Hilbert Boundary Value Problem Research Articles

Clogging of toe drain drastically affects phreatic seepage in earth dams

In aged levees, toe (blanket) drains get clogged with time due to seepage-induced suffusion and translocation of fine soil fractions from the upstream to the downstream part of the embankment. These particles deposit on the top of the drain (usually, Terzhagi's graded gravel) as a cake. Also, high hydraulic gradients in the vicinity of the drain move the fine particles into the body of the coarse filter material such that “internal colmation” takes place. In this paper 2-D seepage to a clogged drain is studied experimentally, analytically and numerically. In a sandbox, we illustrate the difference in the position of a phreatic surface and the seepage flow rate between an equipotential toe drain and a clogged one. In the analytical solution, a potential flow model is used and the Neumann (Kirkham-Brock) boundary condition on the clogged drain surface (horizontal segment) is imposed. A circular triangle is mapped conformally onto a reference half-plane, where Hilbert's boundary value problem for a holomorphic function is solved. For a given size of the levee, clogging causes a significant rise of the phreatic surface, although the seepage flow rate drops. In HYDRUS2-D simulations, a FEM-meshed Richards’ equation for a saturated-unsaturated 2-D flow is used for solving in a composite polygon, which mimics a vertical cross-section of a rectangular levee and a clogged-colmated blanket drain. Numerical results are in rough agreement with the analytical model.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

Rymano–Hilberto kraštinis uždavinys daugiamatėms elipsinėms sistemoms

Multidimensional analogues of Cauchy–Riemann quations and of Riemann–Hilbert boundary value problem are studied. Their relation to the scalar boundary value problem is demonstrated.

The Problem Solution on the Propagation of a Griffith Crack Based on the Equations of a Nonlinear Model

On the basis of a nonlinear model of deformation of a crystalline medium with a complex lattice, the problem of the stationary propagation of a Griffith crack under the action of homogeneous expanding stresses is posed and solved. It is shown that the stressed and deformed states of the medium are determined both by external influences on the medium and by the gradients of the optical mode (mutual displacement of atoms). The contributions from these factors are separated. Finding the components of the stress tensor and macro-displacement vector is reduced to solving Riemann–Hilbert boundary value problems. Their exact analytical solutions are obtained.

Workload analysis of a two-queue fluid polling model

AbstractIn this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in $${\mathbb {R}}^{n+1}$$

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems $$\big ($$ for short RHBVPs $$\big )$$ for axially monogenic functions on Euclidean space $${\mathbb {R}}^{n+1},n\in {\mathbb {N}}$$ . With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to $$\big ({\mathcal {D}}-\alpha \big )\phi =0,\alpha \in {\mathbb {R}}$$ .

On the Hilbert problem for semi-linear Beltrami equations

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors-Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincare boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media.

Phreatic seepage around a rectilinear cutoff wall: The Zhukovsky-Vedernikov-Polubarinova-Kochina dispute revisited

The problem of steady 2-D potential Darcian flow from a flat-bottomed reservoir of a coffer or other type of dam, separated from an empty tailwater by an impermeable vertical sheet piling, is one of the pillars in groundwater hydrology and soil mechanics. Analytical solutions to this problem are presented in the books by Forchheimer, Zhukovsky, Vedernikov, Polubarinova-Kochina, Harr, among others. For the case of seepage with a phreatic line, descending from the downstream face of the cutoff, the formulae presented in classical textbooks - despite the simplicity of conformal mappings of the complex potential quarter-plane onto the Zhukovsky half-strip - have lacunae and flaws. We discuss and clean the mismatch in the abovementioned formulae. The classical analytical solutions in the textbooks deal with the regime of no backwater, for which the magnitude of the specific discharge vector at infinity coincides with hydraulic conductivity of the soil. We obtain a general solution for most general flows with backwater, for which the specific discharge at infinity is zero and the free surface has an inflection point in the physical plane. The capillary fringe in the hodograph domain is imaged by a circular cut. The pressure head infinitely deep under the cutoff-wall is infinite. A circular tetragon in the hodograph domain is conformally mapped onto the quadrant in the complex potential plane via a reference plane. In the case with backwater, one extra condition is needed to fix a mathematical solution. For this purpose, the location of a point on the free surface is fixed. Mathematically, two affixes in the reference plane are found by solving a system of two nonlinear equations (in the no backwater case, only one affix is found and one equation is solved). We also examine flow with a phreatic line in a low-permeable layer (e.g. concrete apron of a dam) with an Λ-shaped topology of streamlines, bounded from the left by a cutoff. Seepage from a dam reservoir through a subjacent highly-permeable alluvium induces the pore water motion in the upper layer. A Hilbert boundary-value problem is solved for the complex potential in this cell. In the Kirkham-Brock type boundary condition along the interface between the two layers, the horizontal component of the specific discharge is constant. Phreatic surfaces rising on the downstream face of the cutoff caps Λ−shaped streamlines, which turn around a hinge point on the interface between two layers of contrasting permeability.

Краевая задача Гильберта для одного класса обобщенных аналитических функций с сингулярной линией

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.

Hilbert problem with measurable data for semilinear equations of the Vekua type

We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.

