Polynomial Function Research Articles

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Multi-Year Cropland Mapping Based on Remote Sensing Data: A Case Study for the Khabarovsk Territory, Russia

Cropland mapping using remote sensing data is the basis for effective crop monitoring, crop rotation control, and the detection of irrational land use. Classification using Normalized Difference Vegetation Index (NDVI) time series from multi-year data requires additional time costs, especially when sentinel data are sparse. Approximation by nonlinear functions was proposed to solve this problem. Time series of weekly NDVI composites were plotted using multispectral Sentinel-2 (Level-2A) images at a resolution of 10 m for sites in Khabarovsk District from April to October in the years 2021 and 2022. Missing values due to the lack of suitable images for analysis were recovered using cubic polynomial, Fourier series, and double sinusoidal function approximation. The classes that were considered included crops, namely, soybean, buckwheat, oat, and perennial grasses, and fallow. The mean absolute percentage error (MAPE) of each class fitting was calculated. It was found that Fourier series fitting showed the highest accuracy, with a mean error of 8.2%. Different classifiers, such as the support vector machine (SVM), random forest (RF), and gradient boosting (GB), were comparatively evaluated. The overall accuracy (OA) for the site pixels during the cross-validation (Fourier series restored) was 67.3%, 87.2%, and 85.9% for the SVM, RF, and GB classifiers, respectively. Thus, it was established that the best result in terms of combined accuracy, performance, and limitations in cropland mapping was achieved by composite construction using Fourier series and machine learning using GB. Similar results should be expected in regions with similar cropland structures and crop phenological cycles, including other regions of the Far East.

Computations on fall of the leaning tower with considering air resistance

The free fall of a sphere was studied by considering air buoyancy and resistance. After selecting the reasonable drag coefficient formula recommended by the literature, partial differential formulas on the motion of balls falling in the Leaning Tower of Pisa are solved. The variation process of acceleration, velocity and displacement over time during the falling process of two spheres is obtained. The research results indicate that the kinematics of free fall considering air resistance is different from that neglecting air resistance. Air Resistance must be considered in the free fall of the solid ball after 0.3005 seconds. In the free fall of the leaning tower, air resistance makes the solid ball land at 0.4418 seconds which is earlier than the hollow. The variation of the acceleration of a solid ball with time can be described by a second-order function a=-0.115t2 -0.022t+9.801. Velocity does not satisfy the product of acceleration and time but can be described as a polynomial function of the velocity variation with time v = c1t2 + c2t + c3 . About the relationship between displacement and time, for a solid sphere, displacement is proportional to 1.9526 power of time h=5.4945t1.9526 , while for a hollow sphere, it cannot be expressed by a function. The relationship between air resistance and velocity during the falling process of an iron ball satisfies a polynomial function Fd =c1v2 -c2v+c3 rather than a simple relationship where air resistance is directly proportional to the first or second power of speed, which was used in many papers as a reasonable assumption.

Analytical computation of stress intensity factor for multi-physics problems

This work presents a methodology for the analytical calculation of the stress intensity factor when the stress distribution on the crack surfaces is non-homogeneous. At first, a polynomial function is used to express the non-homogenous stress distribution. Subsequently, the principle of superposition of effects is applied, and the stress intensity factor is computed by multiplying each polynomial term by its respective geometric factor. Finite element fracture model is used to compute the geometric factor of the single polynomial grade. To explain the method, a spherical body is considered, with central and superficial cracks. Each geometric factor depends on a normalized geometrical parameter (the ratio between the crack length and sphere radius). The proposed methodology is applied to determine the stress intensity factor in the case of a crack driving force caused by diffusive fields, such as the concentration gradient in particles of electrodes active material in lithium-ion batteries. The methodology allows to speed up the fracture computation, then it is used to give electrode design guidelines to limit the fracture likeliness and mechanical degradation in lithium-ion batteries, as well as it is the basis for the development of algorithms assessing the capacity loss and the remaining useful life of lithium-ion batteries in real-time.

