Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems

Coupled systems with Ambrosetti-Prodi-type differential equations

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters

Monotone iterative methods for solving nonlinear partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand

A key aspect of the simulation process is the formulation of proper mathematical models. The model must be able to emulate the physical phenomena under investigation. Partial differential equations play a major role in the modelling of many processes which arise in physics, chemistry and engineering. Most of these partial differential equations cannot be solved analytically and classical numerical methods are not always applicable. Thus, efficient and stable numerical approaches are needed. A fruitful method for solving the nonlinear difference schemes, which discretize the continuous problems, is the method of upper and lower solutions and its associated monotone iterations. By using upper and lower solutions as two initial iterations, one can construct two monotone sequences which converge monotonically from above and below to a solution of the problem. This monotone property ensures the theorem on existence and uniqueness of a solution. This method can be applied to a wide number of applied problems such as the enzyme-substrate reaction diffusion models, the chemical reactor models, the logistic model, the reactor dynamics of gasses, the Volterra-Lotka competition models in ecology and the Belousov-Zhabotinskii reaction diffusion models. In this thesis, for solving coupled systems of elliptic and parabolic equations with quasi-monotone reaction functions, we construct and investigate block monotone iterative methods incorporated with Jacobi and Gauss--Seidel methods, based on the method of upper and lower solutions. The idea of these methods is the decomposition technique which reduces a computational domain into a series of nonoverlapping one dimensional intervals by slicing the domain into a finite number of thin strips, and then solving a two-point boundary-value problem for each strip by a standard computational method such as the Thomas algorithm. We construct block monotone Jacobi and Gauss-Seidel iterative methods with quasi-monotone reaction functions and investigate their monotone properties. We prove theorems on existence and uniqueness of a solution, based on the monotone properties of iterative sequences. Comparison theorems on the rate of convergence for the block Jacobi and Gauss-Seidel methods are presented. We prove that the numerical solutions converge to the unique solutions of the corresponding continuous problems. We estimate the errors between the numerical and exact solutions of the nonlinear difference schemes, and the errors between the numerical solutions and the exact solutions of the corresponding continuous problems. The methods of construction of initial upper and lower solutions to start the block monotone iterative methods are given

Semidefinite Relaxations for Stochastic Optimal Control Policies

Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation have led to the discovery of a formulation of the value function as a linear Partial Differential Equation (PDE) for stochastic nonlinear systems with a mild constraint on their disturbances. This has yielded promising directions for research in the planning and control of nonlinear systems. This work proposes a new method obtaining approximate solutions to these linear stochastic optimal control (SOC) problems. A candidate polynomial with variable coefficients is proposed as the solution to the SOC problem. A Sum of Squares (SOS) relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in Portland, Oregon. 7 pages, colo

Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio

An adaptive preconditioner for steady incompressible flows

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier--Stokes flows. The preconditioner maps the identity (no preconditioner) to the Stokes preconditioner (preconditioning by Laplacian) through a continuous parameter and is built on a first order Euler time-discretization scheme. The preconditioner is tested onto two fluid configurations: three-dimensional doubly diffusive convection and a reduced model of shear flows. In the former case, Stokes preconditioning works but a mixed preconditioner is preferred. In the latter case, the system of equation is split and solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications, this preconditioner is expected to help a wide range of studies