The Comparison of Parameter Identification Methods for Fractional, Partial Differential Equation

2013 ◽  
Vol 210 ◽  
pp. 265-270 ◽  
Author(s):  
Anna Obrączka ◽  
Wojciech Mitkowski
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Parameter Identification ◽  
Numerical Experiments ◽  
Nonlinear Models ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Identification Methods ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt

In this paper the parameter identification methods for nonlinear models were compared for fractional, partial differential equation. The compared three methods are: the Levenberg-Marquardt algorithm, the Gauss-Newton algorithm and Nelder-Mead Simplex method. The series of numerical experiments were performed to test their robustness and calculation speed. The result of this tests were presented and described.

scholarly journals Modification of Levenberg-Marquardt Algorithm for Solve Two Dimension Partial Differential Equation

2018 ◽  
Vol 26 (7) ◽  
pp. 107-117
Author(s):  
Khalid Mindeel M. Al-Abrahemee ◽  
Rana T. Shwayaa
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Partial Differential ◽  
Two Dimension ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt ◽  
In The Beginning ◽  
Lm Algorithm

In this paper we presented a new way based on neural network has been developed for solutione of two dimension  partial differential equations . A modified neural network use to over passing the Disadvantages of LM algorithm, in the beginning we suggest signaler value decompositions of Jacobin matrix (J) and inverse of Jacobin matrix( J-1), if a matrix rectangular or singular  Secondly, we suggest new calculation of μk , that ismk=|| E (w)||2    look the nonlinear execution equations E(w) = 0 has not empty solution W* and we refer   to the second norm in all cases ,whereE(w):  is continuously differentiable and E(x) is Lipeschitz  continuous, that is=|| E(w 2)- E(w 1)||£ L|| w  2- w  1|| ,where L  is Lipeschitz  constant.

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

scholarly journals Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190564
Author(s):  
Zhi-Yong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Algebra ◽  
Infinitesimal Generator ◽  
Linear Time ◽  
Lie Symmetry ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Pde ◽  
Leading Position

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

scholarly journals Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Yi-Fei Pu ◽  
Ji-Liu Zhou ◽  
Patrick Siarry ◽  
Ni Zhang ◽  
Yi-Guang Liu
Keyword(s):  
Differential Equation ◽  
Image Processing ◽  
Partial Differential Equation ◽  
Fractional Order ◽  
Differential Calculus ◽  
Order Partial Differential Equation ◽  
Texture Image ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Integer Order

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

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