Dissertations / Theses on the topic 'Integro-Differential'

Thesis (Ph.D.) -- University of Maryland, College Park, 2009.
Thesis research directed by: Applied Mathematics & Statistics, and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2009.
"May 15th, 2009."
Includes bibliographical references (leaves 246-249).
Differential equations are one of the primary tools for modeling phenomena in chemical engineering. While solution methods for many of these types of problems are well-established, there is growing class of problems that lack standard solution methods: partial integro-differential equations. The primary challenges in solving these problems are due to several factors, such as large range of variables, non-local phenomena, multi-dimensionality, and physical constraints. All of these issues ultimately determine the accuracy and solution time for a given problem. Typical solution techniques are designed to handle every system using the same methods. And often the physical constraints of the problem are not addressed until after the solution is completed if at all. In the worst case this can lead to some problems being over-simplified and results that provide little physical insight. The general concept of exploiting solution domain knowledge can address these issues. Positivity and mass-conservation of certain quantities are two conditions that are difficult to achieve in standard numerical solution methods. However, careful design of the discretizations can achieve these properties with a negligible performance penalty. Another important consideration is the stability domain. The eigenvalues of the discretized problem put restrictions on the size of the time step. For "stiff' systems implicit methods are generally used but the necessary matrix inversions are costly, especially for equations with integral components. By better characterizing the system it is possible to use more efficient explicit methods.
(cont.) This work improves upon and combines several methods to develop more efficient methods. There are a vast number of systems that be solved using the methods developed in this work. The examples considered include population balances, neural models, radiative heat transfer models, among others. For the capstone portion, financial option pricing models using "jump-diffusion" motion are considered. Overall, gains in accuracy and efficiency were demonstrated across many conditions.
by Alexander M. Lewis.
Ph.D.

A state space model is developed for a class of integro-partial differential equations of hyperbolic type which arise in viscoelasticity. An approximation scheme is developed based on a spline approximation in the spatial variable and an averaging approximation in the de1ay variable. Techniques from linear semigroup theory are used to discuss the well-posedness of the state space model and the convergence properties of the approximation scheme. We give numerical results for a sample problem to illustrate some properties of the approximation scheme.
Ph. D.

