Approximate analytical solution of coupled fractional order reaction-advection-diffusion equations

Pandey, Prashant; Kumar, Sachin; Das, S.

doi:10.1140/epjp/i2019-12727-6

Cited by 20 publications

(16 citation statements)

References 38 publications

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“…The involvement of Laguerre polynomial in several aspects of engineering and applied mathematics seeks the attention of many researchers and establishes it as a strong tool for finding the numerical solution of integer as well as fractional order partial differential equations (PDEs). The l th degree Laguerre polynomial is defined as [40] L l ðtÞ ¼ 1 l! e t o l t ðt l e Àt Þ; l ¼ 0; 1; :::; ð19Þ…”

Section: Laguerre Polynomials and Its Some Propertiesmentioning

confidence: 99%

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Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

et al. 2020

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In the present article, an efficient operational matrix based on the famous Laguerre polynomials is applied for the numerical solution of two-dimensional non-linear time fractional order reaction–diffusion equation. An operational matrix is constructed for fractional order differentiation and this operational matrix converts our proposed model into a system of non-linear algebraic equations through collocation which can be solved by using the Newton Iteration method. Assuming the surface layers are thermodynamically variant under some specified conditions, many insights and properties are deduced e.g., nonlocal diffusion equations and mass conservation of the binary species which are relevant to many engineering and physical problems. The salient features of present manuscript are finding the convergence analysis of the proposed scheme and also the validation and the exhibitions of effectiveness of the method using the order of convergence through the error analysis between the numerical solutions applying the proposed method and the analytical results for two existing problems. The prominent feature of the present article is the graphical presentations of the effect of reaction term on the behavior of solute profile of the considered model for different particular cases.

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Section: Laguerre Polynomials and Its Some Propertiesmentioning

confidence: 99%

“…We are going to derive the operational matrix of orthogonal Laguerre polynomials [40]. On solving the equations (28)- (29) and equation 20, we get the following expression as…”

Section: Laguerre Operational Matrix For Fractional Order Differentiamentioning

confidence: 99%

Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

et al. 2020

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“…Now a days the Chebyshev polynomials are worldwide useful in various area of applied and engineering mathematics [40, 41]. The l th degree fifth‐kind Chebyshev polynomials are defined in the interval [−1,1] as:

ω_{l} false(x false) = \frac{1}{\sqrt{δ_{l}}} {\overset{true‾}{normalϒ}}_{l}^{- 3, 2, - 1, 1} false(x false), - 1 \leq x \leq 1;

In the definition of fifth‐kind Chebyshev polynomials

{\overset{true‾}{normalϒ}}_{l}^{- 3, 2, - 1, 1} false(x false)

is given by following general formula

{\overset{true‾}{normalϒ}}_{l}^{f, g, h, i} false(x false) = ({false\prod}_{s = 0}^{lfloor\frac{l}{2}rfloor - 1} \frac{false(2 s + {(- 1)}^{l + 1} + 2 false) i + g}{(2 s + {(- 1)}^{l + 1} + 2 lfloor\frac{l}{2}rfloor) h + f}) {normalϒ}_{l}^{f, g, h, i} false(x false),

and

{normalϒ}_{l}^{f, g, h, i} false(x false) = {false\sum}_{u = 0}^{(lfloorrfloor, \frac{l}{2})} ((\begin{matrix} center \\ lfloor\frac{l}{2}rfloor \\ u \end{matrix}) ({false\prod}_{s = 0}^{lfloor\frac{l}{2}rfloor - u - 1} ()))

…”

Section: Basic Properties and Definitions Of The Fifth‐kind Chebyshevmentioning

confidence: 99%

On solution of a class of nonlinear variable order fractional reaction–diffusion equation with Mittag–Leffler kernel

Pandey

Gómez‐Aguilar

2020

Numerical Methods Partial

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In this article, an efficient variable‐order Chebyshev collocation method which is based on shifted fifth‐kind Chebyshev polynomials is applied to solve a nonlinear variable‐order fractional reaction–diffusion equation with Mittag–Leffler kernel. The operational matrix of shifted fifth‐kind Chebyshev polynomials is derived for variable‐order ABC derivatives. The Chebyshev operational matrix together with the collocation method are applied to concerned nonlinear physical model with Mittag–Leffler kernel which is converted into a system of nonlinear algebraic equations, this system can be solved by using Newton method. The main focus of this paper is finding the convergence analysis of the approximation and high convergence order for small grid approximation. Few test examples with a comparison of maximum absolute error between the obtained numerical solution and existing known solution are being reported to show the accuracy and stability of the scheme. The physical presentation of the absolute errors for considered nonlinear variable‐order reaction–diffusion equations involving the Mittag–Leffler kernel with their exact solutions shows that the method is good for finding the solution of these kind of problems.

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“…In the end of nineteenth century basic theory of fractional calculus was developed with the studies of Liouville, Grünwald, Letnikov, and Riemann. It has been shown that fractional derivative operators are useful in describing dynamical processes with memory or hereditary properties such as creep or relaxation processes in viscoelastoplastic materials [3] , [4] , impact problem [5] , plasma physics [1] , diffusion process models [6] , [7] , [8] , [9] , chaotic systems [10] , control problems [11] , [12] , dynamics modeling of coronavirus (2019-nCov) [13] , etc .…”

Section: Introductionmentioning

confidence: 99%

A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel

Tuấn¹,

Ganji

Jafari

2020

Chinese Journal of Physics

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Approximate analytical solution of coupled fractional order reaction-advection-diffusion equations

Cited by 20 publications

References 38 publications

Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

Two-dimensional nonlinear time fractional reaction–diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media

On solution of a class of nonlinear variable order fractional reaction–diffusion equation with Mittag–Leffler kernel

A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel

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