A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model

Zhang, H.; Liu, F.; Phanikumar, Mantha S.; Meerschaert, Mark M.

doi:10.1016/j.camwa.2013.01.031

Cited by 174 publications

(100 citation statements)

References 25 publications

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“…An advantage of using the Hahn polynomials is that the numerical results achieved only using a small number of bases are accurate in a larger intervals and significant results are achieved. A comparison of the numerical solutions by FDM [39] and RKM [20] shows that this technique is accurate enough to be known as a powerful device. This method can be applied to solve other types of VO fractional functional equations.…”

Section: Discussionmentioning

confidence: 99%

“…A comparison of the numerical solutions is made by the results reported in [39] and [20] at T ¼ 1, by Finite Difference Method (FD) and Reproducing Kernel Method (RKM), respectively. Also, the results of the presented method at T ¼ 2 and T ¼ 4 show that the method is accurate even for larger domains.…”

Section: Mathematical Sciencesmentioning

confidence: 99%

“…Example 2 Consider another time VO fractional mobileimmobile advection-dispersion model as follows [39]: D qðx;tÞ t uðx; tÞ þ ouðx; tÞ ot þ ouðx; tÞ ox À o 2 uðx; tÞ ox 2 ¼ f ðx; tÞ; ðx; tÞ 2 X; ð42Þ where X ¼ ½0; 1 Â ½0; T , with the following initial and boundary conditions: f 1 ðx; tÞ ¼ 5xð1 À xÞ þ 5xð1 À xÞt 1Àqðx;tÞ Cð2 À qðx; tÞÞ ;…”

Section: Mathematical Sciencesmentioning

confidence: 99%

“…In this work, we investigate the VO fractional problem arises in a mobile-immobile advection-dispersion equation (VOFMIE) [39]. This model is obtained from the standard advection-dispersion equation by adding the variable timefractional derivative in the Caputo sense of order 0\qðx; tÞ 1.…”

Section: Introductionmentioning

confidence: 99%

“…Authors of [20] are used reproducing kernel theory and collocation method for solving the VOFMIE. An implicit Euler approximation for the VOFMIE has been explained in [39] Also, Chebyshev wavelets method [16] is employed to solve this VO equation. In [34], the VOFMIE in 2-D arbitrary domains is introduced and a meshless method based on the MLS approach is proposed to solve it.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

2018

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In this paper, we consider the discrete Hahn polynomials fH n g and investigate their application for numerical solutions of the time fractional variable order mobile-immobile advection-dispersion model which is advantageous for modeling dynamical systems. This paper presented the operational matrix of derivative of discrete Hahn polynomials. The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy in the interval [0, N]. Furthermore, for computing the coefficients of the expansion uðxÞ ¼ P 1 n¼0 c n H n ðxÞ, we have to only compute a summation and the calculation of coefficients is exact. Also an upper bound for the error of the presented method, with equidistant nodes, is investigated. Illustrative examples are provided to show the accuracy and efficiency of the presented method. Using a small number of Hahn polynomials, significant results are achieved which are compared to other methods.

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Section: Discussionmentioning

confidence: 99%

Section: Mathematical Sciencesmentioning

confidence: 99%

Section: Mathematical Sciencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

2018

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An efficient high‐order two‐level explicit/implicit numerical scheme for two‐dimensional time fractional mobile/immobile advection‐dispersion model

Ngondiep

2024

Numerical Methods in Fluids

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This article constructs a new two‐level explicit/implicit numerical scheme in an approximate solution for the two‐dimensional time fractional mobile/immobile advection‐dispersion problem. The stability and error estimates of the proposed technique are deeply analyzed in the ‐norm. The developed approach is less time consuming, fourth‐order in space and temporal accurate of order , where is the time step and denotes a positive parameter less than 1. This result shows that the two‐level explicit/implicit formulation is faster and more efficient than a large class of numerical schemes widely discussed in the literature for the considered problem. Numerical experiments are performed to verify the theoretical studies and to demonstrate the efficiency of the new numerical method.

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Bidimensional Gegenbauer Polynomials for Variable‐Order Time‐Fractional Integro‐Partial Differential Equation With a Weakly Singular Kernel

Yaghoubi,

Aminikhah,

Sadri

2024

Math Methods in App Sciences

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In this paper, a pseudo‐operational collocation method based on Gegenbauer polynomials is presented to solve a category of variable‐order time‐fractional integro‐partial differential equations with singular kernels. The applications of these functional equations can be revealed in the theory of elasticity, hydrodynamics, heat conduction, and nuclear reactor theory. The pseudo‐operational matrices are constructed utilizing bivariate Gegenbauer polynomials to approximate the solution of the mentioned equation. Then, using the collocation method and resultant matrices, the main equation is converted into a system of algebraic equations that can be solved by Newton's iteration method. Besides presenting a fast and accurate method, an error bound is determined in a Gegenbauer‐weighted space for the residual function obtained from the proposed approach. Finally, several test examples are performed to confirm the reliability and efficiency of the proposed method.

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A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model

Cited by 174 publications

References 25 publications

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation

An efficient high‐order two‐level explicit/implicit numerical scheme for two‐dimensional time fractional mobile/immobile advection‐dispersion model

Bidimensional Gegenbauer Polynomials for Variable‐Order Time‐Fractional Integro‐Partial Differential Equation With a Weakly Singular Kernel

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