Abstract In this paper, we consider a system of Klein-Gordon equations with vari- able exponents. The first part of the manuscript is devoted to the proof of the blow up of solutions with negative initial energy under suitable conditions on variable ex- ponents and initial data. The theoretical part is supported by numerical experiments based on P 1-finite element method in space and the BDF and the Generalized-alpha methods in time illustrated in the second part. The numerical and analytical results of the blow up solutions agree with each other.

Abstract The main purpose of this chapter is to provide a brief review of Hilbert space with its fundamental features and introduce reproducing kernels of the corresponding spaces. We separate our analysis into two parts. In the first part, the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and finally Hilbert spaces are presented. The second part is devoted to the reproducing kernels and the related Hilbert spaces which is called the reproducing kernel Hilbert spaces (RKHS) in the complex plane. The operations on reproducing kernels with some important theorems on the Bergman kernel for different domains are analyzed in this part.

Abstract We present a theoretical aspect of conservation laws by using simplest scalar models with essential properties. We start by rewriting the general scalar conservation law as a quasilinear partial differential equation and solve it by method of characteristics. Here we come across with the notion of strong and weak solutions depending on the initial value of the problem. Taking into account a special initial data for the left and right side of a discontinuity point, we get the related Riemann problem. An illustration of this problem is provided by some examples. In the remaining part of the chapter, we extend this analysis to the gas dynamics given in the Euler system of equations in one dimension. The transformations of this system into the Lagrangian coordinates follow by applying a suitable change of coordinates which is one of the main issues of this section. We next introduce a first-order Godunov finite volume scheme for scalar conservation laws which leads us to write Godunov schemes in both Eulerian and Lagrangian coordinates in one dimension where, in particular, the Lagrangian scheme is reformulated as a finite volume method. Finally, we end up the chapter by providing a comparison of Eulerian and Lagrangian approaches.

Abstract Relativistic Burgers model and its generalization to various spacetime geometries are recently studied both theoretically and numerically. The numeric implementation is based on finite volume and finite difference approximation techniques designed for the corresponding model on the related geometry. In this work, we provide a summary of several versions of these models on the Schwarzschild, de Sitter, Schwarzschild-de Sitter, FLRW and Reissner-Nordström spacetime geometries with their particular properties.

Abstract Recently several relativistic versions of Burgers equations are derived and developed on different spacetime geometries. In the current work, we apply the technique used in the recent works to the Reissner-Nordström spacetime geometries. As a result, by using the energy-momentum tensor equations, we obtained the Euler system and then the desired relativistic Burgers models. In this article we observed that the relativistic model, obtained from spherically symmetric, electrically charged Reissner-Nordström metric, contains static solutions. We examined these static solutions and their behaviors in detail. Besides, using the finite volume methods, we analyzed shock and rarefaction wave propagation by several numerical tests.

Abstract A relativistic generalization of the inviscid Burgers equation was introduced by LeFloch and co-authors and was recently investigated numerically on a Schwarzschild background. We extend this analysis to a Friedmann-Lemaˆýtre-Robertson-Walker (FLRW) background,which ismore challenging due to the existence of time-dependent, spatially homogeneous solutions. We present a derivation of the model of interest and we study its basic properties, including the class of spatially homogeneous solutions. Then, we design a second-order accurate scheme based on the finite volume methodology, which provides us with a tool for investigating the properties of solutions. Computational experiments demonstrate the efficiency of the proposed scheme for numerically capturing weak solutions.

Abstract The relativistic versions of Burgers equations on the Schwarzschild, FLRW and de Sitter backgrounds have recently been derived and analyzed numerically via finite volume approximation based on the concerning models. In this work, we derive the relativistic Burgers equation on a Schwarzschild-(Anti-)de Sitter spacetime and introduce a second-order Godunov-type finite volume scheme for the approximation of discontinuous solutions to the model of interest. The effect of the cosmological constant is also taken into account both theoretically and numerically. The efficiency of the method for solutions containing shock and rarefaction waves are presented by several numerical experiments.

Abstract Several generalizations of the relativistic models of Burgers equations have recently been established and developed on different spacetime geometries. In this work, we take into account the de Sitter spacetime geometry, introduce our relativistic model by a technique based on the vanishing pressure Euler equations of relativistic compressible fluids on a (1+1)-dimensional background and construct a second order Godunov type finite volume scheme to examine numerical experiments within an analysis of the cosmological constant. Numerical results demonstrate the efficiency of the method for solutions containing shock and rarefaction waves.

Abstract Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler system of relativistic compressible flows on a curved background, we derive, both, the standard inviscid Burgers equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed. Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly, is well--balanced in the sense that it preserves static equilibrium solutions. Numerical experiments are presented which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy solutions on a curved background.

Abstract Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h^1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch’s theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

Keywords Hyperbolic conservation law, entropy solution, finite volume scheme, error estimate, discrete entropy inequality, convergence rate.

Abstract We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry and were originally discovered by Cockburn, Coquel, and LeFloch in the (flat) Euclidian setting.

Keywords Conservation law, Lorenzian manifold, entropy condition, measure-valued solution, finite volume scheme, convergence analysis.

Abstract We consider nonlinear hyperbolic conservation laws, posed on a differential (n + 1)-manifold with boundary referred to as a spacetime, and in which the “flux” is defined as a flux field of n-forms depending on a parameter (the unknown variable). We introduce a formulation of the initial and boundary value problem which is geometric in nature and is more natural than the vector field approach recently developed for Riemannian manifolds. Our main assumption on the manifold and the flux field is a global hyperbolicity condition, which provides a global time-orientation as is standard in Lorentzian geometry and general relativity. Assuming that the manifold admits a foliation by compact slices, we establish the existence of a semi-group of entropy solutions. Moreover, given any two hypersurfaces with one lying in the future of the other, we establish a “contraction” property which compares two entropy solutions, in a (geometrically natural) distance equivalent to the L1 distance. To carry out the proofs, we rely on a new version of the finite volume method, which only requires the knowledge of the given n-volume form structure on the (n + 1)-manifold and involves the total flux across faces of the elements of the triangulations, only, rather than the product of a numerical flux times the measure of that face.

