Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Two novel numerical methods for solving the two-dimensional fractional Fitzhugh-Nagumo-monodomain model

Fractional mathematical models represent a new approach to modelling complex spatial problems in which there is heterogeneity at many spatial and temporal scales. In this paper, a two-dimensional fractional Fitzhugh-Nagumo-monodomain model with zero Dirichlet boundary conditions is considered. The model consists of a coupled space fractional diffusion equation (SFDE) and an ordinary differential equation. For the SFDE, we first consider the numerical solution of the Riesz fractional nonlinear reaction-diffusion model and compare it to the solution of a fractional in space nonlinear reaction-diffusion model. We present two novel numerical methods for the two-dimensional fractional Fitzhugh-Nagumo-monodomain model using the shifted Grunwald-Letnikov method and the matrix transform method, respectively. Finally, some numerical examples are given to exhibit the consistency of our computational solution methodologies. The numerical results demonstrate the effectiveness of the methods.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Numerical methods for ill-posed, linear problems

A means of assessing the effectiveness of methods used in the numerical solution of various linear ill-posed problems is outlined. Two methods: Tikhonov' s method of regularization and the quasireversibility method of Lattès and Lions are appraised from this point of view.

In the former method, Tikhonov provides a useful means for incorporating a constraint into numerical algorithms. The analysis suggests that the approach can be generalized to embody constraints other than those employed by Tikhonov. This is effected and the general "T-method" is the result.

A T-method is used on an extended version of the backwards heat equation with spatially variable coefficients. Numerical computations based upon it are performed.

The statistical method developed by Franklin is shown to have an interpretation as a T-method. This interpretation, although somewhat loose, does explain some empirical convergence properties which are difficult to pin down via a purely statistical argument.

Fast numerical methods for mixed, singular Helmholtz boundary value problems and Laplace eigenvalue problems - with applications to antenna design, sloshing, electromagnetic scattering and spectral geometry

This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

Hybrid numerical methods for exponential models of growth

A series of novel numerical methods for the exponential models of growth are proposed. Based on these methods, hybrid predictor-corrector methods are constructed. The hybrid numerical methods can increase the accuracy and the computing speed obviously, as well as enlarge the stability domain greatly. (c) 2005 Published by Elsevier Inc.

Numerical Methods for Non-Stationary Stokes Flow

We consider a first order implicit time stepping procedure (Euler scheme) for the non-stationary Stokes equations in smoothly bounded domains of R3. Using energy estimates we can prove optimal convergence properties in the Sobolev spaces Hm(G) (m = 0;1;2) uniformly in time, provided that the solution of the Stokes equations has a certain degree of regularity. For the solution of the resulting Stokes resolvent boundary value problems we use a representation in form of hydrodynamical volume and boundary layer potentials, where the unknown source densities of the latter can be determined from uniquely solvable boundary integral equations’ systems. For the numerical computation of the potentials and the solution of the boundary integral equations a boundary element method of collocation type is used. Some simulations of a model problem are carried out and illustrate the efficiency of the method.

Analysis and Simulation of Transverse Random Fracture of Long Fibre Reinforced Composites

La present tesi proposa una metodología per a la simulació probabilística de la fallada de la matriu en materials compòsits reforçats amb fibres de carboni, basant-se en l'anàlisi de la distribució aleatòria de les fibres. En els primers capítols es revisa l'estat de l'art sobre modelització matemàtica de materials aleatoris, càlcul de propietats efectives i criteris de fallada transversal en materials compòsits. El primer pas en la metodologia proposada és la definició de la determinació del tamany mínim d'un Element de Volum Representatiu Estadístic (SRVE) . Aquesta determinació es du a terme analitzant el volum de fibra, les propietats elàstiques efectives, la condició de Hill, els estadístics de les components de tensió i defromació, la funció de densitat de probabilitat i les funcions estadístiques de distància entre fibres de models d'elements de la microestructura, de diferent tamany. Un cop s'ha determinat aquest tamany mínim, es comparen un model periòdic i un model aleatori, per constatar la magnitud de les diferències que s'hi observen. Es defineix, també, una metodologia per a l'anàlisi estadístic de la distribució de la fibra en el compòsit, a partir d'imatges digitals de la secció transversal. Aquest anàlisi s'aplica a quatre materials diferents. Finalment, es proposa un mètode computacional de dues escales per a simular la fallada transversal de làmines unidireccionals, que permet obtenir funcions de densitat de probabilitat per a les variables mecàniques. Es descriuen algunes aplicacions i possibilitats d'aquest mètode i es comparen els resultats obtinguts de la simulació amb valors experimentals.

