scholarly journals Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

2022 ◽  
Vol 2022 ◽  
pp. 1-20
Author(s):  
Khadijeh Sadri ◽  
Hossein Aminikhah
Keyword(s):  
Chebyshev Polynomials ◽  
Climate Models ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Proportional Delay ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Residual Function ◽  
Fractional Pdes ◽  
The One

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

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 Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

New operational matrices approach for optimal control based on modified Chebyshev polynomials

2021 ◽  
Vol 2 (2) ◽  
pp. 68-78
Author(s):  
Anam Alwan Salih ◽  
Suha SHIHAB
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Chebyshev Polynomials ◽  
Optimization Technique ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Quadratic Optimal Control ◽  
Chebyshev Orthogonal Polynomial

The purpose of this paper is to introduce interesting modified Chebyshev orthogonal polynomial. Then, their new operational matrices of derivative and integration or modified Chebyshev polynomials of the first kind are introduced with explicit formulas. A direct computational method for solving a special class of optimal control problem, named, the quadratic optimal control problem is proposed using the obtained operational matrices. More precisely, this method is based on a state parameterization scheme, which gives an accurate approximation of the exact solution by utilizing a small number of unknown coefficients with the aid of modified Chebyshev polynomials. In addition, the constraint is reduced to some algebraic equations and the original optimal control problem reduces to optimization technique, which can be solved easily, and the approximate value of the performance index is calculated. Moreover, special attention is presented to discuss the convergence analysis and an upper bound of the error for the presented approximate solution is derived. Finally, some important illustrative examples of obtained results are shown and proved that powerful method in a simple way to get an optimal control of the considered.

scholarly journals A MUNTZ-LEGENDRE APPROACH TO OBTAIN SOLUTIONS OF SINGULAR PERTURBED PROBLEMS

2020 ◽  
Vol 20 (3) ◽  
pp. 537-544
Author(s):  
SUAYIP YUZBASI ◽  
EMRAH GOK ◽  
MEHMET SEZER
Keyword(s):  
Error Function ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Algebraic Equations ◽  
Residual Function ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Collocation Approach ◽  
Estimated Error

Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

scholarly journals Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Y. Ordokhani ◽  
S. Davaei far
Keyword(s):  
Differential Equations ◽  
Bernstein Polynomial ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

scholarly journals Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals Monte carlo estimation of the solution of fractional partial differential equations

2021 ◽  
Vol 24 (1) ◽  
pp. 278-306
Author(s):  
Vassili Kolokoltsov ◽  
Feng Lin ◽  
Aleksandar Mijatović
Keyword(s):  
Exact Solution ◽  
Monte Carlo ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rates Of Convergence ◽  
Probabilistic Interpretation ◽  
Fractional Partial Differential Equations ◽  
Fractional Pdes ◽  
Monte Carlo Approximation ◽  
Monte Carlo Estimation

Abstract The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem (CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.

scholarly journals Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

2019 ◽  
Vol 24 (2) ◽  
pp. 176-188 ◽  
Author(s):  
Eid H. Doha ◽  
Mohamed A. Abdelkawy ◽  
Ahmed Z.M. Amin ◽  
Dumitru Baleanu
Keyword(s):  
Chebyshev Polynomials ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Riccati Differential Equations ◽  
Spectral Technique ◽  
Expansion Coefficients

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

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.

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