Wavelets Galerkin Method for the Fractional Subdiffusion Equation

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
M. H. Heydari
Keyword(s):  
Galerkin Method ◽  
Fractional Derivatives ◽  
Operational Matrix ◽  
Practical Importance ◽  
Time Derivative ◽  
Algebraic Equations ◽  
Legendre Wavelets ◽  
Equivalent Problem ◽  
Subdiffusion Equation ◽  
The Galerkin Method

The time fractional subdiffusion equation (FSDE) as a class of anomalous diffusive systems has obtained by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0<α<1. Since analytically solving this problem is often impossible, proposing numerical methods for its solution has practical importance. In this paper, an efficient and accurate Galerkin method based on the Legendre wavelets (LWs) is proposed for solving this equation. The time fractional derivatives are described in the Riemann–Liouville sense. To do this, we first transform the original subdiffusion problem into an equivalent problem with fractional derivatives in the Caputo sense. The LWs and their fractional operational matrix (FOM) of integration together with the Galerkin method are used to transform the problem under consideration into the corresponding linear system of algebraic equations, which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account, automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

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 The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 486 ◽  
Author(s):  
Neslihan Ozdemir ◽  
Aydin Secer ◽  
Mustafa Bayram
Keyword(s):  
Fractional Derivative ◽  
Galerkin Method ◽  
Collocation Method ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Coupled System ◽  
Algebraic Equations ◽  
Burgers Equations ◽  
Time Fractional Derivative ◽  
The Galerkin Method

In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods.

scholarly journals An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Galerkin Method ◽  
Operational Matrix ◽  
Computational Method ◽  
Test Problems ◽  
Algebraic Equations ◽  
Dispersion Parameters ◽  
Wavelet Galerkin Method ◽  
Operational Matrix Of Integration ◽  
Nonlinear Physical Phenomena ◽  
The Galerkin Method

Abstract In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

2003 ◽  
Author(s):  
Jose´ Antunes ◽  
Miguel Moreira ◽  
Philippe Piteau
Keyword(s):  
Fourier Series ◽  
Galerkin Method ◽  
Finite Length ◽  
Reynolds Equation ◽  
Double Fourier Series ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Squeeze Film Dampers ◽  
Galerkin Solution ◽  
The Galerkin Method

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.

scholarly journals New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
MohammadHossein Derakhshan
Keyword(s):  
Fractional Derivatives ◽  
Numerical Approximation ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Integrodifferential Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Lagrange Multiplier Technique ◽  
Derivatives Of ◽  
Variable Order Fractional Derivative

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

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 Dynamic Behavior of a Cylindrical Shell with a Liquid under the Action of Nonstationary Pressure Wave

TEM Journal ◽  
2021 ◽  
pp. 815-819
Author(s):  
Boris A. Antufev ◽  
Vasiliy N. Dobryanskiy ◽  
Olga V. Egorova ◽  
Eduard I. Starovoitov
Keyword(s):  
Cylindrical Shell ◽  
Galerkin Method ◽  
Pressure Wave ◽  
Moving Load ◽  
Algebraic Equations ◽  
Thin Cylindrical Shell ◽  
Incompressibility Condition ◽  
Nonlinear Algebraic Equations ◽  
Deformed State ◽  
The Galerkin Method

The problem of axisymmetric hydroelastic deformation of a thin cylindrical shell containing a liquid under the action of a moving load is approximately solved. It is reduced to the equation of bending of the shell and the condition of incompressibility of the liquid in the cylinder. The deflections of the shell and the level of lowering of the liquid are unknown. For solution, the Galerkin method is used and the problem is reduced to a system of nonlinear algebraic equations. A simpler solution is considered without taking into account the incompressibility condition. Here, in addition to the deformed state of the shell, the critical speeds of the moving load are determined analytically.

scholarly journals Multi-Wavelets Galerkin Method for Solving the System of Volterra Integral Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1369
Author(s):  
Hoang Viet Long ◽  
Haifa Bin Jebreen ◽  
Stefania Tomasiello
Keyword(s):  
Wavelet Transform ◽  
Galerkin Method ◽  
Integral Equations ◽  
Operational Matrix ◽  
Computational Effort ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Linear Type ◽  
Numerical Tests ◽  
Operational Matrix Of Integration

In this work, an efficient algorithm is proposed for solving the system of Volterra integral equations based on wavelet Galerkin method. This problem is reduced to a set of algebraic equations using the operational matrix of integration and wavelet transform matrix. For linear type, the computational effort decreases by thresholding. The convergence analysis of the proposed scheme has been investigated and it is shown that its convergence is of order O(2−Jr), where J is the refinement level and r is the multiplicity of multi-wavelets. Several numerical tests are provided to illustrate the ability and efficiency of the method.

scholarly journals A Novel Technique for Predicting the Thermal Behavior of Stratospheric Balloon

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Yunpeng Ma ◽  
Jun Huang ◽  
Mingxu Yi
Keyword(s):  
Thermal Behavior ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Gas Temperature ◽  
Equilibrium Equations ◽  
Legendre Wavelets ◽  
Novel Technique ◽  
Stratospheric Balloon ◽  
Novel Method ◽  
Operational Matrix Of Integration

This paper is devoted to introduce a novel method of the operational matrix of integration for Legendre wavelets in order to predict the thermal behavior of stratospheric balloons on float at high altitude in the stratosphere. Radiative and convective heat transfer models are also developed to calculate absorption and emission heat of the balloon film and lifting gas within the balloon. Thermal equilibrium equations (TEE) for the balloon system at daytime and nighttime are shown to predict the thermal behavior of stratospheric balloons. The properties of Legendre wavelets are used to reduce the TEE to a nonlinear system of algebraic equations which is solved by using a suitable numerical method. The approximations of the thermal behavior of the balloon film and lifting gas within the balloon are derived. The diurnal variations of the film and lifting gas temperature at float conditions are investigated, and the efficiency of the proposed method is also confirmed.

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