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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed by Kassab and Divo [Engineering Analysis with Boundary Elements 1996;18:273]. The goal of this communication is to revisit and clarify two key features of the formulation of Kassab and Divo. First, the contention that the integral equation formulation proposed by them does not possess the desired boundary-only character is made and substantiated; it is shown in particular that Eq. (10) therein does not hold owing to the fact that a crucial requirement for the fundamental solution, Eqs. (5c) and (5d) therein, is actually not met. Second, the limiting process associated with a vanishing neighbourhood in connection with the particular kernel function used therein is revisited.

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$ ∫ S exp ( t Δ ) ( f 1 S ) d V = ∫ S f d V - t π ∫ ∂ S f d A + o ( t ) , t → 0 + , and explicit expressions for similar expansions involving other powers of $$\sqrt{t}$$ t . By the same method, we also obtain short-time asymptotics of $$\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$$ ∫ S exp ( t m Δ m ) ( f 1 S ) d V , $$m \in \mathbb N$$ m ∈ N , and more generally for one-parameter families of operators $$t \mapsto k(\sqrt{-t\Delta })$$ t ↦ k ( - t Δ ) defined by an even Schwartz function k.

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

The sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction., Comment: 29 pages

Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability., Comment: Appears in the 14th Conference on Web and Internet Economics (WINE), 2018

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche’s method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid., Comment: This is a corrected version of the published paper https://doi.org/10.1007/s11118-021-09947-8. Proofs of Lemmas 9.3 and 9.4 had minor gaps (estimates of integrals I_2 and IV in Lemma 9.3 and integral IV in Lemma 9.4 were incorrect). These are now fixed. We also corrected some typos

We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs.arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), theasymptotic homogeneouscase as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or thepower-lawcase as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

PurposeThe purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodologyFor the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

The availability of engineering models to estimate the risk from space debris is essential for space missions. According to current research, cumulative flux calculation is mostly carried out based on the equal-width interval discretization. The method discretizes the volume around the Earth into cells defined in earth centered inertial coordinates. The resulting debris flux onto a target object is shown to depend on the chosen size of the cells. To avoid a discretization error, this must be accounted for. In order to present reliable flux predictions for space mission, the algorithm improvement is an ongoing topic for the related research field. The aim of this study was to examine the discretization error during the cumulative flux determination process. Both the effect of interval step length and the orbital boundary are under investigation. Several typical orbits are selected as examples here and the 2018/01/03 TLE (Two Line Element) data published by the US Space Surveillance Network is used as the debris background in this paper. Furthermore, the Interval Distance-Based method for Discretization (IDD) is adopted in this paper. A position-centered flux determination method is introduced based on the IDD method. According to the example analysis, the IDD used in the flux calculation process provides results which are less affected by the interval step-size setup; and the orbital boundary has no effect on the calculation process. In other words, the discretization error is significantly reduced. The position-centered method provided a possible suggestion for the improvement of space debris environment models.

The method of fundamental solutions is a more and more popular meshless method for solving boundary or initial–boundary value problems. The most important issue in this method is the determination of the positions of the source points. The accuracy of the method depends strongly on the distribution of the source points. In this paper placement of these points for transient heat conduction problems is studied. The problems are initial–boundary value problems and they are considered in a time-space domain. Because of that, the placement of the source points differs from the classical distribution of the source points for boundary values problems. In the paper, four different possible sources distributions are considered for 1D, 2D and 3D transient heat conduction problems. The results show very good accuracy in case of the source points placed in a space much bigger than the considered region, additionally with the negative time coordinate.

The paper presents the problem of modeling the uncertainty of the boundary shape in boundary problems (described by Laplace’s equation) and proposes a method for solving so-defined problems. The uncertainty of the boundary shape is modeled using interval numbers. The interval coordinates of control points, defined using classical and directed interval arithmetic, are considered in this paper. However, because of some disadvantages of well-known interval arithmetic, the authors propose a modification of directed interval arithmetic. This arithmetic is also used in the proposed modification of the traditional parametric integral equation system (PIES) method (previously used for exactly defined problems) to solve so-defined problems. Control points of the appropriate curves, necessary to define the boundary shape, are directly included in the mathematical formalism of the mentioned method. The developed interval method is tested on problems of various shapes, modeled with linear and curvilinear segments. The correctness of the obtained solutions is verified using proposed alternative methods. The obtained solutions indicate a very high potential of the proposed method in solving problems with an uncertainly defined boundary shape.

