Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets

Irandoust‐pakchin, Safar; Dehghan, Mehdi; Abdi-Mazraeh, Somayeh; Lakestani, Mehrdad

doi:10.1177/1077546312470473

Cited by 37 publications

(23 citation statements)

References 53 publications

(47 reference statements)

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“…In the aforementioned results, we get approximate solution of the problem that is very close to the exact solution. To make a comparison, in Table I, we bring results of applying biorthogonal flatlet multiwavelets scheme [41] for numerical solution of the problem with D 0.5 by taking different values of m and J in time t D 0.25.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…A truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations is introduced in [40]. A numerical scheme for solving the fractional convection-diffusion equation presented in [41] that is applied biorthogonal multiwavelet basis to construct operational matrix of fractional derivative. The main advantage of spectral methods lies in their accuracy for given number of unknowns.…”

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confidence: 99%

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Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Izadkhah

Saberi‐Nadjafi

2014

Math Methods in App Sciences

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In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. IntroductionRecently, fractional differential operators are indisputably found to play a fundamental role in the modeling of a considerable number of phenomena. Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diffusion [1][2][3][4]. They have been used in modeling turbulent flow [5,6], chaotic dynamics of classical conservation systems [7], and even finance [8,9] (see [1] for more information).Also, fractional calculus emerged as an important and efficient tool for the study of dynamical systems where classical methods reveal strong limitations. In [10], a fractional advection-dispersion equation is derived by extending Fick's first law from isotropic media to heterogeneous media and is particularly suitable for description of the highly skewed and heavy-tailed dispersion processes observed in rivers and other natural media.In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11][12][13], finite element methods [14,15], and spectral methods [16]. In [17][18][19], the authors used their proposed numerical schemes to solve Bagley-Torvik equation and other ordinary fractional differential equations. Gegenbauer polynomials have received much attention for their fundamental properties as well as for their use in applied mathematics. The described functions are a key ingredient to the implementation of spectral and pseudo-spectral methods to solve certain types of differential equations. Gegenbauer polynomials are a convenient basis for polynomial approximations because they are eigenfunctions of corresponding differential operators. For numerical methods, it is usually as convenient as efficient to convert between representations of a polynomial by expansion coefficients or by function values, respectively.Spectral approximations, such as the Fourier approximation based upon trigonometric polynomials for periodic problems, and the Chebyshev, Legendre, or the general Gegenbauer approximation based upon polynomials for nonperiodic problems are exponentially accurate for analytic functions [20][21][22][23]. In [24], the authors provided collocation method for natural convection heat transfer equations embedded in porous medium by using the rational Gegenbauer polynomials. Gottlieb and Shu in [25] used the Geg...

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Section: Numerical Illustrationsmentioning

confidence: 99%

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confidence: 99%

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Izadkhah

Saberi‐Nadjafi

2014

Math Methods in App Sciences

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“…FCDEs can express physical problems more accurately as compared to ordinary CDEs. In this regard, the numerical and analytical solutions for FCDEs are the focus point for the researchers, and therefore different techniques have been established such as adomian decomposition method [27], Sumudu transform method and homotopy analysis transform method were used by Singh et al [28]; HPM was applied by Yildrim and Momani [29]; variational iteration technique was used by Merdan [30]; and Irandoust-pakchin et al successfully implemented the flatlet oblique multiwavelet and found a mathematical approach for the class of FCDEs [31].…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical fractional view of convection–diffusion equations

et al. 2020

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In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection–diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.

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“…Recently, existence, uniqueness, and numerical solutions of fractional differential equations have been investigated because of application of these equations in mathematical modeling of systems and processes in the field of physics, chemistry, engineering, and so on. There are many papers that discuss about existence, uniqueness, and numerical solutions of fractional differential equations with classic boundary conditions, for example , but there exist a few papers about fractional differential equations with integral boundary conditions. Now, we introduce some of these research works.…”

Section: Introductionmentioning

confidence: 99%

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

Parvizi

Eslahchi

2015

Math Methods in App Sciences

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In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L 2 norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced.

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Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets

Cited by 37 publications

References 53 publications

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Approximate analytical fractional view of convection–diffusion equations

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

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