Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / Ph.D.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18042号 / 理博第3920号 / 新制||理||1566(附属図書館) / 30900 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 加藤 毅, 教授 上田 哲生 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

We look at the Cauchy problem with an accretive Operator on a Banach space. We give an upper bound for the norm of the difference of two solutions of Euler schemes with this accretive operator. This concrete estimate also works for the problem with a non-zero right-hand side in the Cauchy problem and is a generalization of a famous result by Kobayashi. We also show, how this result gives direct proofs for existence, uniqueness, stability and regularity of Euler solutions of the Cauchy problem and also the rate of convergence of solutions of Euler schemes. The results concerning regularity and rate of convergence are generalized for problem data in interpolation sets.:1. Accretive operators 1.1. Thebracket. 1.2. Accretive operators 1.3. The Cauchy problem and Euler solutions 2. A priori estimates for solutions of implicit Euler schemes 2.1. An implicit upper bound 2.2. Properties of the density 2.3. An explicit upper bound 3. Applications 3.1. Wellposedness of the Cauchy problem 3.2. Interpolation theory A. Functions of bounded variation

In this paper we consider approximation schemes and questions of well-posedness for a general class of functional differential equations of neutral-type (NFDE) where the difference operator does not have an atom at zero. Equations of this type occur in the modeling of certain aeroelastic control problems and include many singular integro-differential equations. We obtain general necessary and sufficient conditions for the well-posedness of functional differential equations of neutral-type on the Banach-spaces R<sup>n</sup?xL_p. As an example of the well-posedness of the non-atomic NFDE-system that arises in the study of aeroelasticity is established on R<sup>n</sup?xL_p, 1≤p<2. Employing the equivalence between generalized solutions of NFDEs and mild solutions of the “corresponding” abstract Cauchy-problems, we make use of general approximation results for well-posed Cauchy-problems to establish and analyze the convergence of the “averaging projection” scheme on the Banach spaces R<sup>n</sup?xL_p, 1<p<∞, for a class of problems with atomic difference operators. / Ph. D.

The present work in the area of soliton theory studies two problems which arise when seeking analytic solutions to certain nonlinear partial differential equations. In the first one, Lax pairs associated with prolonged eigenfunctions and prolonged squared eigenfunctions (prolonged squares) are derived for a Schrödinger equation with a potential depending polynomially on the spectral parameter (of degree N) and its respective hierarchy of nonlinear evolution equations (here named generalized polynomial Korteweg-de Vries equations or GPKdV). It is shown that the prolonged squares satisfy the linearized GPKdV equations. On that basis, the Cauchy problem for the linearized GPKdV has been solved by finding a complete set of such prolonged squares and applying an expansion formula derived in another work by the author. The results are a generalization of the ones by Sachs (SIAM J. Math. Anal. 14, 1983, 674-683). Moreover, a condition on the so-called recursion operator A is derived which generates the whole hierarchy of Lax pairs associated with the prolonged squares. As for the second part of the work, it developed a scheme for deriving gauge transformations between different linear spectral problems. Then the scheme is applied to obtain all known Darboux transformations for a quadratic pencil (the spectral problem considered in the first part at N = 2), Schrödinger equation (N = 1), Ablowitz-Kaup-Newell-Segur (AKNS) system and also derive the Jaulent-Miodek transformation. Moreover, the scheme yields a large family of new transformations of the above types. It also gives some insight on the structure of the transformations and emphasizes the symmetry with respect to inversion that they possess. / Ph. D.

In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed. / Ph. D.

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-25Bitstream added on 2014-06-13T20:48:30Z : No. of bitstreams: 1 rosa_ra_me_sjrp.pdf: 528158 bytes, checksum: 87eb91b0d9f48ee60092159a596eccf5 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho vamos estudar a existência e unicidade de solução para equações da forma { u + Au = f(t,u) u(t0)= u0 ∈ X, (I) onde X é um espaço de Banach, A : D(A) ⊂ X → X é um operador linear, f é uma função não linear conhecida, u0 ∈ X é um dado inical conhecido e u : I ⊂ R → X é uma função desconhecida e t0 ∈ I. Faremos este estudo usando a Teoria dos Semigrupos de Operadores Lineares. Para melhor entendimento do estudo das equações (I), faremos duas aplicações. A primeira tratando de um modelo (linear) de divisão celular e a segunda, do modelo (não linear) de condução do calor. / In this work we will study the existence and uniqueness of the solutions for the following equation { u + Au = f(t,u) u(t0)= u0 ∈ X, (I) where X is a Banach space, A : D(A) ⊂ X → X is a linear operator, f is a nonlinear function, u : I ⊂ R → X is unknown function. In this study we will use the theory of semigroup of linear operators. For a best understanding of the study of equations (I), we will do two applications. The first one, is a (linear) model of cellular division and the second one, is about the (nonlinear) model od conduction of the heat.

<p> The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, u_x]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.</p><p>The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.</p><p>The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.</p>

The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, ux]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0. The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense. The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

We consider bounded weak solutions u of a degenerate parabolic-hyperbolic equation defined in a subset [mathematical symbols]. We define strong notion of trace at the boundary [mathematical symbols] reached by L¹ convergence for a large class of functionals of u. Such functionals depend on the flux function of the degenerate parabolic-hyperbolic equation and on the boundary. We also prove the well-posedness of the entropy solution for scalar conservation laws with a strong boundary condition with the above trace result as applications. / text