The thesis is devoted to the analysis of elliptic PDEs and related problems. It is mainly focused on the study of qualitative and regularity properties of solutions to integro-differential equations. The study of such equations has attracted much attention recently since they arise naturally in different areas when dealing with processes where long range interaction phenomena appear. The canonical example of integro-differential operators is the fractional Laplacian, which is translation, rotation, and scale invariant. The thesis is divided into three parts. Part I concerns the study of uniqueness and regularity properties of solutions to integro-differential linear problems. First, we prove, by following a nonlocal Liouville-type method, the uniqueness of solutions in the one-dimensional case, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to semilinear problems of Allen-Cahn type. Next, we establish the first boundary regularity result for the Neumann problem associated to the fractional Laplacian. We prove that weak solutions are Hölder continuous up to the boundary by developing a delicate Moser iteration with logarithmic corrections on the boundary. We also establish a Neumann Liouville-type theorem in a half-space, which is used together with a blow-up argument to show higher regularity of solutions. Part II of the thesis is focused on the study of the saddle-shaped solution to the integro-differential Allen-Cahn equation. These solutions, whose zero level set is the Simons cone, are expected to be the simplest minimizer which is not one-dimensional to the local and nonlocal Allen-Cahn equation in high enough dimensions. It plays, thus, the same role as the Simons cone in the theory of minimal surfaces. First, we study the saddle-shaped solution for the fractional problem by using the extension problem. We establish its uniqueness and, in dimensions greater or equal than 14, its stability. As a byproduct, we give the first analytical proof of a stability result for the Simons cone in the nonlocal setting for such dimensions. The key ingredient to prove these results is a maximum principle for the linearized operator. Next, we study saddle-shaped solutions for any rotation invariant and uniformly elliptic integro-differential operator. In this scenario, we need to develop some new nonlocal techniques since the extension approach is not available. In this respect, our main contribution is a characterization of the kernels for which one can develop a theory of existence and uniqueness of saddle-shaped solutions. Under these assumptions, we establish an energy estimate for doubly radial odd minimizers and some properties of the saddle-shaped solution, namely: existence, uniqueness, asymptotic behavior, and a maximum principle for the linearized operator. Finally, in Part III we develop a nonlocal Weirstrass extremal field theory. In analogy to the local theory, we construct a calibration for the nonlocal functional in the presence of a foliation made of solutions when the nonlocal Lagrangian satisfies an ellipticity condition. The model case in our setting corresponds to the energy functional for the fractional Laplacian, for which such a calibration was still unknown. The existence of such a calibration allows us to prove that any leaf of the foliation is automatically a minimizer for its own exterior datum, with no need to have an existence result of minimizers, neither to know their regularity.
La tesis está dedicada al análisis de EDPs elípticas y problemas relacionados. Se centra principalmente en el estudio de propiedades cualitativas y de regularidad de soluciones de ecuaciones integro-diferenciales. El estudio de estas ecuaciones ha recibido mucho interés en los últimos tiempos, ya que aparecen de forma natural en diferentes áreas cuando se tratan fenómenos que involucran interacciones de largo alcance. El operador integro-diferencial canónico es el laplaciano fraccionario, que es invariante por traslaciones, rotaciones y cambios de escala. La tesis se divide en tres partes. La primera trata el estudio de propiedades de unicidad y regularidad para soluciones de problemas lineales integro-diferenciales. En primer lugar, probamos, siguiendo un método no local de tipo Liouville, la unicidad de soluciones en el caso unidimensional, en presencia de una solución positiva o de una solución impar que se anula solo en el origen. Como aplicación, deducimos la no degeneración de soluciones 'layer' (soluciones acotadas y monótonas) de problemas semilineales de tipo Allen-Cahn. A continuación, establecemos el primer resultado de regularidad en la frontera para el problema de Neumann asociado al laplaciano fraccionario. Demostramos que las soluciones débiles son Hölder continuas hasta el borde mediante una delicada iteración de Moser con correcciones logarítmicas en la frontera. También establecemos un teorema de tipo Liouville con condiciones de Neumann en un semiespacio, que se usa junto con un argumento de 'blow-up' para demostrar regularidad de orden superior para las soluciones. La parte II de la tesis se centra en el estudio de la solución de tipo silla para la ecuación integro-diferencial de Allen-Cahn. Se espera que estas soluciones, cuyo conjunto de nivel cero es el cono de Simons, sean el minimizante más simple que no unidimensional para la ecuación local y no local de Allen-Cahn en dimensiones suficientemente altas. Juegan, por tanto, el mismo papel que el cono de Simons en la teoría de superficies mínimas. Primero, estudiamos la solución de tipo silla para el problema fraccionario utilizando el problema de extensión. Establecemos su unicidad y, en dimensiones mayores o iguales a 14, su estabilidad. Como consecuencia, damos la primera prueba analítica de un resultado de estabilidad para el cono de Simons en el marco no local para tales dimensiones. El ingrediente clave para probar estos resultados es un principio del máximo para el operador linealizado. A continuación, estudiamos soluciones de tipo silla para cualquier operador integro-diferencial que sea invariante por rotaciones y uniformemente elíptico. En este escenario necesitamos desarrollar nuevas técnicas no locales, ya que el problema de extensión no está disponible. En este sentido, nuestra principal contribución es la caracterización de los núcleos para los que se puede desarrollar una teoría de existencia y unicidad para las soluciones de tipo silla. Bajo esa condición, establecemos una estimación de energía para minimizantes impares y doblemente radiales así como algunas propiedades para la solución de tipo silla, como existencia, unicidad, comportamiento asintótico y un principio máximo para el operador linealizado. Finalmente, en la Parte III desarrollamos una teoría de campo de extremales de Weirstrass nolocal. En analogía con la teoría local, construimos una calibración para funcionales no locales en presencia de una foliación por soluciones cuando el lagrangiano no local satisface una condición de elipticidad. El caso modelo en este marco es el funcional de energía asociado al laplaciano fraccionario, para el cual aún se desconocía tal calibración. La existencia de una calibración nos permite probar que cualquier hoja de la foliación es automáticamente minimizante para su propio dato exterior, sin necesidad de tener un resultado de existencia de minimizantes, ni conocer su regularidad
Matemàtica aplicada

This article deals with numerical solutions of nonlinear integro-differential convection-diffusion equations using spectral methods. More specifically, the spectral vanishing viscosity method is introduced and analyzed to show that its family of numerical solutions is compact, and that its solutions converge to the vanishing viscosity solutions. The method is implemented in code, and numerical results including qualitative plots and convergence estimates are given. The article concludes with a discussion of some important implementation concerns and recommendations for further work related to the topic.

