Governing Equations Research Articles

Mathematical Modelling of the Removal of Basic Blue Dye from Effluent by a Mineral Membrane

Water contamination is a major challenge due to the discharging of wastes into the natural resources. In this study, a novel branched pore adsorption model was developed by utilizing external mass transfer coefficients, Kf , effective diffusivity, and lumped microspore diffusion rate parameters. This describes sorption kinetics from long-term adsorption that occurs between an adsorbent and an adsorbate. An investigation of the effects of dye concentration on Basic Blue 9 removal was conducted. Then by considering of initial and boundary conditions the government equations were solved. Using this model, the Basic Blue 9 (or methylene blue is an industrial inhibitor) was removed from a bentonite sample polluted by Basic Blue 9. The maximum adsorption capacity was achieved by using of Freundlich analysis. Based on the experimental data, theoretical concentration - time adjusted models were developed using a FORTRAN computer program to fit the experimental data more precisely and confirm the precision and accuracy of this research.

Multidimensional Governing Equations of Matrix Flow Component of Subsurface Stormflow as Function of Bedrock Surface Geometry

AbstractSubsurface flow is a critical component in the hydrological cycle, since it controls the quantity and timing of surface runoff and groundwater flow. Field studies have shown the fundamental influence of the bedrock surface geometry on subsurface stormflow (SSSF) and that the SSSF process consists of at least two major components: the matrix flow component and the macropore flow component that are in dynamical interaction toward forming the SSSF. Furthermore, field studies have shown that the bedrock surface that underlies the SSSF has essentially a two‐dimensional (2D) geometry, where not only the longitudinal profile along the hillslope but also the profile in the transverse direction influence the subsurface stormflow over the bedrock. The macropore flow itself being a multidimensional flow process, requires a multidimensional matrix flow component in order to quantify realistic dynamical interactions between these components within the subsurface stormflow module of a watershed hydrology model. Within this framework, the critical research question is how to extend the existing 2D Boussinesq model for the matrix flow over an inclined flat 2D surface in the longitudinal hillslope direction by developing the governing equations of matrix flow at hillslope scale over various 2D bedrock surface geometries that are reported in the field studies of SSSF. Then, numerical experiments are performed by applying the developed governing equations to various bedrock surface geometries. The results of the numerical simulations show (a) the proposed governing equations can simulate various two‐dimensional bedrock surface geometries, and are capable of incorporating the upward or downward concavity of bedrock surface in longitudinal and transverse directions into SSSF by modulating the timing and magnitude of the flows and (b) the proposed governing equations for matrix flow provide an appropriate setting for the dynamical interaction of the matrix flows with the macropore flows, creating a promising modeling framework for SSSF.

On Derivations of Basic Governing Equations for Solid Bodies in Bi-spherical Coordinate System

Traditionally solid mechanics problems are formulated in Cartesian, cylindrical and spherical coordinate system. Using such formulation and coordinate system, solutions of solid mechanics problems are obtained for specific geometries such as straight boundary, circular, cylindrical and spherical boundaries. However, such available coordinate system cannot describe many geometries in spherical or cylindrical coordinate system with inclusion of eccentricities, two spherical or cylindrical bodies in contact and parallel. In this article, author address this issue by giving complete original formulations and derivations of basic governing partial differential equations in Bi-spherical coordinate system. Bi-spherical coordinate system allow us to solve problems such as eccentric spheres, two parallel spheres (intersecting or non-intersecting) in solid mechanics which traditional spherical coordinate system cannot handle. Author develop the original equations in Bi-spherical coordinates in terms of useful quantities of stresses and strains components in three dimensions. The paper is limited to basic formulations and practical applications of such formulations can easily be expanded for getting solutions using any known analytical or numerical methods.

