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A boundary element method is proposed for solving an inhomogeneous biharmonic equation with a right-hand side containing the sought function and its derivatives. The accuracy of numerical results obtained for a test problem is analyzed by comparing them with its analytical solution. The flow through a porous medium with a nonuniform permeability distribution based on the Brinkman model is computed. [ABSTRACT FROM AUTHOR]

The error caused by inaccuracy in solving systems of equations by iterative methods is investigated. An upper estimate for an axially symmetric heat conduction equation is found for the error accumulated in several time steps. The estimate shows a linear dependence of the error on the threshold value of a criterion for limiting the number of iterations, a quadratic growth of the error depending on the number of points in space, and its independence of the number of steps in time. A computational experiment shows good agreement of the estimate with real errors at boundary and initial conditions of various types. A quadratic growth for Laplace's equation of the error caused by an accuracy limitation in using an iterative method, depending on the number of points in space , is found empirically. A growth of for the similar error in the biharmonic equation is found. [ABSTRACT FROM AUTHOR]

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost- C 1 , Analysis-Suitable G 1 and the Approximate C 1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C 1 and Analysis-Suitable G 1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost- C 1 and D-Patch provide relatively easy construction on complex geometries. The Almost- C 1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C 1 and Analysis-Suitable G 1 applicable to more complex geometries. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous biharmonic equation. The problem is severely ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose two regularization methods to solve the problem. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in L2 $\mathcal{L}^{2}$ uniformly with respect to the space coordinate under some a priori assumptions on the solution. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability.

Necessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green’s functions.

The biharmonic equation, as well as its nonlinear and inhomogeneous generalizations, plays an important role in engineering and physics. In particular the focusing biharmonic nonlinear Schrödinger equation, and its standing wave solutions, have been intensively investigated. In the present paper we consider the applications of the Laplace-Adomian and Adomian Decomposition Methods for obtaining semi-analytical solutions of the generalized biharmonic equations of the type Δ 2y+α Δ y+ω y+b2+g( y) =f, where α , ω and b are constants, and g and f are arbitrary functions of y and the independent variable, respectively. After introducing the general algorithm for the solution of the biharmonic equation, as an application we consider the solutions of the one-dimensional and radially symmetric biharmonic standing wave equation Δ 2R+R-R2σ +1=0, with σ =constant. The one-dimensional case is analyzed by using both the Laplace-Adomian and the Adomian Decomposition Methods, respectively, and the truncated series solutions are compared with the exact numerical solution. The power series solution of the radial biharmonic standing wave equation is also obtained, and compared with the numerical solution.

Abstract We consider the following nonlinear biharmonic equations: Δ2u−Δu+Vλ(x)u=f(x,u),in RN, $$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$ where Vλ(x) $V_{\lambda }(x)$ is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.

Natural vibrations of a rectangular plate with two clamped and two freely supported edges are under consideration. The Bubnov — Galerkin method is used to calculate the first eigenvalues, and the first eigenvalue in the case of a single trial function is calculated with an error smaller than 1%. The results are compared with known data, and eigenforms are given. [ABSTRACT FROM AUTHOR]

• Violation of the Saint–Venant compatibility condition is considered. • Equation for the Airy stress function in the non-Euclidean model is obtained. • The Airy stress function is calculated through the incompatibility function. • Residual stress are determined in the framework of the non-Euclidean model. We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green's function estimates for the continuous problem which has independent interest. [ABSTRACT FROM AUTHOR]

One of the reasons for the success of the finite element method in Solid Mechanics, among other Applied Sciences, is its versatility to deal with bodies of arbitrary shape. In case the problem at hand is modeled by second‐order partial differential equations with Dirichlet conditions prescribed on a curvilinear boundary, method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for polygonal or polyhedral domains and methods of order greater than one based on standard straight‐edged elements. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. In previous work the author introduced a simple alternative, which bypasses the above drawbacks without eroding qualitative approximation properties. Among other advantages, it can do without curved elements and is based only on polynomial Algebra. The purpose of this paper is two‐fold: On the one hand the mathematical foundations of this method are established, in the framework of a model Poisson problem solved with Lagrange elements; in particular optimal‐rate error estimates both in energy norm and in the mean‐square sense are demonstrated and a numerical comparison with the isoparametric technique is carried out. On the other hand this study is extended to Hermite elements with normal‐derivative degrees of freedom for solving the equation modeling the bending of clamped thin plates with a curvilinear edge. [ABSTRACT FROM AUTHOR]

