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doubly nonlinear parabolic equations, positive solutions, Robin boundary condition

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. The authors acknowledge the financial support from the Ministerio de Ciencia e Innovación (MCIN/AEI/10.13039/501100011033) through the grants PID2020-113463RB-C33 and CEX2018-000797-S and the Italian Ministry of Education, University and Research (Grant number 1613 FISR2019_03221, CECOMES)

In this paper, we develop a nonoverlapping domain decomposition method for Stokes equations by mixed finite elements with discontinuous pressures. Both conforming and nonconforming finite element spaces are considered for velocities. With Robin boundary condition set on the interface, the indefinite Stokes problem is reduced to a positive definite problem for the interface Robin transmission data by a Schur complement procedure. Choosing an appropriate relaxation parameter and two parameters in the Robin boundary conditions, the algorithm may be proved optimal. Based on the Robin-type domain decomposition method, a new preconditioner for the Stokes problem is proposed. Numerical results are given to support our theoretical findings.

A boundary optimal control problem arising in time-space SIR epidemic models is treated. In this work we aim with the control of the flux of infected individuals crossing part of boundary. On the other side of the domain, we suppose a nonlinear boundary condition of third kind: nonlinear Robin boundary condition, this condition models immersing individual crossing this part of the boundary of the domain of study. We give the existence and uniqueness of the solution of both state and optimal control problem ending some numerical tests throughout a simple example.

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L 1 , L 2 , and L ∞ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary.

We consider the problem of contact interaction between two elastic bodies one of which has a coating in the form of a thin Timoshenko-type shell connected with the body through a nonlinear Winkler layer. We present a weak formulation of this problem in the form of a nonlinear variational equation. We propose a class of iterative domain decomposition methods that reduce the solution of this equation to the solution, in each iteration, of independent linear variational equations corresponding to problems of elasticity for massive bodies and a problem of the Timoshenko theory of shells for the coating with Robin boundary conditions imposed on the contact boundaries. The conditions of weak convergence of these methods are established. We analyze the numerical efficiency of the obtained algorithms with the use of finite-element approximations.

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger (increasing) convex means. We prove similar results on balls. We also prove a comparison principle on generalized cylinders with mixed boundary conditions (Neumann and Robin)., Improved notation, new mixed results added, typos corrected

Let $n\ge2$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $\Omega$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse H\"older inequality with exponent $p$ or weighted $W^{1,q}$ estimates of solutions with $q\in(n/(n-1),p]$ and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ coefficients, respectively, on bounded Lipschitz domains, $C^1$ domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when $n=3$ in case of bounded $C^1$ domains, also gives an alternative correct proof of some know result. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak--)Orlicz spaces and variable Lebesgue spaces, Comment: 54 pages; Submitted

In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

Purpose The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction. Design/methodology/approach The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence. Findings The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency. Originality/value Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

This article presents a third-order accurate finite difference approximation for a general nonlinear second-order ordinary differential equation subject to Robin boundary conditions. A quasi-variable mesh with three grid points is constructed by introducing a successive mesh ratio parameter $$\eta$$ . For $$\eta = 1$$ , the mesh turns uniform, and the scheme transforms into a fourth-order one. First, a third-order discretization for the given differential equation is derived at each internal grid point, and further, two fourth-order discretizations are designed at the boundary points. The convergence analysis of the proposed method is discussed in the entire solution domain including two boundary points. Since the scheme involves ‘off-step’ mesh points, it is directly applicable to singular problems, which is its primary advantage. Moreover, the flexibility in the choice of $$\eta$$ enables us to construct a denser grid in the region of boundary layer and a sparser one outside it; hence, the boundary layer problems are effectively dealt with. This is further demonstrated by successful application of the proposed technique over twelve problems of physical significance including singularly perturbed, singular, nonlinear viscous Burgers’ equations and boundary layer problems. Computational results validate the order of the proposed technique, and a comparison with the already existing results in the recent past reveals its superiority.

The mathematical model of the transport and diffusion of ions in biological channels is considered. It is described by the three-dimensional nonlinear evolution classical Poisson–Nernst–Planck (cPNP) system of partial differential equations with nonlinear coupled boundary conditions. In particular the Chang–Jaffe (CJ) conditions are given on the input and output of a channel. The Robin boundary conditions on a potential are taken. Theorems on the existence, uniqueness and nonnegativity of local weak solutions, in the suitable Sobolev spaces, are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

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.

We focus on a system of two competing species with cross-diffusion and self-diffusion. By constructing an appropriate auxiliary function, we derive the sufficient conditions such that there are no coexisting steady-state solutions to the model. It is worth noting that the auxiliary function constructed above is applicable to Dirichlet, Neumann and Robin boundary conditions.