Buckling load optimization of sandwich plates with trapezoidal corrugated core and elliptical cutout using vibration correlation techniques and artificial neural network; experimental and numerical analysis

The present research was carried out in order to estimate the buckling load of a sandwich plate with a trapezoidal corrugated core with cut-out using three vibration correlation techniques (VCT). The structure investigated in this research is a sandwich plate with a trapezoidal core and an oval cut-out, which is made of thermoplastic polymer using a 3D printer. By using artificial neural network (ANN) and genetic algorithm (GA), the specific critical buckling load (p¯cr) which is defined as the ratio of critical buckling load (Pcr) to plate weight (gr) has been minimized and the optimal dimensions for cut-out parameters including the ratio of the large diameter of the ellipse (z=20mm) to the small diameter of the ellipse (b) and oval angles with the horizontal axis (a∘) are obtained. Using the method of equivalent properties, the trapezoidal core is transformed into an equivalent orthotropic core and finally the desired structure is transformed into a three-layer sandwich plate including homogeneous upper and lower layers and an orthotropic middle layer. By using the meshless method based on radial and polynomial functions and with the help of the finite element method, the numerical simulation of buckling and free vibration of the sandwich plate has been done. The results showed that the estimation of the buckling load with the vibration correlation technique (VCT) corresponds to the actual condition of the structure, so that due to the high quality of sandwich plate construction and its integrity, the results obtained from the first, second, and third methods of the VCT show the effectiveness of this method for predicting of the buckling load of the structure.

Classification of traffic accidents’ factors using TrafficRiskClassifier

The TrafficRiskClassifier model introduced in this study adopts an innovative approach that incorporates migration learning, image classification, and self-supervised learning, aiming to significantly improve the accuracy and efficiency of traffic accident risk analysis. Compared with traditional traffic safety analysis techniques, this model focuses on utilizing contextual information and situational data of traffic accidents to achieve a higher level of risk classification accuracy. The core of this approach is to deeply mine and analyze the detailed information in the accident environment, to provide more scientific and effective support for traffic accident risk prevention and response. Initially, by integrating migration learning with image classification techniques, the model proficiently extracts pivotal features from intricate traffic scenarios and formulates initial assessments of accident risks. Subsequently, self-supervised learning is incorporated in this study, augmenting the model's capability to comprehend and categorize accident imagery. The TrafficRiskClassifier model exhibits a generalization ability of 91.82%, 85.16%, and 80.92% on individual classification tasks, respectively, signifying its robust learning capacity and proficiency in managing unseen data. Furthermore, the TrafficRiskClassifier model delineates a functional nexus between accident risk and variables such as weather, road conditions, and personal factors, employing a polynomial regression approach. This methodology not only amplifies the predictive precision of the model but also renders it versatile across diverse scenarios. Through the analysis of various polynomial functions, the model achieves enhanced accuracy in classifying disparate risk levels. The outcomes demonstrate that the TrafficRiskClassifier model can efficaciously amalgamate contextual information within traffic scenarios, thereby achieving more precise classification of traffic accident risks, and consequently serving as an invaluable instrument for urban traffic safety management.

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

A hybrid dynamics modeling method for micro-turbine engine thrust vectoring

Thrust vector control technology is widely used in modern advanced aircrafts, space shuttles and unmanned aerial vehicles, etc. An accurate thrust vector mathematical model is of great significance to enhance the thrust vector control effect. In this paper, the P200 micro-turbine engine with an axisymmetric thrust vector nozzle placed at the rear is taken as an example, A hybrid thrust vectoring dynamics modeling method consisting of engine speed model, engine thrust model, actuator dynamics model, airflow angle model and thrust loss coefficients is proposed. The engine speed model is of Linear Parameter Varying (LPV) model with each segment is characterized by a First Oder Plus Dead Time (FOPDT) model. The engine thrust model is characterized as a cubic polynomial function of engine speed. The actuator’s dynamic model is a FOPDT model. The nozzle airflow deflection angle is proportional to the airflow deflection angle. the thrust loss coefficient is approximated by a cosine relationship with the airflow deflection angle. The experimental results show that the established model can well characterize the dynamics of the thrust vector system.

