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This paper numerically retrieves cubic-quartic solitons having power law of nonlinearity refractive index. An improvement of the Adomian decomposition scheme is the adopted algorithm of this work. The results are displayed along with the established error analysis.
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Interferometry sensors are frequently analyzed by applying the Fourier transform because the transformation separates all frequency components of its signal, making its study on a complex plane feasible. In this work, we study the relation between the optical path difference (OPD) and poles location theoretically and experimentally, using the Laplace transform and a pole-zero map. Theory and experiments are in concordance. For our study, only the cosine function was considered, which is filtered from the interference pattern. In experimental work, two unperturbed low-finesse Fabry-Pérot interferometers were used. First, a Fabry-Pérot interferometer that has a cavity length of ~1.6 mm was used. Its optical path difference was 2.33 mm and the poles were localized at points ±i12. rad/nm. Secondly, a Fabry-Pérot interferometer with a cavity length of ~5.2 mm was used, and its optical path difference was 7.59 mm and the poles were localized at points ±i40.4 rad/nm. Experimental results confirmed the theoretical analysis. Our proposal finds practical application for interferometer analysis, signal processing of optical fiber sensors, communication system analysis, and multiplexing systems based on interferometers.
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The transverse relaxation time (T2), measured with Carr-Purcell-Meiboom-Gill (CPMG) sequence, has been widely used to obtain the direct dimension data in two-dimension time domain NMR (2D TD-NMR). In this paper we are demonstrating that Continuous Wave Free Precession sequence, with low flip angle (CWFP-T1), can be an alternative to CPMG as direct detection dimension. CWFP-T1 is a fast single shot sequence, like CPMG, and yields an exponential signal governed predominantly by the longitudinal (T1) relaxation time. To obtain the correlations between T1 and T2 (T1-T2 maps) we are proposing the use of CPMG-CWFP-T1 pulse sequence. In this sequence CPMG encodes T2 information (indirect dimension) that modulates the CWFP-T1 (direct dimension) signal amplitudes. CPMG-CWFP-T1 experiments were compared with classical 2D sequences such as Saturation-Recovery-CPMG (SR-CPMG) and Inversion-Recovery-CPMG (IR-CPMG) sequence and yields similar results in phantom sample. The experimental time for the 2D sequences, using single scan, shows that SR-CPMG ≤ CPMG-CWFP-T1 < IR-CPMG. Experimental and simulated results demonstrated that 2D-CPMG-CWFP-T1 maps have higher resolution in T1 dimension than the techniques that uses CPMG as direct dimension. CPMG-CWFP-T1 sequence was also applied to study beef samples, and 2D maps showed higher resolution in the two fat signals than the classical IR-CPMG method.
Assuntos
Análise de Alimentos/métodos , Espectroscopia de Ressonância Magnética/métodos , Tecido Adiposo/química , Algoritmos , Animais , Bovinos , Simulação por Computador , Processamento Eletrônico de Dados , Carne/análise , Conformação Molecular , Razão Sinal-RuídoRESUMO
RESUMEN El campo de las ecuaciones diferenciales ha cobrado auge en la actualidad por el desarrollo científico y tecnológico. Por esta situación, el estudio de nuevas metodologías para solucionarlas se ha vuelto importante. A partir de la combinación del método de Laplace Transform (LT) y el método de perturbación (PM) este trabajo presenta el método LT-PM, y su motivación se encuentra en la aplicación conocida de la LT a ecuaciones diferenciales ordinarias lineales. El objetivo de este trabajo fue presentar una modificación del método de perturbación (PM), el método de perturbación con transformada de Laplace (LT-PM), con el fin de resolver problemas perturbativos no lineales, con condiciones a la frontera definidas en intervalos finitos. La metodología consistió en aplicar LT a la ecuación diferencial por resolver y después de asumir que la solución de la misma se puede expresar como una serie de potencias de un parámetro perturbativo, se obtiene la solución del problema aplicando sistemáticamente la transformada inversa de Laplace. Los principales resultados de este trabajo se muestran a partir de dos casos de estudio presentados, donde se observa que LT-PM es potencialmente útil para encontrar soluciones múltiples de problemas no lineales. Además, LT-PM mejora la aplicabilidad del método de perturbación en algunos casos de condiciones a la frontera mixtas y de Neumann, donde PM simplemente no funciona. Con el fin de verificar la exactitud de los resultados obtenidos, se calculó su error residual cuadrático (SRE), el cual resultó muy bajo, de donde se dedujo su precisión y la potencialidad de LT-PM. Se concluye que si bien el método propuesto resulta eficiente en los casos particulares presentados, se espera que sea una herramienta potencialmente eficiente y útil para otros casos de estudio, particularmente, en aquellos relacionados con aplicaciones prácticas en ciencias e ingeniería.
ABSTRACT The field of differential equations has recently gained attention due to recent developments in science and technology. For this reason, the analysis for the use of new methodologies to solve them has become important. Based on the combination of Laplace Transform method (LT) and Perturbation Method (PM) this article pro- poses the Laplace transform-Perturbation Method (LT-PM) which finds its motivation on the application of LT to linear ordinary differential equations. The goal of this work is to propose a modification of PM - the LT-PM), in order to solve nonlinear perturbative problems with boundary conditions defined on finite intervals. The proposed methodology consisted on the application of LT to the differential equation to solve and then, assuming that its solutions can be expressed as a series of perturbative parameter powers. Thus, the solution of the problem is obtained by systematically applying the transformed inverse LT. The main results of this paper were shown through two case studies, where LT-PM is identified as potentially useful for finding multiple solutions to nonlinear problems. Additionally, the LT-PM enhances the applicability of PM, in some cases of mixed and Neumann boundary conditions, where PM is unsuitable to provide the results. With the purpose of verifying the accuracy of the obtained results, the Square Residual Error (SRE) was calculated. The resulting value was extremely low, which showed the precision and potential of LT-PM. We conclude that, although the proposed method resulted efficient for the case studies presented in this article, it is expected that LT-PM can be a potentially useful tool for other case studies. Particularly those related to the practical applications of science and engineering.
