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.

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.

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 Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 224 ◽  
Author(s):  
Harendra Singh ◽  
Rajesh Pandey ◽  
Hari Srivastava
Keyword(s):  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Variational Problems ◽  
Algebraic Equations ◽  
The Third ◽  
Fractional Variational Problems ◽  
Nonlinear Algebraic Equations ◽  
Non Linear ◽  
Fourth Kind

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

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>

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.

Exponentially Accurate Rayleigh–Ritz Method for Fractional Variational Problems

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Dongling Wang ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Exponential Decay ◽  
Numerical Approximation ◽  
Exponential Convergence ◽  
Ritz Method ◽  
Variational Problems ◽  
Basis Functions ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Fractional Variational Problems ◽  
Sturm Liouville

A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

scholarly journals New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 592
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Lagrange Function ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Final State ◽  
Fractional Variational Problems ◽  
Generalized Fractional Derivatives ◽  
Arbitrary Kernel

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

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