In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ℝ + n for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in T n and ℝ n respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

The paper is concerned with the classical Flamant problem of the theory of elasticity. The classical solution of this problem obtained by A. Flamant in the nineteenth century specifies the stresses and the displacements induced in a half-plane by a concentrated force applied to the half-plane boundary in the normal direction. Being presented in numerous textbooks in the theory of elasticity, this solution provides results that can be qualified as paradoxical and which traditionally are left without proper comments in the literature. Particularly, the half-plane displacements are singular at the point of the force application and are infinitely increasing with a distance from this point. The displacement of the boundary in the boundary direction is not continuous and experiences a jump at the point of the force application. A boundary element under any force, irrespective of how low it is, rotates at the point of the force application by 90 $$^{\circ }$$ which does not correspond to basic assumptions of the linear theory of elasticity; hence, the classical solution cannot be qualified as consistent. The consistent solution of the problem is constructed in the paper on the basis of the generalized theory of elasticity the equations of which are obtained for the solid element that has small but not infinitesimal dimensions. As a result, these equations provide regular solutions of the problems that are singular in the classical theory of elasticity. The obtained generalized solution of the Flamant problem demonstrates regular behavior of the displacements which are not singular at the point of the force application. The solution is supported by experimental results obtained for the plate of silicon rubber simulating the half-plane.

We assume that $$n\ge 3$$ , $$u \in C^{2}( \mathbb {B}^{n},\mathbb {R}^{n}) \cap C(\overline{\mathbb {B}}^{n},\mathbb {R}^{n})$$ is a solution to the hyperbolic Poisson equation $$\Delta _{h}u=\psi $$ in $$\mathbb {B}^{n}$$ with the boundary condition $$u|_{\mathbb {S}^{n-1}}=\phi $$ , where $$\Delta _{h}$$ is the hyperbolic Laplace operator and $$\psi \in C( \mathbb {B}^{n},\mathbb {R}^{n})$$ . In Chen et al. (Calc Var Partial Differ Equ 57:32, 2018), the first, the second, and the last author of this paper, together with Rasila, studied expressions of u, and proved that $$u=P_{h}[\phi ]-G_{h}[\psi ]$$ , where $$P_{h}[\phi ]$$ and $$G_{h}[\psi ]$$ denote the Poisson integral of $$\phi $$ and the Green integral of $$\psi $$ with respect to $$\Delta _h$$ , respectively. With the assumption $$|\psi (x)|\le M(1-|x|^{2})$$ $$(M\ge 0$$ is a constant), the Lipschitz-type continuity of u was also investigated. As a continuation, in this paper, we first consider the existence of the solutions, and demonstrate that if $$\phi \in L^{\infty }(\mathbb {S}^{n-1}, \mathbb {R}^{n})$$ , $$\psi \in L^{\infty }(\mathbb {B}^{n}, \mathbb {R}^{n})$$ , $$\int _{\mathbb {B}^{n}} (1-|x|^2)^{n-1} |\psi (x)|\mathrm{d}\tau (x)

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { ( − Δ ) p s u ( x ) = f ( u ( x ) ) , x ∈ Ω , u ( x ) > 0 , x ∈ Ω , u ( x ) = 0 , x ∈ R n ∖ Ω , where ( − Δ ) p s is the fractional p-Laplacian defined as (0.1) ( − Δ ) p s u ( x ) = C n , s , p P . V . ∫ R n | u ( x ) − u ( y ) | p − 2 [ u ( x ) − u ( y ) ] | x − y | n + s p d y with 0 s 1 and p ≥ 2 . We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f ( ⋅ ) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.

In this paper, we investigate steady incompressible Euler flows with nonvanishing vorticity in a planar bounded domain. Let q be a harmonic function that corresponds to an irrotational flow. This paper proves that if q has k isolated local extremum points on the boundary, then there exist two kinds of steady Euler flows with small vorticity supported near these k points. For the first kind, near each maximum point the vorticity is positive and near each minimum point the vorticity is negative. For the second kind, near each minimum point the vorticity is positive and near each maximum point the vorticity is negative. Moreover, near these k points, the flow is characterized by a semilinear elliptic equation with a given profile function in terms of the stream function. The results are achieved by solving a certain variational problem for the vorticity and studying the limiting behavior of the extremizers.

This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages

In this paper stability of numerical solution of Fredholm integral equation of the first kind is studied for radial basis kernels which possess positive Fourier transform. As a result, the equivalence relation between strong and weak forms of partial differential equations (PDEs) is proved for some special radial test functions. Also the stability of boundary elements method (BEM) is proved analytically for Laplace and Helmholtz equations by obtaining Fourier transform of singular fundamental solutions applied in BEM. Analytical result presented in this paper is an extension of stability idea of radial basis functions (RBFs) used to interpolate scattered data described by Wendland in [51] . Similar to the interpolation, it is proved here mathematically that integral operators which have radial kernels with positive Fourier transform are strictly positive definite. Thanks to the stability idea presented in [51] , a positive lower bound for eigenvalues of these integral operators is found here, explicitly.

