Solving fractional Fredholm integro–differential equations using Legendre wavelets

2021 ◽  
Vol 166 ◽  
pp. 168-185
Author(s):  
D. Abbaszadeh ◽  
M. Tavassoli Kajani ◽  
M. Momeni ◽  
M. Zahraei ◽  
M. Maleki
Keyword(s):  
Differential Equations ◽  
Legendre Wavelets

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

A study of nonlinear biochemical reaction model

2016 ◽  
Vol 09 (05) ◽  
pp. 1650071 ◽  
Author(s):  
Muhammad Asad Iqbal ◽  
Syed Tauseef Mohyud-Din ◽  
Bandar Bin-Mohsin
Keyword(s):  
Differential Equations ◽  
Reaction Model ◽  
Numerical Results ◽  
Picard Iteration ◽  
System Of Differential Equations ◽  
Legendre Wavelets ◽  
Enzyme Substrate ◽  
Runge Kutta Method ◽  
Picard Method ◽  
Order Four

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

scholarly journals A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Pseudospectral Method ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Derivative Operator ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

scholarly journals Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Xiaoqun Cao ◽  
Fengshun Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Nonlinear Partial Differential Equations ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Legendre Wavelets ◽  
Partial Differential ◽  
Variational Iteration ◽  
Block Pulse Functions

This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.

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