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Publication# The Adomian Decomposition Method for Nonlinear Partial Differential Equations

Abstract

The goal of this report is to study the method introduced by Adomian known as the Adomian Decomposition Method (ADM), which is used to find an approximate solution to nonlinear partial differential equations (PDEs) as a series expansion involving the recursive solution of linear PDEs. We first describe the method, giving two specific examples with different nonlinearities and show exactly how the method works for these problems. Some analytical convergence results are then given, along with numerical solutions to the examples demonstrating these convergence results. A discussion of parameters inside of these nonlinearities follows, both for polynomial nonlinearities and for the more complicated hyperbolic sine nonlinearity problem. Finally, we compare the ADM with the Picard method, pointing out some important differences and demonstrating them by solving the given examples with both methods and comparing the results.

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In this work, two problems linked to glacier modeling are investigated. We propose an optimisation method for studying the flow of the ice and we present a numerical study about glacier thermal phenomena. In the first chapter of this thesis, we expose the models of these two problems. On one hand, we note that the boundary conditions on the bedrock are misunderstood, which explains it is difficult to obtain an accurate simulation of the motion of the ice. Also we establish a mathematical model where the bedrock boundary conditions depend on a control parameter. The aim of this study is to minimize a cost functional describing the difference between the computed velocity at the surface and the measure done. We study the cost functional with respect to the control parameter and we detail an optimisation method to solve the optimal control problem. On the other hand, we introduce two thermodynamical model governing the temperature and the water content field. The models correspond to a Stefan problem for the temperature and a convection-diffusion equation for the water content. The second chapter deals with the numerical resolution of the optimisation problem. First, a Finite Element Method (FEM) is described to solve the partial differential equations. Then, the algorithms used for the optimal control problem are detailed. Finally, this techniques are applied on two glaciers : Griesgletcher for 2D and Storglaciaren for 3D. The third chapter deals with the numerical resolution of the temperature and the water content models. A FEM is used for each problem. Concerning the temperature problem, the Stefan problem is numerically solved and the results allow to detect a free surface between the temperated ice and the cold ice. The water content field is also simulated. Numerical results are discussed on the Storglaciaren.

New technologies in computer science applied to numerical computations open the door to alternative approaches to mechanical problems using the finite element method. In classical approaches, theoretical developments often become cumbersome and the computer model which follows shows resemblance with the initial problem statement. The first step in the development consists usually in the analysis of the physics of the problem to simulate. The problem is generally described by a set of equations including partial differential equations. This first model is then replaced by successive equivalent or approximated models. The final result consists in a mathematical description of elemental matrices and algorithms describing the matrix form of the problem. The traditional approach consists then in constructing a computer model, generally complex and often quite different from the original mathematical description, thus making further corrections difficult. Therefore, the crucial problem of both the software architecture and the choice of the appropriate programming language is raised. Partially breaking with this approach, we propose a new approach to develop and program finite element formulations. The approach is based on a hybrid symbolic/numerical approach on the one hand, and on a high level software tool, object-oriented programming (supported here by the languages Smalltalk and C++) on the other hand. The aim of this work is to develop an appropriate environment for the algebraic manipulations needed for a finite element formulation applied to an initial boundary value problem, and also to perform efficient numerical computations. The new environment should make it possible to manage al1 the concepts necessary to solve a physical problem: manipulation of partial differential equations, variational formulations, integration by parts, weak forms, finite element approximations… The concepts manipulated therefore remain closely related to the original mathematical framework. The result of these symbolic manipulations is a set of elemental data (mass matrix, stiffness matrix, tangent stiffness matrix,…) to be introduced in a classical numerical code. The object-oriented paradigm is essential to the success of the implementation. In the context of the finite element codes, the object-oriented approach has already proved its capacity to represent and handle complex structures and phenomena. This is confirmed here with the symbolic environment for derivation of finite element formulations in which objects such as expression, integral and variational formulation appear. The link between both the numerical world and the symbolic world is based on an object-oriented concept for automatic programmation of matrix forms derived from the finite element method. As a result, a global environment in which the numerical is capable of evolving, using a language close to the natural mathematical one, is achieved. The potential of the approach is further demonstrated, on the one hand, by the wide range of problems solved in linear mechanics (electrodynamics in 1 and 2D, heat diffusion,…) as well as in nonlinear mechanics (advection dominated 1D problem, Navier Stokes problem), and, on the other hand by the diversity of the formulations manipulated (Galerkin formulations, space-time Galerkin formulations continuous in space and discontinuous in time, generalized Galerkin least-squares formulations).

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.