Technology Focus: Hydraulic Fracturing Modeling (November 2022)

Hydraulic fracture stimulation is proven to be the economic way to develop tight unconventional reservoirs. From an operational perspective, hydraulic stimulation is well established and widely applied to most unconventional plays. There is still room for improvement, however, to increase the efficiency and deepen the understanding of the detailed mechanism in the process. Particularly in the domain of fracture simulation, significant effort has been devoted to enhance model complexity and accuracy. Recent simulation studies have covered sophisticated mechanisms such as hydraulic fractures—natural fracture interaction, proppant transportation and settlement, casing friction, new government equations of fracture propagation, and fracture roughness. Fully coupled fracture modeling work flows with stress variation, solid transportation, and fluid flow become the mainstream, while model complexity and budget tradeoff remains challenging. Interestingly, fracture modeling has been applied beyond hydrocarbon production. New applications include cutting injection, coal-seam-gas production, and recently evolving enhanced geothermal systems (EGS). Both industrial and academic researchers have made progress on EGS by introducing another dimension of complexity in that thermal effects alter hydraulic fracture modeling behavior. With the advancement of powerful measurement techniques in the past decade, the industry has seen novel ways to calibrate fracture models. Other than microseismicity and pumping-schedule history matching, data collected from downhole tools such as distributed temperature and acoustic sensing and fiber-optic sensors have broader applications and show promising results. By leveraging computational power and machine-learning approaches, scaled-up modeling on the field level energizes the study of well interference, the depletion effect, and frac-hit-damage mechanisms. Recommended additional reading at OnePetro: www.onepetro.org SPE 205896 - Offshore Cuttings Reinjection Well-Performance Diagnostics and Fractured Domain Mapping Using Injection Data Analytics and Hydraulic Fracturing Simulation, Verified Through 4D Seismic and Wireline Logging by Franz Marketz, Sakhalin Energy Investment Company, et al. URTeC 2021-5414 - Modeling of Distributed Strain Sensing and Distributed Acoustic Sensing Incorporating Hydraulic and Natural Fractures Interaction by Kildare George Ramos Gurjao, Texas A&M University, et al. URTeC 2021-5526 - Efficient Modeling of Enhanced Geothermal System With 3D Complex Hydraulic and Natural Fractures by Hongbing Xie, Sim Tech, et al.

Analysis of Radiation Creep Problems Considering Pore Growth Ductile Fracture by Rice–Tracey–Huang Models and Kachanov’s Solution for a Spherical Cavity. Part 1. Initial Preconditions and Models for the Formulation of the Governing Equations

Analysis of Radiation Creep Problems Considering Pore Growth Ductile Fracture by Rice–Tracey–Huang Models and Kachanov’s Solution for a Spherical Cavity. Part 1. Initial Preconditions and Models for the Formulation of the Governing Equations

Specialized Local Government and Water Conservation Policy in the United States

Special districts are an increasingly important part of the local government equation in the United States, representing over forty percent of local governments. The spread of these governments is controversial, however, as some argue that they will have a negative impact on service delivery, due to a perceived lack of political accountability. Others argue that their focus on single policy issues allow them to more efficiently respond to the citizens they serve. Despite the controversy, only a few studies have quantitatively investigated the differences in service delivery between special district and general purpose governments. Building on Mullin's earlier work, in this research note we investigate the relationship between specialized local government and water utility rates. We find little direct difference between special districts and general-purpose governments, with some minimal support for a conditional relationship between special districts and scarcity.

The unsteady nonlinear convective flow of a nanofluid due to impulsive motion: Triple diffusion and magnetic effects

• The unsteady nonlinear convection due to impulsive motion is investigated. • Governing equations have been transformed using Mangler's transformations. • Quasi-linearization technique is utilized for solution of the problem. • The surface drag coefficient increases with magnetic parameter. • Heat transport rate rises only when the Eckert number is positive. By a semi-similar approach, the paper aims to study the unsteady nonlinear convection due to impulsive motion with triple diffusion and magnetic effects. Since the mainstream has been set into impulsive motion, the fluid motion is unsteady. Simultaneously, the temperature of the surface is abruptly changed from its ambient fluid temperature. The influence of chemical reactions and convective boundary constraints are also considered in this investigation. The nonlinear coupled partial differential equations (PDEs) with appropriate boundary conditions that govern the given problem are transformed into dimensionless equations by using Mangler's transformations. For mathematical computations, the technique of Quasi-linearization and second order implicit finite difference method are utilized, and so obtained results are semi-similar solutions that are discussed with the aid of graphs. The results reveal that for the case of aiding buoyancy, the surface drag coefficient increases about 5% when the nonlinear convection parameter increases from 0 to 3 and about 61% when the magnetic parameter increases from 0 to 0.5. The heat transfer rate enhances about 65% for variation of the magnetic parameter from 0 to 0.3 for a positive Eckert number. The mass transfer rate is pronounced more for linear chemical reaction than the nonlinear chemical reaction for the constructive chemical reaction parameter. It is found at about 7% for Schmidt number 240. The current findings are compared to previously reported literature and are found to be in good agreement.

