Finite difference method for a fractional porous medium equation

Teso, Félix del

doi:10.1007/s10092-013-0103-7

Cited by 35 publications

(13 citation statements)

References 23 publications

(38 reference statements)

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“…Based on the cited reference [6] some Barenblatt profiles as in Vázquez [16] are expected. In Figure 2…”

Section: Example 1: Barenblatt Profilesmentioning

confidence: 99%

“…(see [6,20]). The errors of the numerical method are shown in Table 1 for times t = 3, 5 and 10 s. In order to obtain a general idea of the performance of the scheme, in [6] the author obtained an error of order 4.329 × 10 −4 in our second example.…”

Section: Examplementioning

confidence: 99%

“…See [3,6,7] for more details about the Silvestre-Caffarelli extension. The reader can also see [8,9] for more information about the method.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical results for the discrete problem have been obtained using the finite difference method in [6]. In [7], authors used the standart finite difference method to solve a Caputo-type parabolic equation with a fractional laplacian.…”

Section: Introductionmentioning

confidence: 99%

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A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

García

Negreanu²,

Ureña

et al. 2021

Mathematics

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The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

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“…Based on the cited reference [6] some Barenblatt profiles as in Vázquez [16] are expected. In Figure 2…”

Section: Example 1: Barenblatt Profilesmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

“…See [3,6,7] for more details about the Silvestre-Caffarelli extension. The reader can also see [8,9] for more information about the method.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

García

Negreanu²,

Ureña

et al. 2021

Mathematics

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“…In this process, one must be aware of two fundamental points: first, it does not calculate exact solutions but approximate ones; second, it discretizes the problem by representing functions by a finite number of values, that is, to move from the “continuous” to the “discrete”. There are numerous methods for the numerical approximation of PDEs, among them, popularly adopted are finite element method (Lewis and Garner, 1972; Strada and Lewis, 1980; Morgan et al , 1984; Tadayon et al ,1987; Ahmed et al , 2011; Ahmed et al , 2009; Badruddin et al , 2006a, 2006b, 2007a, 2007b, 2012a, 2012b, 2012c; Li and Rui, 2015; Sajid et al , 2008; Balla and Kishan, 2015; Wansophark et al , 2005) finite difference method (Achemlal and Sriti, 2015; del Teso, 2014; Oka et al , 1994; Rui and Liu, 2015; Liu and Yuan, 2008; Sheremet and Pop, 2014; Chamkha and Muneer, 2013; Sheremet, 2015) and finite volume method (Dotlić, 2014; Kumar, 2012). These finite element, finite difference and finite volume methods require that each PDE be converted into its equivalent set of algebraic equations that depends on the number of elements into which the physical domain is divided.…”

Section: Introductionmentioning

confidence: 99%

Simplified finite element algorithm to solve conjugate heat and mass transfer in porous medium

Badruddin

Khan

Idris

et al. 2017

HFF

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Purpose The purpose of this paper is to highlight the advantages of a simplified algorithm to solve the problem of heat and mass transfer in porous medium by reducing the number of partial differential equations from four to three. Design/methodology/approach The approach of the present paper is to develop a simplified algorithm to reduce the number of equations involved in conjugate heat transfer in porous medium. Findings Developed algorithm/method has many advantages over conventional method of solution for conjugate heat transfer in porous medium. Research limitations/implications The current work is applicable to conjugate heat transfer problem. Practical implications The developed algorithm is useful in reducing the number of equations to be solved, thus reducing the computational resources required. Originality/value Development of simplified algorithm and comparison with conventional method.

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The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

Vázquez

2017

Lecture Notes in Mathematics

103

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We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic differential equations gives a foundation to the probabilistic study of diffusion.Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research experience, we use the porous medium and fast diffusion equations as case examples.A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

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Finite difference method for a fractional porous medium equation

Cited by 35 publications

References 23 publications

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

Simplified finite element algorithm to solve conjugate heat and mass transfer in porous medium

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

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