In this paper we study a new problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain. There are many works devoted to study problems for the second order mixed parabolic-hyperbolic and elliptic-hyperbolic type equations in rectangular domains with two gluing conditions with respect to second argument and with boundary value conditions on all borders of the domain. In studying the unique solvability of this problem, it becomes necessary to specify an additional condition on the hyperbolic boundary of the domain. For this reason, the considering problem became unresolved in an arbitrary rectangular domain. In this paper, we were able to remove this restriction by setting three gluing conditions for the second argument.

The work is devoted to the fundamental problem of studying the solvability of the initial-boundary value problem for a pseudo-parabolic equation (also called Sobolev type equations) with a fairly smooth boundary. In this paper, the initial-boundary value problem for a quasilinear equation of a pseudoparabolic type with a nonlinear Neumann–Dirichlet boundary condition is studied. From a physical point of view, the initial-boundary-value problem we are considering is a mathematical model of quasi-stationary processes in semiconductors and magnetics, taking into account the most diverse physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions of tasks boundary conditions of which are linear with respect to the function and its derivatives. Among these methods, Galerkin’s method leads to the simplest calculations. In the paper, by means of the Galerkin method the existence of a weak solution of a pseudoparabolic equation in a bounded domain is proved. The use of the Galerkin approximations allows us to get an estimate above the time of the solution existence. Using Sobolev ’s attachment theorem, a priori solution estimates are obtained. The local theorem of the existence of the solution has been proved. The uniqueness of the weak generalized solution of the initial-boundary value problem of quasi-linear equations of pseudoparabolic type is proved on the basis of a priori estimates.A special place in the theory of nonlinear equations is taken by the range of studies of unlimited solutions, or, as they are otherwise called, modes with exacerbation. Nonlinear evolutionary problems that allow unlimited solutions are globally intractable: solutions increase indefinitely over a finite period of time. Sufficient conditions have been obtained for the destruction of its solution over finite time in a limited area with a nonlinear Neumann–Dirichle boundary condition.

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

This paper is related to some dynamical aspects of a class of predator–prey interactions incorporating cannibalism and Allee effects for non-overlapping generations. Cannibalism has been frequently observed in natural populations, and it has an ability to alter the functional response concerning prey–predator interactions. On the other hand, from dynamical point of view cannibalism is considered as a procedure of stabilization or destabilization within predator–prey models. Taking into account the cannibalism in prey population and with addition of Allee effects, a new discrete-time system is proposed and studied in this paper. Moreover, existence of fixed points and their local dynamics are carried out. It is verified that the proposed model undergoes transcritical bifurcation about its trivial fixed point and period-doubling bifurcation around its boundary fixed point. Furthermore, it is also proved that the proposed system undergoes both period-doubling and Neimark–Sacker bifurcations (NSB) around its interior fixed point. Our study demonstrates that outbreaks of periodic nature may appear due to implementation of cannibalism in prey population, and these periodic oscillations are limited to prey density only without leaving an influence on predation. To restrain this periodic disturbance in prey population density, and other fluctuating and bifurcating behaviors of the model, various chaos control methods are applied. At the end, numerical simulations are presented to illustrate the effectiveness of our theoretical findings.

This paper uses Lie symmetry methods to analyze boundary crossing probabilities for a large class of diffusion processes. We show that if the Fokker–Planck–Kolmogorov equation has non-trivial Lie symmetry, then the boundary crossing identity exists and depends only on parameters of process and symmetry. For time-homogeneous diffusion processes we found the necessary and sufficient conditions of the symmetries’ existence. This paper shows that if a drift function satisfies one of a family of Riccati equations, then the problem has nontrivial Lie symmetries. For each case we present symmetries in explicit form. Based on obtained results, we derive two-parametric boundary crossing identities and prove its uniqueness. Further, we present boundary crossing identities between different process. We show, that if the problem has 6 or 4 group of symmetries then the first passage time density to any boundary can be explicitly represented in terms of the first passage time by a Brownian motion or a Bessel process. Many examples are presented to illustrate the method.

In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator, , 2018) to parabolic case and also the theory for the parabolic single obstacle problem in Caffarelli et al. (J Am Math Soc 17(4):827–869, 2004) to double obstacle case. New difficulties in the theory which are generated by the characteristic of parabolic PDEs and the existence of the upper obstacle are discussed in detail. Furthermore, the thickness assumptions to have the regularity of the free boundary are carefully considered.