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems. [ABSTRACT FROM AUTHOR]

Missing data recovery is an important and yet challenging problem in imaging and data science. Successful models often adopt certain carefully chosen regularization. Recently, the low dimensional manifold model (LDMM) was introduced by [S. Osher, Z. Shi, and W. Zhu, Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] and shown to be effective in image inpainting. The authors of [Low Dimensional Manifold Model for Image Processing, Technical report, cam report 16-04, UCLA, Los Angeles, CA, 2016] observed that enforcing low dimensionality on the image patch manifold serves as a good image regularizer. In this paper, we observe that having only the low dimensional manifold regularization is not enough sometimes, and we need smoothness as well. For that, we introduce a new regularization by combining the low dimensional manifold regularization with a higher order CU rvature RE gularization, which we call new regularization CURE for short. The key step of CURE is to solve a biharmonic equation on a manifold. We further introduce a weighted version of CURE, called WeCURE, in a similar manner as the weighted nonlocal Laplacian (WNLL) method [Z. Shi, S. Osher, and W. Zhu, Weighted nonlocal Laplacian on interpolation from sparse data, J. Sci. Comput., 73 (2017), pp. 1164-1177]. Numerical experiments for image inpainting and semisupervised learning show that the proposed CURE and WeCURE significantly outperform LDMM and WNLL, respectively. [ABSTRACT FROM AUTHOR]

A numerical technique is applied to solve a biharmonic equation problem of a steady-state seepage flow of water through a thin layer of soil, under an impermeable dam. Due to the existence of a point boundary singularity and particular boundary conditions in the problem of the present study, the biharmonic equation is selected to be the governing equation. The potential function of the problem is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing biharmonic equation in the Galerkin sense. Implementing Green’s theorem twice, the discretized equations are reduced to boundary integrals. The system of linear equations is completed by weakly enforcing the Dirichlet boundary conditions by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients. The numerical results obtained with the method, indicate high accuracy and very fast convergence with the number of singular functions and the number of Lagrange multipliers. The volume of water per unit time q flowing through the soil in the horizontal direction, under the dam, which is directly proportional to the directional derivative of the potential function, is approximated by the derivatives of the leading terms of the local asymptotic solution expansion. Values of q and the corresponding discharge velocity, obtained with the method, compare favourably with the values given by FEM and soil Mechanics theory. The favourable numerical behaviour of this numerical technique, which in the bibliography is known as singular function boundary integral method (SFBIM), can be exploited in numerous ways. Combining the SFBIM with standard schemes, such as the FEM, appears to be an interesting extension of the method for future work, aiming at enforcing the strengths of the SFBIM in problems with boundary singularities.

Abstract In this work, we implement the mortar spectral element method for the biharmonic problem with a homogeneous boundary condition. We consider a polygonal domain with corners which relies on the mortar decomposition domain technique. We propose the Strang and Fix algorithm, which permits to enlarge the discrete space of the solution by the first singular function. The interest of this algorithm is the approximation of the solution and the leading singular coefficient which has a physical significance in the propagation of cracks. We give some numerical results which confirm the optimality of the order of the error.

Abstract In this paper, we study the following biharmonic equation: {Δ2u−Δu+λV(x)u=α(x)f(u)+μK(x)|u|q−2uin RN,u∈H2(RN), $$ \textstyle\begin{cases} \Delta^{2}u - \Delta u + \lambda V( x)u = \alpha( x) f(u) + \mu K(x) \vert u \vert ^{q-2}u \quad\text{in } \mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{cases} $$ where Δ2u=Δ(Δu) $\Delta^{2}u=\Delta(\Delta u)$, N>4 $N>4$, λ>0 $\lambda>0$, 1

The spatial properties of solutions for a class of thermoelastic plate with biharmonic operator were studied. The energy method was used. We constructed an energy expression. A differential inequality which the energy expression was controlled by a second-order differential inequality is deduced. The Phragme´n-Lindelo¨f alternative results of the solutions were obtained by solving the inequality. These results show that the Saint-Venant principle is also valid for the hyperbolic–hyperbolic coupling equations. Our results can been seen as a version of symmetry in inequality for studying the Phragme´n-Lindelo¨f alternative results.

