scholarly journals Numerical approach for approximating the Caputo fractional-order derivative operator

2021 ◽  
Vol 6 (11) ◽  
pp. 12743-12756
Author(s):  
Ramzi B. Albadarneh ◽  
◽  
Iqbal Batiha ◽  
A. K. Alomari ◽  
Nedal Tahat ◽  
...  
Keyword(s):  
Fractional Order ◽  
Truncation Error ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Analytic Solutions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Linear And Nonlinear ◽  
Caputo Fractional Order Derivative

<abstract><p>This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator $ D_{a}^{\alpha}y(t) $ of order $ m-1 &lt; \alpha &lt; m $, where $ m\in\mathbb{N} $. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.</p></abstract>

scholarly journals Numerical approach of riemann-liouville fractional derivative operator

2021 ◽  
Vol 11 (6) ◽  
pp. 5367
Author(s):  
Ramzi B. Albadarneh ◽  
Iqbal M. Batiha ◽  
Ahmad Adwai ◽  
Nedal Tahat ◽  
A. K. Alomari
Keyword(s):  
Fractional Derivative ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Weighted Mean ◽  
Derivative Operator ◽  
Liouville Fractional Derivative ◽  
Linear And Nonlinear ◽  
Residual Errors

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

scholarly journals A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 275-283
Author(s):  
Kubra Bicer ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Mean Value ◽  
Computational Method ◽  
Mean Value Theorem ◽  
Residual Functions ◽  
Non Linear ◽  
Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

scholarly journals Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons

2020 ◽  
pp. 66-76
Author(s):  
Falade Kazeem Iyanda ◽  
Tiamiyu Abd`gafar Tunde
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Analytical Approach ◽  
Computational Algorithm ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Linear And Nonlinear ◽  
Variational Iterative Method

In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.

Neural Approximation-Based Adaptive Control for Pure-Feedback Fractional-Order Systems With Output Constraints and Actuator Nonlinearities

2018 ◽  
pp. 468-495 ◽  
Author(s):  
Farouk Zouari ◽  
Amina Boubellouta
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Fractional Order Systems ◽  
Control Approach ◽  
Output Constraints ◽  
Actuator Nonlinearities ◽  
Unknown Control ◽  
The Stability

In this chapter, an adaptive control approach-based neural approximation is developed for a category of uncertain fractional-order systems with actuator nonlinearities and output constraints. First, to overcome the difficulties arising from the actuator nonlinearities and nonaffine structures, the mean value theorem is introduced. Second, to deal with the uncertain nonlinear dynamics, the unknown control directions and the output constraints, neural networks, smooth Nussbaum-type functions, and asymmetric barrier Lyapunov functions are employed, respectively. Moreover, for satisfactorily designing the control updating laws and to carry out the stability analysis of the overall closed-loop system, the Backstepping technique is used. The main advantage about this research is that (1) the number of parameters to be adapted is much reduced, (2) the tracking errors converge to zero, and (3) the output constraints are not transgressed. At last, simulation results demonstrate the feasibility of the newly presented design techniques.

scholarly journals Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 597 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Wave Equations ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
High Rate ◽  
Numerical Examples ◽  
Linear And Nonlinear

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

scholarly journals Non-standard Volterra integral equations: a mean-value theorem numerical approach

2020 ◽  
Vol 14 (9) ◽  
pp. 423-432
Author(s):  
Paolo De Angelis ◽  
Roberto De Marchis ◽  
Antonio Luciano Martire ◽  
Stefano Patri
Keyword(s):  
Integral Equations ◽  
Numerical Approach ◽  
Mean Value ◽  
Volterra Integral Equations ◽  
Mean Value Theorem

scholarly journals Fractional-Order Investigation of Diffusion Equations via Analytical Approach

2021 ◽  
Vol 8 ◽  
Author(s):  
Haobin Liu ◽  
Hassan Khan ◽  
Saima Mustafa ◽  
Lianming Mou ◽  
Dumitru Baleanu
Keyword(s):  
Applied Sciences ◽  
Fractional Order ◽  
Closed Form Solution ◽  
Approximate Solutions ◽  
Form Solution ◽  
Diffusion Equations ◽  
Physical Problems ◽  
Research Article ◽  
Closed Contact ◽  
Caputo Fractional Order Derivative

This research article is mainly concerned with the analytical solution of diffusion equations within a Caputo fractional-order derivative. The motivation and novelty behind the present work are the application of a sophisticated and straight forward procedure to solve diffusion equations containing a derivative of a fractional-order. The solutions of some illustrative examples are calculated to confirm the closed contact between the actual and the approximate solutions of the targeted problems. Through analysis it is shown that the proposed solution has a higher rate of convergence and provides a closed-form solution. The small number of calculations is the main advantage of the proposed method. Due to a comfortable and straight forward implementation, the suggested method can be utilized to nonlinear fractional-order problems in various applied science branches. It can be extended to solve other physical problems of fractional-order in multiple areas of applied sciences.

scholarly journals Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Excellent Agreement ◽  
Analytical Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Small Parameters

We apply an analytical method called the Optimal Parametric Iteration Method (OPIM) to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

scholarly journals Adaptation of Differential Transform Method for the Numeric-Analytic Solution of Fractional-Order Rössler Chaotic and Hyperchaotic Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Asad Freihat ◽  
Shaher Momani
Keyword(s):  
Fractional Order ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Time Step ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential ◽  
Hyperchaotic Systems

A new reliable algorithm based on an adaptation of the standard generalized differential transform method (GDTM) is presented. The GDTM is treated as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of fractional-order Rössler chaotic and hyperchaotic systems. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

scholarly journals The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2218
Author(s):  
Mohd Taib Shatnawi ◽  
Adel Ouannas ◽  
Ghenaiet Bahia ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Approximate Methods ◽  
Homotopy Analysis ◽  
Optimal Homotopy Asymptotic Method ◽  
Relativistic Oscillator ◽  
Convergence Of Approximate Solutions

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

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