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 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.

Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics

2017 ◽  
Vol 06 (02) ◽  
pp. 1750016
Author(s):  
Firdous A. Shah ◽  
R. Abass
Keyword(s):  
Fluid Mechanics ◽  
Relaxation Oscillation ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Specific Test ◽  
Fractional Relaxation ◽  
Oscillation Equation

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Reliable Method ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Implementation Process ◽  
Haar Wavelet ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

scholarly journals Numerical Method for Inverse Laplace Transform with Haar Wavelet Operational Matrix

2014 ◽  
Vol 8 (4) ◽  
Author(s):  
Suazlan Mt Aznam ◽  
Amran Hussin
Keyword(s):  
Laplace Transform ◽  
Numerical Method ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Wavelet Function ◽  
Inverse Laplace Transform ◽  
Orthogonal Wavelet ◽  
Transform Method ◽  
Operational Matrix Method ◽  
Made In

Wavelets have been applied successfully in signal and image processing. Many attempts have been made in mathematics to use orthogonal wavelet function as numerical computational tool. In this work, an orthogonal wavelet function namely Haar wavelet function is considered. We present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix for integration. We proved the method for the cases of the irrational transfer function using the extension of Riemenn-Liouville fractional integral. The proposed method extends the work of J.L.Wu et al. (2001) to cover the whole of time domain. Moreover, this work gives an alternative way to find the solution for inversion of Laplace transform in a faster way. The use of numerical Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Additionally, few benefits come from its great features such as faster computation and attractiveness. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to analytical solution. Examples on solving differential equation by Laplace transform method are also given.

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.

Haar Wavelet Operational Matrix Method for the Numerical Solution of Fractional Order Differential Equations

2015 ◽  
Vol 4 (4) ◽  
Author(s):  
Firdous A. Shah ◽  
R. Abbas
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Matrix Method ◽  
Present Method ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Operational Matrix Method ◽  
Fractional Order Differential Equations ◽  
Block Pulse Functions

AbstractIn this paper, we propose a new operational matrix method of fractional order integration based on Haar wavelets to solve fractional order differential equations numerically. The properties of Haar wavelets are first presented. The properties of Haar wavelets are used to reduce the system of fractional order differential equations to a systemof algebraic equationswhich can be solved numerically byNewton’s method.Moreover, the proposed method is derived without using the block pulse functions considered in open literature and does not require the inverse of the Haar matrices. Numerical examples are included to demonstrate the validity and applicability of the present method.

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