Modeling of slurry-type photocatalytic reactors containing core-shell particles for predicting transient behaviours based on Langmuir-Hinshelwood kinetics

Governing equations were derived for photocatalytic batch reactors and CSTR (Continuously Stirred Tank Reactor) containing photocatalytic pellets to obtain numerical solutions of the concentration in bulk medium and intra-particle concentration based on Langmuir-Hinshelwood kinetics. Numerical solutions could be obtained by finite element method under proper initial and boundary conditions to solve coupled reaction-diffusion equations using partial and ordinary differential equations. Effects of Thiele modulus, Biot number, Langmuir—Hinshelwood parameter and photocatalyst loading were investigated by predicting the performance of batch reactor. Effects of residence time and initial reactant concentration were studied for CSTR. In addition to conventional morphologies of pellets such as sphere, cylinder and slab, mathematical modeling of core-shell particles was also conducted to study the effect of thickness of inert core. Although the performance of spherical pellets was the best, cylindrical pellets also showed good activity for removing reactant by photocatalytic decomposition. In core-shell spherical pellets, there was little effect of thickness of inert core on the reduction rate of reactant concentration and steady-state concentration in batch and CSTR, respectively, compared to core-shell cylindrical and slab-type particles. The cascade connection of photocatalytic CSTRs and fixed bed reactors with back mixing could be also analysed by finite element method. Experimental data about the removal of methylene blue using silica-titania core-shell particles in batch photocatalytic reactor were compared with numerical solutions to estimate reaction parameters.

THE NUMERICAL MESHLESS APPROACH FOR SOLVING THE 2D TIME NONLINEAR MULTI-TERM FRACTIONAL CABLE EQUATION IN COMPLEX GEOMETRIES

The cable equation plays a prominent role in biological neuron models, for instance, in spiking neuron models and electrophysiology. Thus, the current investigation scrutinizes the 2D time nonlinear multi-term fractional cable equation. By adopting a valid meshfree technique, the nonlinear multi-term time-fractional cable equations (NM-TTFCEs) that consist of government equations and their boundary conditions are transformed into the boundary value problems. For this purpose, the finite difference method is derived for temporal discretization such that the considered NM-TTFCEs can be transformed into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. The dual reciprocity method (DRM) is implemented to obtain a particular solution and the improved singular boundary method (ISBM) is employed to evaluate the homogeneous solution. Moreover, we apply the meshless method for solving two-dimensional NM-TTFCEs on regular and irregular distribution points with several computational domains. The numerical results vouch for the accuracy and high efficiency of the proposed method. Finally, we will conclude that the ISBM/DRM method can be considered a potential alternative to existing meshless strong form approaches in solving multi-term fractional equation problems with complex geometries.

Genesis of Antibiotic Resistance (AR) LXXXVIII: Turbulence Modeling(TM) of Hemodynamics in Simplified Severe Sepsis Protocol‐2 (SSSP‐2)‐NCT01663701: “Hypnotic Trance Ameliorate Turbulence to Laminar Flow (Homeostasis)”: A Neuropsychiatric Basis

