Approximate solution for kinetic differential equations

Summary In the present work we show the analysis of an approximate solution for the coupled kinetic differential equations of a defect involving processes of untrapping, retrapping and recombination annihilation, using the Poincaré-Dulac theorem. The authors of this paper have agreed to not receive the proofs for correction.     

Path integral solution of linear second order partial differential equations I: the general construction

A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schrödinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette.     

RLC electrical circuit of non-integer order

In this work a fractional differential equation for the electrical RLC circuit is studied. The order of the derivative being considered is 0 < γ ≤ 1. To keep the dimensionality of the physical quantities R, L and C an auxiliary parameter γ is introduced. This parameter characterizes the existence of fractional components in the system. It is shown that there is a relation between and σ through the physical parameters RLC of the circuit. Due to this relation, the analytical solution is given in terms of the Mittag-Leffler function depending on the order of the fractional differential equation.     

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττ_k:T ^k+1(M)=T(T ^k(M))→T ^k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT ⁰(M)=M. LetV represent a possibly nonstationary vector field overT ^k(M), and letQ be a subset/submanifold inT ^k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f ^[k], wheref ^[k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail. To the memory of my teacher and friend M. Kuga (1928–1990).     

On fractional order differential equations model for nonlocal epidemics

A fractional order model for nonlocal epidemics is given. Stability of fractional order equations is studied. The results are expected to be relevant to foot-and-mouth disease, SARS and avian flu.     

Macroscopic Einstein equations to second order in the interaction constant

The present paper is a direct continuation of an earlier paper [JETP 83, 1 (1996)] devoted to the derivation of the macroscopic Einstein equations to within terms of second order in the interaction constant. Ensemble averaging of the microscopic Einstein equations and the Liouville equation for the random functions leads to a closed system of macroscopic Einstein equations and kinetic equations for one-particle distribution functions. The macroscopic Einstein equations differ from the classical equations in that their left-hand side contains additional terms due to particle interaction. The terms are traceless tensors with zero divergence. An explicit covariant expression for these terms is given in the form of momentum-space integrals of expressions dependent on one-particle distribution functions of the interacting particles of the medium. The given expressions are proportional to the cube of the Einstein constant and the square of the particle number density. The latter relationship implies that interaction effects manifest themselves in systems of very high density (the universe in the early stages of its evolution, dense objects close to gravitational collapse, etc.) Zh. éksp. Teor. Fiz. 112, 1153–1166 (October 1997)     

Limit cycles for a class of second order differential equations

We study the limit cycles of a wide class of second order differential equations, which can be seen as a particular perturbation of the harmonic oscillator. In particular, by choosing adequately the perturbed function we show, using the averaging theory, that it is possible to obtain as many limit cycles as we want.     

Higher order differential lift equations over manifolds

LetM be a differentiable manifold modeled on a Banach space overK=R or C. LetT ^k(M) be thekth iterated tangential extension ofM, and letk ^M be thekth Bowman (=restricted tangential) extension ofM. It is shown that there is an embedding ϕ_k:k ^{→T k}(M), and that such embeddings constitute a natural transformation of functors. LetQ be a subset/submanifold inT ^k(M), and letV:Q→T(Q) be a differentiable vector field. CallV k-suitable if everyK-curveg inQ satisfyingg′=V° g has the formg=f ^[k], wheref ^[k] denotes thekth iterated differential lift of aK-curvef inM. It is shown thatV isk-suitable if and only if: (a) , where is a subset/submanifold ink ^M, and (b) , where isk-suitable relative to restricted tangentialK-curve liftsf ^(k). Interpretive consequences for motion problems are discussed.     

Parametric solution method for self-consistency equations and order parameter equations derived from nonlinear Fokker-Planck equations

We propose a parametric approach to solve self-consistency equations that naturally arise in many-body systems described by nonlinear Fokker-Planck equations in general and nonlinear Vlasov-Fokker-Planck equations of Haissinski type in particular. We demonstrate for the Hess-Doi-Edwards model and the McMillan model of nematic and smectic liquid crystals that the parametric approach can be used to compute bifurcation diagrams and critical order parameters for systems exhibiting one or more than one order parameters. In addition, we show that in the context of the parametric approach solutions of the Haissinski model can be studied from the perspective of a pseudo order parameter.     

Methods of analytical mechanics for solving differential equations of first order

A differential equation of first order can be expressed by the equation of motion of a mechanical system. In this paper, three methods of analytical mechanics, i.e. the Hamilton--Noether method, the Lagrange--Noether method and the Poisson method, are given to solve a differential equation of first order, of which the way may be called the mechanical methodology in mathematics.     

Graphical solution of special kinetic differential equations

A system of homogeneous linear differential equations describes the time evolution of many dynamic systems in physics. chemistry, and biology (e.g. radioactive decay, chemical kinetics of monomolecular reactions, etc.). A graph-theory approach for the direct solution of this system represented by an acyclic reaction graph is elaborated. Applying this simple method one can construct the time-dependent solution immediately from the corresponding reaction graph.I would like to express my thanks to Drs. Z. Slanina and P. Hadrava for many helpful discussions and critical comments concerning the subject of this communication. I am greatly indebted to Prof. V. Kvasnika for all the support and understanding he has given to my work in this field.     

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.     

Numerical solution of differential equations with colored noise

Using the general theory of numerical integration of stochastic differential equations, a constructive approach to numerical methods for a system with colored noise is proposed. Efficient methods up to the 5/2 strong order and up to the third weak order, including Runge-Kutta and implicit schemes, are presented. The algorithms are tested on the Kubo oscillator.     

变质量完整力学系统的高阶运动微分方程

从Мещерский方程出发,建立变质量力学系统的高阶D'Alembert-Lagrange原理,导出变质量完整力学系统的各类高阶运动微分方程.结果表明,它们扩充和优化了完整力学的相关理论. 关键词： 变质量完整力学系统 高阶力变率 高阶D'Alembert-Lagrange原理 高阶运动微分方程     

Path integral solution of linear second order partial differential equations II: elliptic, parabolic, and hyperbolic cases

The general theorem of LaChapelle [Path Integral Solution of Linear Second Order Partial Differential Equations. I. The General Case, preprint (2003)] is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.     

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.     

Existence and approximation of solutions of fractional order iterative differential equations

In this paper, we investigate existence and approximation of solutions of fractional order iterative differential equations by virtue of nonexpansive mappings, fractional calculus and fixed point methods. Three existence theorems as well as convergence theorems for a fixed point iterative method designed to approximate these solutions are obtained in two different work spaces via Chebyshev’s norm, Bielecki’s norm and β norm. Finally, an example is given to illustrate the obtained results.     

A higher order lattice BGK model for simulating some nonlinear partial differential equations

In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.