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The famous Black-Scholes partial differential equation is one of the most widely used and researched equations in modern financial engineering to address the complex evaluations in the financial markets. This thesis investigates a numerical technique, using a fourth-order discretization in time and space, to solve a generalized version of the classical Black-Scholes partial differential equation. The numerical discretization in space consists of a fourth order centered difference approximation in the interior points of the spatial domain along with a fourth order left and right sided approximation for the points near the boundary. On the other hand, the temporal discretization is made by implementing a Runge-Kutta order four (RK4) method. The designed approximations are analyzed numerically with respect to stability and convergence properties.

Cette thèse concerne l'étude de certains problèmes elliptiques d'ordre 4 et, notamment, des propriétés qualitatives des solutions. Ces problèmes apparaissent dans de nombreux domaines, par exemple dans la théorie des plaques et dans la géométrie conforme, et, comparés à leurs homologues du deuxième ordre, ils présentent des difficultés intrinsèques, surtout liées à l'absence de principe de maximum. Premièrement on étudie la positivité des solutions dans le cas des conditions au bord de Steklov, qui sont intermédiaires entre les conditions de Dirichlet et de Navier. Elles apparaissent naturellement dans l'étude des minimiseurs de la fonctionnelle de Kirchhoff-Love, qui représente l'énergie d'une plaque encastrée soumise à l'action d'une force extérieure, en fonction d'un paramètre $\sigma$. On trouve des conditions suffisantes sur le domaine pour que les minimiseurs de la fonctionnelle soient positifs. De plus, pour ces domaines on étudie une version généralisée de la fonctionnelle. En utilisant des techniques variationnelles, on examine l'existence et la positivité des états fondamentaux, ainsi que leur comportement asymptotique pour les valeurs pertinentes de $\sigma$. Dans la deuxième partie de la thèse on établit des estimations uniformes a priori pour des problèmes semi linéaires du quatrième ordre dans $\mathbb R^4$, et donc avec des non linéarités exponentielles. On considère des conditions au bord soit de Dirichlet soit de Navier et on suppose que les non linéarités sont positives et sous-critiques. Nos arguments combinent des estimations uniformes près du bord et une analyse de blow-up. Enfin, en utilisant la théorie du degré, on obtient l'existence d'une solution. / This thesis concerns the study of fourth-order elliptic boundary value problems and, in particular, qualitative properties of solutions. Such problems arise in various fields, from plate theory to conformal geometry and, compared to their second-order counterparts, they present intrinsic difficulties, mainly due to the lack of the maximum principle. In the first part of the thesis, we study the positivity of solutions in case of Steklov boundary conditions, which are intermediate between Dirichlet and Navier boundary conditions. They naturally appear in the study of the minimizers of the Kirchhoff-Love functional, which represents the energy of a hinged thin and loaded plate in dependence of a parameter $\sigma$. We establish sufficient conditions on the domain to obtain the positivity of the minimizers of the functional. Then, for such domains, we study a generalized version of the functional. Using variational techniques, we investigate existence and positivity of the ground states, as well as their asymptotic behaviour for the relevant values of $\sigma$. In the second part of the thesis we establish uniform a-priori bounds for a class of fourth-order semi linear problems in $\mathbb R^4$, and thus with exponential non linearities. We considered both Dirichlet and Navier boundary conditions and we suppose our non linearities positive and subcritical. Our arguments combine uniform estimates near the boundary and a blow-up analysis. Finally, by means of the degree theory, we obtain the existence of a positive solution.

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

On étudie des flots géométriques d'ordre quatre sur des variétés riemanniennes compactes, qui apparaissent naturellement comme flots de gradient de fonctionnelles quadratiques en la courbure. Lorsque la constante de Yamabe reste minorée par une constante strictement positive le long du flot, on montre que la variété ne s'effondre pas, et qu'une suite de métriques dilatées au voisinage d'un temps singulier converge vers une variété complète qui modélise la singularité. En particulier, en dimension quatre, cette hypothèse est vérifiée pour une certaine classe de flots de gradients, du moment que l'énergie initiale est inférieure à une constante explicite. Les singularités de ces flots sont alors modélisées par des variétés complètes et non compactes, dont le tenseur de Bach et la courbure scalaire s'annulent. En combinant une formule de Weitzenböck avec l'inégalité de Sobolev induite par la positivité de la constante de Yamabe, on montre une série de résultats de rigidité pour des métriques dont la courbure est intégralement pincée. En particulier, on prouve un théorème de rigidité pour les variétés de dimension quatre à tenseur de Bach et à courbure scalaire nuls, qui implique que les singularités de notre classe de flots de gradient ne peuvent exister que si l'énergie initiale est supérieure à une certaine constante. Dans le cas contraire, ces flots existent pour tous temps positifs et convergent vers une métrique à courbure sectionnelle constante et positive. On retrouve ainsi un "théorème de la sphère" pour les variétés compactes de dimension quatre dont la courbure est intégralement pincée. En appliquant cette même méthode aux formes harmoniques d'une variété à courbure intégralement pincée, on démontre une version intégrale du théorème de Bochner-Weitzenböck. On en déduit l'annulation des nombres de Betti sous diverses conditions de pincement intégral, et on caractérise les cas d'égalité. / We study fourth-order geometric flows on compact Riemannian manifolds, which naturally appear as gradient flows of quadratic curvature functionals. When the Yamabe constant remains bounded from below by a positive constant along the flow, we show that the manifold doesn't collapse, and that a sequence of dilated metrics near a singular time converges to a singularity model. In particular, in dimension four, this assumption is satisfied by a class of gradient flows, provided that the initial energy is less than an explicit constant. The singularities of these flows are then modeled by complete non-compact manifolds, which are Bach-flat and scalar-flat. By combining a Weitzenböck formula with the Sobolev inequality induced by the positivity of the Yamabe constant, we prove several rigidity results for metrics with integral pinched curvature. In particular, we prove a rigidity result for Bach-flat and scalar-flat manifolds in dimension four, which implies that the singularities of our gradient flows can only exist when the initial energy is bigger than a given constant. When this is not the case, these flows exist for all time, and converge to a metric with constant positive curvature. It provides a proof of a "sphere theorem" for closed four-dimensional manifolds with integral pinched curvature. Applying the same method to harmonic forms on an integral pinched manifold, we prove an integral version of the Bochner-Weitzenböck theorem. As a corollary, we obtain the vanishing of Betti numbers under various integral pinching conditions, and we characterize the equality cases.