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2 + 1 Levin–Wen model to obtain 1 + 1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

This paper presents a novel SPH scheme for modelling incompressible and divergence-free flow with a free surface (IDFSPH) associated with semi-analytical wall boundary conditions. In line with the projection method, the velocity field is decoupled from the pressure field in the momentum equation. A Poisson equation, serving as the pressure solver, is obtained by which pressure field is decoupled completely from the velocity field. In particular, an exact projection scheme is deployed to fulfil the requirement of the divergence-free velocity field. The condition of incompressibility is satisfied by iteratively updating the density field till the convergence. The two-equation k–e model is employed to describe the turbulence effects in Newtonian flows. It is shown that the discretised SPH schemes have the feature of both linear and angular momentum conservations. The semi-analytical wall method implements the appropriate integrals to evaluate the boundary contributions to the mass and momentum equations. In comparison to the boundary particle methods, it can greatly enhance the feasibility and efficiency with the complex geometries. The algorithm presented within this paper is applied to several academic test cases for which either analytical results or simulations with other methods are available. The comparisons verify that this scheme is provided with convincing efficiency and extensive applicability.

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included.

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartlomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.

The current paper considers the boundedness of solutions to the following quasilinear Keller–Segel model (with logistic source) (KS) u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + μ ( u − u 2 ) , x ∈ Ω , t > 0 , v t − Δ v = u − v , x ∈ Ω , t > 0 , ( D ( u ) ∇ u − χ u ⋅ ∇ v ) ⋅ ν = ∂ v ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , where Ω ⊂ R N ( N ≥ 1 ) is a bounded domain with smooth boundary ∂ Ω , χ > 0 and μ ≥ 0 . One novelty of this paper is that we find a new a-priori estimate ∫ Ω u χ max { 1 , λ 0 } ( χ max { 1 , λ 0 } − μ ) + − e ( x , t ) d x , so that, we develop new L p -estimate techniques and thereby obtain the boundedness results, where C G N and λ 0 ≔ λ 0 ( γ ) are the constants which are corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.2 ) and the maximal Sobolev regularity (see Lemma 2.3). To our best knowledge, this seems to be the first rigorous mathematical result which indicates the relationship between m and μ χ that yields the boundedness of the solutions, where m is the exponent of diffusion term D ( u ) . The above-mentioned results have significantly improved and extended previous results of several authors.

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: { div ( A ( x , ∇ u ) ) = div ( | F | p − 2 F ) in Ω , u = σ on ∂ Ω , where Ω ⊂ R n ( n ≥ 2 ), the nonlinearity A is a monotone Caratheodory vector valued function defined on W 0 1 , p ( Ω ) for p > 1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F ∈ L p ( Ω ; R n ) , the problem is set up with general Dirichlet boundary data σ ∈ W 1 , p ( Ω ) . In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works [45] , [52] , [53] and references therein.

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u] s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m 0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u 0) < 0, where J(u 0) denotes the initial energy. The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in (1.13).

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

In this paper, we consider convergence rates to solutions for the damped system of compressible adiabatic flow through porous media with boundary effect. Compared with the results obtained by Pan, the better convergence rates are obtained in this paper. Our approach is based on the technical time-weighted energy estimates.

Accurate computation of stress intensity factors for two-dimensional cracks by a dual interpolation boundary face method (DiBFM) is presented in this paper. Dual interpolation method combines traditional element polynomial interpolation and moving least-squares approximation. Dual interpolation elements are obtained by adding virtual nodes on the boundary of traditional discontinuous elements, and they unify traditional continuous and discontinuous elements. With the dual interpolation elements, both the continuous and discontinuous fields can be approximated, and the accuracy of interpolation increases by two orders compared with the corresponding discontinuous elements. In addition, a new singular element based on the dual interpolation method for modeling displacement fields around the crack tip is proposed in this paper. The stress intensity factors are extracted from the relative displacements between the crack surfaces. Using the DiBFM combined with the proposed singular element, accurate results of the stress intensity factors can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