An effort is made to substantiate that vascular abnormalities (fluid dynamic/mechanical properties) could be restored to homeostasis by an appropriate hypnotic trance. Clinical Background:Sepsis: Clinical data/Record(observation): decreased capillary density ref Fig 4 c., PMID: 34055944, hypovolemia, endothelial dysfunction, increased WBC accumulation, thrombi deposition, altered CCP, NFP gradient, shunting, degradation of endothelial glycocalyx, vascular hyper permeability (PMID: 23566729; PMID: 20179590; PMID: 24067428), as an indicator of sepsis progress to septic shock. Neuropsychiatry diagnostic index: HAND (HIV associate neurodegenerative disorder), PTSD, psychosis, depression, mania, ANI, MND, HIV associated dementia (HAD), possible multifocal leukoencephalopathy (CMV origin, Toxoplasmaencephalitis, diffused myelin pallor, opportunistic infections, and neurosyphilis (Jarisch Herxheimer reaction). Geometry: Part I: RBC, Platelets, WBC: p47‐56; Part II: blood velocity, and function, Table 2.1; 2,2; 2.3; p63; ISBN 9780128024089; Governing Equations: Part A: CD, https://www.grc.nasa.gov/www/k‐12/airplane/shaped.html;; Part B: Keulegan – Carpenter Number equation: □=Vm/ƒoscD; Clinical Data: Hemodynamics Hypnotic Trance:Before hypnotic induction: 120.65 mmHg / 79.84 mmHg (Derived →PP= 41 mmHg Normal;Derived →MAP=93mmHg); After hypnotic induction: 115.92 mmHg /76.16 mmHg; (Derived →PP= 40 mmHg Normal;Derived →MAP= 89 mmHG)).Heart rate before hypnotic induction: 86.16 / beats per minute (Derived →CO= 86.16/ beats per minute x 70ml/min=6029.8 ml/min); After hypnotic induction, to 78.89 beats per minute (Derived →CO=78.89/ beats per minute x 70 ml/min=5522.3ml/min). (Kurnia S, et.al., (2021). Blood Pressure Control and Heart Rate: Effectiveness Brief Hypnotic Induction Methods on Adults. 10.2991/assehr.k.210423.004). Spirometry (Homeostasis): VT: 500 ml; IRV: 3100 ml; ERV: 1200 ml; VC: 4800 ml; RV: 1200 ml; A respiratory rate of 12 breaths/ minute, the minute ventilation = 12 x 500 = 6000 ml. alveoli alveolar ventilation (volume of air/min that actually reaches the alveoli) = 12 x 350 = 4200ml. Clinical correlation:The basis for correlating the sepsis, neurological(physiological/psychiatric) and hypnotic trance is that the hypnotic trance subject recovers to homeostasis provided abreaction not encountered by the subject as much as the sepsis, neurological ‐ physiological/psychiatric patients with appropriate Dx, and Tx. Taken together, it is suggested that a decrease in capillary density observed in sepsis and en route to septic shock is a transient phenomenon implicit of a reversible loss of hemodynamic function, for patients Txwith SSSP‐ERP‐EGDT similar to that of SSSP‐2 NCT0166370. Such patient cohort could possibly be reverted to sustain homeostasis in a patient‐specific hypnotic suggestion(s), re‐alert concluding with a reassuring‐post hypnotic trans conversation (devoid of abreaction).

Dynamic response of a transversely isotropic and multilayered poroelastic medium subjected to a moving load

The dynamic response of a transversely isotropic and multilayered poroelastic medium over an impermeable rigid base due to a surface moving strip load with constant speed is investigated. First, the moving load and induced response of any field quantity are expanded in the form of Fourier series. Then, based on the Biot's poroelastodynamic theory, the governing equations in terms of the displacement and pore pressure components are decoupled via a potential-function method, and the corresponding general solutions in each layer are derived. A recursive relation for the field quantities is established by utilizing the powerful dual variable and position method along with the continuity conditions on the layer interface. By virtue of the boundary conditions, semi-analytical solutions for the transversely isotropic and multilayered poroelastic medium with different surface hydraulic conditions are finally derived. After verifying the accuracy of the developed solutions by comparison with existing solutions, the effects of surface hydraulic condition, material anisotropy, layer thickness, material layering and load speed on the dynamic response are analyzed in detail. Results show that the effects of material anisotropy and layering are significant for the accurate assessment of the dynamic response induced by the moving strip load over a transversely isotropic and poroelastic medium. • Analytical method for anisotropic layered poroelastic media under moving load. • Governing equations in terms of Fourier expansion and potential-function method. • Layered poroelastic media via the stable dual variable and position method. • Effect of anisotropy, layering, and loading speed on the dynamic response.

Anomaly Format of Atmospheric Governing Equations with Climate as a Reference Atmosphere