The laminar viscous fluid flow past a circular cylinder placed in a square cavity of a uniform cross-section is generated by applying injection/suction at the adjacent sidewalls. The temperature on the side walls without suction and on the boundary of the cylinder is kept constant, and constant heat flux is maintained on the walls with suction. The streamline flow pattern and isothermal lines are drawn. The flow is assumed to be Stokesian. Hence, the resulting biharmonic equation is solved for stream function by expressing it in two coupled equations, and a 5-point formula is used to solve these equations. Fictitious nodes are introduced for derivative boundary conditions for stream function by using a central difference scheme, and a 3-point backward difference formula is used for derivative boundary conditions on temperature. [ABSTRACT FROM AUTHOR]

In this work, we focus on the Cauchy problem for the biharmonic equation associated with random data. In general, the problem is severely ill-posed in the sense of Hadamard, i.e, the solution does not depend continuously on the data. To regularize the instable solution of the problem, we apply a nonparametric regression associated with the Fourier truncation method. Also we will present a convergence result. [ABSTRACT FROM AUTHOR]

In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained. [ABSTRACT FROM AUTHOR]

In various fields of applied mathematics, e.g., electrostatics, heat conduction, fluid mechanics, elastostatics, etc., boundary-value problems involving regions described in spheroidal, toroidal, and bispherical coordinate systems reduce to a system of second-order difference equations, whose solution, {xn}T0 with xn E should vanish asymptotically, i.e., limn--xn = 0. There are several methods for constructing such {xn}T0. However, in general, those methods do not guarantee limn--xn = 0. Moreover, in actual computations, they yield an approximate solution {xn}=0 different from the truncated true solution {xn}=0 and coinciding with the solution of the system being truncated at N with xjv+i set to 0. This work establishes sufficient conditions for the existence of an asymptotically vanishing solution to the system and provides the rate of convergence of the solution to the truncated system. Those results are used to analyze systems of second-order difference equations arising in the boundary-value problems in electrostatics, heat conduction, fluid mechanics, and elastostatics when a medium contains an inhomogeneity having the shape of either a torus or two unequal spheres. In particular, when m = 1 in those problems, ||xn -- xn|| decays exponentially as N oc. [ABSTRACT FROM AUTHOR]

We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor. [ABSTRACT FROM AUTHOR]

This study presents a finite element space comprising piecewise quadratic polynomials on quadrilateral grids. This space provides a minimal-degree consistent discretisation for the biharmonic equation. [ABSTRACT FROM AUTHOR]

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G = (V , E) , which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation Δ 2 u − Δ u + (λ a + 1) u = | u | p − 2 u on G = (V , E). Under some suitable assumptions, we prove that for any λ > 1 and p > 2 , the equation admits a ground state solution u λ. Moreover, we prove that as λ → + ∞ , the solutions u λ converge to a solution of the equation { Δ 2 u − Δ u + u = | u | p − 2 u , in Ω , u = 0 , on ∂ Ω , where Ω = { x ∈ V : a (x) = 0 } is the potential well and ∂Ω denotes the boundary of Ω. [ABSTRACT FROM AUTHOR]

• A novel scheme is developed to simulate the thin plate bending problems. • The proposed method is meshless without numerical integration and iteration. • The presented method is quite accurate. This study develops a simple and effective numerical technique, which aims to accurately and quickly address thin plate bending problems. Based on the given boundary conditions, the thin plate homogenization function is constructed and a family of two-parameter homogenization functions are derived. Then, the superposition of homogenization functions method for the thin plate, the clamped plate, and the simply supported plate is obtained, which is meshless without numerical integration and iteration with the merits of easy-to-program and easy-to-implement. Six numerical experiments are employed to verify the effectiveness, accuracy and convergence of the proposed novel strategy. The proposed method is evaluated by the comparisons with the analytical solutions and the referenced solutions. It can be observed that the proposed method is quite accurate for the thin plate, the clamped plate, and the simply supported plate problems. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill‐posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well‐posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A new method based on quadratic spline collocation is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves the solution of an auxiliary biharmonic problem using fast Fourier transforms and the solution of a nonsymmetric Schur complement system using preconditioned BICGSTAB, at a total cost of N 2 log N on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimality of the global accuracy of the method and also superconvergence results, in particular, third-order accuracy in the L ∞ norm of the solution and its fourth-order accuracy at the partition nodes and the collocation points. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

