The Richards equation is widely used in the simulation of unsaturated flow and related fields. In numerical solution process, finite volume method can be to carry out discretization, Picard for iterative solution. However, order obtain a more reliable solution, space step size conventional uniform grid often small. Especially under some unfavorable conditions, such as infiltration into dry soil...

The purpose of this paper is to introduce a new class of quasi-contractive operators and to show that the most used fixed point iterative methods, that is, the Picard and Mann iterations, are convergent to the unique fixed point. The comparison of these methods with respect to their convergence rate is obtained.

In this article, we discuss the existence and uniqueness of proportional Itô–Doob stochastic fractional order systems (PIDSFOS) by using Picard iteration method. The paper provides new results integral calculus techniques. We have shown convergence solution averaged PIDSFOS to that standard in context mean square also probability. One example is given illustrate our results.

Let {Xn} be a sequence of i.i.d. random vectors with values in a separable Banach space. Moderate deviation principles for trajectories of sums of {Xn} are proved, which generalize related results of Borovkov and Mogulskii (1980) and Deshayes and Picard (1979). As an application, functional laws of the iterated logarithm are given. The paper also contains concluding remarks, with examples, on e...

In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We desig...

This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal ex...

Received: December 21, 2010 Accepted: January 7, 2011 doi:10.5539/jmr.v3n2p151 Abstract In this paper, we study fixed point theorems for multi-valued weak contractions. We show that the Picard projection iteration converges to a fixed point, give a rate of convergence and generalize Collage theorem. This work includes results on multi-valued contraction mappings studied by (Kunze, H.E., La Torr...

This paper gives some results on almost automorphic strong solutions to time-fractional partial differential equations by employing a mix o thef Galerkin method, Fourier series, and Picard iteration. As an application, the existence, uniqueness, global Mittag–Leffler convergence of solution are discussed concrete parabolic equations. To best our knowledge, this is first study subject.

This thesis investigates some effective and quantitative aspects of metric fixed point theory in the light of methods from proof theory. The thesis consists of contributions to the program of proof mining, as developed by Kohlenbach and various collaborators since the early 1990s (but with roots back to Kreisel’s program “unwinding of proofs” from the 1950s). The contributions involve both case...