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.

scholarly journals Φ -Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing Φ -Caputo Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Pongsakorn Sunthrayuth ◽  
Noufe H. Aljahdaly ◽  
Amjid Ali ◽  
Rasool Shah ◽  
Ibrahim Mahariq ◽  
...  
Keyword(s):  
Numerical Method ◽  
Matrix Method ◽  
Relaxation Oscillation ◽  
Operational Matrix ◽  
Relaxation Oscillator ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Linear Algebraic Equations ◽  
Fractional Relaxation

This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in the Φ -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the Φ -Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

2019 ◽  
Vol 17 (04) ◽  
pp. 1950026 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Type Equation ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
Quasilinearization Technique ◽  
Distribution Equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

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.

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 Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms

2020 ◽  
Vol 10 (2) ◽  
pp. 218-225
Author(s):  
Burcu Gürbüz ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Matrix Method ◽  
Laguerre Polynomials ◽  
Operational Matrix ◽  
Second Order ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
The Matrix

In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.

scholarly journals A new numerical approach for solving high-order linear and non-linear differantial equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 67-77
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Matrix Method ◽  
Operational Matrix ◽  
Numerical Approach ◽  
High Order ◽  
Algebraic Equations ◽  
Practical Application ◽  
Order Linear ◽  
Operational Matrix Method ◽  
Non Linear ◽  
Linear Problems

In this paper, the Legendre wavelet operational matrix method has been introduced for solving high-order linear and non-linear multi-point: initial and boundary value problems. It has been suggested that the technique is rest upon practical application of the operational matrix and its derivatives. The differential equation is presented that it is converted to a system of algebraic equations via the properties of Legendre wavelet together with the operational matrix method. As a result of this study, the scheme has been tested on five linear and non-linear problems. The results have demonstrated that this method is a very effective and advantageous tool in solving such problems.

scholarly journals Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
S. Balaji
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Matrix Method ◽  
Operational Matrix ◽  
Computational Effort ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Operational Matrix Method ◽  
Riccati Differential Equations

A Legendre wavelet operational matrix method (LWM) is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. The fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified.

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.

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