Energy Balance Theory Research Articles

Dual effects of pressure difference for long-wave stratified flow in horizontal microchannels

In this work, the linear stability analysis combining with energy balance theory is performed to clarify the suppressing and boosting long-wave instabilities driven by pressure difference in horizontal microchannels as well as its corresponding physical meanings. It is found that, for a specific fluid combination, there exists a certain relative liquid film thickness (hN,c) at which the disturbance work done by interfacial shear stress over unit wavelength (denoted by ISS) is exactly equal to the total viscous dissipation rate over unit wavelength within both phase bulk flows (denoted by DIStot). When the liquid film thickness (hN) exceeds hN,c, ISS is smaller than DIStot, indicating that the initial disturbance energy cannot be replenished from the primary flow until its completely disappear due to the viscosity dissipation. Conversely, ISS is larger than DIStot, indicating that the initial disturbance can acquire energy from the primary flow and grow up. The strength of suppressing and boosting effect can be influenced by liquid film thickness, flow rate, and channel height. However, the demarcation of the dual effects remains unchanged. An analytical expression of hN,c with respect to the viscosity ratio is established by means of long-wave approximation method, which matches well with the numerical results. The energy balance mechanisms revealed in the present study provide a new insight into the wave mode in the confined space and help the design of microfluidic systems.

Research on the relationships between river erosion and sedimentation and energy loss in the Yangtze River Basin of China

River scour and siltation, inherently prolonged and multifaceted phenomena, profoundly impact the hydraulic conductivity and navigational safety of waterways, thereby rendering the precise forecasting of their fluctuations a paramount and unresolved challenge. This study embarks on a novel investigative trajectory by examining energy dissipation in rivers, specifically through the scrutiny of empirical hydrological records. Employing the principles of energy balance theory, this study elucidates the intricate connection between energy losses and sedimentation volumes in the upper reaches of the Yangtze River and at its confluence with the significant tributary, the Jialing River.Our findings diverge from conventional wisdom, revealing that standard metrics of energy loss, encompassing kinetic, potential, and total mechanical energies, along with momentum losses, inadequately characterize the complexity of sediment deposition processes. In contrast, we demonstrate that the loss of suspension power is a more indicative measure of sedimentation quantity. Through meticulous comparisons of various formulations for calculating suspension power, we identified a formula grounded in the concept of minimal available power dissipation in river systems that exhibits superior accuracy in correlating suspension power losses with sedimentation patterns in river sections. This formula's efficacy is further enhanced when considering aggregate suspension power dissipation under the influence of tributary inflows, enabling refined predictions of sedimentation variations along river segments. In conclusion, the insights derived from this meticulous research contribute significantly to the realm of river resource management and the pursuit of sustainable, high-quality aquatic ecosystem development.

Effects of Dietary Changes on Feline Obesity in Cats in the United States

Purpose: To aim of the study was to analyze the effects of dietary changes on feline obesity in cats in the United States. Methodology: This study adopted a desk methodology. A desk study research design is commonly known as secondary data collection. This is basically collecting data from existing resources preferably because of its low cost advantage as compared to a field research. Our current study looked into already published studies and reports as the data was easily accessed through online journals and libraries. Findings: Recent studies indicate that dietary changes play a crucial role in addressing feline obesity in the United States. Research shows that transitioning to high-protein, low-carbohydrate diets can significantly reduce weight gain and promote healthier body conditions in cats. Additionally, the incorporation of portion control and feeding schedules, alongside the use of specialized weight management foods, has been linked to effective weight loss in overweight felines. The findings also highlight the importance of increasing fiber content in cat diets, which aids in satiety and reduces overall calorie intake. Furthermore, educating pet owners about the impact of treats and the need for regular veterinary consultations can enhance dietary compliance and improve feline health outcomes. Unique Contribution to Theory, Practice and Policy: Energy balance theory, obesity hypoventilation syndrome (OHS) theory & behavioral theory of eating may be used to anchor future studies on the effects of dietary changes on feline obesity in cats in the United States. Practically, to address feline obesity, pet owners and veterinarians should adopt several evidence-based dietary strategies. From a policy perspective, several key actions are recommended to address feline obesity on a broader scale.

