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.

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.

scholarly journals Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Nonlinear Oscillation ◽  
Matrix Method ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
The Other ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Operational Matrix Method

We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution.

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.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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