In this paper, we study a free boundary problem modeling solid tumor growth with vasculature which supplies nutrients to the tumor; this is characterized in the Robin boundary condition. It was recently established [Discrete Cont. Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold value μ ⁎ such that the unique radially symmetric stationary solution is linearly stable under non-radial perturbations for 0 μ μ ⁎ and linearly unstable for μ > μ ⁎ . In this paper we further study the nonlinear stability of the radially symmetric stationary solution, which introduces a significant mathematical difficulty: the center of the limiting sphere is not known in advance owing to the perturbation of mode 1 terms. We prove a new fixed point theorem to solve this problem, and finally obtain that the radially symmetric stationary solution is nonlinearly stable for 0 μ μ ⁎ when neglecting translations.

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

An isogeometric boundary element method (IGABEM) is proposed for solving the steady-state heat transfer problems with concentrated/surface internal heat sources. The isogeometric boundary element method (IGABEM) possesses the advantages of both the isogeometric analysis (IGA) and the boundary element method (BEM), the non-uniform rational B-spline (NURBS) basis functions used in the (computer-aided design) CAD system are flexible and stable in dealing with irregular boundaries, due to which the NURBS basis functions are applied to exactly reconstruct the boundary geometry of the analysis domain, and the physical variables are also approximated by the NURBS as the standard IGA does. Dirac delta function is introduced in the present IGABEM for dealing with the concentrated point heat sources, and a local coordinate system is proposed for the domain integration over line heat source based on the NURBS basis functions. As to the domain integrals caused by the surface heat sources, the radial integration method (RIM) is used to transform the domain integrals to boundary ones without any internal cells for the original version of IGABEM. Several numerical examples involving Dirichlet, Neumann and Robin boundary conditions are analyzed, and circular region with point/line heat source(sources), global and local surface heat sources within multiply connected regions are also considered. Verification of the proposed method has been gained by comparing the present results with those obtained by several other researchers, and solutions achieved by analytical expression together with the fine-meshed ANSYS model are also utilized for the purpose of comparison. Meanwhile, the computational efficiency and convergence of the present method is evaluated by comparing the IGABEM with the finite element method (FEM) as well as the traditional BEM. Good performance of the IGABEM is observed. Finally, heat transfer within a simplified printed circuit board (PCB) with both concentrated (line and point) and surface heat generations is studied, the temperature distribution is achieved, which verifies the applicability of the proposed technoque in practical engineering.

In this paper, we consider Carleman estimates for the damped fourth-order plate operators ∂ t t − ρ ∂ t Δ + Δ 2 in a bounded smooth domain with Robin boundary conditions. Because the appearance of ...

To quantify fluorescence imaging of biological tissues, we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissues. We begin by giving a concise mathematical argument to derive coupled diffusion equations with the Robin boundary condition as an approximation of the radiative transfer system. Then by using this coupled system of equations as a model for the fluorescence imaging, we have a nonlinear inverse problem to identify the absorption coefficient in this system. The associated linearized inverse problem is to ignore the absorbing effect on the excitation field. We firstly establish the estimates of errors on the excitation field and the solution to the inverse problem, which ensures the reasonability of the model approximation quantitatively. Some numerical verification is presented to show the validity of such a linearizing process quantitatively. Then, based on the analytic expressions of excitation and emission fields, the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for the absorption coefficient in the special form, revealing the physical difficulty of the 3-dimensional imaging model by the back scattering diffusive system.

A mathematical model and numerical method for simulation of the continuous casting process in a variable in time domain are presented. The variable geometry of the slab is caused by the change in time of the width of the mould. The mathematical model of the process is a Stefan problem with prescribed convection and non-linear Robin boundary condition. Considered differential equation is approximated by a finite difference scheme, which is constructed in several steps. First, a semi-discrete problem is constructed using the method of characteristics with respect to the time variable. Then, at each time level, the current elliptic problem in a curvilinear domain is replaced by a problem in the parallelepiped domain using the fictitious domain method. Finally, the boundary-value problem in the parallelepiped domain is approximated by a finite-difference scheme. The constructed fully discrete problem in algebraic form is a system of nonlinear equations containing a diagonal monotone operator and a linear part with a symmetric and positive definite $$M$$ -matrix. To solve the resulting system of nonlinear algebraic equations, well-known iterative solution methods can be applied.

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $\delta$ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to $\infty$ in $\mathbb{C}$: for each vertex $v$ with a Robin parameter $\alpha \in \mathbb{C}$ for which $\mathrm{\Re}\,\alpha \to -\infty$ sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like $-\alpha^2/\mathrm{deg}\,v^2$, while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at $v$; if $\mathrm{Re}\,\alpha$ remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh--Weyl M-functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s)., Comment: 16 pages, 1 figure. Revised version: the proof of Lemma 6.2 has been expanded and a mistake in Theorem 4.2 and Corollary 4.3 has been corrected

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem:−div∇v1+|∇v|2=f(x,v,∇v)inΩ,a0v+a1∂v∂ν=0on∂Ω,\left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right.withΩ\text{Ω}an open ball inℝN{{\mathbb{R}}}^{N}, in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the functionfallow us to complement or improve several results in the literature.