Analysing Double Shearing Mechanism in Fiberglass Rock Bolting Systems: a Comprehensive Analytical Model and Numerical Simulation Approach

An analytical method and numerical simulation were developed to investigate the shear performance of fiberglass rock bolts (20-tonne and 30-tonne) by conducting sixteen double-shearing tests with both clean and infilled shear interfaces. Following the preparation of the required samples, each test set-up was subjected to different ranges of pretension values. The infilled scenario involved 5 mm thick sandy clay infilled shear interfaces. The results of the double shearing tests unveiled that as pretension increased, so did the confining pressures at the shear interfaces for both clean and infilled joints. Also, an analytical model was developed utilising the Fourier transform, energy balance theory, and linear elastic theory. The result was an empirical relationship that could determine the double shear performance of fibreglass rock bolts in close agreement with the experimental data. Coefficients were incorporated to facilitate model calibration and tuning. Eventually, fast Lagrangian analysis of continua (FLAC) three-dimensional (3D) modelling was utilised to conduct numerical simulations of fibreglass rock bolts subjected to double shearing scenarios. The numerical model was calibrated against experimental data and then extended to conduct a sensitivity analysis on fibreglass rock bolts subjected to double shear test setup variations. Scenarios included rock bolt installation angles, shearing rates, and various host rock strengths. The results revealed that increasing the shear speed from the experimental test baseline yielded substantial displacement increases in the post-failure residual performance of the rock bolts. Changing the installation angle resulted in greater peak shear forces and extended residual zones. The least significant impacts were observed when changing the host rock UCS, suggesting neither rock bolt was drastically impacted by weak or strong host rocks.

A frost heave pressure model for fractured rocks subjected to repeated freeze-thaw deterioration

Rock engineering in the Tibetan Plateau is usually suffered severe freeze-thaw (F-T) deterioration, triggering numerous geotechnical engineering disasters. As the essential triggering factor of F-T damage, the frost heave pressure (FHP) in rocks should be quantitatively described and predicted. In this study, a novel theoretical model is established and validated to estimate the critical entire evolution process of FHP in fractured rocks subjected to repeated F-T deterioration, in which the influence of fracture geometric configuration, rock mechanical properties and F-T conditions are comprehensively considered. Besides, given the obvious five-stage manner of FHP evolution, the combination effect of different dominant mechanisms in different stages is theoretically addressed. During freezing, after a silence stage induced by temperature above freezing point of water, volumetric expansion and frost-heave cracking separately controls the generation and reduction stages of FHP, and the solutions are obtained based on elastic mechanics and Griffith energy balance theory, respectively. During thawing, ice segregation dominates the second-arising stage of FHP that is quantitatively described via segregation potential and water migration velocity, and then the FHP enters the dissipation stage owing to complete melting of fracture ice. In addition, a decay coefficient is introduced to estimate the influence of F-T cycle number on FHP, and a piecewise FHP model is finally constructed for fractured rocks subjected to repeated F-T deterioration. To validate our new model, a total of 12 F-T cycle tests with real-time FHP monitoring are conducted on granite and sandstone specimens considering three typical influencing factors, i.e., fracture width, rock types and freezing temperatures. A rational consistency is identified between the theoretical and testing results in terms of the FHP evolution curves and the peak FHPs in rock fracture, demonstrating the generalizability and robustness of our proposed model.

Persistent urban heat.

Urban surface and near-surface air temperatures are known to be often higher than their rural counterparts, a phenomenon now labeled as the urban heat island effect. However, whether the elevated urban temperatures are more persistent than rural temperatures at timescales commensurate to heat waves has not been addressed despite its importance for human health. Combining numerical simulations by a global climate model with a surface energy balance theory, it is demonstrated here that urban surface and near-surface air temperatures are significantly more persistent than their rural counterparts in cities dominated by impervious materials with large thermal inertia. Further use of these materials will result in even stronger urban temperature persistence, especially for tropical cities. The present findings help pinpoint mitigation strategies that can simultaneously ameliorate the larger magnitude and stronger persistence of urban temperatures.

