Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL)

Kazem, Saeed; Dehghan, Mehdi

doi:10.1007/s00366-018-0595-5

Cited by 19 publications

(8 citation statements)

References 22 publications

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“…The MOL discretizes the spatial dimensions by different methods, for example: finite volume, finite element, finite difference, spectral and meshless methods (Bratsos, 2007;Dehghan and Shakeri, 2009;Haq et al, 2010;Shakeri and Dehghan, 2008;Shen, 2009;Voss and Khaliq, 1996). The MOL has been used for different types of problems for example: three-dimensional heat equation (Kazem and Dehghan, 2018), dispersive nonlinear wave equations (Saucez et al, 2004), three-dimensional time-fractional diffusion equation (Kazem and Dehghan, 2019), the conservation laws problem (Hyman, 1979), onedimensional wave equation with respect to an integral conservation condition (Shakeri and Dehghan, 2008), the inverse parabolic problem with an over specification at a point (Dehghan and Shakeri, 2009), the extended Boussinesq equations , biomedical sciences and engineering (Schiesser, 2016), parabolic PDEs which appear in chemical engineering (White and Subramanian, 2010) and parabolic equations via successive convolution (Causley et al, 2016).…”

Section: Introductionmentioning

confidence: 99%

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Mesgarani

Kermani

Abbaszadeh

2021

HFF

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Purpose The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients. Design/methodology/approach The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well. Findings Several test problems are provided to confirm the validity and efficiently of the proposed method. Originality/value For the first time, some famous examples are solved by using the proposed high-order technique.

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Section: Introductionmentioning

confidence: 99%

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Mesgarani

Kermani

Abbaszadeh

2021

HFF

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“…Fractional calculus is an attractive branch of pure mathematics to study the integrals or derivatives of real or complex order. Recently, fractional calculus has been applied in the context of many sciences [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In this study, the sub-fractional diffusion equations (SFDEs) [18] are considered as follow:…”

Section: Introduction and Basic Notationsmentioning

confidence: 99%

“…Fractional calculus is an attractive branch of pure mathematics to study the integrals or derivatives of real or complex order. Recently, fractional calculus has been applied in the context of many sciences [1–17]. In this study, the sub‐fractional diffusion equations (SFDEs) [18] are considered as follow:

\frac{\partial^{α} ξ false(y, t false)}{\partial t^{α}} + μ false(y false) \frac{\partial ξ false(y, t false)}{\partial y} + η false(y false) \frac{\partial^{2} ξ false(y, t false)}{\partial y^{2}} = f false(y, t false), 0 < α \leq 2,

with initial condition

ξ false(y, 0 false) = g false(y false), a < y < b,

and boundary conditions

ξ false(a, t false) = ξ false(b, t false) = 0, 0 < t \leq τ,

where f is the source term.…”

Section: Introduction and Basic Notationsmentioning

confidence: 99%

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

Erfanifar

Sayevand

Ghanbari

et al. 2020

Numerical Methods Partial

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This study presents a robust modification of Chebyshev ϑ-weighted Crank-Nicolson method for analyzing the sub-diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub-fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub-diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.

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“…Other, different methods for the evaluation of matrix functions can be found in works by Dehghan & Hajarian. 3,4 Furthermore, recent works by Kazem & Dehghan 5,6 demonstrate how matrix functions can be used to approximate solutions in problems of heat conduction.…”

Section: Introductionmentioning

confidence: 99%

Simulation of harmonic oscillators on the lattice

Tung

Ibáñez

Defez

et al. 2020

Math Methods in App Sciences

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This work deals with the simulation of a two-dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second-order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state-of-the-art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two-dimensional lattice and check its energy conservation.

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Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL)

Cited by 19 publications

References 22 publications

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

Simulation of harmonic oscillators on the lattice

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