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

Noor‐ul‐Amin, Muhammad; Abbas, Muhammad; Iqbal, Muhammad Kashif; Ismail, Ahmad Izani Md.; Băleanu, Dumitru

doi:10.1186/s13662-019-2442-4

Cited by 16 publications

(9 citation statements)

References 20 publications

(31 reference statements)

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“…T is required to commence the iterative process, which can be obtained using the IC and the derivatives of IC at the two boundaries as follows [30][31][32][33][34][35][36][37][38]:…”

Section: Initial Vectormentioning

confidence: 99%

A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

2021

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Fractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.

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“…T is required to commence the iterative process, which can be obtained using the IC and the derivatives of IC at the two boundaries as follows [30][31][32][33][34][35][36][37][38]:…”

Section: Initial Vectormentioning

confidence: 99%

A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

2021

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“…Let the time domain [0, T] be divided into R subintervals of equal length t = T R with endpoints 0 = t 0 < t 1 < • • • < t R = T, where t r = r t and r = 0 : 1 : R. We first discretize the Caputo fractional derivative at t = t r+1 as [46]…”

Section: Time Discretizationmentioning

confidence: 99%

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

et al. 2020

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In this article we develop a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution of the time-fractional Klein-Gordon equation. The proposed technique employs the finite difference formulation to discretize the Caputo fractional time derivative of order α ∈ (1, 2] and uses redefined extended cubic B-spline functions to interpolate the solution curve over a spatial grid. A stability analysis of the scheme is conducted, which confirms that the errors do not amplify during execution of the numerical procedure. The derivation of a uniform convergence result reveals that the scheme is O(h 2 + t 2−α) accurate. Some computational experiments are carried out to verify the theoretical results. Numerical simulations comparing the proposed method with existing techniques demonstrate that our scheme yields superior outcomes.

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Section: Introductionmentioning

confidence: 99%

“…Owing to the fact that both fractional definitions and spectral methods perform the same global feature, the spectral method using global basis functions seems suitable for fractional problems. [21][22][23][24][25][26][27][28][29] Li and Xu 30,31 proposed a space-time spectral method based on a classical combination of Jacobi polynomials. Very recently, Chen et al 32,33 put forward generalized Jacobi functions (GJFs) and Laguerre functions to apply for fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A space–time spectral Petrov–Galerkin method for nonlinear time fractional Korteweg–de Vries–Burgers equations

Sun

2020

Math Methods in App Sciences

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In this work, we study a space–time Petrov–Galerkin method for third‐ and fifth‐order time fractional Korteweg–de Vries–Burgers equations. The method is based on the framework of Legendre and Jacobi polynomials. The basis functions of the fractional part are constructed by the generalized Jacobi functions, which contained the singularity of weak solutions. The numerical schemes of the problems are transformed into the nonlinear schemes constructed by matrices. Based on the orthogonality of ideal basis functions, we get the optimal estimation under the specific weighted Sobolev spaces. Numerical experiments confirm the expected convergence.

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

Cited by 16 publications

References 20 publications

A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

A space–time spectral Petrov–Galerkin method for nonlinear time fractional Korteweg–de Vries–Burgers equations

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