A two-dimensional model for the annular two-phase flow of water and steam, along with the dryout, in steam generating pipes of a liquid metal fast breeder reactor is proposed. The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for in the model. The mass exchange rate depends on the details of the flow conditions and it is calculated using some simple thermodynamic models. The change of phase at the free surface between the liquid layer and the vapour core is modelled by proposing a suitable Stefan problem. Appropriate boundary conditions for the model, at the onset of the annular flow region and at the dryout point, are stated and discussed. The resulting unsteady nonlinear singular integro-differential equation for the liquid film free surface is solved asymptotically and numerically (using some regularisation techniques) in the steady state case, for a number of industrially relevant cases. Predictions for the length to the dryout point from the entry of the annular regime are made. The influence of the constant parameter values in the model (e.g. the traction r provided by the fast flowing vapour core on the liquid layer and the mass transfer parameter 77) on the length to the dryout point is investigated. The linear stability of the problem where the temperature of the pipe wall is assumed to be a constant is investigated numerically. It is found that steady state solutions to this problem are always unstable to small perturbations. From the linear stability results, the influence on the instability of the problem by each of the constant parameter values in the model is investigated. In order to provide a benchmark against which the results for this problem may be compared, the linear stability of some related but simpler problems is analysed. The results reinforce our conclusions for the full problem.

In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (SDEs) with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (SPDEs) by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.

Mestrado em Matemática Financeira
This dissertation discusses under which conditions we can express the function that represents the option price as the solution of a certain partial integro-differential equation (PIDE) in a exponential Lévy model. The main difference between this case and the Black Scholes case is that there is a non-local term in the equation, which makes the analysis more complicated. Also, we discuss under which conditions we can obtain a Feynman-Kac formula for the case of a pure jump process and discuss the conditions under which option prices are classical solutions of the PIDEs. When such conditions are not verified, we consider the concept of viscosity solutions which only requires that the function representing the option price is continuous. Continuity results for option prices of barrier options are presented for some types of Lévy processes. In addition, we show the same continuity results for processes of finite variation and with no diffusion component. Also, we present some examples in which the function that represents the option price is discontinuous. Moreover, we present a numerical scheme that gives the price of an European put option for the Variance Gamma process. This finite difference scheme was initially proposed by Cont and Voltchkova, to solve numerically the associated PIDE.
Este trabalho discute sob que condições se pode expressar a função que representa o preço de uma opção como solução de uma determinada equação integro-diferencial parcial num modelo exponencial de Lévy. A grande diferença entre o caso aqui considerado e o de Black-Scholes é que existe na equação um termo não local, o que faz com que a análise seja mais complexa. Também é discutido sob que condições se pode obter a fórmula de Feynman Kac para o caso de um processo de saltos puros e sob que condições o preço de uma opção é solução clássica de uma equação integro-diferencial. Quando tais condições não são verificadas, considera-se o conceito de solução de viscosidade, que apenas exige que a função que representa o preço da opção seja contínua. Para alguns tipos de processos de Lévy são apresentados resultados de continuidade para os preços de opções barreira. Para além disso demonstram-se os mesmos resultados para processos de variação finita e sem componente de difusão. Também são apresentados alguns exemplos em que a função que representa o preço da opção é descontínua. É apresentado um esquema numérico que permite obter o preço de uma opção de venda Europeia para o caso do processo "Variance Gamma". Este esquema de diferenças finitas foi proposto inicialmente por Cont e Voltchkova para resolver numericamente a equação integro-diferencial parcial associada.

Magister Scientiae - MSc
Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

e will evaluate forward contracts in the electricity market. A thorough presentation of stochastic analysis for processes with discontinuous paths are provided, and some results concerning these from mathematical finance are stated. Using a Feynman-Kac-type theorem by Pham we derive a partial integro-differential equation giving the forward price from the spot dynamics taken from Geman and Roncoroni. This spot model is regime switching, so we get two equations. These equations are then attempted solved numerically. We suggest the following approach: When implementing boundary-conditions numerically we use values obtained from a Monte Carlo simulation of the spot dynamics to calibrate the boundary.

Traveling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing (decreasing) traveling wave solutions are established. Some faults in previous studies are corrected.

