scholarly journals Image Enhancement of Cross-Border E-Commerce Logistics Video Surveillance Based on Partial Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Xiaosheng Yu ◽  
Peili Wu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Image Enhancement ◽  
Video Surveillance ◽  
Empirical Formula ◽  
Histogram Equalization ◽  
Partial Differential ◽  
Cross Border ◽  
Enhancement Method

Although the development time of cross-border e-commerce in China is very short, the scale of its transactions and the speed of development are amazing, and as a supporting foundation for promoting economic and trade globalization, cross-border e-commerce has an extremely important strategy for this guiding role. This not only brings new opportunities to cross-border e-commerce companies but also excavates a huge potential market for the logistics industry. Cross-border e-commerce not only breaks through the trade barriers between countries; it makes trade move towards borderlessness and at the same time triggers major changes in international trade. This paper introduces partial differential equations into the video surveillance image enhancement system of cross-border e-commerce logistics. Aiming at the shortcomings of the contrast enhancement method based on gradient field equalization, this paper proposes a partial differential enhancement method based on histogram equalization. By proposing a gradient transformation function, the edges and textures with relatively small gradient values are enhanced to make the original weaker texture details clearer. In order to better adjust the brightness and contrast of the image, combined with histogram equalization, we propose an inverse equalization transformation. When the histogram equalization and the inverse equalization transform are combined reasonably, the brightness and contrast of the image can be adjusted very well. In this paper, the finite difference method is used for discretization when solving partial differential equations, and Euler’s equation is obtained by applying the principle of least squares. By introducing the heat equation, the direct solution of Euler’s equation is converted into an iterative form, which greatly reduces the amount of calculation. This article uses statistical methods to obtain the empirical formula of the fractional differential order. This empirical formula makes the calculation of the order of the fractional derivative easy and can be extended to other fractional image enhancement models and overcomes the shortcomings of the traditional fractional derivative order obtained through experience or a large number of experiments. Experiments show that the proposed algorithm not only enhances detailed texture information but also improves image clarity, overall brightness, and contrast without color distortion. The objective evaluation indicators also show the superiority of the algorithm.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

2021 ◽  
pp. 2150492
Author(s):  
Delmar Sherriffe ◽  
Diptiranjan Behera ◽  
P. Nagarani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Function Method ◽  
Visual Representations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Varying Parameters ◽  
Wave Solutions ◽  
Soliton Wave Solutions

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

scholarly journals The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 771
Author(s):  
Yusuf Gurefe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method

In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

scholarly journals A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

2021 ◽  
Vol 9 ◽  
Author(s):  
Ahmad El-Ajou ◽  
Zeyad Al-Zhour
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Hyperbolic System ◽  
Series Solution ◽  
Fractional Power ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

Infrared image enhancement based on variational partial differential equations

2014 ◽  
Vol 29 (2) ◽  
pp. 281-285 ◽  
Author(s):  
赵文达 ZHAO Wen-da ◽  
赵建 ZHAO Jian ◽  
韩希珍 HAN Xi-zhen ◽  
续志军 XU Zhi-jun
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Image Enhancement ◽  
Infrared Image ◽  
Partial Differential

scholarly journals The need for the fractional operators

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
E. A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Time ◽  
Space Time ◽  
Fractional Operators ◽  
Partial Differential ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

AbstractIn this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

scholarly journals Numerical simulation method of lines together with a pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mushtaq Ali ◽  
Mohammed Almoaeet ◽  
Basim Karim Albuohimad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Method Of Lines ◽  
Pseudospectral Method ◽  
Partial Differential ◽  
Content Type ◽  
Jacobi Functions ◽  
Left And Right

PurposeThis study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.Design/methodology/approachIn this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.FindingsApplying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.Originality/valueThis paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.

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