A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

Oruç, Ömer; Esen, Alaattin; Bulut, Fatih

doi:10.1007/s00366-018-0584-8

Cited by 59 publications

(26 citation statements)

References 63 publications

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“…Thus, the wavelet expansion is applied to derivatives of integer order. An alternate approach was employed in [44,53,60,62] where the fractional derivatives are expanded directly to wavelet series.…”

Section: Introductionmentioning

confidence: 99%

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Ratas

Salupere

2020

Mathematical Modelling and Analysis

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Section: Introductionmentioning

confidence: 99%

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Ratas

Salupere

2020

Mathematical Modelling and Analysis

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“…Nevertheless, new reliable mathematical algorithms including numerical ones, solitons, and approximations via polynomials and wavelets are found more effective. Several attempts have been made to handle nonlinear problems numerically and a wide range of these efforts have been reported earlier in literature [6][7][8][9][10][11][12]. The approximation through polynomials and wavelet-based algorithms is an emerging area and has attracted the attention from research community.…”

Section: Introductionmentioning

confidence: 99%

“…The listed techniques cover some areas from chemical physics, quantum chemistry, and many engineering disciplines as well but still need further investigations to make progress in other fields of science. Studies in the recent past [6–8, 10, 18–23] reveal that wavelets methods are more precise in numerous situations to tackle problems containing higher nonlinearity factor. Alongside, various attempts [11, 14, 24–27] in order to improve the efficiency and accuracy levels of these methods have been reported, but such methods also carry certain disadvantages that need to be overcome.…”

Section: Introductionmentioning

confidence: 99%

“…Alongside, various attempts [11, 14, 24–27] in order to improve the efficiency and accuracy levels of these methods have been reported, but such methods also carry certain disadvantages that need to be overcome. Different wavelet methods; named as Chebyshev, CAS (cosine and sine), Gegenbauer, Haar, Laguerre, Legendre and so on, have been proposed or extended to attain approximate solutions for dynamical or other relevant physical problems [6–12, 20, 28].…”

Section: Introductionmentioning

confidence: 99%

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A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

Usman

Hamid

Khalid

et al. 2020

Numerical Methods Partial

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In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative (D x and D t), fractional-order derivative (P x and P t), fractional-order integration (J x and J t) and delay terms (D a b How to cite this article: Usman M, Hamid M, Khalid MSU, Haq RU, Liu M. A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics.

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“…In [19], Yang developed novel numerical techniques for the solution of the two-dimensional fractional sub-diffusion equation, which is based on the orthogonal spline collocation method in space and a finite difference method (FDM) in time. Ömer et al [20] established a wavelet method, based on Haar wavelets and a finite difference scheme for the two-dimensional time fractional reaction-subdiffusion equation. Li [21] proposed a numerical treatment for two-dimensional fractional subdiffusion equations using the parametric quintic spline.…”

Section: Introductionmentioning

confidence: 99%

A high-order numerical scheme using orthogonal spline collocation for solving the two-dimensional fractional reaction–subdiffusion equation

2019

Adv Differ Equ

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In this paper, a high-order numerical scheme is proposed for solving the two-dimensional fractional reaction-subdiffusion equation. The method is based on adopting a third-order weighted and shifted Grünwald difference (WSGD) operator to approximate the time Caputo fractional derivative and applying the orthogonal spline collocation (OSC) method to approximate the spatial derivative. Stability and convergence analysis of the proposed method are rigorously proved. Several numerical examples in one variable and in two space variables are presented to validate our theoretical analysis.Recently, Tian et al. [28] proposed second-and third-order approximations for the Riemann-Liouville fractional derivative with order α (0 < α < 1) by using the weighted and shifted Grünwald difference (WSGD) operators. Thereafter, some related discussion problems covering the WSGD idea were addressed by many scholars. Following the idea of the WSGD operator, Wang and Vong [29] used compact finite difference schemes for the modified time sub-diffusion equation with Riemann-Liouville fractional derivative and the temporal Caputo fractional diffusion-wave equation. Ji and Sun [30] applied a thirdorder approximation for the Caputo fractional derivative and used compact difference scheme to solve fractional sub-diffusion equation. In [31], Liu et al. developed a highorder local discontinuous Galerkin method combined with WSGD approximation for a Caputo time-fractional sub-diffusion equation. In [32], a temporal second-order fully discrete two-grid finite element (FE) scheme is presented to solve nonlinear fractional Cable equation, in which the spatial direction is approximated by two-grid FE method and Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

Cited by 59 publications

References 63 publications

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

Application of Higher Order Haar Wavelet Method for Solving Nonlinear Evolution Equations

A robust scheme based on novel‐operational matrices for some classes of time‐fractional nonlinear problems arising in mechanics and mathematical physics

A high-order numerical scheme using orthogonal spline collocation for solving the two-dimensional fractional reaction–subdiffusion equation

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