О решениях эллиптических дифференциальных уравнений в частных производных второго порядка с неограниченным интегралом Дирихле

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

Magnetic resonance electrical properties tomography (MREPT) aims to visualize the internal high-frequency conductivity distribution at Larmor frequency using the B1 transceive phase data. From the magnetic field perturbation by the electrical field associated with the radiofrequency (RF) magnetic field, the high-frequency conductivity and permittivity distributions inside the human brain have been reconstructed based on the Maxwell’s equation. Starting from the Maxwell’s equation, the complex permittivity can be described as a second order elliptic partial differential equation. The established reconstruction algorithms have focused on simplifying and/or regularizing the elliptic partial differential equation to reduce the noise artifact. Using the nonlinear relationship between the Maxwell’s equation, measured magnetic field, and conductivity distribution, we design a deep learning model to visualize the high-frequency conductivity in the brain, directly derived from measured magnetic flux density. The designed moving local window multi-layer perceptron (MLW-MLP) neural network by sliding local window consisting of neighboring voxels around each voxel predicts the high-frequency conductivity distribution in each local window. The designed MLW-MLP uses a family of multiple groups, consisting of the gradients and Laplacian of measured B1 phase data, as the input layer in a local window. The output layer of MLW-MLP returns the conductivity values in each local window. By taking a non-local mean filtering approach in the local window, we reconstruct a noise suppressed conductivity image while maintaining spatial resolution. To verify the proposed method, we used B1 phase datasets acquired from eight human subjects (five subjects for training procedure and three subjects for predicting the conductivity in the brain).

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

We compare a block successive overrelaxation method (BSOR) with a direct method for solving difference equations derived from fourth order elliptic partial differential equations. The direct method appears to be competitive in time and accuracy for a general problem.

Optimal Alternating Direction Implicit Preconditioners for Conjugate Gradient methods

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

A method for the numerical solution of the time dependent two-dimensional magneto-plasma dynamic coupled problem in the low magnetic Reynolds number case, is presented. The method is based on a time dependent two-dimensional description of the fluid dynamics iteratively coupled to a steady state two-dimensional description of the electrodynamics. This is allowed as the characteristic times of variation of the electrodynamic quantities are much lower than those of the fluid dynamic quantities. The fluid is assumed to be compressible and viscous, and governed by the Navier-Stokes equations. A semi-implicit finite difference pressure scheme has been utilised for the solution of the fluid dynamic problem. The electrodynamics is described by a second order elliptic partial differential equation obtained from the Maxwell equations and generalised Ohm's law. The elliptic equation is solved by means of an exponential fitting finite difference method. In order to test the algorithm described above, the transient of a plasma flow in a channel, caused by the variation of the externally applied magnetic field, has been studied.

The presence of noise in images is unavoidable.It may be formed by image formation process,image recording,image transmission etc.These random distortions make it difficult to perform any required image processing task.High quality noise free remote sensing images are necessary for various applications.Therefore, removal of noise is necessary. The purpose of image denoising is to preserve edges as far as possible while removing noise, making the resulting images approximate the ideal image.This paper proposes a method for denoising remote sensing images using a combination of second order and fourth order partial differential equations.The advantages of both second order and fourth order partial differential equations are utilized here.The image is denoised using second order partial differential equations , fourth order partial differential equations and the combination of both.A comparison of the three methods is also presented.The proposed algorithm smooth out more noise and conserve more detail in the denoising process.

The strong maximum principle revisited

Coal fired power plants contribute significantly to greenhouse gas emission, notably CO2. A novel method to effectively reduce the amount of CO2 emission is to cofire a high fraction of biomass and coal at oxygen enriched environments. However, this technique has not been demonstrated on a large-scale yet. The objective of this PhD research is to perform a detailed modeling study of combustion of a single biomass particle and to establish reduced models which can be used for describing the main characteristics of pyrolyzing and combusting single biomass particles to be employed in design codes of industrial furnaces. To accomplish the goals of this PhD thesis, the research has been carried out in two main stages. First, the sub-processes involved in the biomass combustion are identified and described with a one-dimensional mathematical model based on conservation of mass, energy and momentum. The model encompasses the kinetics of biomass pyrolysis, homogeneous reactions and heterogeneous char oxidation and gasification reactions, coupled with transient transport equations. Subsequently, this model is implemented in an in-house code and a comprehensive numerical study on pyrolysis and combustion of single biomass particles is conducted. The accuracy of the model is examined by comparing its predictions with several experimental data obtained from the literature on pyrolysis and combustion of various types of single biomass particles. The computer code based on the detailed model allows one to observe time and space evolution of several parameters including biomass and char densities, gaseous species mass fractions, porosity, internal pressure, mass flux of volatiles within the pores of the solid matrix, and temperature. The model is used for simulation of combustion of particles of three common shapes; i.e. slab, cylinder and sphere. The results of the detailed modeling study reveal that the combustion of a single biomass particle at the conditions of industrial furnaces (small particles and high heating conditions) consists of three main sub-processes: preheating, pyrolysis, and char oxidation. Therefore, any simplified/reduced particle model should account for these three processes. In the next stage of the project, simplified models are developed to predict the main characteristics of pyrolyzing and combusting single biomass particles. Initially, the preheating stage is modeled using a time and space integral method, which allows one to convert the partial differential form of the heat transfer equation into an algebraic equation. This treatment is then applied to model the pyrolysis process. Two possible regimes are identified: thermally thin and thermally thick particles. A model is established for both regimes, which consists of a set of algebraic equations. This treatment highly simplifies the pyrolysis model so that it can be used in practical applications, which may involve thousands or even millions of particles. The validation of the simplified preheating and pyrolysis models is carried out using various experiments and the results of the detailed model based on partial differential equations (PDEs). The char particle oxidation and gasification processes as the last stages of the particle combustion process, are modeled using the shrinking core approximation. The accuracy of this model is assessed using the experiments reported in past studies. The model is used to study the dynamics of biomass char combustion at oxy-fuel conditions. The effects of the main process parameters on the maximum particle temperature and burnout time are examined. It is found that oxy-fuel combustion with an oxygen mass fraction of 0.3 and higher may lead to a considerable reduction in particle temperature and burnout time compared to the conventional operating case with air as the gasifying agent. In the last stage of this PhD research, the simplified models of biomass particle pyrolysis and char combustion are combined to establish a reduced model for combustion of a single biomass particle. The accuracy of this simplified combustion model is examined using the measured data available in the literature as well as the results of the detailed model. As a conclusion, the simplified models developed in this thesis for pyrolysis and combustion of single biomass particles are efficient enough to capture the main process parameters, and computationally cheaper than the PDE -based models so that they can be used in the design codes of biomass furnaces.

We propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of a corresponding discrete problem and a so-called interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a Cassini oval.

A classical problem in materials science involves the determination of effective transport properties in composites. The simplest problem is perhaps described by the second order elliptic partial differential equation where σ is piecewise constant, supplemented by suitable boundary conditions or by a radiation condition at infinity. This is a standard model for steady-state heat conduction and conductivity. For our purposes, it will be more convenient to rewrite (1) as an interface problem; we seek a potential function u which is harmonic in each component, continuous, and satisfies the continuity of flux condition

In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with C1,1 boundary. We assume that at least one of the material parameters is W1,p for some p > 3. Using regularity theory for second order elliptic partial differential equations, we derive W1,p estimates and Hölder estimates for electric and magnetic fields up to the boundary, together with their higher regularity counterparts. We also derive interior estimates in bianisotropic media.