Dimensional Nonlinear Diffusion Research Articles

The law of large numbers for quantum stochastic filtering and control of many-particle systems

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles inside a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Scrodinger equations or Hartree equations, or Gross-Pitaevski equations. In this paper we extend some of these convergence results to a stochastic framework. Concretely we work with the Belavkin stochastic filtering of many particle quantum systems. The resulting limiting equation is an equation of a new type, which can be seen as a complex-valued infinite dimensional nonlinear diffusion of McKean-Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.

The law of large numbers for quantum stochastic filtering and control of many-particle systems

Существует обширная литература, где рассматривается динамический закон больших чисел для систем квантовых частиц, т. е. выводится уравнение, описывающее предельное поведение отдельных частиц внутри большого ансамбля одинаковых взаимодействующих частиц. Полученные уравнения обычно называют нелинейными уравнениями Шредингера, или уравнениями Хартри, или уравнениями Гросса-Питаевского. Дано стохастическое обобщение некоторых из этих результатов, касающихся сходимости. Рассмотрена стохастическая фильтрация Белавкина для многочастичных квантовых систем. Полученное предельное уравнение - это уравнение нового типа, которое можно трактовать как комплекснозначную бесконечномерную нелинейную диффузию типа Маккина-Власова. Этот результат является ключевым для теории квантовых игр среднего поля, разработанной автором в предыдущей работе.

Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity

We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion–reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.

On the equivalence groups for (2+1) dimensional nonlinear diffusion equation

(2+1) dimensional diffusion equation is considered within the framework of the group of equivalence transformations. Generators for the group are obtained and admissible transformations between linear and nonlinear equations are examined. It is shown that transformations between linear and nonlinear equations are possible provided that the generators of independent variables depend on the dependent variable. Exact solutions for some nonlinear equations are obtained.

Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L2, L∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

A study of one dimensional nonlinear diffusion equations by Bernstein polynomial based differential quadrature method

The purpose of this work is to study numerical solutions of nonlinear diffusion equations such as Fisher’s equation, Burgers’ equation and modified Burgers’ equation by applying Bernstein differential quadrature method (BDQM). These nonlinear diffusion equations occur in many applications of theoretical, engineering and environmental sciences. Therefore, finding numerical solutions of these equations is very important. In BDQM, Bernstein polynomials are used as base functions to find weighting coefficients of differential quadrature method. We have applied BDQM on eleven different test problems taken from the literature and the computed results confirm that BDQM is an efficient method for finding solution of nonlinear partial differential equations. It is found that BDQM produces very good results even at small number of grid points.

An integrable nonlinear diffusion hierarchy with a two-dimensional arbitrary function

By introducing the hereditary condition to a general first order differential strong symmetry operator, we obtain general 1+1 and 2+1 dimensional integrable nonlinear diffusion hierarchies with infinitely many symmetries and Lax pairs. For a special example the infinitely many nonlocal conservation laws and some explicit and implicit exact solutions are also given.

Quasi-solution approach for a two dimensional nonlinear inverse diffusion problem

This paper is related to a class of inverse problems for identification of the coefficient in a square porous medium. The unknown coefficient depends on the solution and belongs the class of admissible coefficients. Using continuous dependence of solutions on the coefficient convergence in the corresponding direct problems, the existence of the quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

在广义条件对称下的一类非线性扩散方程的精确解

本文利用广义条件对称法讨论了一类(1 + 1)维非线性扩散方程 的精确解问题。其中，对流项 与源项 都显示的依赖于变量x，本文针对方程的扩散项 这一重要的情形，对该方程进行对称约化、分类，进而给出方程的精确解。 This paper will discuss the exact solution of (1 + 1) dimensional nonlinear diffusion equation by using the generalized conditional symmetries method. The convection term and source term are dependent on the variable x. This paper mainly discusses diffusion term and finally it will give the exact solutions by using method of symmetrically reduced and classified.

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.

Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.

Some results for an $${\mathcal{N}}$$-dimensional nonlinear diffusion equation with radial symmetry

The solutions of a nonlinear diffusion equation by considering the radially symmetric \({\mathcal{N}}\)-dimensional case are investigated. This equation has the nonlinearity present in the diffusive term and external force. The solutions are obtained by using a similarity method and connected to the q-exponential and q-logarithmic functions which emerge from the Tsallis formalism. In addition, the results obtained here may be useful to investigate a rich class of situations related to anomalous diffusion.

Solutions for a fractional nonlinear diffusion equation with external force and absorbentterm

We devote this work to investigate the solutions of a -dimensional nonlinear fractional diffusion equation which emerges from the continuityequation by considering a nonlinear fractional generalization of Darcy law andincorporating an absorbent term. The solutions obtained show a nonusual spreading of thedistribution and a compact or long tail behavior which can be related to the anomalousdiffusion.

On the integrability of the generalized Fisher-type nonlinear diffusion equations

In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature −1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2.

Conditional Lie Bäcklund symmetries and solutions to (n+1)-dimensional nonlinear diffusion equations

This paper discusses a class of (n+1)-dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non-Newtonian fluids. It is shown that some radially symmetric equations admit certain types of conditional Lie Bäcklund symmetries. As a result, exact solutions to the resulting equations are obtained. Those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term. The phenomena of extinction and blow up and behavior to many of the solutions are described.

Application of finite difference method to estimation of diffusion coefficient in a one dimensional nonlinear inverse diffusion problem

Application of finite difference method to estimation of diffusion coefficient in a one dimensional nonlinear inverse diffusion problem

Exact solutions and generalized conditional symmetries to [formula omitted]-dimensional nonlinear diffusion equations with source term

It is shown that the ( n + 1 ) -dimensional nonlinear diffusion equations with source term admit a class of second-order generalized conditional symmetry depending on the space variable. Symmetry reductions and exact solutions are obtained through the generalized conditional symmetry approach. The results extend some known exact solutions such as the instantaneous source solution, dipole solution and certain polynomial type solutions of nonlinear diffusion equations without source.

Functional Variable Separation for Generalized(1+2)-Dimensional Nonlinear Diffusion Equations

The functional variable separation approach is applied to study thegeneralized (1+2)-dimensional nonlinear diffusion equations. Completeclassification for those equations admitting the functional separablesolutions and some such exact solutions are obtained. Consequently, theresults reported previously are widely extended.

Experimental test of the self-organized criticality of vortices in superconductors

Magneto-optical studies of the vortex dynamics in Nb films reveal stochastic jumps of flux bundles forming a rough penetration front. The scaling analysis of the width, correlation function, and the length of flux profiles behind the front retrieves the roughness exponents and Housdorff dimension characterizing the self-organized critical state. The exponents correspond to (1+1)-dimensional nonlinear diffusion in systems with quenched disorder and long-range correlations. The power spectra of the profiles confirm the self-affine character of the dynamically formed critical state.

Approximate Lie group analysis and solutions of 2D nonlinear diffusion convection equations

Approximate Lie symmetries of the (2 + 1)-dimensional nonlinear diffusion equation with a small convection are completely classified. It is known that the invariance principle furnishes a systematic method of solving initial-value problems. The solutions of instantaneous source type of the 2D diffusion–convection equation are obtained for the case of power-law diffusivity, using a symmetry reduction.