scholarly journals Convergence Analysis from the Solution of Riccati’s Fractional Differential Equation by Using Polynomial Least Squares Method

2020 ◽  
Vol 21 (1) ◽  
pp. 7-14
Author(s):  
Dian Nuryani ◽  
Endang Rusyaman ◽  
Betty Subartini
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Least Squares ◽  
Fractional Differential Equation ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Percentage Error ◽  
Mathematical Form ◽  
Fractional Differential

Riccati's Fractional Differential Equation (RFDE) has become a topic of study for researchers because RFDE can model variety of phenomenon in science such as random processes, optimal control and diffusion problems. Phenomena that can be modeled in a mathematical form can make it easier for humans to analyze several things from that phenomenon. RFDE generally does not have an exact solution, therefore a numerical approach solution is needed, one of the methods that gives good accuracy to the actual or exact solution is Polynomial Least Squares, where the errors calculated based on mean absolute percentage error (MAPE) produce a percentage below 1%. In addition, the convergence of a sequence from approximate solutions indicates that the sequence will converge to a solution.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

scholarly journals The Exact Iterative Solution of Fractional Differential Equation with Nonlocal Boundary Value Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jinxiu Mao ◽  
Zengqin Zhao ◽  
Chenguang Wang
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Differential Equation ◽  
Nonlocal Boundary ◽  
Approximation Error ◽  
Iterative Solution ◽  
Stieltjes Integral ◽  
Iterative Technique ◽  
Integral Conditions ◽  
Fractional Differential

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.

scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

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.

A numerical approach for analyzing the transverse vibrations of an axially moving viscoelastic string

2014 ◽  
Vol 05 (03) ◽  
pp. 1450005
Author(s):  
Ying Wu ◽  
Weijia Zhao ◽  
Jiang Zhu
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Memory Effect ◽  
Dynamical Behavior ◽  
Numerical Approach ◽  
Computation Cost ◽  
Viscoelastic Parameters ◽  
Transverse Vibrations ◽  
Fractional Differential ◽  
Axially Moving

In this paper, a fractional differential equation is introduced to describe the transverse vibrations of an axially moving viscoelastic string. An iterative algorithm is constructed to analyze the dynamical behavior. By conveying the memory effect of the fractional differential terms step by step, the computation cost can be greatly reduced. As a numerical example, the effects of the viscoelastic parameters on a moving string are investigated.

scholarly journals DGJ Method for Linear and Nonlinear Third order Fractional Differential Equation

2018 ◽  
Author(s):  
Mahmut Modanli
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Differential Equation ◽  
Initial Boundary ◽  
Third Order ◽  
Partial Differential ◽  
Approximation Solution ◽  
The Third ◽  
Numerical Result ◽  
Fractional Differential

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

scholarly journals What is the best fractional derivative to fit data?

2017 ◽  
Vol 11 (2) ◽  
pp. 358-368 ◽  
Author(s):  
Ricardo Almeida
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Fractional Derivative ◽  
Least Squares ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Least Squares Fitting ◽  
Different Types ◽  
Fractional Differential

The aim of this work is to show, based on concrete data observation, that the choice of the fractional derivative when modelling a problem is relevant for the accuracy of a method. Using the least squares fitting technique, we determine the order of the fractional differential equation that better describes the experimental data, for different types of fractional derivatives.

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