Cumulative-impact method for the assessment of dynamic crush curve of crashworthy devices

Crashworthy devices are integral for protecting bridges from vessel collisions, typically evaluated through finite-element methods. However, experimental verification is challenging owing to the impracticality of using large-scale, high-velocity vessels. In this study, a cumulative-impact method is proposed, which is theoretically and experimentally validated. It uses a small-scale impactor in multiple impacts to determine crashworthy device dynamic crush curve based on energy balance theory. The accuracy of the proposed method depends on the number of impacts and the impact velocity. The relative errors are < 5% for the peak impact force and residual deformation, if the number of impacts is < 5 and the impact velocity is > 3 m/s. A parametric study explores application conditions for the cumulative-impact method through horizontal-impact tests, indicating that small-scale multi-impacts and large-scale single impacts have consistent dynamic crush curves. The proposed method is sufficiently accurate to be used to determine the safety assessment of bridges under vessel collisions.

Evolution of interfacial defects and energy losses during aging of organic photovoltaics

As the power conversion efficiencies of Organic Solar Cells (OSCs) approach 20 %, the stability of the device becomes an increasingly urgent issue. To enhance device stability, it is crucial to identify potential loss mechanisms. In this study, we investigated the trap-state-dependent degradation mechanism of OSCs by directly comparing devices with different hole transport layers (HTLs) that introduce distinct interfacial defect distributions. Employing electrochemical impedance spectroscopy (EIS), Fourier transform photocurrent spectroscopy (FTPS), electroluminescence quantum efficiency (EQEEL), and temperature-dependent J-V techniques, we unraveled the relationship between device degradation and interfacial trap states and energy loss in PM6:Y6 devices. A lower density of interfacial deep traps is evidently correlated with smaller non-radiative recombination losses during the aging of devices based on PEDOT:PSS. Conversely, a higher density of deep traps in aged devices with WS2 interlayers and HTL-free configurations is presumed to be responsible for a significant increase in non-radiative recombination losses. The escalating deep-trap-state density in aged devices is observed to elevate carrier recombination, consequently deteriorating device performance. Beyond the scope of the energy balance theory, an additional factor, probably attributed to the change in the work function of ITO, was found to contribute significantly to energy loss in aged cells, particularly in HTL-free devices. These results highlight the potential for improving device stability via interface engineering.

Energy balance support method in soft rock tunnel with energy absorbing anchor cable

Deformation and failure phenomena in rock mass after excavation can be described from the energy perspective, i.e. the work done by external forces increases the energy, thereby exceeding the energy storage limit of rock mass, resulting in the release of residual energy. Currently, energy-absorbing anchor supports have been used for stability control of surrounding rocks to avoid engineering failure by large deformations of soft rocks. However, the design of energy-absorbing anchor support still depends on field experience and engineering tests. In particular, under the influence of structural plane, the characteristics of energy transfer of surrounding rock with energy-absorbing anchorage are still unclear. In this paper, a design method of energy anchorage considering the influence of structural plane based on the theory of energy balance is proposed, which is corroborated by numerical simulation, similar simulation, and engineering application. The study demonstrates that under the effect of ground stresses, the arch support alone cannot absorb the deformation energy produced by the surrounding rock, leading to a significant region of the plastic zone and deformation instability. Under the control of ordinary anchor cable, the deformation of surrounding rock and the extension of the plastic zone are further alleviated; however, some anchor cable tensile fractures still occur. The energy-absorbing anchor cable scheme is created following the energy balancing concept. The surrounding rock deformation is maintained in an ideal range, while the plastic zone area is substantially reduced. The final engineering application also validates the accuracy of the energy-absorption anchorage design based on the energy balance theory, which provides a theoretical and practical basis for the application in similar projects.

UNIVERSAL HYDROGAS-DYNAMIC PATTERNS OF THE DRAINAGE ZONE OF OIL AND GAS WELLS

Large-scale experimental hydrogasodynamic studies of the flow of a two-phase gas-liquid mixture in a vertical pipeline were carried out. Based on the analysis of the results of these experimental studies, it was found that among the results obtained there are dependencies with theoretical bases by which theoretical hydrogasodynamic methods for studying multiphase flow in a vertical pipeline can be developed. The article provides a theoretical hydrogasodynamic study using dependencies describing a two-phase flow in a vertical pipeline of a round section, and it is confirmed by existing experiments. As a result of the gas-hydrodynamic study of pumping wells, in order to confirm the rational use of formation gas energy, a positive effect due to an increase in oil production rate and well process parameters, a relationship was established between well Q production rate, bottom-hole pressure of well P and production of formation gas V. Parameters can be modeled using three-term polynomials. During operation of wells in three modes (Qi, Vi or ∆Pi,Vi, where i = 1,2,3), processing of data obtained as a result of studies, indicators determined by existing research methods were obtained, and the technological mode ensuring layer-bylayer cooperation of the well system was selected.&#x0D; It is known that the motion of a two-phase flow of gas-liquid mixtures in a vertical pipe string is described on the basis of energy balance theory. The hydrogasodynamic characteristics of the drainage zone can be obtained in a real oil and gas well, as an experimental installation in the form of