Keywords Hyperbolic conservation law, differential manifold, flux field of forms, entropy solution, finite volume method.

Abstract We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions.

Abstract The first part of this thesis is devoted to the study of finite volume methods for conservation laws on manifolds. We study first an approach based on a metric on Lorentzian manifolds. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. Next, we consider another approach based on di erential forms. We establish a new version of the finite volume methods which only requires the knowledge of family of n-volume form on an (n + 1)-manifold. The second part is concerned with error estimates for finite volume methods and the implementation of a model of relativistic compressible fluids. We consider first nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. Next, we consider the hyperbolic balance laws posed on a curved spacetime endowed with a volume form, and, after imposing a natural Lorentz invariance property we identify a unique balance law which can be viewed as a relativistic version of Burgers equation

Keywords finite volume method, conservation law, manifolds, entropy, measure-valued solution, differential forms, convergence analysis.

Abstract In this thesis we make a survey of the theory of reproducing kernel Hilbert spaces associated with positive definite kernels and we illustrate their applications for interpolation problems of Nevanlinna-Pick type. Firstly we focus on the properties of reproducing kernel Hilbert spaces, generation of new spaces and relationships between their kernels and some theorems on extensions of functions and kernels. One of the most useful reproducing kernel Hilbert spaces, the Bergman space, is studied in details in chapter 3. After giving a brief definition of Hardy spaces, we dedicate the last part for applications of interpolation problems of Nevanlinna-Pick type with three main theorems: interpolation with a finite number of points, interpolation with an infinite number of points and interpolation with points on the boundary. Finally we include an Appendix that contains a brief recall of the main results from functional analysis and operator theory.

Keywords Reproducing kernel, Reproducing kernel Hilbert spaces, Bergman spaces, Hardy spaces, Interpolation, Riesz theorem.

Abstract An Introduction to Sobolev Spaces provides a brief introduction to Sobolev spaces at a simple level with illustrated examples. Readers will learn about the properties of these types of vector spaces and gain an understanding of advanced differential calculus and partial difference equations that are related to this topic. The contents of the book are suitable for undergraduate and graduate students, mathematicians, and engineers who have an interest in getting a quick, but carefully presented, mathematically sound, basic knowledge about Sobolev Spaces.

Keywords Banach space, dual, Hilbert space, fixed point theorems, Minkowski inequality, Gronwall inequality, Komornik inequality, Nakao inequality, Green’s identities, Riesz-Fischer theorem, Sobolev space, Sobolev space of real order, Sobolev space of negative order, weighted Lebesgue space, Schwartz space, Plancherel theorem, weight function, Embedding property of normed spaces, Sobolev embedding theorems, Sobolev-Gagliardo-Nirenberg inequality, Poincare inequality, Variable exponent, variable exponent Lebesgue space, Luxemburg norm, Holder inequality, variable exponent Sobolev space

Abstract This book has been prepared in accordance with the programs of Basic Mathematics I, Analysis I or Calculus I courses which are taught at the undergraduate level of the Faculties of Science and Engineering as well as the Faculties of Economics and Administrative Sciences. The book covers brief descriptions of the theorems, and focusses particularly on exercises with their solutions. While these exercises support the relevant topics and theorems, they are prepared to support the readers on related courses and help them prepare for exams. The content of the book is as follows: The concepts of limit and continuity for functions of one (real) variable, derivative and differentiability, applications of derivative, optimization problems, inverse-derivative concept and sketching graphs of functions.

Keywords Functions of one variable, limit, derivative, differentiablity, related rates problems, optimization, sketching graphs.

Abstract The purpose of this book is to lay out a mathematical framework for the convergence and error analysis of the finite volume method for the discretization of hyperbolic conservation laws on manifolds. Finite Volume Method (FVM) is a discretization approach for the numerical simulation of a wide variety physical processes described by conservation law systems. It is extensively employed in fluid mechanics, meteorology, heat and mass transfer, electromagnetic, models of biological processes and many other engineering applications formed by conservative systems. In this book, from one point of view, we provide a brief description for the convergence of the FVM by approaches based on metric and differential forms. The latter can be viewed as a generalization of the formulation and convergence of the method for general conservation laws on curved manifolds. On the other hand, we carried over the error estimate for FVM that is established for the Euclidean setting to the curved manifolds and obtained an expected rate of error in the L1-norm.

Keywords finite volume method, conservation law, manifolds, entropy, measure-valued solution, differential forms, convergence analysis.

Abstract The theory of reproducing kernel Hilbert spaces has important applications to boundary value problems, integral operators, harmonic and analytic functions, in conformal mappings of simply- and multiply-connected domains, in pseudo-conformal mappings, in the study of invariant Riemann metrics, in probability theory, interpolation of functions, and in many other subjects. In this short presentation, we consider an introduction to this subject by emphasizing first the abstract theory, the Bergman kernels, and some of their applications to interpolation of functions in the unit disc. The book is aimed to a broader audience of graduate students, mathematicians, physicists, and engineers, and all those having an interest in getting a quick, but carefully presented, mathematically sound basic knowledge on this domain.

Keywords Reproducing kernel, Reproducing kernel Hilbert spaces, Bergman spaces, Hardy spaces, Interpolation, Riesz theorem.