We prove an optimal-order error bound in the discrete H2() norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in n space dimensions, with n 2 f2; : : : ; 7g, whose generalized solution belongs to the Sobolev space Hs()\H2 0 () for 1 2 max(5; n) < s 4, where = (0; 1)n. The result extends the range of the Sobolev index s in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of Hs() into C() in n space dimensions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of biharmonic elliptic differential inclusion in RN. Under various growth conditions on the nonlinearity, new results on existence and multiplicity of solutions are derived. The main tools used in this paper are the nonsmooth version of mountain pass theorem and the generalized gradient for locally Lipschitz functionals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in ℒ2 uniformly with respect to the space coordinate under some a priori assumptions on the solution. [ABSTRACT FROM AUTHOR]

In the paper the analitical expressions for computing stresses in a watered layer have been obtained. It is not required to solve endless systems of equations. [ABSTRACT FROM AUTHOR]

The purpose for this paper is to study heat transfer in a rectangular channel with suction applied at the adjacent two side walls. A two‐dimensional laminar viscous fluid flow is generated due to the application of suction/injection. The other two opposite sides are kept at a constant temperatures, and the walls with suction are maintained at a constant heat flux. The streamlines thus obtained due to the flow and isothermal lines and heat function are analyzed. The regions of high and low frictions are found by drawing contours of the entropy generation number and the Bejan number. The biharmonic equation for stream function is numerically solved by the Finite Difference Method (FDM) using a 13‐point formula, and a five‐point formula is used to solve for all other harmonic equations for temperature, heat function, and pressure. For derivative boundary conditions, the central difference formula with fictitious nodes is used. In the analysis, we note that the corner points are regions of high energy dissipation. The least dissipation of energy is near the wall, where the nondimensional temperature is 1. This paper analyzes heat transfer in the rectangular channel through the heat function and entropy generation number. [ABSTRACT FROM AUTHOR]

The methods involving the functions analytic in a complex plane for plane potential fields inspire the search for the analogous efficient methods for solving the spatial and multidimensional problems of mathematical physics. Many such methods are based on the mappings of hypercomplex algebras. The essence of the algebraic-analytic approach to elliptic equations of mathematical physics consists in the finding of a commutative Banach algebra such that the differentiable functions with values in this algebra have components satisfying the given equation with partial derivatives. The use of differentiable functions given in commutative Banach algebras combines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs. The paper contains the review of results reflecting the formation and the development of the mentioned approach. [ABSTRACT FROM AUTHOR]

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H 2 norm and a L 2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation. [ABSTRACT FROM AUTHOR]

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

We study some boundary-value problems for inhomogeneous biharmonic equation with periodic boundary conditions. These problems are generalization to periodic data of the Neumann-type boundary-value problems considered before by the authors. We obtain existence and uniqueness of solutions for the problems under consideration.

In this work, we are interested in a class of problems of great importance in many areas of industry and engineering. It is the invese problem for the biharmonic equation. It consists to complete the missing data on the inaccessible part from the measured data on the accessible part of the boundary. To solve this ill-posed problem, we opted for the alternative iterative method developed by Kozlov, Mazya and Fomin which is a convergent method for the elliptical Cauchy problems in general. The numerical implementation of the iterative algorithm is based on the application of the boundary element method (BEM) for a sequence of mixed well-posed direct problems. Numerical results are performed for a square domain showing the effectiveness of the algorithm by BEM to produce accurate and stable numerical results.

The model of elastic porous solid medium for simulation of fluid penetration in an elastic porous body has been investigated. Similar processes can occur as a result of exposure of, for example, hard coal seams or deep solid type concrete, glass, etc. to fluid under high pressure. Assuming that an elastic body is a bundle of capillaries, a linear relationship between the filtration coefficient and the first invariant of the stress tensor has been revealed. Therefore, the problem of filtration through a deformable elastic medium has been reduced to two tasks: finding the stress tensor and solving the problem of filtration with the known filtration coefficient. In the general case, both tasks are complicated for analytical studies, but they can be solved numerically, for example, by the method of finite element analysis. The problem is significantly simplified for infinitely long cylindrical bodies. In the present work, the simplest mathematical model of plane elastic stress state in the transverse weightness field within both Hooke's law and Darcy's law with a constant filtration coefficient has been considered. In this case, the elastic deformation is described by the Airy biharmonic function, while filtration by the harmonic function. Based on Green's integral formula, integral relations combined into a single system have been found for the desired functions. The numerical solution has been performed using the method of boundary elements, with the help of which the problem has been reduced to a system of linear equations. Using a round tube, comparative analysis of the numerical and analytical solutions has been carried out. Numerical values have been obtained for the required parameters of an elliptical tube.