Comparison of Four Numerical Schemes for Isoperimetric Constraint Fractional Variational Problems With A-Operator

2015 ◽  
Author(s):  
Rajesh K. Pandey ◽  
Om P. Agrawal
Keyword(s):  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Numerical Schemes ◽  
Linear Quadratic ◽  
Step Size ◽  
Special Cases ◽  
Fractional Variational Problems ◽  
Isoperimetric Constraint ◽  
Fractional Orders

This paper presents a comparative study of four numerical schemes for a class of Isoperimetric Constraint Fractional Variational Problems (ICFVPs) defined in terms of an A-operator introduced recently. The A-operator is defined in a more general way which in special cases reduces to Riemann-Liouville, Caputo, Riesz-Riemann-Liouville and Riesz-Caputo, and several other fractional derivatives defined in the literature. Four different schemes, namely linear, quadratic, quadratic-linear and Bernsteins polynomials approximations, are used to obtain approximate solutions of an ICFVP. All four schemes work well, and when the number of terms approximating the solution are increased, the desired solution is achieved. Results for a modified power kernel in A-operator for different fractional orders are presented to demonstrate the effectiveness of the proposed schemes. The accuracy of the numerical schemes with respect to parameters such as fractional order α and step size h are analyzed and illustrated in detail through various figures and tables. Finally, comparative performances of the schemes are discussed.

Numerical Scheme for Generalized Isoparametric Constraint Variational Problems With A-Operator

2013 ◽  
Author(s):  
Rajesh K. Pandey ◽  
Om P. Agrawal
Keyword(s):  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
General Scheme ◽  
Functional Space ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Approximation Schemes ◽  
Finite Dimensional Space ◽  
Special Cases

This paper presents a numerical scheme for a class of Isoperimetric Constraint Variational Problems (ICVPs) defined in terms of an A-operator introduced recently. In this scheme, Bernstein’s polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases the A-operator reduce to Riemann-Liouville, Caputo, Riesz-Riemann-Liouville and Riesz-Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein’s polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future.

A Development in the Extended Ritz Method for Solving a General Class of Fractional Variational Problems

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
Ali Lotfi
Keyword(s):  
General Class ◽  
Fractional Derivatives ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Polynomial Functions ◽  
Great Flexibility ◽  
Fractional Polynomial ◽  
Fractional Variational Problems ◽  
Free Parameters

Abstract In this paper, based on the idea of the extended Ritz method, we introduce an efficient approximate technique for solving a general class of fractional variational problems. In the discussed problem, the fractional derivatives are considered in the Caputo sense. First, we introduce a family of fractional polynomial functions with a free parameter in the exponent. With the aid of the presented fractional polynomials, we construct a family of functions with free parameters, which provides the extended Ritz method with a great flexibility in searching for the approximate solution of the problem. The approximate solutions satisfy all the initial and the boundary conditions of the problem. The convergence of the method is analytically studied and some test examples are included to demonstrate the superiority of the new technique over the ordinary Ritz method.

Numerical Scheme for a Quadratic Type Generalized Isoperimetric Constraint Variational Problems With A-Operator

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Rajesh K. Pandey ◽  
Om P. Agrawal
Keyword(s):  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Dimensional Space ◽  
General Scheme ◽  
Functional Space ◽  
Variational Problems ◽  
Approximation Schemes ◽  
Finite Dimensional Space ◽  
Special Cases ◽  
Isoperimetric Constraint

This paper presents a numerical scheme for a class of isoperimetric constraint variational problems (ICVPs) defined in terms of an A-operator introduced recently. In this scheme, Bernstein's polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases, the A-operator reduces to Riemann–Liouville, Caputo, Riesz–Riemann–Liouville, and Riesz–Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein's polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future. While the presented numerical scheme is applied to a quadratic type generalized ICVPs, it can also be applied to other types of problems.

Modified wavelet method for solving fractional variational problems

2020 ◽  
pp. 107754632093202
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani ◽  
Mohsen Razzaghi
Keyword(s):  
Error Analysis ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variational Problems ◽  
High Accuracy ◽  
Construction Method ◽  
Computational Method ◽  
Wavelet Method ◽  
Fractional Variational Problems ◽  
Operational Matrix Of Integration

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

scholarly journals Fractional Variational Problems of Euler-Lagrange Equations with Holonomic Constrained Systems

2016 ◽  
Vol 8 (3) ◽  
pp. 60 ◽  
Author(s):  
Eyad Hasan Hasan
Keyword(s):  
Variational Problem ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Constrained Systems ◽  
Lagrange Equations ◽  
Integer Order ◽  
Fractional Variational Problems ◽  
Fractional Variational Problem ◽  
Problem Of Lagrange

<p class="1Body">In this paper, we examined the fractional Euler-Lagrange equations for Holonomic constrained systems. The Euler-Lagrange equations are derived using the fractional variational problem of Lagrange. In addition, we achieved that the classical results were obtained are agreement when fractional derivatives are replaced with the integer order derivatives. Two physical examples are discussed to demonstrate the formalism.</p>

scholarly journals Optimum solutions of space fractional order diffusion equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 329-339
Author(s):  
Sajjad Ali ◽  
Kamal Shah ◽  
Yongjin Li ◽  
Muhammad Arif
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Approximate Analytical Solutions ◽  
Special Cases ◽  
Fractional Differential ◽  
Optimal Homotopy Asymptotic Method ◽  
Excellent Accuracy

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional order partial differential equations. Fractional differential equations have great importance regarding distinct fields of science and engineering. Approximate solutions of space fractional order diffusion model and its various special cases are handled using the innovative proposed method. The space fractional derivatives are described in the Caputo sense. The results obtained by the proposed method are compared with various methods. The proposed method demonstrates excellent accuracy and strength over various methods.

scholarly journals HPM for Solving the Time-fractional Coupled Burgers Equations

2016 ◽  
Vol 12 (4) ◽  
pp. 6133-6138
Author(s):  
Khadijah Abu Alnaja
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Burgers Equations ◽  
Homotopy Perturbation ◽  
Special Cases ◽  
Burger's Equations

In this paper, the homotopy perturbation method is implemented to derive the explicit approximate solutions for the time-fractional coupled Burger's equations. The including fractional derivative is in the Caputo sense. Special attention is given to prove the convergence of the method. The results are compared with those obtained by the exact at special cases of the fractional derivatives. The results reveal that the proposed method is very effective and simple.

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