Cette thèse porte sur l'étude qualitative de plusieurs équations paraboliques de type Lotka-Volterra issues de la biologie évolutive et de l'écologie, équations qui prennent en compte un taux de croissance périodique en temps et un phénomène de compétition non locale. Dans une première partie nous étudions d'abord la dynamique des populations phénotypiquement structurées sous l'effet des mutations et de la sélection dans des environnements qui varient périodiquement en temps, puis nous étudions l'impact d'un changement climatique sur ces populations, en considérant que les conditions environnementales varient selon une tendance linéaire, mais de manière oscillatoire. Dans les deux problèmes nous commençons par étudier le comportement en temps long des solutions. Ensuite nous utilisons une approche basée sur les équations de Hamilton-Jacobi pour l'étude asymptotique de ces solutions en temps long lorsque l'effet des mutations est petit. Nous prouvons que lorsque l'effet des mutations disparaît, la densité phénotypique de la population se concentre sur un seul trait (qui varie linéairement avec le temps dans le deuxième modèle), tandis que la taille de la population oscille périodiquement. Pour le modèle de changement climatique nous fournissons également un développement asymptotique de la taille moyenne de la population et de la vitesse critique menant à l'extinction de la population, ce qui est lié à la dérivation d'un développement asymptotique de la valeur propre de Floquet en fonction du taux de diffusion. Dans la deuxième partie, nous étudions quelques exemples particuliers de taux de croissance en donnant des solutions explicites et semi-explicites au problème, et nous présentons quelques illustrations numériques pour le modèle périodique. De plus, étant motivés par une expérience biologique, nous comparons deux populations évoluant dans des environnements différents (constants ou périodiques). En outre, nous présentons une comparaison numérique entre les modèles stochastiques et déterministes pour le phénomène de transfert horizontal des gènes. Dans un contexte Hamilton-Jacobi, nous parvenons à reproduire numériquement le sauvetage évolutif d'une petite population que nous observons dans le modèle stochastique
This thesis focuses on the qualitative study of several parabolic equations of the Lotka-Volterra type from evolutionary biology and ecology taking into account a time-periodic growth rate and a non-local competition term. In the initial part we first study the dynamics of phenotypically structured populations under the effect of mutations and selection in environments that vary periodically in time and then the impact of a climate change on such population considering environmental conditions which vary according to a linear trend, but in an oscillatory manner. In both problems we first study the long-time behaviour of the solutions. Then we use an approach based on Hamilton-Jacobi equations to study these long-time solutions asymptotically when the effect of mutations is small. We prove that when the effect of mutations vanishes, the phenotypic density of the population is concentrated on a single trait (which varies linearly over time in the second model), while the population size oscillates periodically. For the climate change model we also provide an asymptotic expansion of the mean population size and of the critical speed leading to the extinction of the population, which is closely related to the derivation of an asymptotic expansion of the Floquet eigenvalue in terms of the diffusion rate. In the second part we study some particular examples of growth rates by providing explicit and semi-explicit solutions to the problem and present some numerical illustrations for the periodic model. In addition, being motivated by a biological experiment, we compare two populations evolved in different environments (constant or periodic). In addition, we present a numerical comparison between stochastic and deterministic models modelling the horizontal gene transfer phenomenon. In a Hamilton-Jacobi context, we are able to numerically reproduce the evolutionary rescue of a small population that we observe in the stochastic model

