Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Plasma Confined Ground and Excited State Helium Atom: A Comparison Theorem Study Using Variational Monte Carlo and Lagrange Mesh Method

The energy eigenvalues of the ground state helium atom and lowest two excited states corresponding to the configurations 1s2s embedded in the plasma environment using Hulthén, Debye–Hückel and exponential cosine screened Coulomb model potentials are investigated within the variational Monte Carlo method, starting with the ultracompact trial wave functions in the form of generalized Hylleraas–Kinoshita functions and Guevara–Harris–Turbiner functions. The Lagrange mesh method calculations of energy are reported for the He atom in the ground and excited 1S and 3S states, which are in excellent agreement with the variational Monte Carlo results. Interesting relative ordering of eigenvalues are reported corresponding to the different screened Coulomb potentials in the He ground and excited electronic states, which are rationalized in terms of the comparison theorem of quantum mechanics.

Equations of Heat Conduction with Slow Combustion

<p>A study is made of the equations of heat conduction with slow combustion. A mathematical model is established from an interpretation of the physical model, with a few simplifying assumptions. This gives rise to a coupled pair of partial differential equations which are the direct concern of this thesis, the dependent variables being the temperature and reactant concentration as functions of position and time. The model is shown to possess a unique solution for which some properties, such as Lipschitz conditions etc., are established. An investigation into the use of a comparison theorem is given, in which it is shown that no direct comparison theorem is possible for this and related systems. However, it is also shown that it is possible to obtain upper and lower estimates by appealing to the physical model. A discussion of the boundary layer is given and this is followed by a detailed discussion of stability. The latter has been one of the main concerns of earlier authors on this system. Their use of a space-averaging process to establish a criterion for stability is also discussed. Probably one of the most interesting features of this system is the subclass of problems for which the reactant is exhausted in a finite time. These hare been named the "cut-off" problems and they can be likened to the free boundary problems in fluid dynamics. A discussion of the cut-off problem is given with particular examples chosen to illustrate the main features. This thesis, contains no material which has been accepted for the award of any other degree or diploma in any University and to the best of my knowledge and belief, the thesis contains no material previously published or written by another person, except where due reference is made in the text of the thesis.</p>

Equations of Heat Conduction with Slow Combustion

<p>A study is made of the equations of heat conduction with slow combustion. A mathematical model is established from an interpretation of the physical model, with a few simplifying assumptions. This gives rise to a coupled pair of partial differential equations which are the direct concern of this thesis, the dependent variables being the temperature and reactant concentration as functions of position and time. The model is shown to possess a unique solution for which some properties, such as Lipschitz conditions etc., are established. An investigation into the use of a comparison theorem is given, in which it is shown that no direct comparison theorem is possible for this and related systems. However, it is also shown that it is possible to obtain upper and lower estimates by appealing to the physical model. A discussion of the boundary layer is given and this is followed by a detailed discussion of stability. The latter has been one of the main concerns of earlier authors on this system. Their use of a space-averaging process to establish a criterion for stability is also discussed. Probably one of the most interesting features of this system is the subclass of problems for which the reactant is exhausted in a finite time. These hare been named the "cut-off" problems and they can be likened to the free boundary problems in fluid dynamics. A discussion of the cut-off problem is given with particular examples chosen to illustrate the main features. This thesis, contains no material which has been accepted for the award of any other degree or diploma in any University and to the best of my knowledge and belief, the thesis contains no material previously published or written by another person, except where due reference is made in the text of the thesis.</p>

Convergence of approximate solutions for singular difference systems with maxima

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

Hilbert Implication Algebra and Some Properties

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.