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Two-dimension (2D) Nuclear Magnetic Resonance relaxometry experiments are a powerful tool extensively used to probe the interaction among different pore structures, mostly in inorganic systems. The analysis of the collected experimental data generally consists of a 2D numerical inversion of time-domain data where T2-T2 maps are generated. Through the years, different algorithms for the numerical inversion have been proposed. In this paper, two different algorithms for numerical inversion are tested and compared under different conditions of exchange dynamics; the method based on Butler-Reeds-Dawson (BRD) algorithm and the fast-iterative shrinkage-thresholding algorithm (FISTA) method. By constructing a theoretical model, the algorithms were tested for a two- and three-site porous media, varying the exchange rates parameters, the pore sizes and the signal to noise ratio. In order to test the methods under realistic experimental conditions, a challenging organic system was chosen. The molecular exchange rates of water confined in hierarchical porous polymeric networks were obtained, for a two- and three-site porous media. Data processed with the BRD method was found to be accurate only under certain conditions of the exchange parameters, while data processed with the FISTA method is precise for all the studied parameters, except when SNR conditions are extreme.
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Many rural communities in the developing world practice small-scale harvesting of marine species of various kinds (fish, crustaceans, mollusks) as their main economic activity. Lack of regulation or enforcement or both has led in many cases to overexploitation of resources and consequent decline of the populations of organisms on which the communities depend. This situation, common along the north-central Chilean coast, served as the starting point of the present work. Recent advances in biotechnology and aquaculture, however, have made plausible halting or even reversing the trend towards resource depletion, favorably affecting the economic wellbeing of the communities. In this paper the theoretical implications of seeding the habitual harvesting grounds of a hypothetical community with individuals produced in the laboratory is explored. In particular, we examine the effect of periodic seeding of hatchlings or juveniles on the population of an economically important species harvested under either one of two modes of extraction, constant harvest or harvest proportional to the stock level. The population dynamics is modeled by a first order ordinary differential equation containing a series of delta functions, and which in words reads as follows: the population growth rate is equal to the natural growth rate minus the harvest rate plus the seeding rate. It was found that, at least from a theoretical standpoint, seeding could potentially lead to a notable increase in the productivity and well-being of these sea-based economies.
Muchas comunidades rurales en el mundo en desarrollo practican la recolección de especies marinas de varios tipos (peces, crustáceos, moluscos) como actividad económica principal. La falta de regulación o de aplicación de leyes ha conllevado en muchos casos a la sobreexplotación de los recursos y disminución de poblaciones de organismos de los que dependen estas comunidades. Esta situación, común en las zonas norte y centro del litoral chileno, sirvió como punto de partida para este trabajo. Avances recientes en biotecnología y acuicultura han hecho creíble la posibilidad de detener e incluso revertir la tendencia hacia el agotamiento del recurso y así afectar favorablemente el bienestar de la comunidad. En este trabajo se explora las implicaciones teóricas de la siembra con individuos producidos en el laboratorio en sectores usuales de captura de una comunidad hipotética; en particular, el efecto de la siembra periódica de crías o juveniles sobre la población de una especie económicamente importante cuya recolección se hace bajo uno de dos modos de extracción, constante o proporcional a la población. La dinámica poblacional se modela por una ecuación diferencial ordinaria de primer orden que contiene una serie de funciones delta y que, en palabras, dice que la tasa de crecimiento de la población es igual a la tasa natural de crecimiento menos la tasa de extracción más la tasa de siembra. Se encontró que, al menos desde el punto de vista teórico, la siembra podría potencialmente llevar a un incremento notable en la productividad y bienestar de estas comunidades marino-dependientes.
Muitas comunidades rurais no mundo em desenvolvimento praticam a colheita de espécies marinhas de vários tipos (peixes, crustáceos, moluscos) como atividade econômica principal. A falta de regulamentação ou de aplicação de leis tem conduzido em muitos casos à superexploração dos recursos e diminuição de populações de organismos dos quais dependem estas comunidades. Esta situação, comum nas zonas norte e centro do litoral chileno, serviu como ponto de partida para este trabalho. Avanços recentes em biotecnologia e aqüicultura têm tornado aceitável a possibilidade de deter e inclusive reverter a tendência para o esgotamento do recurso e assim afetar favoravelmente o bem-estar da comunidade. Neste trabalho se explora as implicações teóricas do plantio com indivíduos produzidos no laboratório em setores usuais de captura de uma comunidade hipotética; em particular, o efeito do plantio periódico de filhotes ou juvenis sobre a população de uma espécie economicamente importante cuja colheita se faz sob um de dois modos de extração, constante ou proporcional à população. A dinâmica populacional se modela por uma equação diferencial ordinária de primeira ordem que se representa por uma série de funções delta e que, em palavras, diz que a taxa de crescimento da população é igual à taxa natural de crescimento menos taxa de extração mais taxa de semeadura. Encontrou-se que, pelo menos desde o ponto de vista teórico, o plantio poderia potencialmente levar a um incremento notável na produtividade e bem-estar destas comunidades marinho-dependentes.