scholarly journals The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1454
Author(s):  
Richard Paris
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Expansion ◽  
Entire Function ◽  
Fractional Order ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Hypergeometric Type ◽  
Finite Sum ◽  
Special Case

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

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.

scholarly journals Solution of Fractional Partial Differential Equations Using Fractional Power Series Method

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Asif Iqbal Ali ◽  
Muhammad Kalim ◽  
Adnan Khan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Power Series ◽  
Fractional Order ◽  
Order Partial Differential Equation ◽  
Series Method ◽  
Partial Differential ◽  
Power Series Method ◽  
Power Series Technique ◽  
Series Technique

In this paper, we are presenting our work where the noninteger order partial differential equation is studied analytically and numerically using the noninteger power series technique, proposed to solve a noninteger differential equation. We are familiar with a coupled system of the nonlinear partial differential equation (NLPDE). Noninteger derivatives are considered in the Caputo operator. The fractional-order power series technique for finding the nonlinear fractional-order partial differential equation is found to be relatively simple in implementation with an application of the direct power series method. We obtained the solution of nonlinear dispersive equations which are used in electromagnetic and optics signal transformation. The proposed approach of using the noninteger power series technique appears to have a good chance of lowering the computational cost of solving such problems significantly. How to paradigm an initial representation plays an important role in the subsequent process, and a few examples are provided to clarify the initial solution collection.

scholarly journals Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 24
Author(s):  
Muhammad Shakeel ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Fractional Order ◽  
Soliton Solutions ◽  
Wave Transformation ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Closed Form Solutions

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

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