Made available in DSpace on 2015-05-15T11:46:10Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 517535 bytes, checksum: 44009b0bc09a5af772f82b9303aa5e7b (MD5) Previous issue date: 2010-11-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Here we study a class of semilinear elliptic equations with nonlinearity of an inverse square type. This equations arise, in applications, on the modeling of certain electrostatic devices from microtechnology, MEMS - Micro Electro Mechanical Systems. More precisely, these equations characterizes the function that represents the deformation of a deformable capacitor under the influence of an applied voltage. The Mathematical tools used on the study of such problems involve a bit of Nonlinear Analysis and Partial Differential Equations' methods as sub and supersolutions, sign preserving Theorems (Maximum Principle, Boggio's Principle), energy estimates via Sobolev spaces, etc. In a parallel way we wish to emphasize the importance of this investigation, in Mathematics, on helping the understanding on the class of singular problems in Partial Differential Equations. / Estudamos aqui uma classe de equações elípticas semilineares com singularidade do tipo inverso do quadrado. Estas equações aparecem, na modelagem de certos dispositivos eletrostáticos da microtecnologia, MEMS - Micro Electro Mechanical Systems (sistemas microeletromecânicos). Mais precisamente tais equações caracterizam a função que descreve a deformação de um capacitor deformável sob a influência de uma voltagem aplicada. A Matemática necessária ao estudo de tais problemas envolve um bom aparato de métodos da Análise não Linear e das Equações Diferenciais Parciais tais como Método de Sub- e Supersolução, Teoremas de Preservação de Sinal (Princípio do Máximo, Princípio de Boggio), estimativas de Energia via Espaços de Sobolev, entre outros. Em paralelo destacamos a importância desta investigação em Matemática, para entendermos como se comportam as soluções de problemas supercríticos em Equações Diferenciais Parciais.

Neste trabalho estudamos a existência e a multiplicidade de soluções clássicas positivas para uma classe de problemas de quarta-ordem sob a condição de fronteira de Navier, relacionando o número de soluções com a topologia do domínio, mais precisamente, com sua categoria de Lusternik-Schnirelman. Introduzimos também uma noção de regiões crítica e não-crítica associadas a um de nossos problemas, a fim de garantir condições para existência de solução / In this work we study the existence and multiplicity of positive classical solutions for a class of fourth-order problems under Navier boundary condition, relating the number of solutions to the domain topology, more specifically, to its Lusternik-Schnirelman category. We also introduce the notion of critical and noncritical regions related to one of our problems, in order to ensure conditions to existence of solutions

In this thesis the spectral properties of differential operators generated by the formally self-adjoint differential expression Τy = w⁻₁[(ry″)″ - (py′)′ + qy] are investigated. The main tools to be used are the theory of asymptotic integration and the Titchmarsh--Weyl M-matrix. Subject to certain regularity conditions on the coefficients asymptotic integration leads to estimates for the eigenfunctions of the corresponding differential equation Τy = zy. According to the theory of asymptotic integration the regularity conditions combine smoothness with decay, i.e. admissible coefficients are (in an appropriate sense) either short range or slowly varying. Knowledge of the asymptotics (x → ∞) of the solutions will then be used to determine the deficiency index and to derive properties of the M-matrix which is closely related to the spectral measure of an associated self-adjoint realization Τ. Consequently we can compute the multiplicity of the spectrum, locate the absolutely continuous spectrum and give conditions for the singular continuous spectrum to be empty. This generalizes classical results on second order operators.

This study focuses on the numerical solution of a fourth-order cell problem obtained through a two- scale expansion approach applied to a higher gradient heat equation microscopic problem involving temperature distributions. The main objective is to investigate the temperature field within the macroscale domain and compute the effective conductivity using finite element methods. The research utilizes numerical techniques, specifically finite element methods, to solve the fourth-order cell problem and obtain the temperature distribution.

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