Comparison of methodologies for degree-day estimation using numerical methods

O desenvolvimento de projetos relacionados ao desempenho de diversas culturas tem recebido aperfeiçoamento cada vez maior, incorporado a modelos matemáticos sendo indispensável à utilização de equações cada vez mais consistentes que possibilitem previsão e maior aproximação do comportamento real, diminuindo o erro na obtenção das estimativas. Entre as operações unitárias que demandam maior estudo estão aquelas relacionadas com o crescimento da cultura, caracterizadas pela temperatura ideal para o acréscimo de matéria seca. Pelo amplo uso dos métodos matemáticos na representação, análise e obtenção de estimativas de graus-dia, juntamente com a grande importância que a cultura da cana-de-açúcar tem para a economia brasileira, foi realizada uma avaliação dos modelos matemáticos comumente usados e dos métodos numéricos de integração na estimativa da disponibilidade de graus-dia para essa cultura, na região de Botucatu, Estado de São Paulo. Os modelos de integração, com discretização de 6 em 6 h, apresentaram resultados satisfatórios na estimativa de graus-dia. As metodologias tradicionais apresentaram desempenhos satisfatórios quanto à estimativa de grausdia com base na curva de temperatura horária para cada dia e para os agrupamentos de três, sete, 15 e 30 dias. Pelo método numérico de integração, a região de Botucatu, Estado de São Paulo, apresentou disponibilidade térmica anual média de 1.070,6 GD para a cultura da cana-de-açúcar.

An introduction to numerical methods in low-dimensional quantum systems

This is an introductory course to the Lanczos Method and Density Matrix Renormalization Group Algorithms (DMRG), two among the leading numerical techniques applied in studies of low-dimensional quantum models. The idea of studying the models on clusters of a finite size in order to extract their physical properties is briefly discussed. The important role played by the model symmetries is also examined. Special emphasis is given to the DMRG.

Variance Reduction in Analytical Chemistry : New Numerical Methods in Chemometrics and Molecular Simulation

This thesis is based on five papers addressing variance reduction in different ways. The papers have in common that they all present new numerical methods. Paper I investigates quantitative structure-retention relationships from an image processing perspective, using an artificial neural network to preprocess three-dimensional structural descriptions of the studied steroid molecules. Paper II presents a new method for computing free energies. Free energy is the quantity that determines chemical equilibria and partition coefficients. The proposed method may be used for estimating, e.g., chromatographic retention without performing experiments. Two papers (III and IV) deal with correcting deviations from bilinearity by so-called peak alignment. Bilinearity is a theoretical assumption about the distribution of instrumental data that is often violated by measured data. Deviations from bilinearity lead to increased variance, both in the data and in inferences from the data, unless invariance to the deviations is built into the model, e.g., by the use of the method proposed in paper III and extended in paper IV. Paper V addresses a generic problem in classification; namely, how to measure the goodness of different data representations, so that the best classifier may be constructed. Variance reduction is one of the pillars on which analytical chemistry rests. This thesis considers two aspects on variance reduction: before and after experiments are performed. Before experimenting, theoretical predictions of experimental outcomes may be used to direct which experiments to perform, and how to perform them (papers I and II). After experiments are performed, the variance of inferences from the measured data are affected by the method of data analysis (papers III-V).

