К ЧИСЛЕННОМУ РЕШЕНИЮ ОБРАТНЫХ ЗАДАЧ ДЛЯ ЛИНЕЙНОГО ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

В статье рассматриваются обратные задачи для линейного параболического уравнения с неизвестными коэффициентами в правой части. К данным задачам, в частности, приводятся краевые задачи, с нелокальными условиями. Отдельно рассмотрены случаи, когда идентифицируемые коэффициенты зависят либо только от временной переменной, либо только от пространственной координаты. Предлагается методика численного решения задач с использованием метода прямых. Приводятся результаты численных экспериментов, проведенных на тестовых задача

Efficient low rank approximations for parabolic control problems with unknown heat source

An inverse problem of finding an unknown heat source for a class of linear parabolic equations is considered. Such problems can typically be converted to a direct problem with non-local conditions in time instead of an initial value problem. Standard ways of solving these non-local problems include direct temporal and spatial discretization as well as the shooting method, which may be computationally expensive in higher dimensions. In the present article, we present approaches based on low-rank approximation via Arnoldi algorithm to bypass the computational limitations of the mentioned classical methods. Regardless of the dimension of the problem, we prove that the Arnoldi approach can be effectively used to turn the inverse problem into a simple initial value problem at the cost of only computing one-dimensional matrix functions while still retaining the same accuracy as the classical approaches. Numerical results in dimensions d=1,2,3 are provided to validate the theoretical findings and to demonstrate the efficiency of the method for growing dimensions

Numerical solution of inverse problems in mechanics using the boundary element method

Due to the ill-posed nature of inverse problems, it is difficult to obtain solutions using well known analytical and numerical techniques. The use of boundary element method as a numerical technique to solve inverse problems is quite new. In this work, the algorithms for the solution of two kinds of inverse problems are examined in detail. For the first kind, the shape and location of a part of the boundary is unknown; and for the second kind, the boundary condition is not specified on a part of the boundary;Boundary value problems with partially unknown boundary are ill-posed. To solve these problems additional information is necessary. Over-specified boundary data in the form of experimentally measured quantities can be used as additional information for solving the problem. An algorithm, based on the boundary element method and non-linear optimization techniques, is proposed to solve this inverse problem. Using the overspecified boundary data, a functional is formed which involves parameters describing the unknown boundary. Minimization of this functional with respect to these parameters determines the unknown boundary. The performance of this scheme is examined through two problems. It is shown that the algorithm performs well even for complex shapes of the unknown boundary;For the problems in which the specified boundary conditions are insufficient, experimentally obtained data at some internal points are used as additional conditions. The boundary is divided into straight boundary elements and the unknown boundary conditions are represented as unknowns at the nodes of the boundary elements. It is shown that, for practical reasons, the number of nodes where the boundary condition is not specified is usually larger than the number of probes used for obtaining interior data. This results in an under-determined system of linear equations. A regularization method is used to solve these equations. The scheme, when applied to several example problems, showed satisfactory performance. Few guidelines for the placement of the temperature probes in the interior of the domain are developed through numerical experiments

Adaptive Methods for Modeling Chemical Transport in the Earth's atmosphere

This work is devoted to analysis and development of efficient adaptive algorithms for problems related to the transport of chemical species in the Earth’s atmosphere from data of remote-sensing instruments. Focal point of the thesis is the assessment of different types of errors by a posteriori error analysis. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to discretization within a succesive mesh adaptation process. The presented adaptive algorithms are applicable for a wide range of problems

Certain Developments of Laguerre Equation and Laguerre Polynomials via Fractional Calculus

Recently, much interests have been paid in studying fractional calculus due to its effectiveness in modeling many of the natural phenomena. Motivated essentially by the success of the applications of the orthogonal polynomials, this paper is mainly devoted to developing Laguerre equation and Laguerre polynomials in the fractional calculus setting. We provide some type of generalizations of the classical Laguerre polynomials, via conformable fractional calculus. We start by solving the fractional Laguerre equation in the sense of conformable calculus about the fractional regular singular point. Next, we write the conformable fractional Laguerre polynomials (CFLPs), through various generating functions. Subsequently, Rodrigues’ type representation formula of fractional order is reported, besides certain types of recurrence relations are then developed. The conformable fractional integral and the fractional Laplace transform, and the orthogonal property of CFLPs, are established. As an application, we present a numerical technique to obtain solutions of interesting differential equations in the frame of conformable derivative. For this purpose, a new operational matrix of the fractional derivative of arbitrary order for CFLPs is derived. This operational matrix is applied together with the generalized Laguerre tau method for solving general linear multi-term fractional differential equations (FDEs). The method has the advantage of obtaining the solution in terms of the CFLPs’ parameters. Finally, some examples are given to illustrate the applicability and efficiency of the proposed method