The Problem of Wedge Indenter with Flat-Rounded Bottom Indenting Half-Plane Elastic Body

In this paper, the problem of a wedge indenter with flat-rounded bottom indenting an infinite half-plane elastic body is solved. It can be used to study the penetration force or the surface stress distribution of the cutter when the tool intrudes into an elastic body. First, a first-order continuity method is proposed to eliminate the stress singularity caused by the sharp corners of the wedge indenter with flat bottom. The two-dimensional elastic contact problem is transformed into a Riemann–Hilbert boundary value problem with discontinuous coefficients expressed in the form of complex variables. For the frictionless contact problem, the explicit analytical solution is obtained for contact stress on the half-plane surface. Furthermore, the expression of the stress potential function is also derived to solve the stress field in the half-plane. Verification indicates that the wedge indenter with flat-rounded bottom can degenerate to several well-studied shaped indenters, including the Hertz model and flat-rounded model. The model in this paper is also validated with its numerical counterpart built in the finite element program LS-DYNA. Finally, the sensitivity of different configuration parameters on the normal contact stress and interior stress field in numerical model was investigated.

On a Riemann--Hilbert boundary value problem for (ϕ,ψ)-harmonic functions in ℝ m

Abstract The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m {\Omega\subset\mathbb{R}^{m}} with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ ( 1 + α , Γ ) {\operatorname{Lip}(1+\alpha,\Gamma)} .

An efficient method for singular integral equations of non-normal type with two convolution kernels

This article deals with the solvability and explicit solutions of one class of singular integral equations with two convolution kernels in the non-normal type case. Via using techniques of complex analysis, such equations are transformed into Riemann–Hilbert boundary value problems (R-HPs) with the discontinuous property. We establish a regularity theory and existence of solutions, and obtain the general solutions and the conditions of Noether solvability. Especially, we investigate the asymptotic behaviours of solutions at nodes. Thus, this paper generalizes the theories of integral equations and Riemann–Hilbert boundary value problems.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb{D}$ is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with H\"{o}lder continuous data for generalized analytic functions, i.e., continuous complex-valued functions $f(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial _{\bar{z}}f\,+\,af\,+\ b{% \overline{f}}\,=\,c\,$, where the complex-valued functions $a,b$, and $c$ are assumed to belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb{C}$. Our last paper \cite{GNRY} contained theorems on the existence of nonclassical solutions of the Hilbert boundary-value problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions $f:D\to\mathbb{C}$ such that $\partial_{\bar z}f = g$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincar\'e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G\in L^p, p>2$, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article \cite{GNRY} and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $\partial _{\bar{z}% }f(z)=h(z)q(f(z))$. On this basis, existence theorems were also established for the Poincar\'{e} boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form $\triangle \,U(z)=H(z)Q(U(z))$ with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Propagation of conductive crack along interface of piezoelectric/piezomagnetic bimaterials

This paper investigates the fracture characteristics of a Yoffe conductive crack moving along the interface of piezoelectric (PE)/piezomagnetic (PM) bimaterials. By assuming that the tangential electric- and magnetic-fields along the crack surface are zero and that the speed of the moving crack is lower than the minimum shear wave speed of the bimaterial system, the considered problem can be transformed into a Riemann–Hilbert boundary value problem of vector form. Then, the singularity parameters are exactly solved for different speed regions. In contrast to the anti-plane moving crack model including impermeable and permeable crack-face assumptions along the interface of magnetoelectroelastic (MEE) bimaterials studied before, which shows inverse square-root singularity, three novel kinds of singularities are found as the speed of the moving crack is varied for the present PE/PM interface model, which can be defined as δ1,2 = − 1/2 ± ie (Case 1), δ1,2 = − 1 ± ie (Case 2) and δ1,2 = − 1/2 ± κ (Case 3), and the third parameter δ3 = − 1/2 always holds true for all three cases. Two bimaterial combinations, i.e., BaTiO3/CoFe2O4 and BaTiO3/Terfenol-D, are numerically examined. Different from the piezoelectric case, Case 3 does not appear for BaTiO3/CoFe2O4 bimaterial combination. Above all, the singularity parameters significantly depend on the speed of the moving crack and the material properties of bimaterial systems.

On the Hilbert boundary-value problem for Beltrami equations with singularities

We investigate the Hilbert boundary-value problem for Beltrami equations \( \overline{\partial}f \) = μ∂f with singularities in generalized quasidisks D whose Jordan boundary ∂D consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves.Generally speaking, generalized quasidisks do not satisfy the standard (A)–condition in PDE by Ladyzhenskaya–Ural’tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring–Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations.In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain D of a general nature. As usual, a point in \( \overline{D} \) is called a singularity of the Beltrami equation, if the dilatation quotient Kμ := (1 + |μ|)/(1 – |μ|) is not essentially bounded in all its neighborhoods.Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincaré boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

Logarithmic potential and generalized analytic functions

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Hilbert boundary value problem for generalized analytic functions with a singular line

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.

Неоднородная краевая задача Гильберта с конечным числом точек завихрения логарифмического порядка

Неоднородная краевая задача Гильберта с конечным числом точек завихрения логарифмического порядка