ABSTRACT The quest to model wave propagation in various physical systems has produced a large set of diverse nonlinear equations. Nonlinear singular integro-differential equations rank amongst the intricate nonlinear wave equations available to study the classical problem of wave propagation in physical systems. Integro-differential equations are characterized by the simultaneous presence of integration and differentiation in a single equation. Substantial interest exists in nonlinear wave equations that are amenable to the Inverse Scattering Transform (IST). The IST is an adroit mathematical technique that delivers analytical solutions of a certain type of nonlinear equation: soliton equation. Initial value problems of numerous physically significant nonlinear equations have now been solved through elegant and novel implementations of the IST. The prototype nonlinear singular integro-differential equation receptive to the IST is the Intermediate Long Wave (ILW) equation, which models one-dimensional weakly nonlinear internal wave propagation in a density stratified fluid of finite total depth. In the deep water limit the ILW equation bifurcates into a physically significant nonlinear singular integro-differential equation known as the 'Benjamin-Ono' (BO) equation; the shallow water limit of the ILW equation is the famous Korteweg-de Vries (KdV) equation. Both the KdV and BO equations have been solved by dissimilar implementations of the IST. The Modified Korteweg-de Vries (MKdV) equation is a nonlinear partial differential equation, which was significant in the historical development of the IST. Solutions of the MKdV equation are mapped by an explicit nonlinear transformation known as the 'Miura transformation' into solutions of the KdV equation. Historically, the Miura transformation manifested the intimate connection between solutions of the KdV equation and the inverse problem for the one-dimensional time independent Schroedinger equation. In light of the MKdV equation's significance, it is natural to seek 'modified' versions of the ILW and BO equations. Solutions of each modified nonlinear singular integro-differential equation should be mapped by an analogue of the original Miura transformation into solutions of the 'unmodified' equation. In parallel with the limiting cases of the ILW equation, the modified version of the ILW equation should reduce to the MKdV equation in the shallow water limit and to the modified version of the BO equation in the deep water limit. The Modified Intermediate Long Wave (MILW) and Modified Benjamin-Ono (MBO) equations are the two nonlinear singular integro-differential equations that display all the required attributes. Several researchers have shown that the MILW and MBO equations exhibit the signature characteristic of soliton equations. Despite the significance of the MILW and MBO equations to soliton theory, and the possible physical applications of the MILW and MBO equations, the initial value problems for these equations have not been solved. In this thesis we use the IST to solve the initial value problems for the MILW and MBO equations on the real-line. The only restrictions that we place on the initial values for the MILW and MBO equations are that they be real-valued, sufficiently smooth and decay to zero as the absolute value of the spatial variable approaches large values.

Thesis (Ph. D.)--University of Sydney, 1999.
Title from title screen (viewed Apr. 21, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.

Le manuscrit est constitué de deux parties indépendantes.Propriétés des Solutions de Viscosité des Equations Integro-Différentielles.Nous considérons des équations intégro-différentielles elliptiques et paraboliques non-linéaires (EID), où les termes non-locaux sont associés à des processus de Lévy. Ce travail est motivé par l'étude du Comportement en temps long des solutions de viscosité des EID, dans le cas périodique. Le résultat classique nous dit que la solution u(¢, t ) du problème de Dirichlet pour EID se comporte comme ?t Åv(x)Åo(1) quand t !1, où v est la solution du problème ergodique stationaire qui correspond à une unique constante ergodique ?.En général, l'étude du comportement asymptotique est basé sur deux arguments: la régularité de solutions et le principe de maximumfort.Dans un premier temps, nous étudions le Principe de Maximum Fort pour les solutions de viscosité semicontinues des équations intégro-différentielles non-linéaires. Nous l'utilisons ensuite pour déduire un résultat de comparaison fort entre sous et sur-solutions des équations intégro-différentielles, qui va assurer l'unicité des solutions du problème ergodique à une constante additive près. De plus, pour des équationssuper-quadratiques le principe de maximum fort et en conséquence le comportement en temps grand exige la régularité Lipschitzienne.Dans une deuxième partie, nous établissons de nouvelles estimations Hölderiennes et Lipschitziennes pour les solutions de viscosité d'une large classe d'équations intégro-différentielles non-linéaires, par la méthode classique de Ishii-Lions. Les résultats de régularité aident de plus à la résolution du problème ergodique et sont utilisés pour fournir existence des solutions périodiques des EID.Nos résultats s'appliquent à une nouvelle classe d'équations non-locales que nous appelons équations intégro-différentielles mixtes. Ces équations sont particulièrement intéressantes, car elles sont dégénérées à la fois dans le terme local et non-local, mais leur comportement global est conduit par l'interaction locale - non-locale, par exemple la diffusion fractionnaire peut donner l'ellipticité dans une direction et la diffusion classique dans la direction orthogonale.Visualisation et Restauration d'Images par Mouvements de CourbureLe rôle de la courbure dans la perception visuelle remonte à 1954, et on le doit à Attneave. Des arguments neurologiques expliquent que le cerveau humain ne pourrait pas possiblement utiliser toutes les informations fournies par des états de simulation. Mais en réalité on enregistre des régions où la couleur change brusquement (des contours) et en outre les angles et les extremas de courbure. Pourtant, un calcul direct de courbures sur une image est impossible. Nous montrons comment les courbures peuvent être précisément évaluées, à résolution sous-pixelique par un calcul sur les lignes de niveau après leur lissage indépendant.Pour cela, nous construisons un algorithme que nous appelons Level Lines (Affine) Shortening, simulant une évolution sous-pixelique d'une image par mouvement de courbure moyenne ou affine. Aussi bien dans le cadre analytique que numérique, LLS (respectivement LLAS) extrait toutes les lignes de niveau d'une image, lisse indépendamment et simultanément toutes ces lignes de niveau par Curve Shortening(CS) (respectivement Affine Shortening (AS)) et reconstruit une nouvelle image. Nousmontrons que LL(A)S calcule explicitement une solution de viscosité pour le le Mouvement de Courbure Moyenne (respectivement Mouvement par Courbure Affine), ce qui donne une équivalence avec le mouvement géométrique.Basé sur le raccourcissement de lignes de niveau simultané, nous fournissons un outil de visualisation précis des courbures d'une image, que nous appelons un Microscope de Courbure d'Image. En tant que application, nous donnons quelques exemples explicatifs de visualisation et restauration d'image : du bruit, des artefacts JPEG, de l'aliasing seront atténués par un mouvement de courbure sous-pixelique

