An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

Patra, Asim

doi:10.1140/epjp/i2018-11922-3

Cited by 7 publications

(3 citation statements)

References 16 publications

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“…The fractional Schrӧdinger equation is employed in the quantum field theory and quantum mechanics for the purpose of evolution of wave packets. There are some literatures wherein the fractional Schrӧdinger equation has been be tackled by some numerical methodologies 14 . So it will be a worthy and interesting task to handle the fractional version of the Schrӧdinger equation via a novel analytical approach.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, Elsaid et al, 16 in a different paper, have tackled with the fractional heat equation in context with the similarity solutions. Patra 14 treated the fractional Schrӧdinger equation via a semi‐numerical method. Djordjevic et al 17 utilized the similarity method for the nonlinear fractional Korteweg–de Vries (KdV) equation.…”

Section: Introductionmentioning

confidence: 99%

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Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

Patra

2020

Math Methods in App Sciences

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The present article deals with the similarity method to tackle the fractional Schrdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore, the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives α.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

Patra

2020

Math Methods in App Sciences

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“…It should also be noted that there are several definitions for fractional derivatives. These definitions include Riemann–Liouville, Grunwald–Letnikov, Weyl, Caputo, Marchaud, Riesz, Jumarie, and Nishimoto 11‐14 . All these fractional derivatives definitions have their advantages and disadvantages.…”

Section: Introductionmentioning

confidence: 99%

On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method

Moradi

Sayevand

2020

Math Methods in App Sciences

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In this study, a reliable implicit finite difference method based on the modified trapezoidal quadrature rule, backward Euler differences, nonstandard central approximations, and the Hadamard finite‐part integral is being considered to solve a viscous asymptotical model named as fractional Kakutani–Matsuuchi water wave model. The fractional derivative is used in the Riemann–Liouville sense. Based on the properties of Brouwer's fixed‐point theorem, the existence, uniqueness, convergence, and stability of the proposed method are proved. Furthermore, we show that the global convergence order of the method in maximum norm is O(τ, hmin{β,3 − α}), where 0 < α ≤ 1 and β > 0 are the order of fractional derivative and the Lipschitz constant. Also, τ and h are the time step and space step, respectively. Finally, several examples are used to illustrate the accuracy and performance of the method.

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Soliton propagation through three types of Fibonacci-ordered photonic multilayers in the fractional medium

2022

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An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

Cited by 7 publications

References 16 publications

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

On fractional Kakutani–Matsuuchi water wave model: Implementing a reliable implicit finite difference method

Soliton propagation through three types of Fibonacci-ordered photonic multilayers in the fractional medium

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