Numerical solution of Riccati equation using operational matrix method with Chebyshev polynomials

2015 ◽  
Vol 08 (02) ◽  
pp. 1550020
Author(s):  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Approximate Solution ◽  
Riccati Equation ◽  
Chebyshev Polynomials ◽  
Matrix Method ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Numerical Examples ◽  
Operational Matrix Method

In this paper, we present numerical technique for solving the Riccati equation by using operational matrix method with Chebyshev polynomials. The method consists of expanding the required approximate solution as truncated Chebyshev series. Using operational matrix method, we reduce the problem to a set of algebraic equations. Some numerical examples are given to demonstrate the validity and applicability of the method. The method is easy to implement and produces very accurate results.

Numerical solution of Abel equation using operational matrix method with Chebyshev polynomials

2017 ◽  
Vol 10 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Chebyshev Polynomials ◽  
Matrix Method ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Abel Equation ◽  
Operational Method ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Nonlinear Algebraic Equations

In this paper, we present a numerical scheme for solving the Abel equation. The approach is based on the shifted Chebyshev polynomials together with operational method. We reduce the problem to a set of nonlinear algebraic equations using operational matrix method. In addition, convergence analysis of the method is presented. Some numerical examples are given to demonstrate the validity and applicability of the method. The only a small number of Chebyshev polynomials is needed to obtain a satisfactory result.

scholarly journals A Reliable Method for Solving Fractional Sturm–Liouville Problems

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 176 ◽  
Author(s):  
M. Khashshan ◽  
Muhammed Syam ◽  
Ahlam Al Mokhmari
Keyword(s):  
Matrix Method ◽  
Reliable Method ◽  
Operational Matrix ◽  
Liouville Problem ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Sturm Liouville

In this paper, a reliable method for solving fractional Sturm–Liouville problem based on the operational matrix method is presented. Some of our numerical examples are presented.

Fractional Chebyshev Finite Difference Method for Solving the Fractional-Order Delay BVPs

2015 ◽  
Vol 12 (06) ◽  
pp. 1550033 ◽  
Author(s):  
M. M. Khader
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Fractional Order ◽  
Difference Method ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Operational Matrix Method ◽  
Chebyshev Finite Difference Method

In this paper, we implement an efficient numerical technique which we call fractional Chebyshev finite difference method (FChFDM). The fractional derivatives are presented in terms of Caputo sense. The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a nonuniform finite difference scheme. The error bound for the fractional derivatives is introduced. We used the introduced technique to solve numerically the fractional-order delay BVPs. The application of the proposed method to introduced problem leads to algebraic systems which can be solved by an appropriate numerical method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

scholarly journals Fractional Order Operational Matrix Method for Solving Two-Dimensional Nonlinear Fractional Volterra Integro-Differential Equations

2021 ◽  
Vol 45 (4) ◽  
pp. 571-585
Author(s):  
AMIRAHMAD KHAJEHNASIRI ◽  
◽  
M. AFSHAR KERMANI ◽  
REZZA EZZATI ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Volterra Integral Equation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Basis Functions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Operational Matrix Method

This article presents a numerical method for solving nonlinear two-dimensional fractional Volterra integral equation. We derive the Hat basis functions operational matrix of the fractional order integration and use it to solve the two-dimensional fractional Volterra integro-differential equations. The method is described and illustrated with numerical examples. Also, we give the error analysis.

A New Bernoulli Wavelet Operational Matrix of Derivative Method for the Solution of Nonlinear Singular Lane–Emden Type Equations Arising in Astrophysics

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
S. Balaji
Keyword(s):  
Approximate Solution ◽  
Operational Matrix ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Wavelet Expansions ◽  
Collocation Points ◽  
Derivative Method ◽  
Bernoulli Wavelet ◽  
The Given

In this paper, a new method is presented for solving generalized nonlinear singular Lane–Emden type equations arising in the field of astrophysics, by introducing Bernoulli wavelet operational matrix of derivative (BWOMD). Bernoulli wavelet expansions together with this operational matrix method, by taking suitable collocation points, converts the given Lane–Emden type equations into a system of algebraic equations. Solution to the problem is identified by solving this system of equations. Further applicability and simplicity of the proposed method has been demonstrated by some examples and comparison with other recent methods. The obtained results guarantee that the proposed BWOMD method provides the good approximate solution to the generalized nonlinear singular Lane–Emden type equations.

scholarly journals A New Operational Matrix of Fractional Derivatives to Solve Systems of Fractional Differential Equations via Legendre Wavelets

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 238 ◽  
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Matrix Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Operational Matrix Method ◽  
Fractional Differential

This paper introduces a new numerical approach to solving a system of fractional differential equations (FDEs) using the Legendre wavelet operational matrix method (LWOMM). We first formulated the operational matrix of fractional derivatives in some special conditions using some notable characteristics of Legendre wavelets and shifted Legendre polynomials. Then, the system of fractional differential equations was transformed into a system of algebraic equations by using these operational matrices. At the end of this paper, several examples are presented to illustrate the effectivity and correctness of the proposed approach. Comparing the methodology with several recognized methods demonstrates that the advantages of the Legendre wavelet operational matrix method are its accuracy and the understandability of the calculations.

scholarly journals Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 181
Author(s):  
Yalçın ÖZTÜRK
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Matrix Equation ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Solution Function ◽  
The Given

In this paper, we use the collocation method together with Chebyshev polynomials to solve system of Lane–Emden type (SLE) equations. We first transform the given SLE equation to a matrix equation by means of a truncated Chebyshev series with unknown coefficients. Then, the numerical method reduces each SLE equation to a nonlinear system of algebraic equations. The solution of this matrix equation yields the unknown coefficients of the solution function. Hence, an approximate solution is obtained by means of a truncated Chebyshev series. Also, to show the applicability, usefulness, and accuracy of the method, some examples are solved numerically and the errors of the solutions are compared with existing solutions.

scholarly journals An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering

2019 ◽  
Vol 8 (1) ◽  
pp. 429-437 ◽  
Author(s):  
Yalçın Öztürk
Keyword(s):  
Numerical Method ◽  
Numerical Algorithm ◽  
Matrix Method ◽  
Operational Matrix ◽  
Type Equation ◽  
Algebraic Equations ◽  
Operational Matrix Method

Abstract The purpose of this paper is to propose an efficient numerical method for solving system of Lane–Emden type equations using Chebyshev operational matrix method. This method transforms the system of Lane-Emden type equation into the system of algebraic equations with unknown Chebyshev coefficients. Some illustrative examples are given to demonstrate the efficiency and validity of the proposed algorithm.

Sign in / Sign up

Export Citation Format

Share Document