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ ( 0 , 1 ) . A ( 3 − α ) th-order numerical formula (called the L2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of ( 3 − α ) th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α ∈ ( 0 , 1 ) . Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results.

We consider a spectral homogenization problem for the linear elasticity system posed in a domain $\varOmega $ of the upper half-space $\mathbb{R}^{3+}$ , a part of its boundary $\varSigma $ being in contact with the plane $\{x_{3}=0\}$ . We assume that the surface $\varSigma $ is traction-free out of small regions $T^{\varepsilon }$ , where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function $M(x)$ and a reaction parameter $\beta (\varepsilon )$ that can be very large when $\varepsilon \to 0$ . The size of the regions $T^{\varepsilon }$ is $O(r_{\varepsilon })$ , where $r_{\varepsilon }\ll \varepsilon $ , and they are placed at a distance $\varepsilon $ between them. We provide all the possible spectral homogenized problems depending on the relations between $\varepsilon $ , $r_{\varepsilon }$ and $\beta (\varepsilon )$ , while we address the convergence, as $\varepsilon \to 0$ , of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on $\varSigma $ . New capacity matrices are introduced to define these strange terms.

Two new variants of approximate analytical solution of the one-phase Stefan problem with a convective boundary condition at a fixed boundary are proposed. These approaches are based on the use of new integral relations forming infinite sequences. It is shown that the most exact variant of solving the Stefan problem with a convective boundary condition is to refuse from the classical Stefan condition at the free boundary and to replace it with its integral relation. By the example of solving the test Stefan problem with a Robin boundary condition, having an exact analytical solution, it is shown that the proposed approach is much more exact and efficient compared to the known variants of the integral computational scheme, including the heat-balance integral method allowing the Stefan condition at the free boundary to be satisfied. The solutions obtained with the use of the square-law and cubic polynomials are presented. As for the test problem using the cubic polynomial, the relative error in determining the free boundary comprises hundredths and thousandths of percent. In this case, at the time instant t = 1, the relative error in determining the temperature profile is e T = 0.075 %.

In the present work, an inverse problem for the heat equation in two dimensional space with Robin boundary condition that involving a new fractional derivative, namely, Atangana-Baleanu approach with non-local and non-singular kernel is considered. An explicit solution set { u ( x , y , t ) , a ( t ) } of the given inverse problem is obtained by using the eigenfunctions expansion method and the integral overdetermination condition. Under some assumptions the existence, uniqueness of the suggested solution, and its continuous dependence on the data are proved.

For the nonlinear Emden–Fowler equation, a singular Cauchy problem and singular two-point boundary value problem on the half-line $$r \in [0, + \infty )$$ and on an interval $$r \in [0,R]$$ with a Dirichlet boundary condition at the origin and with a Robin boundary condition at the right endpoint of the interval are considered. For special parameter values, the given boundary value problem corresponds to the Thomas–Fermi model of the charge density distribution inside a spherically symmetric cooled heavy atom occupying a confined or infinite space, where $$R$$ denotes the boundary of a compressed atom and grows to infinity for a free atom. For the boundary value problem on the half-line, a new parametric representation of the solution is obtained that covers the entire range of argument values, i.e., the half-line $$r \in [0, + \infty )$$ , with the parameter $$t$$ running over the unit interval. For analytic functions involved in this representation, an algorithm for explicit computation of their Taylor coefficients at $$t = 0$$ is described. As applied to the Thomas–Fermi problem for a free atom, corresponding Taylor series expansions are given and they are shown to converge exponentially on the unit interval $$t \in [0,1]$$ at a rate higher than that for an earlier constructed similar representation. An efficient analytical-numerical method is presented that computes the solution of the Thomas–Fermi problem on the half-line with any prescribed accuracy not only in an neighborhood of $$r = + \infty $$ , but also at any point of the half-line $$r \in [0, + \infty )$$ . For the Cauchy problem set up at the origin, a new formula for the critical value of the derivative that corresponds to the solution of the problem on the half-line is derived. It is shown in a numerical experiment that this formula is more efficient than the Majorana formula. For the solution of the Cauchy problem with a positive derivative at the origin, a parametrization is obtained that ensures that the boundary conditions of the singular boundary value problem on the interval $$r \in [0,R]$$ are satisfied with a suitable $$R > 0$$ . An efficient analytical-numerical method for solving this Cauchy problem is constructed and numerically implemented.

In this paper, a second order numerical scheme for solving a singularly perturbed convection diffusion problem with Robin boundary conditions is proposed. The numerical scheme is a combination of the fitted operator finite difference and the backward Euler finite difference methods. These are designed in order to solve respectively the spatial derivatives and the time derivative. Using some properties of the discrete problem the methods are analysed for convergence. Richardson extrapolation technique is used to improve the accuracy and also accelerate the convergence of the method. Numerical simulations are carried out to confirm the theoretical findings in the analysis before and after extrapolation.