In this paper, a boundary node method is presented to study wave propagation in laminated composite plates. The Zig-Zag theory of Cho and Parmerter is used for deriving the governing equations of laminated plates. To the best of the authors’ knowledge, this study is the first one employing the theory in non-stationary dynamic plate problems addressing wave propagation issues. For this theory, there is no information about the Green’s functions and thus the presented method can be considered as an alternative to the boundary integral method. With the use of exponential basis functions (EBFs), the response of the structure is first found in the frequency domain and finally the time-domain response is obtained using inverse Fourier transformation. The EBFs are found so that they satisfy the governing equations in the frequency domain. For the first time, in this paper, we shall present explicit relations for the EBFs in the frequency domain for Cho and Parmerter’s Zig-Zag theory. The coefficients of the EBFs are found through the collocation of the boundary conditions. The dynamic analysis of some composite laminated plates is presented, and the results are compared to those obtained from the dynamic analysis of two-/three-dimensional finite element method (FEM). We shall discuss the capabilities and limitations of the theory and the solution method. The capability of the method, in the analysis of problems excited by high-frequency loads, is shown in the solution of wave propagation problems for which the FEM needs excessive number of elements and thus it is practically impossible to be applied.

For the exterior Dirichlet problems (EDP) of the Helmholtz equation in 2D, when the Sommerfeld radiation condition is satisfied, there always exists the unique solution. Denote the unbounded domain S ∞ outside of a bounded simply-connected domain, where the interior boundary Γ i n is non-circular. The standard MFS is first studied by using the complex Hankel function H 0 ( 1 ) ( k r ) as the fundamental solutions (FS). However, some solutions can not be obtained correctly (e.g., the spurious eigenvalues called). This paper is devoted to explore the spurious eigenvalues and their removals. For the non-circular interior boundary Γ i n , the error analysis is briefly made for the standard MFS, and some guidance is provided to bypass the spurious eigenvalues. Denote two nodes: P = ( ρ , θ ) and Q = ( R , ϕ ) , and r = | P Q ‾ | = R 2 + ρ 2 − 2 R ρ cos ⁡ ( θ − ϕ ) . To completely eliminate all spurious eigenvalues, we solicit the new combined FS: ( ∂ ∂ R ± i k ) H 0 ( 1 ) ( k r ) with i = − 1 , and propose the new modified MFS. The new algorithms are simple, and the strict analysis may be made. The bounds of errors are derived, and the polynomial convergence rates can be achieved. The bounds of condition numbers are derived for circular Γ ∘ i n only, to display the exponential growth via the number of the new FS used. Numerical experiments are carried to support the analysis made. This paper is the first time for the MFS to explore the spurious eigenvalues and their removals, accompanied with strict analysis of errors and stability.

The paper presents an efficient numerical technique for 3D-modeling of seismic stability of large buried structures. This technique allows considering the subsoil-structure contact interaction, gravity field effects, the inhomogeneous structure of soil and the varieties location of the earthquake hypocenter. As part of this technique is substantiated sizing of the soil-structure computational domain and continuum model for hard and soft soil foundations describing. The numerical technique for determining the kinematic conditions at the lower boundary of the computational domain from the experimental accelerations at the soil surface is given, offered special non-reflecting waves boundary conditions. The methods described above and algorithms for seismic resistance solving have been implemented in certified software package “Dynamica-3”, parallelization of the algorithm has been held according to the spatial domain decomposition principle. The developed numerical technique allows correctly posing the problem of seismic vibrations of buried structure, reducing computing costs and increasing the efficiency of numerical studies of Earthquake Engineering. Through this, the multiple recalculation of the task with different action scenarios generated from experimental seismograms by probabilistic methods became technically possible. The results of these calculations allow reflecting the experience of many earthquakes, which increases reliability of the estimates. The paper presents the model calculations results of seismic stability of large-sized buried structure—the mutual vertical and horizontal displacement of soil and building walls—which can then be used to assess strength of adjacent underground pipelines. A number of numerical experiments on seismic vibrations of nearby large-sized structures have been carried out to assess the mutual influence of structures on their behavior in an earthquake. It is established that the influence of nearby large-sized objects on seismic vibrations of structures differs for different buildings.

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal $$\mathrm {SLE}_\kappa ( \kappa -6)$$ with $$\kappa =16/3$$. Our proof follows the classical excursion construction of $$\mathrm {SLE}_\kappa (\kappa -6)$$ processes in the continuum, and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from that of Kemppainen and Smirnov (Conformal invariance of boundary touching loops of FK–Ising model. arXiv:1509.08858, 2015; Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527, 2016) as it only relies on the convergence to the chordal $$\mathrm {SLE}_{\kappa }$$ process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: One important emphasis of this paper is to carefully write down some properties which are often considered folklore in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: We end the paper with a detailed sketch of the convergence to radial $$\mathrm {SLE}_\kappa ( \kappa -6)$$ when $$\kappa =16/3$$ as well as the derivation of Onsager’s one-arm exponent 1 / 8.

Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Findings Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Practical implications Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. Originality/value In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.