scholarly journals The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations with the Riemann-Liouville Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Dumitru Baleanu ◽  
Mohsen Alipour ◽  
Hossein Jafari
Keyword(s):  
Differential Equation ◽  
Approximate Analytical Solution ◽  
Fractional Integration ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Liouville Derivative

We obtain the approximate analytical solution for the fractional quadratic Riccati differential equation with the Riemann-Liouville derivative by using the Bernstein polynomials (BPs) operational matrices. In this method, we use the operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduce the problem to a system of algebraic equations that can be solved easily. The efficiency and accuracy of the proposed method are illustrated by several examples.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

2021 ◽  
Vol 5 (3) ◽  
pp. 100
Author(s):  
Youssri Hassan Youssri
Keyword(s):  
Differential Equation ◽  
Operational Matrix ◽  
Initial Guess ◽  
Test Problems ◽  
Tau Method ◽  
Algebraic Equations ◽  
Differential Problem ◽  
Riccati Differential Equation ◽  
Ultraspherical Polynomials ◽  
Expansion Coefficients

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Chandrali Baishya ◽  
P. Veeresha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Laguerre Polynomial ◽  
Fractional Integration ◽  
Laguerre Polynomials ◽  
Operational Matrix ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differential

The Atangana–Baleanu derivative and the Laguerre polynomial are used in this analysis to define a new computational technique for solving fractional differential equations. To serve this purpose, we have derived the operational matrices of fractional integration and fractional integro-differentiation via Laguerre polynomials. Using the derived operational matrices and collocation points, we reduce the fractional differential equations to a system of linear or nonlinear algebraic equations. For the error of the operational matrix of the fractional integration, an error bound is derived. To illustrate the accuracy and the reliability of the projected algorithm, numerical simulation is presented, and the nature of attained results is captured in diverse order. Finally, the achieved consequences enlighten that the solutions obtained by the proposed scheme give better convergence to the actual solution than the results available in the literature.

scholarly journals New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Main Idea ◽  
Initial Condition ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Expansion Coefficients

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented.

Direct numerical method for isoperimetric fractional variational problems based on operational matrix

2017 ◽  
Vol 24 (14) ◽  
pp. 3063-3076 ◽  
Author(s):  
Samer S Ezz–Eldien ◽  
Ali H Bhrawy ◽  
Ahmed A El–Kalaawy
Keyword(s):  
Direct Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Variational Problems ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Direct Numerical Method ◽  
Fractional Variational Problems ◽  
A Direct Method

In this paper, we applied a direct method for a solution of isoperimetric fractional variational problems. We use shifted Legendre orthonormal polynomials as basis function of operational matrices of fractional differentiation and fractional integration in combination with the Lagrange multipliers technique for converting such isoperimetric fractional variational problems into solving a system of algebraic equations. Also, we show the convergence analysis of the presented technique and introduce some test problems with comparisons between our numerical results with those introduced using different methods.

scholarly journals A fast numerical method for fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
S. Mockary ◽  
E. Babolian ◽  
A. R. Vahidi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations

Abstract In this paper, we use operational matrices of Chebyshev polynomials to solve fractional partial differential equations (FPDEs). We approximate the second partial derivative of the solution of linear FPDEs by operational matrices of shifted Chebyshev polynomials. We apply the operational matrix of integration and fractional integration to obtain approximations of (fractional) partial derivatives of the solution and the approximation of the solution. Then we substitute the operational matrix approximations in the FPDEs to obtain a system of linear algebraic equations. Finally, solving this system, we obtain the approximate solution. Numerical experiments show an exponential rate of convergence and hence the efficiency and effectiveness of the method.

scholarly journals Solving a Quadratic Riccati Differential Equation, Multi-Pantograph Delay Differential Equations, and Optimal Control Systems with Pantograph Delays

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 82
Author(s):  
Fateme Ghomanjani ◽  
Stanford Shateyi
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Differential Equations ◽  
Control Systems ◽  
Delay Differential Equations ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Riccati Differential Equation ◽  
Genocchi Polynomials ◽  
Delay Differential

An effective algorithm for solving quadratic Riccati differential equation (QRDE), multipantograph delay differential equations (MPDDEs), and optimal control systems (OCSs) with pantograph delays is presented in this paper. This technique is based on Genocchi polynomials (GPs). The properties of Genocchi polynomials are stated, and operational matrices of derivative are constructed. A collocation method based on this operational matrix is used. The findings show that the technique is accurate and simple to use.

scholarly journals Bernstein polynomials method and it’s error analysis for solving nonlinear problems in the calculus of variations: Convergence analysis via residual function

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1379-1393 ◽  
Author(s):  
Ahmad Bataineh
Keyword(s):  
Convergence Analysis ◽  
Calculus Of Variations ◽  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Residual Function ◽  
Approximate Analytical Solutions

In this paper, Bernstein polynomials method (BPM) and their operational matrices are adopted to obtain approximate analytical solutions of variational problems. The operational matrix of differentiation is introduced and utilized to reduce the calculus of variations problems to the solution of system of algebraic equations. The solutions are obtained in the form of rapidly convergent finite series with easily computable terms. Comparison between the present method and the homotopy perturbation method (HPM), the non-polynomial spline method and the B-spline collocation method are made to show the effectiveness and efficiency for obtaining approximate solutions of the calculus of variations problems. Moreover, convergence analysis based on residual function is investigated to verified the numerical results.

Numerical solution for fractional optimal control problems by Hermite polynomials

2020 ◽  
pp. 107754632093312
Author(s):  
Ayatollah Yari
Keyword(s):  
Optimal Control ◽  
Numerical Method ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

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