The Relationship of Social Economic Status and Obesity Among Adolescents

This paper discusses the impact on adolescent obesity from the perspectives of subjective social economic status (SSES) and objective social economic status (OSES). It is found that both types of social economic statuses were negatively associated with obesity, which is mean that higher social economic status (SES) is a protective factor of obesity. By reviewing previous relevant literature, this paper further found that: (1) External environment (e.g., community) and personal SES can interacted predict the obesity; (2) Childhood social economic status may lead to obesity by affecting cognitive function; (3) Life history theory and energy balance theory can explain the association between subjective social economic status and obesity. There is a certain difference between SSES and OSES, and it is recommended both of them need to be used at the same time in one research. Further research is needed on the underlying mechanisms of social economic status and obesity, as well as longitudinal research and intervention experiments. This paper provides methods and reference for researchers to understand the association between obesity and two types of social economic status among adolescents.

Underwater propeller turbine blade redesign based on developed inverse design method for energy performance improvement and cavitation suppression

This paper conducts a systematic study to optimize the high-speed propeller turbine blade with high energy losses and severe cavitation. The trubine blade takes the role to provide the sustainable power for the underwater vehicle, whose structure is complex and crucial. To redesign the blade effectively, a developed inverse design method to generate various design schemes has been adopted with the union of the derived constrained zone and the one-dimensional theory. Established on the design method, a design procedure comprised of the CFD simulation, the LHS sampling, the multi-source constrained screening, the Kriging model, and the MOPSO algorithm was proposed. By setting energy performances (thrust, consuming power) and the cavity size as the objectives, the adaptive optimization of the runner blade installed on the underwater vehicle has been implemented, and two optimized models were finally designed out. Via further comparative analysis of the original and the redesigned blades, the maximum thrust increasement is 6.6%, the maximum power reduction is 3.1%, while the maximum cavity size reduction can be as much as 2.67%. Moreover, there exists a large structural difference for the runner blade before and after optimization, which could provide multiple options for further application. Finally, the specific inner flow analysis consisted of the performance calculation at different cruising speeds, and 3D analysis of key flows fields has been applied to reveal the reason to improve the performance and suppress the cavitation. Further, the energy balance theory was adopted to extract the specific losses at different hydraulic components, which demonstrates that the optimized propeller turbine blades can improve the energy performance. The study is expected to provide a technical reference for the energy performance improvement and cavitation suppression of the high-speed turbine machinery.

Chronic positive mass balance is the actual etiology of obesity: A living review

The fundamental cause of obesity is widely assumed to be an energy imbalance between calories consumed and calories expended (i.e., the energy balance theory). However, this century-old obesity paradigm is fallacious. According to known laws of physics, the actual etiology of obesity is chronic positive mass balance, not positive energy balance. Furthermore, the relevant physical law in body mass regulation is the Law of Conservation of Mass, not the Law of Conservation of Energy. It is important to understand that energy balance and mass balance are separate balances in the human body. Since calories simply represent the heat released on food oxidation, they have no impact on body mass. Body mass can only change as a result of net mass flow; thus, the only food property that can augment body mass is its nutrient mass, not its energy content. The recently proposed mass balance model describes the temporal evolution of body weight and body composition under a wide variety of feeding experiments, and it seems to provide a highly accurate description of the very best experimental human feeding data. By shifting to a mass balance paradigm of obesity, a deeper understanding of this condition may follow in the near future. The purpose of this living review is to present the core issues of the upcoming paradigm shift and some practical applications related to the subject.

Dynamic properties of the attachment oscillator arising in the nanophysics

Abstract The attachment oscillator, which plays an important role in the nanophysics such as nano/microelectromechanical systems, molecular devices, and nanofibers, is studied in this work. With the help of the semi-inverse method, the variational principle is established, and the Hamiltonian of the system is correspondingly constructed based on the obtained variational principle. Then, according to the principle of energy conservation, the energy balance theory is implemented to seek for the amplitude–frequency relationship. As predicted, the obtained solution has a good agreement with the existing results, which shows that the presented method is simple but effective, and is expected to provide a new idea for the study of the nonlinear oscillator arising in the nanophysics.

