N-dimensional Laplace transformations and their applications in partial differential equations

This dissertation focuses on the theoretical and computation aspects of N-dimensional Laplace transformation pairs, for N ≥ 2. Laplace transforms can be defined either as a unilateral or a bilateral integral. We concentrate on the unilateral integrals. We have successfully developed a number of theorems and corollaries in N-dimensional Laplace transformations and inverse Laplace transformations. We have given numerous illustrative examples on applications of these results in N and particularly in two dimensions. We believe that these results will further enhance the use of N-dimensional Laplace transformation and help further development of more theoretical results;Specifically, we derive several two-dimensional Laplace transforms and inverse Laplace transforms in two-dimension pairs. We believe most of these results are new. However, we have established some of the well-known results for the case of commonly used special functions;Several initial boundary value problems (IBVPs) characterized by non-homogenous linear partial differential equations (PDEs) are explicitly solved in Chapter 4 by means of results developed in Chapters 2 and 3. In the absence of necessitous three and N-dimensional Laplace transformation tables, we solve these IBVPs by the double Laplace transformations. These include non-homogenous linear PDEs of the first order, non-homogenous second order linear PDEs of Hyperbolic and Parabolic types;Even though multi-dimensional Laplace transformations have been studied extensively since the early 1920s, or so, still a table of three or N-dimensional Laplace transforms is not available. To fill this gap much work is left to be done. To this end, we have established several new results on N-dimensional Laplace transformations as well as inverse Laplace transformations and many more are still under our investigation. A successful completion of this task will be a significant endeavor, which will be extremely beneficial to the further research in Applied Mathematics, Engineering and Physical Sciences. Especially, by the use of multi-dimensional Laplace transformations a PDE and its associated boundary conditions can be transformed into an algebraic equation in n-independent variables. This algebraic equation can be used to obtain the solution of the original PDE

Development of reliability methodology for systems engineering. Volume III - Theoretical investigations - An approach to a class of reliability problems Final report

Random quantities from continuous time stochastic process with application to reliability and probabilit

Lyapunov-type inequality and eigenvalue estimates for fractional problems

In this work, we establish the Lyapunov-type inequalities for the fractional boundary value problems with Hilfer derivative for different boundary conditions. We apply this inequality to fractional eigenvalue problems and prove one of the important results of real zeros of certain Mittag-Leffler functions and improve the bound of the eigenvalue using the Cauchy-Schwarz inequality and Semi-maximum norm. We extend it for higher order cases

Fractional evolution equations in Banach spaces

IV+107hlm.;24c

Stability criteria for systems with colored multiplicative noise.

Massachusetts Institute of Technology. Dept. of Electrical Engineering. Thesis. 1974. Ph.D.MICROFICHE COPY ALSO AVAILABLE IN BARKER ENGINEERING LIBRARY.Vita.Bibliography: leaves 165-171.Ph.D

Boundary value problem for PDEs and some clases of L^p bounded pseudodifferential operators

In recent years much attention has been extended in the study of differential equations of non-classical types. These articles need, on one hand, fluid mechanics, hydro-and gas dynamics and other applied disciplines, and on the other hand, the actual needs of the mathematical sciences. One of the most important classes of equations of non-classical type is the third-order equation with multiple characteristics which is a generalization of linear Korteweg-de Vries-Burgers equation, special cases which occur in the dissemination of waves in weakly dispersive media, the propagation of waves in a cold plasma, magneto-hydrodynamics, problems of nonlinear acoustics, the hydrodynamic theory of space plasma. A pioneering work in the theory of odd order partial differential equations with multiple characteristics was done by E.Del Vecchio, H.Block, in which they studied the technique of constructing fundamental solutions of these equations. Consequently, the theory of equations with multiple characteristics has been greatly developed by the Italian mathematician L.Cattabriga. In the first part of Ph.D thesis we develop and study boundary value problems for third-order equations with multiple characteristics in areas with curved boundaries, as well as some properties of the fundamental solutions of the equations, when the transition line is a curve. In addition, we construct a solution of the Cauchy problem in the classes of functions growing at infinity, depending on the behaviour of the right-hand side of the equation. Our thesis explores both linear and nonlinear boundary value problems for linear and non-linear third-order equation with multiple characteristics in the domain with curved boundaries. The main result of the first chapter is to prove the unique solvability of the general boundary value problem for the third-order equation with multiple characteristics in curved domains. The proof of the uniqueness theorem of the solution, we use the method of energy integrals. For the existence theorem, we find equivalent systems of Volterra second type integral equations. The next chapter consists of three sections and it investigates the problem with nonlinear boundary conditions for linear and non-linear equations of the third order with multiple characteristics. To prove the existence and uniqueness theorems, we will use methods of integral energy and theory of integral equations. In the last part of the thesis we analyze basic properties of pseudodifferential operators, such as the behaviour of products and adjoins of such operators, their continuity on L^2, L^p and Sobolev spaces. In the thesis we study the L^p - boundedness of vector weighted pseudodifferential operators with symbols which have derivatives with respect to x only up to order k, in the Holder continuous sense

Local well-posedness of the higher order nonlinear Schr\"odinger equation on the half-line: single boundary condition case

We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to be Dirichlet type. Our functional framework centers around fractional Sobolev spaces. We treat both high regularity and low regularity solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of initial value problems. The linear analysis, which is the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method. In this connection, we note that the higher-order Schr\"odinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivative. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multi-term linear differential operator.Comment: 30 pages, 2 figure