Partial Differential Equations Of Elliptic Type Research Articles

Solving the 2D and 3D nonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

AbstractIn this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single‐parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over‐specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

A homogenization function technique to solve the 3D inverse Cauchy problem of elliptic type equations in a closed walled shell

We solve the inverse Cauchy problem of elliptic type partial differential equations in an arbitrary 3D closed walled shell for recovering unknown data on an inner surface, with the over-specified Cauchy boundary conditions given on an outer surface. We first derive a homogenization function in the 3D domain to annihilate the Dirichlet as well as the Neumann data on the outer surface. Then, we can transform the inverse Cauchy problem to solve a direct problem inside the closed walled shell, using the homogenization technique and the domain type collocation method. The boundary functions are constructed from the 3D Pascal polynomials multiplied by an elementary boundary function, which are adopted as the bases to expand the numerical solution of the transformed elliptic equation. A simple scaling regularization is employed to reduce the condition number of the linear system to determine the expansion coefficients. Several numerical examples are presented to show that the novel method can overcome the highly ill-posed property of the inverse Cauchy problem in the 3D closed walled shell. The proposed algorithm is robust against large noise up to and very time saving to obtain accurate solution.

Collocation method with fractional powers exponential trial functions for singularly perturbed reaction-convection-diffusion equation

We develop a collocation method for solving the singularly perturbed convection-diffusion equation and reaction-diffusion equation of elliptic type partial differential equations. When the singularly perturbed solution is expressed in terms of a set of fractional powers 2D exponential trial functions, and collocate points to satisfy the boundary conditions and the governing equation, we can obtain a small scale normal linear system, which is solved by the Gaussian elimination method or the conjugate gradient method to determine the expansion coefficients. The numerical algorithm with complexity O(n) is very time saving, effective and accurate to find the solutions of highly singularly perturbed reaction-convection-diffusion problems, which are defined in a rectangular domain as well as in an arbitrary domain, and the numerical examples together with an experimental result confirm the assessments.

THE TREFFTZ TEST FUNCTIONS METHOD FOR SOLVING THE GENERALIZED INVERSE BOUNDARY VALUE PROBLEMS OF LAPLACE EQUATION

The issue of data completion is important for the elliptic type partial differential equation. In the inverse Cauchy problem, we need to complete the boundary data by over-specifying Dirichlet and Neumann data on a portion of the boundary. In this paper, we numerically solve the generalized inverse boundary value problems of Laplace equation in a rectangle with one boundary function and two boundary functions missing, which are more difficult than the inverse Cauchy problem. By using the technique of a boundary integral equation method together with a specially designed Trefftz test function, we can complete the boundary data by requiring minimal extra data. Then solving the Laplace equation with the given data and recovered data by the multiple-scale Trefftz method, we can find the numerical solution in the interior nodal points.

A meshless method for solving the nonlinear inverse Cauchy problem of elliptic type equation in a doubly-connected domain

In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the right-hand sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance.

IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

A Residual-Norm Based Algorithm for Solving Determinate/Indeterminate Systems of Non-Linear Algebraic Equations

In this paper, a residual-norm based algorithm (RNBA) is applied to solve determinate/indeterminate systems of nonlinear algebraic equations. The RNBA is derived from a manifold and defined in terms of a squared residual norm and a fictitious time variable, from which a robust iterative algorithm with either fixing or adjusting parameters can be obtained. Besides, some convergent indexes, such as manifold factor A0 and ratio of residual errors S, are defined to indicate the manifold attracting effect. Through the convergent indexes, the convergent mechanism of the RNBA displays a Hopf bifurcation when approximating the true solutions. Several numerical examples, including the root finding in two-, three-variable and in elliptic-type partial differential equations (PDEs), are examined. Comparisons of numerical results and exact solutions show that the proposed algorithm has good computational efficiency and accuracy simultaneously.

ON MATHEMATICAL MODELLING OF METALS DISTRIBUTION IN PEAT LAYERS

In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in multilayered domain. We consider the metals Fe and Ca concentration in the layered peat blocks. Using experimental data the mathematical model for calculation of concentration of metals in different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations (PDEs) of second order with piece-wise diffusion coefficients in the layered domain. We develop here a finite-difference method for solving of a problem of one, two and three peat blocks with periodical boundary condition in x direction. This procedure allows to reduce the 3-D problem to a system of 2-D problems by using circulant matrix.