Neste trabalho estudaremos a existência de soluções fracas, semi-clássicas e clássicas, conceitos introduzidos no texto para uma classe de sistemas integro-diferenciais do tipo neutro com retardamento não limitado modelados na forma d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, onde A é um operador linear fechado densamente definido em um espaço de Banach X, cada B(t) : D(B(t)) ⊂ X → X, t ≥ 0 é um operador linear fechado, a história xt : (-∞, 0] → X, xt(θ) = x(t + θ), pertence a um espaço de fase abstrato B definido axiomaticamente e D, F, g, G : [0, a] × B → X são funções apropriadas. Para obter alguns de nossos resultados, estudamos a existência e propriedades qualitativas de uma família resolvente de operadores lineares limitados (R(t))t≥0, para o sistema integro-diferencial d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, onde (N(t)) t≥0 é uma família de operadores lineares limitados em X. Mencionamos que este tipo de sistemas aparece no estudo da condução de calor em materiais com memória amortecida.
In this work we study the existence of mild, semi-classical and classical solution, concepts introduced be later for a class of abstract neutral functional integrodifferential systems with unbounded delay in the form d/dt D(t, xt) = AD(t, xt) + ∫t0 B(t - s)D(s, xs)ds + g(t, xt), t ∈ (0, a), x0 = φ ∈ B, d/dt (x(t) + F(t, xt)) = Ax(t) + ∫t0 B(t - s)x(s)ds + G(t, xt), t ∈ (0, a), x0 = φ ∈ B, where A : D(A) ⊂ X → X is a closed linear densely defined operator in a Banach space X, each B(t) : D(B(t)) ⊂ X → X, is a closed linear operator, the history xt : (-∞, 0] → X, xt(θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically and D, F, g :[0, a] × B → X are appropriate functions. To establish some of our results, we studied the existence and qualitative properties of a resolvent of bounded linear operators (R(t))t≥0, for a system in the form d/dt (x(t) + ∫t0 N(t - s)x(s)ds) = Ax(t) + ∫t0 B(t - s)x(s) ds, t ∈ (0, a), x(0) = x0, where (N(t)) t≥0 is a family of bounded linear operators on X. We mention that this class of system arise in the study of heat conduction in material with fading memory.

The boundary element method (BEM) has become a powerful method for the numerical solution of boundary-value problems (BVPs), due to its ability (at least for problems with constant coefficients) of reducing a BVP for a linear partial differential equation (PDE) defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution. The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundary-domain integral or integro-differential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zero-order Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundary-only integral which relaxes the “star-shaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, non-homogeneous and variable coefficient diffusion problems.

