Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix

2021 ◽  
Author(s):  
Shyam Lal ◽  
Priya Kumari
Keyword(s):  
Operational Matrix ◽  
Haar Wavelet ◽  
Approximation Of Functions

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 Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
F. M. Alharbi ◽  
A. M. Zidan ◽  
Muhammad Naeem ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Suggested Technique ◽  
Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

Wavelet Method for a Class of Fractional Klein-Gordon Equations

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
G. Hariharan
Keyword(s):  
Applied Mathematics ◽  
Operational Matrix ◽  
Correlation Coefficients ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Algebraic Equations ◽  
Fundamental Idea ◽  
Wavelet Method ◽  
Klein Gordon ◽  
Constant Coefficients

Wavelet analysis is a recently developed mathematical tool in applied mathematics. A wavelet method for a class of space-time fractional Klein–Gordon equations with constant coefficients is proposed, by combining the Haar wavelet and operational matrix together and efficaciously dispersing the coefficients. The behavior of the solutions and the effects of different values of fractional order α are graphically shown. The fundamental idea of the Haar wavelet method is to convert the fractional Klein–Gordon equations into a group of algebraic equations, which involves a finite number of variables. The examples are given to demonstrate that the method is effective, fast, and flexible; in the meantime, it is found that the difficulties of using the Daubechies wavelets for solving the differential equation, which need to calculate the correlation coefficients, are avoided.

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.

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