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.

A reduced spectral function approach for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations with non-Gaussian random fields are considered. A novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated. It is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors. These functions, called as the spectral functions, can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation. Based on the projection in a reduced orthonormal vector basis, a Galerkin error minimization approach is proposed. The constants appearing in the Galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation. A hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution. The method is illustrated using the stochastic nanomechanics of a zinc oxide (ZnO) nanowire deflected under the atomic force microscope (AFM) tip. The results are compared with the results obtained using direct Monte Carlo simulation, classical Neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness.

The stabilized V-cycle method

Algebraic multilevel iterations methods are preconditioning algorithms, to solve elliptic type partial differential equations by iteration which can give both a robust and optimal, or nearly optimal, convergence rate and require per iteration step an arithmetic complexity proportional to the degree of freedoms in the problem. In addition, each iteration step can, in general, be implemented efficiently on massively parallel computers. To stabilize the condition number in the V-cycle version of the method one can use polynomial stabilization or inner solutions at certain, properly chosen levels in the multilevel hierarchy of meshes. The latter is considered here.

Natural convection from an isothermal vertical surface embedded in a thermally stratified high-porosity medium

Non-Darcy effects including convective, boundary and inertia effect on the natural convection boundary layer flow along an isothermal vertical surface embedded in a thermally stratified high-porosity medium are considered and solved. The ambient temperature (less than the vertical surface temperature) increases linearly with the height measured from the leading edge and its slope is proportional to the stratification parameter S. Positive values of S implies a stable, stratified ambient fluid. A flow reversal is observed for S < 1. But it becomes very weak and insignificant when the Darcy and inertia resistances on the flowing fluids in high-porosity medium increase. For S < 1, the flow reversal is evident. Such flow regime, in fact, is modelled by the elliptic type partial differential equations and this situation is not the scope of the present paper. In this study, only S < 1 is considered because the governing boundary layer equations are of parabolic type. It is found that as S increases, a plume flow arises at the end of heating part of the vertical surface where minimum temperature difference (or thermal buoyancy) occurs. For small values of S, the plume flow is not obvious and the flow regime behaves like that of S = 0.

An Inverse Inviscid Method for the Design of Quasi-Three-Dimensional Turbomachinery Cascades

The calculation of the blade shape, when the desired velocity distribution is imposed, has been the object of numerous investigations in the past. The object of this paper is to present a new method suitable for the design of turbomachinery stator and rotor blade sections, lying on an arbitrary axisymmetric stream-surface with varying streamtube width. The flow is considered irrotational in the absolute frame of reference and compressible. The given data are the streamtube geometry, the number of blades, the inlet flow conditions and the suction and pressure side velocity distributions as functions of the normalized arc-length. The output of the computation is the blade shape that satisfies the above data. The method solves an elliptic type partial differential equation for the velocity modulus with Dirichlet and periodic type boundary conditions on the (potential function, stream function)-plane (Φ, Ψ). The flow angle field is subsequently calculated solving an ordinary differential equation along the iso-Φ or iso-Ψ lines. The blade coordinates are, finally, computed by numerical integration. A set of closure conditions has been developed and discussed in the paper. The method is validated on several test cases and a discussion is held concerning its application and limitations.

Numerical calculations on viscous flow fields through cylinder arrays

Abstract The incompressible time independent Navier-Stokes equations for two-dimensional laminar flow through arrays of parallel circular cylinders are solved numerically. A square mesh in the physical field is used to effect a finite difference iterative procedure for solving the elliptic type partial differential equations. Results which exhibit the main characteristics of this type of fluid flow are given for certain regular arrays at cylinder Reynolds numbers of 1 and 20.

Necessary conditions of the extremum in optimal control problems for systems described by elliptic-type partial differential equations

Necessary conditions of the extremum in optimal control problems for systems described by elliptic-type partial differential equations