A Teoria das equações integrais, desde a segunda metade do século XX, tem assumido um papel cada vez maior no âmbito de problemas aplicados. Com isso, surge a necessidade do desenvolvimento de métodos numéricos cada vez mais eficazes para a resolução deste tipo de equação. Isso tem como consequência a possibilidade de resolução de uma gama cada vez maior de problemas. Nesse sentido, outros tipos de equações integrais estão sendo objeto de estudos, dentre elas as chamadas equações integro-diferenciais. O presente trabalho tem como objetivo o estudo das equações integro-diferenciais singulares lineares e não-lineares. Mais especificamente, no caso linear, apresentamos os principais resultados necessários para a obtenção de um método numérico e a formulação de suas propriedades de convergência. O caso não-linear é apresentado através de um modelo matemático para tubulações em um tipo específico de reator nuclear (LMFBR) no qual origina-se a equação integro-diferencial. A partir da equação integro-diferencial um modelo numérico é proposto com base nas condições físicas do problema
The theory of the integral equations, since the second half of the 20th century, has been assuming an ever more important role in the modelling of applied problems. Consequently, the development of new numerical methods for integral equations is called for and a larger range of problems has been possible to be solved by these new techniques. In this sense, many types of integral equations have been derived from applications and been the object of studies, among them the so called singular integro-differential equation. The present work has, as its main objective, the study of singular integrodifferential equations, both linear and non-linear. More specifically, in the linear case, we present our main results regarding the derivation of a numerical method and its uniform convergence properties. The non-linear case is introduced through the mathematical model of boiler tubes in a specific type of nuclear reactor (LMFBR) from which the integro-differential equation originates. For this integro-differential equation a numerical method is proposed based on the physical conditions of the problem

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Esta tese analisa o método de colocação polinomial, para uma classe de equações integro-diferenciais singulares em espaços ponderados de funções contínuas e condições de fronteira não nulas. A convergência do método numérico em espaços com norma uniforme ponderada, é demonstrada, e taxas de convergências são determinadas, usando a suavidade dos dados das funções envolvidas no problema. Exemplos numéricos confirmam as estimativas
This thesis analyses the polynomial collocation method, for a class of singular integro-differential equations in weighted spaces of continuous functions, and non-homogeneous boundary conditions. Convergence of the numerical method, in weighted uniform norm spaces, is demonstrated and convergence rates are determined using the smoothness of the data functions involved in problem. Numerical examples confirm the estimates

We compare an internal state method and a direct Runge-Kutta method for solving Volterra integro-differential equations and Volterra delay differential equations. The internal state method requires the kernel of the Volterra integral to be realizable as an impulse response function. We discover that when applicable, the internal state method is orders of magnitude more efficient than the direct numerical method. However, constructing state representation for realizable kernels can be challenging at times; therefore, we propose a rational approximation approach to avoid the problem. That is, we approximate the transfer function by a rational function, construct the corresponding linear system, and then approximate the Volterra integro-differential equation. We show that our method is convergent for the case where the kernel is nuclear. We focus our attention on time-invariant realizations but the case where the state representation of the kernel is a time-variant linear system is briefly discussed.
Ph. D.

The thesis deals with so-called evolutionary equations, a class of abstract linear operator equations, which cover a huge class of partial differential equation with and without memory. We provide a unified Hilbert space framework for the well-posedness of such equations. Moreover, we inspect the exponential stability of those problems and construct spaces of admissible inital values and pre-histories, on which a strongly continuous semigroup could be associated with the given problem. The theoretical results are illustrated by several examples.

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied.
[ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates.
[CAT] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates.
El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917
TESIS

We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O objetivo deste trabalho é estudar a existência de soluções para equações integrodiferenciais em espaço de Banach. Primeiramente, estudaremos a teoria de Semigrupos de operadores lineares limitados, analisando suas principais propriedades e finalizando com o Teorema de Hille-Yosida, que apresenta condições para que um operador linear seja o gerador infinitesimal de um semigrupo fortemente contínuo. Esta teoria auxiliará no estudo das equações diferenciais abstratas e servirá de motivação para o desenvolvimento de técnicas de resolução para as equações integrodiferenciais, mediante o estudo de uma família de operadores lineares chamados operadores resolventes. Apresentaremos também uma versão do Teorema de Hille-Yosida para os operadores resolventes
The objective of this work is to study the existence of solutions to integrodifferential equations in Banach spaces. First, we will study the theory of Semigroups of bounded linear operators, analyzing their main properties and ending with the Hille-Yosida Theorem, which presents conditions for a linear operator be the infinitesimal generator of a strongly continuous semigroup. This theory will assist in the study of abstract differential equations and will serve as a motivation for the development of techniques for resolution to the integrodifferential equations, through the study of a family of linear operators called resolvent operators. We also have a version of the Hille-Yosida Theorem to resolvent operators