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

Patra, Asim

doi:10.1002/mma.6695

Cited by 8 publications

(4 citation statements)

References 28 publications

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“…Example 1 is presented to verify the convergence order of the MDD approximation and its maximum absolute error (MAE). Other cases illustrate the accuracy and the MAE of our scheme (13). MATLAB R2022a is used for all numerical results.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In this problem, we approximated the MDD term using the numerical approach presented in equation (13). In figure 2, the numerical solution and the exact one are displayed at Δx = π/10, h = 5 × 10 −4 , ω = 0.01, a = 1 and b = 0.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Riemann-Liouville and Caputo fractional derivatives have been frequently employed in problem formulation using the memory effect [9]. Numerical [10], semi-analytical [11], iterative [12], and similarity techniques [13] have been used to solve problems involving fractional derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Khalaf,

Elsaid,

Hammad

et al. 2024

Phys. Scr.

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Although memory-dependent derivative (MDD) has recently been the subject of extensive study, only one numerical approximation has been reported in the literature. Hence, this study introduces a novel approximation for MDD. Moreover, a new form of the kernel function is presented. The convergence order of our approximation is O(h^(3-S) ),0<S<1, where (1-S) is the exponent of the kernel function. The proposed approach is used to numerically solve the memory-dependent advection-diffusion problem, and the numerical scheme’s stability and convergence are discussed. Also, memory-based models have been described to study drug delivery and its diffusion from multi-layer capsules/tablets. These models are based on the fractional derivative and the MDD to solve the paradox of the unphysical feature of the infinite propagation speed of the published Fickian-based models. The theoretical analysis is validated with numerical examples by investigating the convergence order that is (3-S) in time. To illustrate the validity and efficiency of the proposed models, profiles of concentration and drug mass are compared with the observed in vivo data. It is observed to be highly accurate and compatible with the in vivo data when compared with Fick’s model.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Khalaf,

Elsaid,

Hammad

et al. 2024

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…Consequently, mathematicians and physicists have shown a consistent interest in identifying precise wave solutions for nonlinear systems. The ability to discover exact solutions holds substantial significance across various scientific domains [4,5]. Unlike linear theory, obtaining explicit closed-form solutions for even simple NLPDEs is not straightforward.…”

Section: Introductionmentioning

confidence: 99%

Unveiling multi-wave patterns: dynamic characterization and sensitivity analysis of the Yu-Toda-Sasa-Fukuyama model in lattice and liquid

Riaz,

Kazmi,

Jhangeer

2024

Phys. Scr.

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In this study, an examination of the Yu-Toda-Sasa-Fukuyama equation is undertaken, a model that characterizes elastic waves in a lattice or interfacial waves in a two layer liquid. Our emphasis lies in conducting a comprehensive analysis of this equation through various viewpoints, including the examination of soliton dynamics, exploration of bifurcation patterns, investigation of chaotic phenomena, and a thorough evaluation of the model's sensitivity. Utilizing a simplified version of Hirota's approach, multi-soliton pattens, including 1-wave, 2-wave, and 3-wave solitons, are successfully derived. The identified solutions are depicted visually via 3D, 2D, and contour plots using Mathematica software. The dynamic behavior of the discussed equation is explored through the theory of bifurcation and chaos, with phase diagrams of bifurcation observed at the fixed points of a planar system. Introducing a perturbed force to the dynamical system, periodic, quasi-periodic and chaotic patterns are identified using the RK4 method. The chaotic nature of perturbed system is discussed through Lyapunov exponent analysis. Sensitivity and multistability analysis are conducted, considering various initial conditions. The results acquired emphasize the efficacy of the methodologies used in evaluating solitons and phase plots across a broader spectrum of nonlinear models.

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

2022

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

Cited by 8 publications

References 28 publications

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Unveiling multi-wave patterns: dynamic characterization and sensitivity analysis of the Yu-Toda-Sasa-Fukuyama model in lattice and liquid

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

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