A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

Mohyud‐Din, Syed Tauseef; Akram, Tayyaba; Abbas, Muhammad; Ismail, Ahmad Izani; Ali, Norhashidah Hj. Mohd.

doi:10.1186/s13662-018-1537-7

Cited by 33 publications

(23 citation statements)

References 28 publications

(30 reference statements)

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“…Various numerical methods based on spline functions have also been employed by researchers in pursue of reliable solutions for fractional-order differential equations [29][30][31]. The B-spline functions provide decent approximations in contrast with rest of numerical schemes due to the nominal, compact support and C 2 continuity [32]. The approximate solution of a time-fractional DWE via cubic trigonometric B-splines was explored in [33].…”

Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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In this study, we have proposed an efficient numerical algorithm based on third degree modified extended B-spline (EBS) functions for solving time-fractional diffusion wave equation with reaction and damping terms. The Caputo time-fractional derivative has been approximated by means of usual finite difference scheme and the modified EBS functions are used for spatial discretization. The stability analysis and derivation of theoretical convergence validates the authenticity and effectiveness of the proposed algorithm. The numerical experiments show that the computational outcomes are in line with the theoretical expectations. Moreover, the numerical results are proved to be better than other methods on the topic.

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Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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“…Yaseen et al [28] presented a scheme for the numerical solution of fractional diffusion equation using a finite difference method based on cubic trigonometric B-spline basis functions. Mohyud-Din et al [29] constructed a fully implicit finite difference scheme for solving a time fractional diffusion equation by incorporating an extended cubic Bspline (ExCuBs) approach in its formulation. Because of the promising results obtained by this scheme, efforts are now being made in this work extend the formulation to solve the more complicated telegraph equation with fractional order derivatives.…”

Section: Application and Literature Reviewmentioning

confidence: 99%

“…In Table 2, we calculate the L 2 -norm for different spatial and temporal step size h = 5, τ = 1 M , (M = 20, 40, 80). In Table 3, we determine the order of convergence [16,29] from the computed data and present maximum absolute errors at different space-time step sizes. We give L 2 -norm and maximum errors for α = 0.6, 0.7, 0.8, 0.9.…”

Section: Problem 1 Consider the Tfte Of The Formmentioning

confidence: 99%

Extended cubic B-splines in the numerical solution of time fractional telegraph equation

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A finite difference scheme based on extended cubic B-spline method for the solution of time fractional telegraph equation is presented and discussed. The Caputo fractional formula is used in the discretization of the time fractional derivative. A combination of the Caputo fractional derivative together with an extended cubic B-spline is utilized to obtain the computed solutions. The proposed scheme is shown to possess the unconditional stability property with second order convergence. Numerical results demonstrate the applicability, simplicity and the strength of the scheme in solving the time fractional telegraph equation with accuracies very close to the exact solutions.and ∂ α u(x,t) ∂t α denote the Caputo fractional derivative of order 2α and α, respectively, which will be defined in Sect. 2.This model has been developed to overcome the shortcomings of the classical telegraph equation which might not adequately model the abnormal diffusion phenomena during a long transmission process in a transmission line [2].

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“…In [14], the authors introduced new fractional order spline functions to study the numerical solution of fractional Bagely-Torvik equation. Mohyu-Din et al [15] investigated the extended Bspline solution of time fractional advection diffusion equation by means of a fully implicit finite difference scheme. Li et al [16] developed a non-polynomial spline scheme to solve time fractional nonlinear Schrodinger equation.…”

Section: Introductionmentioning

confidence: 99%

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

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The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiffusion equation. The central difference approximation is used to discretize the time derivative, while non-polynomial quintic spline is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O(h 4 + t 2 ) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature.

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A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

Cited by 33 publications

References 28 publications

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

Extended cubic B-splines in the numerical solution of time fractional telegraph equation

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

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