layer adapted mesh, we have achieved a better agreement with the experimental data than some other previous results available in the literature, especially in the closest downstream location. The results of this study are likely to be of interest to understand the basic mechanism of dispersion process of solute in blood through a catheterized artery with an absorptive arterial wall.

For Schrödinger operators on an interval with either convex or symmetric single-well potentials and Robin or Neumann boundary conditions, the gap between the two lowest eigenvalues is minimized when the potential is constant. We also have results for the p p -Laplacian.

We study the spectrum of the Laplacian on the hemisphere with Robin boundary conditions. It is found that the eigenvalues fall into small clusters around the Neumann spectrum, and satisfy a Szeg\H{o} type limit theorem. Sharp upper and lower bounds for the gaps between the Robin and Neumann eigenvalues are derived, showing in particular that these are unbounded. Further, it is shown that except for a systematic double multiplicity, there are no multiplicities in the spectrum as soon as the Robin parameter is positive, unlike the Neumann case which is highly degenerate. Finally, the limiting spacing distribution of the desymmetrized spectrum is proved to be the delta function at the origin., Comment: 5 figures

The eigenvalues and eigenfunctions of the discrete spectrum for Robin Laplacians in an angle are constructively computed by means of the Sommerfeld integral and of the Malyuzhinets functional equations.

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition: \begin{document}$ \begin{equation} \left\{\begin{array}{lll} \varepsilon \Delta w-\lambda w^{1+\chi} = 0, &\text{in} \ \Omega\\ \nabla w \cdot \vec{n}+\gamma w = 0, & \text{on} \ \partial \Omega \end{array}\right. \end{equation} ~~~~~~~~~~~~~~~~~~~~(*)$\end{document} where \begin{document}$ \Omega \subset {\mathbb R}^N (N\geq 1) $\end{document} is a bounded domain with smooth boundary, \begin{document}$ \vec{n} $\end{document} denotes the unit outward normal vector of \begin{document}$ \partial \Omega $\end{document} and \begin{document}$ \gamma \in {\mathbb R}/\{0\} $\end{document} . \begin{document}$ \varepsilon $\end{document} and \begin{document}$ \lambda $\end{document} are positive constants. The problem (*) is derived from the well-known singular Keller-Segel system. When \begin{document}$ \gamma>0 $\end{document} , we show there is only trivial solution \begin{document}$ w = 0 $\end{document} . When \begin{document}$ \gamma and \begin{document}$ \Omega = B_R(0) $\end{document} is a ball, we show that problem (*) has a non-constant solution which converges to zero uniformly as \begin{document}$ \varepsilon $\end{document} tends to zero. The main idea of this paper is to transform the Robin problem (*) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (*), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.

We are primarily concerned with the absence of positive solutions of the following problem, \[ { ∂ u ∂ t = Δ ( u m ) + V ( x ) u m + λ u q a m p ; in Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 a m p ; in Ω , ∂ u ∂ ν = β ( x ) u a m p ; on ∂ Ω × ( 0 , T ) , \begin {cases} \frac {\partial u}{\partial t}=\Delta ( u^m)+V(x)u^{m}+ \lambda u^q & \text {in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text {in} \quad \Omega ,\\ \frac {\partial u}{\partial \nu }=\beta (x) u & \text {on }\partial \Omega \times (0,T), \end {cases} \] where 0 > m > 1 0>m>1 , V ∈ L l o c 1 ( Ω ) V\in L_{loc}^1(\Omega ) , β ∈ L l o c 1 ( ∂ Ω ) \beta \in L_{loc}^1(\partial \Omega ) , λ ∈ R \lambda \in \mathbb {R} , q > 0 q>0 , Ω ⊂ R N \Omega \subset \mathbb {R}^N is a bounded open subset of R N \mathbb {R}^N with smooth boundary ∂ Ω \partial \Omega , and ∂ u ∂ ν \frac {\partial u}{\partial \nu } is the outer normal derivative of u u on ∂ Ω \partial \Omega . Moreover, we also present some new sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation of our interest.