To reduce numerical instability and increase forecast accuracy of a numerical weather prediction (NWP) model, one approach is to subtract a reference atmosphere from atmospheric governing equations. In the past, scientists have proposed one-dimensional, two-dimensional, and three-dimensional static (in time) reference atmospheres with respect to temperature and pressure. These three reference atmospheres were first reviewed, and their corresponding perturbation equations were derived. Then, a new four-dimensional (space and time) all-variable (temperature, pressure, wind, moisture, etc.) reference atmosphere was defined using observed climatic states. Unlike the previous three approaches, the perturbations derived from this new method are actual anomalies relative to climate and directly a part of individual weather systems in both structure and strength. By subtracting climatic states, anomaly equations were derived and analyzed. Finally, the benefits and challenges of the anomaly-equation-based NWP model were discussed. Theoretically, an anomaly model should reduce model systematic errors (bias) and should avoid model climate drift to significantly enhance a model’s performance. An example of tropical cyclone track forecasts using the Beta advection model (vorticity) was demonstrated. The separation of model physics into climatic and anomalous physics is a significant challenge if a pure anomaly-equation-based NWP model is desired. Fortunately, a model including both anomaly and climatic equations should work with current full physics. In an anomaly climate mixed model, the anomaly part needs to be predicted and the climate parts are precalculated constants. It is hoped that this study will inspire model developers to explore the approach, which could be a possible new direction in developing next-generation NWP models. A high-resolution reanalysis is also key to the success of this new approach.

Governing the Corporation: Regulations in the Era of Scandals and Globalization

Purpose: This study evaluates the effects of the global financial crisis rooted in ethical deterioration. However, the worldwide regulatory reform agenda has largely overlooked this factor and focused on the technical needs. Also, look at the relative influence of various players within the corporate governance and regulatory equation to determine the shape that enforcement takes.
 Design/Methodology/Approach: The qualitative method is used in this study by analyzing statutes, local and international protocols, conventions, and treaties.
 Findings: In this work, the magnitude of the global financial disaster has shown how erroneous market ordering optimism. The tangible and conceptual certainty linked with finance capitalism's primacy has vanished. This has created a debate regarding the privileged and the general public. The critical question is how to respond. Rules are too readily transacted around, and principles lack the granularity to be enforceable to address ethical flaws.
 Implications/Originality/Value: We must pay considerably greater attention to the moral dimension of market activity or how people live their lives professionally. It is a chore we should avoid at all costs.

Air-lift Reactor's Characterization via Computational Fluid Dynamic (CFD): Review

Airlift reactors are seen as the most promising reactor for many valuable productions such as algae culturing. However, this kind of reactor still needs more information and data to understand its phenomena due to limited studies. Also, to reduce the time and offers obtained with sufficient reactor design, capable of achieving high productivities, Computational Fluid Dynamics (CFD) could play an important role in optimizing the reactor design by analyzing the interaction of hydro-dynamics. This review presents the literature review on the recent CFD work for such a reactor that addressed the fluid dynamics parameters, such as bubble dynamics. Earlier researches find more reports utilizing uniform bubble diameter in CFD simulations. However, the latest research in the CFD modeling of multi-phase flow reactors showed that the description of the bubble has significant effects on the performance of the simulation. As a result, systematic research into the impact of bubble diameter on the simulation results of the CFD was performed. Finally, we present and discuss the CFD modeling approaches, a Governing equation such as Eulerian-Eulerian (E-E), and closure such as the drag force.

Effectiveness and Productivity Improvement of Conventional Pultrusion Processes.

Pultrusion is a technological process in which fibers impregnated with resin move through the heated die and solidify into a composite profile with a constant cross section, as in the metallic die. The effectiveness and productivity of conventional pultrusion processes, preserving the quality of pultruded profiles, could be improved by process optimization or by the application of new, effective heating sources instead of electrical resistances with high heat losses. Due to the large dimension of the numerical problem and multiple iterations applied for the solution of government equations, an optimization methodology was developed, using the method of experimental design and the response surface technique. To develop microwave-assisted pultrusion processes, as well as pultrusion tooling design and process control, new effective electromagnetic-thermo-chemical finite element models and algorithms were developed by using general-purpose finite element software that results in considerable savings in development time and costs and makes available various modeling features of the finite element packages. The effectiveness and productivity of the optimized conventional pultrusion processes and the developed microwave-assisted pultrusion processes are estimated in comparison with the real pultrusion processes used in laboratory and industrial shops.

