scholarly journals A collocation method for solving the fractional calculus of variation problems

2017 ◽  
Vol 35 (1) ◽  
pp. 163
Author(s):  
Mohammad Reza Ahmadi ◽  
Mitra Nasiri
Keyword(s):  
Fractional Calculus ◽  
Collocation Method ◽  
Fractional Order ◽  
Computational Technique ◽  
Algebraic Equations ◽  
Calculus Of Variation ◽  
Collocation Points ◽  
Müntz Polynomials ◽  
Linear And Nonlinear ◽  
Fractional Calculus Of Variation

In this paper we use a family of Muntz polynomials and a computational technique based on the collocation method to solve the calculus variation problem. This approach is utilizedto reduce the solution of linear and nonlinear fractional order dierential equations to the solution of a system of algebraic equations. Thus we can obtain a good approximation evenby using a smaller of collocation points.

Chaos Analysis and Control in Fractional Order Systems Using Fractional Chebyshev Collocation Method

2016 ◽  
Author(s):  
Arman Dabiri ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Fractional Order System ◽  
Order System ◽  
Integer Number ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Differentiation Matrix ◽  
Chebyshev Collocation ◽  
Collocation Points

In this paper, fractional Chebyshev collocation method is proposed to study Lyapunov exponents (LEs) and chaos in a fractional order system with nonlinearities. For this purpose, the solution of the fractional order system is discretized by N-degree Gauss-Lobatto-Chebyshev (GLC) polynomials where N is an integer number. Then, the discrete orthogonality relationship for the Chebyshev polynomials is used to obtain the fractional Chebyshev differentiation matrix. The differentiation matrix is then used to convert the nonlinear fractional differential equations to a system of nonlinear algebraic equations with the collocation points as the unknowns. The dominant LE (other than the zero LE) that corresponds to the time dimension is then computed by measuring the exponential rate of the trajectory deviations initiated slightly off the attractor point. The proposed technique is implemented to a damped driven pendulum with fractional order damping and the convergence of the dominant LE is studied versus the number of Chebyshev collocation points. The LE analysis is also verified by studying the system time and frequency responses for different values of the bifurcation parameter. Furthermore, the LE obtained by the proposed method for the analogous integer order system is compared with those obtained by the Jacobian technique and Grüwald-Letnikov approximation. Finally a fractional state feedback controller is designed to control the chaotic system to a desired equilibrium or periodic trajectory such that the error dynamics are time invariant or time periodic, respectively. The numerical example studied is the damped driven pendulum with fractional dampers.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Maleki ◽  
M. Tavassoli Kajani ◽  
I. Hashim ◽  
A. Kilicman ◽  
K. A. M. Atan
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Method ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Weighted Interpolation ◽  
Collocation Points ◽  
Present Solution ◽  
Nonlinear Algebraic Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

A collocation method for numerical solutions of fractional-order logistic population model

2016 ◽  
Vol 09 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Şuayip Yüzbaşı
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Fractional Order ◽  
Population Model ◽  
Numerical Solutions ◽  
Model Problem ◽  
Error Function ◽  
Approximate Solutions ◽  
Algebraic Equations

In this study, a collocation technique is presented for approximate solution of the fractional-order logistic population model. Actually, we develop the Bessel collocation method by using the fractional derivative in the Caputo sense to obtain the approximate solutions of this model problem. By means of the fractional derivative in the Caputo sense, the collocation points, the Bessel functions of the first kind, the method transforms the model problem into a system of nonlinear algebraic equations. Numerical applications are given to demonstrate efficiency and accuracy of the method. In applications, the reliability of the scheme is shown by the error function based on the accuracy of the approximate solution.

scholarly journals A CHEBYSHEV PSEUDO-SPECTRAL METHOD FOR SOLVING FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 51 (4) ◽  
pp. 464-475 ◽  
Author(s):  
N. H. SWEILAM ◽  
M. M. KHADER
Keyword(s):  
Differential Equations ◽  
Spectral Method ◽  
Fractional Order ◽  
Algebraic Equations ◽  
Computationally Efficient ◽  
Volterra Type ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractA Chebyshev pseudo-spectral method for solving numerically linear and nonlinear fractional-order integro-differential equations of Volterra type is considered. The fractional derivative is described in the Caputo sense. The suggested method reduces these types of equations to the solution of linear or nonlinear algebraic equations. Special attention is given to study the convergence of the proposed method. Finally, some numerical examples are provided to show that this method is computationally efficient, and a comparison is made with existing results.

The generalized fractional order of the Chebyshev functions on nonlinear boundary value problems in the semi-infinite domain

2017 ◽  
Vol 6 (3) ◽  
Author(s):  
Kourosh Parand ◽  
Mehdi Delkhosh
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Fractional Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Numerical Results ◽  
Algebraic Equations ◽  
Nonlinear Boundary Value Problems ◽  
Infinite Domain ◽  
Nonlinear Boundary Value

AbstractA new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

scholarly journals Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel

2021 ◽  
Vol 6 (1) ◽  
pp. 19
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
António M. Lopes ◽  
Ishak Hashim ◽  
Mohammed M. Babatin
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomials ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Singular Kernel ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples.

scholarly journals A numerical method for solving systems of higher order linear functional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 15-25 ◽  
Author(s):  
Suayip Yüzbasi ◽  
Emrah Gök ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Legendre Polynomials ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Functional Differential

AbstractFunctional differential equations have importance in many areas of science such as mathematical physics. These systems are difficult to solve analytically.In this paper we consider the systems of linear functional differential equations [1-9] including the term y(αx + β) and advance-delay in derivatives of y .To obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown co-efficients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kumbinarasaiah Srinivasa ◽  
Hadi Rezazadeh
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Collocation Method ◽  
Fractional Order ◽  
Numerical Technique ◽  
Telegraph Equation ◽  
Algebraic Equations ◽  
One Dimensional ◽  
Wavelet Collocation Method

AbstractIn this article, we proposed an efficient numerical technique for the solution of fractional-order (1 + 1) dimensional telegraph equation using the Laguerre wavelet collocation method. Some examples are illustrated to inspect the efficiency of the proposed technique and convergence analysis is discussed in terms of a theorem. Here, the fractional-order telegraph equation is converted into a system of algebraic equations using the properties of the Laguerre wavelet, and solutions obtained by the proposed scheme are more accurate and they are compared with the analytical solution and other method existed in the literature.

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