Improved PSO Research for Solving the Inverse Problem of Parabolic Equation

Peng Yamian,Ji Nan,Zhang Huancheng 보안공학연구지원센터 2016 International Journal of Database Theory and Appli Vol.9 No.12

Parameter identification problem has important research background and research value, has become in recent years inverse problem of heat conduction of top priority. This paper studies the Parabolic Equation Inverse Problems of parameter identification problem, and applies PSO to solve research. Firstly, this paper establishes the model of the inverse problem of partial differential equations. The content and classification of the inverse problem of partial differential equations are explained. Frequently, the construction and solution of the finite difference method for parabolic equations are studied, and two stable schemes for one dimensional parabolic equation are given. And two numerical simulations were given. Partial differential equation discretization was with difference quotient instead of partial derivative. The partial differential equations with initial boundary value problem into algebraic equations, and then solving the resulting algebraic equations. Then, the basic principles of PSO and its improved algorithms are studied and compared. Particle swarm optimization algorithm program implementation. Finally, the Parabolic Equation Inverse Problems of particle swarm optimization algorithm performed three simulations. We use a set of basis functions gradually approaching the true solution, selection of initial value. The reaction is converted into direct problem question, then use difference method Solution of the direct problem. The solution of the problem with the additional conditions has being compared. The reaction optimization problem is transformed into the final particle swarm optimization algorithm to solve. Verify the Parabolic Equation Inverse Problems of particle swarm optimization algorithm correctness and applicability.

A Generalized Finite Difference Method for Solving Fokker-Planck-Kolmogorov Equations

Zhao, Li,Yun, Gun Jin The Korean Society for Aeronautical and Space Scie 2017 International Journal of Aeronautical and Space Sc Vol.18 No.4

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

A Generalized Finite Difference Method for Solving Fokker-Planck-Kolmogorov Equations

Li Zhao,윤군진 한국항공우주학회 2017 International Journal of Aeronautical and Space Sc Vol.18 No.4

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

ON THE DIFFERENCE EQUATION $x_{n+1}=\frac{a+bx_{n-k}-cx_{n-m}}{1+g(x_{n-l})}$

Zhang, Guang,Stevic, Stevo 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.25 No.1

In this paper we consider the difference equation $$x_{n+1}=\frac{a+bx_{n-k}\;-\;cx_{n-m}}{1+g(x_{n-l})}$$ where a, b, c are nonegative real numbers, k, l, m are nonnegative integers and g(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.

Dynamics of a higher order rational difference equation

Yanqin Wang 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.3

In this paper, we investigate the invariant interval, periodic character, semicycle and global attractivity of all positive solutions of the equation yn+1 =<수식> , n = 0, 1, . . . , where the parameters p, q, r and the initial conditions y−k, . . . , y−1, y0 are positive real num- bers, k ∈ {1, 2, 3, . . . }.It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002] In this paper, we investigate the invariant interval, periodic character, semicycle and global attractivity of all positive solutions of the equation yn+1 =<수식> , n = 0, 1, . . . , where the parameters p, q, r and the initial conditions y−k, . . . , y−1, y0 are positive real num- bers, k ∈ {1, 2, 3, . . . }.It is worth to mention that our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002]

A multigrid solution for the Cahn-Hilliard equation on nonuniform grids

Choi, Y.,Jeong, D.,Kim, J. Elsevier [etc.] 2017 Applied Mathematics and Computation Vol.293 No.-

We present a nonlinear multigrid method to solve the Cahn-Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.

An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation

Li, Y.,Lee, H.G.,Jeong, D.,Kim, J. Pergamon Press ; Elsevier Science Ltd 2010 COMPUTERS & MATHEMATICS WITH APPLICATIONS - Vol.60 No.6

We present an unconditionally stable second-order hybrid numerical method for solving the Allen-Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen-Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank-Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in both time and space.

The upwind hybrid difference methods for a convection diffusion equation

Jeon, Youngmok,Tran, Mai Lan Elsevier 2018 Applied numerical mathematics Vol.133 No.-

<P><B>Abstract</B></P> <P>We propose the upwind hybrid difference method and its penalized version for the convection dominated diffusion equation. The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells <I>(cell FD)</I> and the other is the <I>interface finite difference (interface FD)</I> on edges of cells. The interface finite difference is derived from continuity of normal fluxes. The penalty method is obtained by adding small diffusion in the interface FD. The penalty term makes it possible to reduce severe numerical oscillations in the upwind hybrid difference solutions. The penalty parameter is designed to be some power of the grid size. A complete stability is provided. Convergence estimates seems to be conservative according to our numerical experiments. To exposit convergence property and controllability of numerical oscillations several numerical tests are provided.</P>

ON THE RATIONAL(${\kappa}+1,\;{\kappa}+1$)-TYPE DIFFERENCE EQUATION

Stevic, Stevo 한국전산응용수학회 2007 Journal of applied mathematics & informatics Vol.24 No.1

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n+1}=\frac{a_0+{{\sum}^k_{j=1}}a_jx_{n-j+1}}{b_0+{{\sum}^k_{j=1}}b_jx_{n-j+1}},\;\;n=0,\;1,...$$ where $k{\in}N,\;and\;a_j,b_j,\;j=0,\;1,...,\;k $, are nonnegative numbers such that $b_0+{{\sum}^k_{j=1}}b_jx_{n-j+1}>0$ for every $n{\in}N{\cup}\{0\}$. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations(part 6), J. Differ. Equations Appl. 11(8)(2005), 759-777.

A Crank@?Nicolson scheme for the Landau@?Lifshitz equation without damping

Jeong, D.,Kim, J. Koninklijke Vlaamse Ingenieursvereniging ; Elsevie 2010 Journal of computational and applied mathematics Vol.234 No.2

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.