Uncertainty Analysis and Experimental Validation of Identifying the Governing Equation of an Oscillator Using Sparse Regression

In recent years, the rapid growth of computing technology has enabled identifying mathematical models for vibration systems using measurement data instead of domain knowledge. Within this category, the method Sparse Identification of Nonlinear Dynamical Systems (SINDy) shows potential for interpretable identification. Therefore, in this work, a procedure of system identification based on the SINDy framework is developed and validated on a single-mass oscillator. To estimate the parameters in the SINDy model, two sparse regression methods are discussed. Compared with the Least Squares method with Sequential Threshold (LSST), which is the original estimation method from SINDy, the Least Squares method Post-LASSO (LSPL) shows better performance in numerical Monte Carlo Simulations (MCSs) of a single-mass oscillator in terms of sparseness, convergence, identified eigenfrequency, and coefficient of determination. Furthermore, the developed method SINDy-LSPL was successfully implemented with real measurement data of a single-mass oscillator with known theoretical parameters. The identified parameters using a sweep signal as excitation are more consistent and accurate than those identified using impulse excitation. In both cases, there exists a dependency of the identified parameter on the excitation amplitude that should be investigated in further research.

Numerical simulation of reynolds number effect of a free incompressible isothermal turbulent coaxial jet

This paper studies the effect of Reynolds number on a two-dimensional free incompressible isothermal coaxial turbulent jet over a range of high Reynolds numbers. This is necessary because of its application in noise control and mixing. The Reynolds numbers at the nozzle exit were 9824, 19648, 29472, 39296 and 49120. The models were designed in ANSYS Design Modeler and the numerical simulation was done using a finite volume based Computational Fluid Dynamics (CFD) in ANSYS FLUENT using the two-dimensional Realizable turbulence model. The Governing equations were discretized using the finite volume method with the solution based on the PISO algorithm. The decay of centerline velocity, turbulent kinetic energy profile, the radial profile of axial velocity and similarity profile were investigated along the flow direction. Contour plot indicates that the velocity is high at the jet exit and decreases downstream due to the rapid mixing of the inner and outer jet and the surrounding fluid. It is found generally that Reynolds number plays significant role especially before self-similarity region. The result shows that increasing the Reynolds number give rise to more turbulence which in turn decreases the potential core length, turbulent kinetic energy and enhances the mixing of the fluid. However, at the jet exit, the flow with the lowest Reynolds number has the highest turbulent kinetic energy because it suffers the greater shear. The spreading of the jet was more or less independent of the Reynolds number beyond the self-similarity region. It is also found that the velocity profile is brought to congruence at about z/D=25 for the Reynolds numbers considered

A Unified Framework for Governing Equations of Hydrologic Flows

AbstractLaws of conservation of mass, momentum, and energy lead, respectively, to the equations of continuity, momentum, and energy, which are used to mathematically represent hydrologic flow syste...

A Toolkit for Data-Driven Discovery of Governing Equations in High-Noise Regimes

We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sparse identification of nonlinear dynamics</i> (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\dot { \boldsymbol x} = { \boldsymbol f} ({ \boldsymbol x})$ </tex-math></inline-formula> . First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\dot { \boldsymbol {x}}$ </tex-math></inline-formula> . This toolkit includes: (regression step) weight timepoints based on estimated noise, use ensembles to estimate coefficients, and regress using FFTs; (culling step) leverage linear dependence of functionals, and restore and protect culled functionals based on Figures of Merit (FoMs). In a novel Assessment step, we define FoMs that compare model predictions to the original time-series (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${ \boldsymbol x}(t)$ </tex-math></inline-formula> rather than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\dot { \boldsymbol {x}}(t)$ </tex-math></inline-formula> ). These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g., 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1%−3% (for 50% noise), 6%−8% (for 100% noise), and 23%−25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\dot { \boldsymbol x} = { \boldsymbol f} ({ \boldsymbol x})$ </tex-math></inline-formula> , to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model’s accuracy and thus a discovery method’s effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">true</i> model, enabling more accurate assessment of a discovered model’s correctness.

Development and Application of an Interactive Coupling Rainfall-Runoff Model According to Soil Texture

In this study, we developed a model that simulates surface and subsurface flows for rainfall-runoff analyses using an interactive coupling method. To characterize the interaction between the surface and subsurface flows, we studied the coupling analysis model. The proposed model was designed following comparisons with existing models. For the analysis of the surface and subsurface flows, a governance equation was constructed. The goodness of fit of the model was also tested. To examine the sensitivity of the input parameters, simulations were performed while changing the major parameters of the model according to the soil texture. The developed model showed high applicability to actual watersheds after adjustment of the parameters. This model can be applied to the extension module for channel analysis; therefore, it can be used efficiently in urbanized watersheds wherein the upstream and downstream parts are pervious and impervious watersheds, respectively.