Estimating Hourly All-Weather Land Surface Temperature From FY-4A/AGRI Imagery Using the Surface Energy Balance Theory

Thermal infrared (TIR) observations from geostationary satellites enable the retrievals of the diurnal variations in land surface temperatures (LST) linked to the land-atmosphere energy exchange and water cycle. However, cloud cover obstructs TIR signals and leads to missing gaps that generally account for more than 50% of TIR LST maps. This study proposed an effective method to estimate hourly all-weather LST from the Advanced Geosynchronous Radiation Imager (AGRI) data under the framework of surface energy balance (SEB) theory. An improved temperature and emissivity separation algorithm was first used to obtain the high-quality clear-sky LST, which plays a decisive role in ensuring the accuracy of recovered cloudy-sky LST. Then we proposed a unique way to solve the temperature difference (ΔLST) between the cloudy-sky LST and hypothetical clear-sky LST caused by cloud radiative effects (CRE). The bias (RMSE) of the estimated AGRI hourly cloudy-sky LST is 0.10 K (3.71 K) during the daytime, and -0.20 K (2.73 K) during the nighttime, respectively. The overall bias (RMSE) of the estimated AGRI all-weather LST is 0.02 K (2.84 K). The estimated hourly all-weather LST not only captures the rapid variation in diurnal LST but is also promising for temporal upscaling. The temporally upscaled daily mean LST shows a bias (RMSE) of 0.03 K (1.35 K). This study provides a promising solution to generate diurnal hourly all-weather LST for AGRI and other geostationary satellites.

SIMPLIFIED ASSESSMENT OF EARTHQUAKE DAMAGE RATIO FOR LOW- AND MID-RISE BUILDINGS BASED ON ENERGY BALANCE CONSIDERING SOIL-STRUCTURE INTERACTION

In order to easily assess the damage caused by earthquakes in low- and mid-rise building complexes, this paper proposes a simplified method for assessing the damage ratio in real building complexes by combining a seismic evaluation method based on the energy balance and reliability theory. This method enables the effect of dissipation damping to be taken into account, and is effective for the evaluation of the largest foreshocks and aftershocks in addition to the main shock. A comparison of the damage ratio calculated using this method with the actual damage ratio from the 1995 Hyogo-ken Nanbu Earthquake showed that the two ratios generally matched with each other. In terms of applications, this method provides a standard for calculating the base shear coefficient of low-rise reinforced concrete buildings for continued use after an earthquake from the viewpoint of performance design.

A physical method for downscaling land surface temperatures using surface energy balance theory

Fine-resolution land surface temperature (LST) derived from thermal infrared remote sensing images is a good indicator of surface water status and plays an essential role in the exchange of energy and water between land and atmosphere. A physical surface energy balance (SEB)-based LST downscaling method (DTsEB) is developed to downscale coarse remotely sensed thermal infrared LST products with fine-resolution visible and near-infrared data. The DTsEB method is advantageous for its ability to mechanically interrelate surface variables contributing to the spatial variation of LST, to quantitatively weigh the contributions of each related variable within a physical framework, and to efficaciously avoid the subjective selection of scaling factors and the establishment of statistical regression relationships. The applicability of the DTsEB method was tested by downscaling 12 scenes of 990 m Moderate Resolution Imaging Spectroradiometer (MODIS) and aggregated Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) LST products to 90 m resolution at six overpass times between 2005 and 2015 over three 9.9 km by 9.9 km cropland (mixed by grass, tree, and built-up land) study areas. Three typical LST downscaling methods, namely the widely applied TsHARP, the later developed least median square regression downscaling (LMS) and the geographically weighted regression (GWR), were introduced for intercomparison. The results showed that the DTsEB method could more effectively reconstruct the subpixel spatial variations in LST within the coarse-resolution pixels and achieve a better downscaling accuracy than the TsHARP, LMS and GWR methods. The DTsEB method yielded, on average, root mean square errors (RMSEs) of 2.01 K and 1.42 K when applied to the MODIS datasets and aggregated ASTER datasets, respectively, which were lower than those obtained with the TsHARP method, with average RMSEs of 2.41 K and 1.71 K, the LMS method, with average RMSEs of 2.35 K and 1.63 K, and the GWR method, with average RMSEs of 2.38 K and 1.64 K, respectively. The contributions of the related surface variables to the subpixel spatial variation in the LST varied both spatially and temporally and were different from each other. In summary, the DTsEB method was demonstrated to outperform the TsHARP, LMS, and GWR methods and could be used as a good alternative for downscaling LST products from coarse to fine resolution with high robustness and accuracy.