Comparison between analytical and numerical methods for the assessment of masonry arch bridges: Case study of Clemente Bridge on Savio river

This research has focused on the study of the behavior and of the collapse of masonry arch bridges. The latest decades have seen an increasing interest in this structural type, that is still present and in use, despite the passage of time and the variation of the transport means. Several strategies have been developed during the time to simulate the response of this type of structures, although even today there is no generally accepted standard one for assessment of masonry arch bridges. The aim of this thesis is to compare the principal analytical and numerical methods existing in literature on case studies, trying to highlight values and weaknesses. The methods taken in exam are mainly three: i) the Thrust Line Analysis Method; ii) the Mechanism Method; iii) the Finite Element Methods. The Thrust Line Analysis Method and the Mechanism Method are analytical methods and derived from two of the fundamental theorems of the Plastic Analysis, while the Finite Element Method is a numerical method, that uses different strategies of discretization to analyze the structure. Every method is applied to the case study through computer-based representations, that allow a friendly-use application of the principles explained. A particular closed-form approach based on an elasto-plastic material model and developed by some Belgian researchers is also studied. To compare the three methods, two different case study have been analyzed: i) a generic masonry arch bridge with a single span; ii) a real masonry arch bridge, the Clemente Bridge, built on Savio River in Cesena. In the analyses performed, all the models are two-dimensional in order to have results comparable between the different methods taken in exam. The different methods have been compared with each other in terms of collapse load and of hinge positions.

Numerical methods for the dispersion analysis of Guided Waves

The use of guided ultrasonic waves (GUW) has increased considerably in the fields of non-destructive (NDE) testing and structural health monitoring (SHM) due to their ability to perform long range inspections, to probe hidden areas as well as to provide a complete monitoring of the entire waveguide. Guided waves can be fully exploited only once their dispersive properties are known for the given waveguide. In this context, well stated analytical and numerical methods are represented by the Matrix family methods and the Semi Analytical Finite Element (SAFE) methods. However, while the former are limited to simple geometries of finite or infinite extent, the latter can model arbitrary cross-section waveguides of finite domain only. This thesis is aimed at developing three different numerical methods for modelling wave propagation in complex translational invariant systems. First, a classical SAFE formulation for viscoelastic waveguides is extended to account for a three dimensional translational invariant static prestress state. The effect of prestress, residual stress and applied loads on the dispersion properties of the guided waves is shown. Next, a two-and-a-half Boundary Element Method (2.5D BEM) for the dispersion analysis of damped guided waves in waveguides and cavities of arbitrary cross-section is proposed. The attenuation dispersive spectrum due to material damping and geometrical spreading of cavities with arbitrary shape is shown for the first time. Finally, a coupled SAFE-2.5D BEM framework is developed to study the dispersion characteristics of waves in viscoelastic waveguides of arbitrary geometry embedded in infinite solid or liquid media. Dispersion of leaky and non-leaky guided waves in terms of speed and attenuation, as well as the radiated wavefields, can be computed. The results obtained in this thesis can be helpful for the design of both actuation and sensing systems in practical application, as well as to tune experimental setup.

Coupling time-stepping numerical methods and standard aerodynamics codes for instability analysis of flows in complex geometries

The development of a global instability analysis code coupling a time-stepping approach, as applied to the solution of BiGlobal and TriGlobal instability analysis 1, 2 and finite-volume-based spatial discretization, as used in standard aerodynamics codes is presented. The key advantage of the time-stepping method over matrix-formulation approaches is that the former provides a solution to the computer-storage issues associated with the latter methodology. To-date both approaches are successfully in use to analyze instability in complex geometries, although their relative advantages have never been quantified. The ultimate goal of the present work is to address this issue in the context of spatial discretization schemes typically used in industry. The time-stepping approach of Chiba 3 has been implemented in conjunction with two direct numerical simulation algorithms, one based on the typically-used in this context high-order method and another based on low-order methods representative of those in common use in industry. The two codes have been validated with solutions of the BiGlobal EVP and it has been showed that small errors in the base flow do not have affect significantly the results. As a result, a three-dimensional compressible unsteady second-order code for global linear stability has been successfully developed based on finite-volume spatial discretization and time-stepping method with the ability to study complex geometries by means of unstructured and hybrid meshes

Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose.