Variable separation method for a nonlinear time fractional partial differential equation with forcing term

2018 ◽  
Vol 339 ◽  
pp. 297-305 ◽  
Author(s):  
Sheng Zhang ◽  
Siyu Hong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Separation Method ◽  
Variable Separation ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Forcing Term ◽  
Variable Separation Method

scholarly journals Solusi Persamaan Burgers Inviscid dengan Metode Pemisahan Variabel

2021 ◽  
Vol 4 (2) ◽  
pp. 88
Author(s):  
Hisyam Ihsan ◽  
Syafruddin Side ◽  
Muhammad Iqbal
Keyword(s):  
Differential Equation ◽  
Stokes Equation ◽  
Burgers Equation ◽  
Equation System ◽  
Separation Method ◽  
Navier Stokes ◽  
Variable Separation ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
Variable Separation Method

Penelitian ini mengkaji tentang solusi persamaan Burgers Inviscid dengan metode pemisahan variabel. Tujuan dari penelitian ini adalah untuk mengetahui penyederhanaan sistem persamaan Navier-Stokes menjadi persamaan Burgers Inviscid, menemukan solusi persamaan Burgers Inviscid dengan metode pemisahan variabel, dan melakukan simulasi solusi persamaan dengan menggunakan software Maple18. Persamaan Burgers muncul sebagai penyederhanaan model yang rumit dari sistem persamaan Navier-Stokes. Persamaan Burgers adalah persamaan diferensial parsial hukum konservasi dan merupakan masalah hiperbolik, yaitu representasi nonlinier paling sederhana dari persamaan Navier-Stokes. Metode pemisahan variabel merupakan salah satu metode klasik yang efektif digunakan dalam menyelesaikan persamaan diferensial parsial dengan mengasumsikan  untuk mendapatkan komponen x dan t. Kemudian akan dilakukan subtitusi pada persamaan diferensial, sehingga dengan cara ini akan didapatkan solusi persamaan diferensial parsial.Kata Kunci: Persamaan Burgers Inviscid, metode pemisahan variabel, persamaan Navier-StokesThis study examines the solution of Burgers Inviscid equation with variable separation method. The purpose of this study was to find out the simplification of the Navier-Stokes equation system into the Burgers Inviscid equation, find a solution to the Burgers Inviscid equation with the variable separation method, and simulate equation solutions using Maple18 software. The Burgers equation emerged as a complicated simplification of the Navier-Stokes equation system. The Burgers equation is a partial differential equation of conservation law and is a hyperbolic problem, i.e. the simplest nonlinear representation of the Navier-Stokes equation. The variable separation method is one of the classic methods that is effectively used in solving partial differential equations assuming  to obtain the x and t components. Then there will be substitutions to differential equations, so that in this way there will be a partial differential equation solution.Keywords: Burgers Inviscid Equation, variable separation method, Navier-Stokes equations.

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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