Integration of flux footprint and physical mechanism into convolutional neural network model for enhanced simulation of urban evapotranspiration

Estimating urban evapotranspiration (ET) is of great significance for urban water resource allocation and assessing the urban heat island effect. However, most current urban ET models are based on the energy balance theory to estimate urban ET. These models lead to significant errors in urban ET simulation due to the surface heterogeneity and the existence of anthropogenic heat fluxes in urban areas. To solve this issue, this study proposes a modified machine learning-based urban ET method that can estimate urban ET at the site and regional scales. To better characterize the heterogeneity of urban surfaces, the flux footprint of in-situ ET and physical mechanism of ET process are integrated into the convolutional neural network (CNN) model. The modified CNN model is tested in a fast-developing city: Shenzhen, China based on two Eddy Correlation (EC) observations. The verification results indicated that coupling flux footprint and physical mechanism into the CNN model could effectively improve the accuracy of urban ET simulation at the site scale. The modified CNN model significantly reduced the root-mean-square-error (RMSE) of 25.8 W/m2 and increased the determination coefficient (R2) of 0.17 compared to the CNN-O model (The CNN-O model is defined as the CNN model do not integrate flux footprint and physical mechanism of ET). Further analyses suggested that fusing flux footprint data into a machine learning model helps enhance ET estimation in regions with high heterogeneity and highly variable wind directions. Moreover, the integration of physical mechanisms significantly enhanced the model capability to simulate extreme ET events. The modified CNN model is further applied to map the spatial distribution of urban ET and reconstruct long-term urban ET changes. The spatial pattern of urban ET exhibited large spatial variability, where the urban ET in water bodies (mean λET larger than 480 W/m2) and vegetation-covered areas (mean λET larger than 260 W/m2) are substantially higher than the impervious surfaces (mean λET less than 30 W/m2). Long-term trend analyses demonstrated that urbanization resulted in decline in urban ET. The average decreasing rate of urban ET is 1.61 mm/yr (P < 0.05), with a 18 % decrease relative to the long-term ET average. The leading causes for the decline of urban ET are the increased impervious surfaces and the decreased radiation. This study improved the simulation accuracy of urban ET and revealed the response of urban ET to urbanization.

Complementarity Characteristics of Actual and Potential Evapotranspiration and Spatiotemporal Changes in Evapotranspiration Drought Index over Ningxia in the Upper Reaches of the Yellow River in China

Based on energy balance theory, using Theil–Sen median trend analysis and the Mann–Kendall test, this research studied the applicability of the complementary theory of evapotranspiration (ET) over Ningxia in the Upper Reaches of the Yellow River with MOD16 ET product and the measured data of meteorological stations, based on which ET drought index (EDI) was proposed for the first time. Moreover, the usability of EDI was also verified and its influencing factors were analyzed. The results revealed that there was a complementary relationship between AET and PET in 91.1% of the area in Ningxia, including strictly complementary and asymmetrically complementary relationships in 69.2% and 21.9% of the total area, respectively. EDI ranged from 0 to 1 and was useful to accurately reflect the degree of drought of the study area on the annual and monthly scales. From 2001 to 2020, the average annual EDI was 0.66, and the smallest monthly EDI was in January and the largest was in May. EDI of different time scales had different influencing factors. Precipitation was the most influencing factor of annual EDI, but the influencing factors of monthly EDI was different over time. However, surface non-precipitation water replenishment, such as irrigation, had great impact both on annual EDI and monthly EDI. The application scope of the theory of ET complementarity was extended to the study area for the first time, and EDI was proposed and applied, which will provide a theoretical basis and empirical reference for drought research based on ET data in arid and semi-arid areas.

PERIODIC WAVE STRUCTURE OF THE FRACTAL GENERALIZED FOURTH-ORDER BOUSSINESQ EQUATION TRAVELING ALONG THE NON-SMOOTH BOUNDARY

In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.