Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation

Alolyan, Ibraheem; Simos, T. E.

doi:10.1007/s10910-015-0522-6

Cited by 54 publications

(3 citation statements)

References 109 publications

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“…The solution from the seed layer was carefully combined with the solution from the growth. The flask was sealed, and the mixture was placed inside an oven for 5 hours at a controlled temperature of 95  C (Alolyan & Simos, 2015). After the thermal treatment, the flask was removed from the oven and immersed in DI water.…”

Section: Devicementioning

confidence: 99%

Efficient Synthesis of Zno as an Electrode of Piezoelectric Nanogenerators

Salh,

Ibrahem

2024

SJUOZ

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Piezoelectric nanogenerators (NGs) hold immense promise as self-powered devices for harvesting mechanical energy from the environment. This study introduces an efficient and scalable synthesis method for zinc oxide (ZnO) nanorods, a pivotal material in piezoelectric nanogenerators (NGs), with several key results. A Chemical Bath Deposition technique is employed, optimizing parameters such as growth time, temperature, and precursor concentrations to achieve well-aligned and high-quality ZnO nanorods. The structural and morphological characteristics of the synthesized nanorods are systematically investigated using advanced characterization techniques. The synthesized ZnO nanorods exhibit an average length of 400nm, demonstrating their slender shape. Furthermore, the study determines an energy gap value of 3.5 eV for multilayer zinc oxide thin films, indicating the transition from the valence band to the conduction band. Notably, thermal annealing at 500°C leads to a substantial increase in average output voltage, reaching 1.95 V, a fourfold improvement compared to as-deposited nanopowders. These findings emphasize the efficiency and potential of the proposed synthesis method and underscore its practical applications in enhancing energy harvesting capabilities for sustainable power generation from mechanical sources in piezoelectric NGs.

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Section: Devicementioning

confidence: 99%

Efficient Synthesis of Zno as an Electrode of Piezoelectric Nanogenerators

Salh,

Ibrahem

2024

SJUOZ

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“…9 Accuracy (digits) for several values of C PU time (in seconds) for the eigenvalue E 2 = 341.495874. The nonexistence of a value of accuracy (digits) indicates that for this value of CPU, accuracy (digits) is less than 0 -The Phase-Fitted Method (Case 2) developed in [48], which is indicated as Method NMPF2 -The Method developed in [52] (Case 2), which is indicated as Method NMC2 -The Method developed in [52] (Case 1), which is indicated as Method NMC1 -The Method developed in [45], which is indicated as Method RKTPLDDEA -The Method developed in [58], which is indicated as Method HYBPLDDDEA -The Hybrid Low Computational Computational Cost Four-Step Method developed in [46], which is indicated as Method HYMETH8 -The New Obtained Three Stages Explicit Symmetric Four-Step Method which is developed in Sect. 5, which is indicated as Method MuSMeth10…”

Section: Remark 16mentioning

confidence: 99%

“…7. The low computational cost hybrid explicit four-step method of eight algebraic order with vanished phase-lag and its first, second, third and fourth derivatives developed in [46], which is indicated as Method HYMETH8, is more efficient than the Method developed in [58], which is indicated as Method HYBPLD-DDEA. 8.…”

Section: Remark 17mentioning

confidence: 99%

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

Alolyan

Simos

2015

J Math Chem

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A new Multistage high algebraic order four-step method is obtained in this paper. It is the first time in the literature that a method of this category is developed and has vanishing of the phase-lag and its first, second, third, fourth and fifth derivatives. We study this new method by investigating: (1) the development of the new method, i.e. the calculation of the coefficients of the method in order the phase-lag and its first, second, third, fourth and fifth derivatives of the phase-lag to be vanished, (2) the determination of the formula of the Local Truncation Error, (3) the comparative analysis of the Local Truncation Error (with this we mean the application of the new method and similar methods on a test problem and the analysis of their behavior), (4) the stability of the new method, by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis and by studying the results of this application i.e. by investigating the interval of periodicity of the new obtained method. We finally study the computational behavior the new Electronic supplementary material The online version of this article (123 J Math Chem developed method by using the application of the new method to the approximate solution of the resonance problem of the radial Schrödinger equation. We prove the effectiveness of the new obtained method by comparing it with (1) well known methods of the literature and (2) very recently obtained methods.

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A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation

Zhou

Simos

2015

J Math Chem

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Efficient low computational cost hybrid explicit four-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the Schrödinger equation

Cited by 54 publications

References 109 publications

Efficient Synthesis of Zno as an Electrode of Piezoelectric Nanogenerators

Efficient Synthesis of Zno as an Electrode of Piezoelectric Nanogenerators

A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation

A new two stage symmetric two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of the radial Schrödinger equation

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