Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

Hosseininia, M.; Heydari, Mohammad Hossein; Avazzadeh, Z.; Ghaini, F. M. Maalek

doi:10.1515/ijnsns-2018-0168

Cited by 37 publications

(19 citation statements)

References 27 publications

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“…In the current work, two important types of radial basis functions, the generalized multiquadric and thin plate spline functions are used as basis functions in the relation (12). The thin plate splines (TPS), (r) = r 2m ln(r), m = 1, 2, 3, … , are strictly conditionally positive definite radial functions and belong to C 2m−1 .…”

Section: Spatial Discretization By the Methods Of Approximate Particulmentioning

confidence: 99%

“…is used as the basis function in (12), too. In the GMQ function, the exponent q and the shape parameter c play important rules to improve the accuracy and stability of the radial basis interpolation or collocation techniques.…”

Section: Spatial Discretization By the Methods Of Approximate Particulmentioning

confidence: 99%

“…The model has been adapted from an existing nonlinear advection-diffusion model with variable coefficients. 12 The exact solution of the problem is u(x, , t) = (cos(x) + cos( ))t 3 and the source function, (x, t), is extracted from the exact solution. The proposed computational approach is used to simulate the model.…”

Section: Figurementioning

confidence: 99%

“…Especially, Tayebi et al introduced a two‐dimensional variable‐order time fractional advection‐diffusion equation with variable coefficients and proposed a meshless method based on the moving least squares approximation to solve the model. Hosseininia et al studied a variable‐order time fractional advection‐diffusion equation with variable coefficients and a nonlinear source term on the 2D spatial domain.…”

Section: Introductionmentioning

confidence: 99%

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The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

Ghehsareh

Zaghian

Majlesi

2020

Math Methods in App Sciences

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In the current work, a generalized mathematical model based on the Coimbra time fractional derivative of variable order, which describes an anomalous mobile‐immobile transport process in complex systems is investigated numerically. A robust numerical technique based on the meshfree strong form method combined with an efficient time‐stepping scheme is performed to compute the approximate solution of the problem with high accuracy. For this purpose, firstly, an effective implicit time discretization approach is used for discretizing the variable‐order time fractional problem in the time direction. Then a global meshless technique based on the method of approximate particular solutions is performed to fully discretize the model in the spatial domain. The validity and performance of the procedure to numerically simulate the proposed generalized solute transport model on regular and irregular domains are demonstrated through some numerical examples.

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Section: Spatial Discretization By the Methods Of Approximate Particulmentioning

confidence: 99%

Section: Spatial Discretization By the Methods Of Approximate Particulmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

Ghehsareh

Zaghian

Majlesi

2020

Math Methods in App Sciences

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“…There have been several studies for solving the variable-order (VO) fractional differential equation (FDE). Li et al [23] solved VO FDE model of shape memory polymers, Dehestani et al [24] proposed pseudo-operational matrix method for the solution of VO FPIDEs, Doha et al [25] employed spectral technique for solving VO fractional Volterra integro-differential equations (VIDEs), Babaei et al [26] suggested Sinc collocation method for the numerical solution of VO FIPDEs, Heydari et al [27] presented LWs optimization method for solving VO fractional Poisson equation, and Hosseininia et al [28] proposed 2-D LWs for solving VO fractional nonlinear advection-diffusion equation with variable coefficients.…”

mentioning

confidence: 99%

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

Ray

2020

Numerical Methods Partial

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In this paper, a new computational scheme based on operational matrices (OMs) of two-dimensional wavelets is proposed for the solution of variable-order (VO) fractional partial integro-differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two-dimensional Legendre wavelets. By implementing two-dimensional wavelets approximations and the OMs of integration and variable-order fractional derivative (VO-FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples. KEYWORDS two-dimensional Legendre wavelets, variable order fractional partial integro-differential equation, variable-order Caputo fractional derivative 1 INTRODUCTION In real life, many phenomena from diverse fields of science and engineering are modeled with nonlinear equations. The modeling of many dynamic systems leads to the integral term containing the unknown function when taking into account of memory effects. The integral and integro-differential

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A numerical method based on the Chebyshev cardinal functions for variable‐order fractional version of the fourth‐order 2D Kuramoto‐Sivashinsky equation

Hosseininia

Heydari

Hooshmandasl

et al. 2020

Math Methods in App Sciences

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In this article, the variable-order (VO) time fractional 2D Kuramoto-Sivashinsky equation is introduced, and a semidiscrete approach is presented through 2D Chebyshev cardinal functions (CCFs) for solving this equation. In the proposed method, we obtain a recurrent algorithm by using the finite difference method to approximate the VO fractional differentiation, the weighted finite difference method with parameter , and the approximation of the unknown function by the 2D CCFs. The differentiation operational matrices and the collocation technique are used to extract a linear system of algebraic equations which can be easily solved. The credibility of the developed method is examined on three numerical examples.

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Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

Cited by 37 publications

References 27 publications

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

A numerical method based on the Chebyshev cardinal functions for variable‐order fractional version of the fourth‐order 2D Kuramoto‐Sivashinsky equation

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