Inverse problems for blood perfusion identification

In this thesis we investigate a sequence of important inverse problems associated with the bio-heat transient flow equation which models the heat transfer within the human body. Given the physical importance of the blood perfusion coefficient that appears in the bio-heat equation, attention is focused on the inverse problems concerning the accurate recovery of this information when exact and noisy measurements are considered in terms of the mass, flux, or temperature, which we sampled over the specific regions of the media under investigation. Five different cases are considered for the retrieval of the perfusion coefficient, namely when this parameter is assumed to be either constant, or dependent on time, space, temperature, or on both space and time. Theanalytica:l and numerical techniques that arc used to investigate the existence and uniqueness of the solution for this inverse coefficient identification are embedded in an extensiveú computational approach for the retrieval of the perfusion coefficient. Boundary integral methods, for the constant and the time-dependent cases, or Crank-Nicolson-type global schemes or local methods based on solutions of the first-kind integral equations, in the space, temperature, or space and time cases, are used in conjunction either with Gaussian mollification or with Tikhonov regularization methods, which arc coupled with optimization techniques. Analytically, a number of uniqueness and existence criteria and structural results are formulated and proved

Nonlinear magnetic reconnection

In many astrophysical problems magnetic reconnection plays a major role. Despite this reconnection theory remains incompletely understood, partly due to the strong non-linearity of the governing equations and the resulting difficulties in demonstrating analytical solutions. This thesis examines some fundamental aspects of reconnection theory: namely, the dynamics of driven and spontaneously reconnecting systems. We first consider the dynamics of a driven reconnecting system by numerically modelling a configuration consisting of two oppositely oriented flux systems with a variety of different boundary conditions and internal parameters. The results indicate that the rate of reconnection is chiefly dependent on the magnetic Reynolds number, but that the plasma flow is weakly dependent on this parameter, being more affected by the curvature of Incoming magnetic field. Scaling laws for the dimensions of the diffusion region are derived, and the existence of several reconnection regimes is shown. Using the same computer code we also simulate tearing modes in Cartesian geometry under different boundary conditions. By imposing a suitable perturbation a magnetic island is generated. The plasma flows show marked differences for the different boundary conditions implemented. Lastly, we examine some aspects of the coalescence instability. The usual flux function taken to represent a tearing node Island in the linear growth phase is shown to be erroneous, and we derive a correct expression. We show that under certain conditions there exists a threshold to coalescence that depends on the island wavenumbers and the associated perturbation

Effect of high order interpolation in the stability and efficiency of the time-integration process in vorticity-velocity CFD algorithms

The numerical solution of the incompressible Navier-Stokes Equations offers an effective alternative to the experimental analysis of Fluid-Structure interaction i.e. dynamical coupling between a fluid and a solid which otherwise is very complex, time consuming and very expensive. To have a method which can accurately model these types of mechanical systems by numerical solutions becomes a great option, since these advantages are even more obvious when considering huge structures like bridges, high rise buildings, or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the KLE to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ODE time integration schemes and thus allows us to tackle the various multiphysics problems as separate modules. The current algorithm for KLE employs a structured or unstructured mesh for spatial discretization and it allows the use of a self-adaptive or fixed time step ODE solver while dealing with unsteady problems. This research deals with the analysis of the effects of the Courant-Friedrichs-Lewy (CFL) condition for KLE when applied to unsteady Stoke’s problem. The objective is to conduct a numerical analysis for stability and, hence, for convergence. Our results confirmthat the time step ∆t is constrained by the CFL-like condition ∆t ≤ const. hα, where h denotes the variable that represents spatial discretization

Numerical Boundary Condition Procedures

Topics include numerical procedures for treating inflow and outflow boundaries, steady and unsteady discontinuous surfaces, far field boundaries, and multiblock grids. In addition, the effects of numerical boundary approximations on stability, accuracy, and convergence rate of the numerical solution are discussed

Effect of mesh distortion on the accuracy of high order vorticity-velocity CFD approaches

The numerical solution of the incompressible Navier-Stokes equations offers an alternative to experimental analysis of fluid-structure interaction (FSI). We would save a lot of time and effort and help cut back on costs, if we are able to accurately model systems by these numerical solutions. These advantages are even more obvious when considering huge structures like bridges, high rise buildings or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the Kinematic Laplacian Equation (KLE) to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ordinary differential equations (ODE) time integration schemes, allowing us to tackle each problem as a separate module. The current algortihm for the KLE uses an unstructured quadrilateral mesh, formed by dividing each triangle of an unstructured triangular mesh into three quadrilaterals for spatial discretization. This research deals with determining a suitable measure of mesh quality based on the physics of the problems being tackled. This is followed by exploring methods to improve the quality of quadrilateral elements obtained from the triangles and thereby improving the overall mesh quality. A series of numerical experiments were designed and conducted for this purpose and the results obtained were tested on different geometries with varying degrees of mesh density