A new algorithm for fractional differential equation based on fractional order reproducing kernel space

2020 ◽  
Vol 44 (2) ◽  
pp. 2171-2182
Author(s):  
Ruimin Zhang ◽  
Yingzhen Lin
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Reproducing Kernel ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Fractional Differential

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Jiang ◽  
Zhong Chen ◽  
Ning Hu ◽  
Yali Chen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Reproducing Kernel ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Orthonormal Bases ◽  
Fractional Differential

AbstractIn recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

2018 ◽  
pp. 515-522
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Expansion Method ◽  
Travelling Wave ◽  
Boussinesq System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Fractional System ◽  
Partial Fractional Differential Equation

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation (FDEs) with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

scholarly journals Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

2021 ◽  
Vol 7 (2) ◽  
pp. 2281-2317
Author(s):  
Yong Xian Ng ◽  
◽  
Chang Phang ◽  
Jian Rong Loh ◽  
Abdulnasir Isah ◽  
...  
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Successive Approximations ◽  
Explicit Analytical Solution ◽  
Special Cases ◽  
Fractional Differential ◽  
Alpha Beta ◽  
Published Paper

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 &lt; \alpha, \beta &lt; 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 &lt; \alpha, \beta &lt; 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

scholarly journals Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xiaoli Zhang ◽  
Haolu Zhang ◽  
Lina Jia ◽  
Yulan Wang ◽  
Wei Zhang
Keyword(s):  
Numerical Solution ◽  
Fractional Order ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Diffusion Equations ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Reproducing Kernel Method ◽  
Reproducing Kernel Spaces ◽  
Methods Numerical

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

scholarly journals Reproducing Kernel Space Method for the Solution of Linear Fredholm Integro-Differential Equations and Analysis of Stability

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Yue Gao
Keyword(s):  
Differential Equation ◽  
Reproducing Kernel ◽  
Simple Algorithm ◽  
Approximate Solutions ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
True Solution ◽  
Integro Differential Equation ◽  
Analysis Of Stability ◽  
Space Method

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.

scholarly journals Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Sameer Qasim Hasan ◽  
Moataz Abbas Holel
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Optimal Control Problems ◽  
Chebyshev Approximation ◽  
Approximate Solutions ◽  
Control Problems ◽  
Fractional Differential ◽  
Working Method

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.

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