Research on the Application of Fractional differential integral equation in the calculation of shooting rate and the Construction of influencing factors

2020 ◽  
Vol 29 (4) ◽  
Author(s):  
Zheng Bo ◽  
Lili Tian ◽  
Zhihua Liu
Keyword(s):  
Integral Equation ◽  
Influencing Factors ◽  
Differential Integral Equation ◽  
Fractional Differential

scholarly journals An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Integral Equation ◽  
Chebyshev Polynomials ◽  
Algebraic System ◽  
Operational Matrix ◽  
Original Equation ◽  
Variable Order ◽  
Computationally Efficient ◽  
Collocation Points ◽  
Differential Integral Equation ◽  
Fractional Differential

An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

Controllability of Abstract Systems of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Therese Mur ◽  
Hernán R. Henríquez
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Control Systems ◽  
Banach Spaces ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
First Order ◽  
Fractional Integral Equation ◽  
Fractional Differential

AbstractIn this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

A NOTE ON DIFFRACTION BY AN INFINITE SLIT

1960 ◽  
Vol 38 (1) ◽  
pp. 38-47 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Integral Equation ◽  
Normal Derivative ◽  
Cylindrical Wave ◽  
Narrow Slit ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Boundary Values ◽  
Transmission Coefficients ◽  
Differential Integral Equation

The two-dimensional problem of diffraction of a plane wave by a narrow slit is considered. The assumed boundary values on the screen are the vanishing of either the total wave function or its normal derivative. In the former case, a differential–integral equation is obtained for the unknown function in the slit; in the latter, a pure integral equation is found. Solutions to these equations are given in the form of series in powers of ε (where ε/π is the ratio of slit width to wavelength), the coefficients of which depend on log ε. Expressions are found for the transmission coefficients as functions of ε and the angle of incidence; these are compared with previous determinations of other authors.A brief outline is given for the treatment of diffraction of a cylindrical wave by the slit.

Numerical solution of fractional differential equations via a Volterra integral equation approach

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Shahrokh Esmaeili ◽  
Mostafa Shamsi ◽  
Mehdi Dehghan
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Volterra Integral Equation ◽  
Equation Approach ◽  
Algebraic Equations ◽  
Integral Equation Approach ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
The Given

AbstractThe main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.

scholarly journals A Model of Batch Grinding of Talc Powders

2011 ◽  
Vol 250-253 ◽  
pp. 4016-4021 ◽  
Author(s):  
Hong Yong Xie ◽  
Hao Yuan ◽  
Jie Guan
Keyword(s):  
Integral Equation ◽  
Particle Size ◽  
Distribution Functions ◽  
Specific Rate ◽  
Rate Process ◽  
First Order ◽  
Measurement Results ◽  
Differential Integral Equation ◽  
Distribution Parameters ◽  
Grinding Time

Grinding of talc powders has been studied both theoretically and experimentally. The specific rates of breakage of talc powders were measured based on the first-order breakage kinetics model and the cumulative breakage distribution parameters of talc powders were measured from primary breakage products. Based on the measurement results, the specific rate of breakage and cumulative breakage distribution functions were correlated with particle size asand , repectively. A differential-integral equation was thus build to describe grinding as a rate process and was integrated numerically. Comparisons on size distribution showed that the specific rate of breakage of talc powders increased with grinding time at an increase rateabout 0.0066min-2.

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One

2005 ◽  
Vol 1 (2) ◽  
pp. 178-185 ◽  
Author(s):  
Pankaj Kumar ◽  
Om P. Agrawal
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Volterra Type ◽  
Entire Domain ◽  
Type Integral ◽  
Fractional Differential

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

Fractional impulsive differential equations: Exact solutions, integral equations and short memory case

2019 ◽  
Vol 22 (1) ◽  
pp. 180-192 ◽  
Author(s):  
Guo–Cheng Wu ◽  
De–Qiang Zeng ◽  
Dumitru Baleanu
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Impulsive Differential Equations ◽  
Fundamental Solutions ◽  
Impulsive Conditions ◽  
Short Memory ◽  
Fractional Differential ◽  
Piecewise Functions

Abstract Fractional impulsive differential equations are revisited first. Some fundamental solutions of linear cases are given in this study. One straightforward technique without using integral equation is adopted to obtain exact solutions which are given by use of piecewise functions. Furthermore, a class of short memory fractional differential equations is proposed and the variable case is discussed. Mittag–Leffler solutions with impulses are derived which both satisfy the equations and impulsive conditions, respectively.

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