Treatment Planning Methods for Dose Painting by Numbers Treatment in Gamma Knife Radiosurgery

PURPOSEDose painting radiotherapy delivers a non-uniform dose to tumours to account for heterogeneous radiosensitivity. With recent and ongoing development of Gamma Knife (GK) machines making large-volume brain tumour treatments more practical, it is increasingly feasible to deliver dose painting treatments. The increased prescription complexity means automated treatment planning is greatly beneficial, and the impact of dose painting on SRS plan quality has not yet been studied. This research investigates the plan quality achievable for GK SRS dose painting treatments when using optimisation techniques and automated isocentre placement in treatment planning. METHODS & MATERIALSDose painting prescription functions with varying parameters are applied to convert voxel image intensities to prescriptions for ten sample cases. To study achievable plan quality and optimisation, clinically-placed isocentres are used with each dose painting prescription, and optimised using a semi-infinite linear programming formulation. To study automated isocentre placement, a grassfire sphere-packing algorithm and a clinically available Leksell Gamma Plan isocentre fill algorithm are used. Plan quality for each optimised treatment plan is measured with dose painting SRS metrics. RESULTSOptimisation can be used to find high quality dose painting plans, and plan quality is affected by the dose painting prescription method. Polynomial function prescriptions show better achievable plan quality than sigmoid function prescriptions even with high mean dose boost. Automated isocentre placement is shown as a feasible method for dose painting SRS treatment, and increasing the number of isocentres improves plan quality. The computational solve time for optimisation is within 5 minutes in most cases, which is suitable for clinical planning. CONCLUSIONSThe impact of dose painting prescription method on achievable plan quality is quantified in this study. Optimisation and automated isocentre placement are shown as possible treatment planning methods to obtain high quality plans.

Energy evolution and strain localization in fractured sandstone under freeze-thaw cycling and uniaxial loading-unloading

In cold regions, combined cyclic freeze-thaw (F-T) and loading-unloading can cause rock slope instability and sliding. The macro-failure of rock masses originates from the expansion of strain localization areas driven by energy. Therefore, a deep understanding of the energy evolution and strain distribution can reveal the instability process. Along these lines, in this work, intact and intermittently jointed sandstones were subjected to a cyclic F-T environment and subsequently carried out uniaxial graded loading-unloading to obtain their mechanical parameters and X-ray computed tomography images. By investigating the energy density, it was found that the input, elastic, dissipated, and damage energies all increased with loading according to a quadratic polynomial function, with the energies obeying a linear allocation law. Meanwhile, F-T changed the energy storage capacity and energy dissipation coefficient of the rock by causing meso-damage. To measure the meso-scale strain localization characteristics, the advanced digital volume correlation technique was also used to obtain three-dimensional strain fields. In terms of local strain, the failure behavior of intact specimens under the application of low and high numbers of F-T cycles was controlled by potential fracture bands and frost-heave cracks, respectively, while fractured specimens were mainly dependent on the macro-fracture arrangement. However, both frost heaving and thawing transformed sandstone from brittle to ductile. In terms of statistical strain, the equivalent strain (εeq) showed a log-normal distribution and continued to deviate to the right due to loading-unloading. Meanwhile, the normal strain (εxx, εyy, εzz) exhibited a normal distribution, where fractured sandstone mainly developed along with the slip direction. Finally, the intrinsic correlations among the energy parameters, meso-damage, and macro-failure were discussed. From our analysis, three stages of energy allocation and transformation were revealed, namely the uniform allocation of damage energy, banded accumulation of damage energy, and fracture caused by the elastic energy release.

The Effects of Propylene oxide Fumigation on the Mortality of Plodia interpunctella (Hubner) (Lepidoptera: Pyralidae)

This study was carried out in 2014-2015 at Kahramanmaraş Sütçü İmam University, Faculty of Agriculture, Department of Plant Protection, Entomology Laboratory. This study aims to evaluate the viability of propylene oxide (PPO) as an alternative fumigant to methyl bromide (MeBr) for the efficient control of Plodia interpunctella (Hubner) (Lepidoptera: Pyralidae) and to model the mortality rates of P. interpunctella insects using different regression techniques. The biological effectiveness of PPO was assessed across all life stages of P. interpunctella during a brief exposure period (4 hours) under three conditions: normal atmospheric pressure (PPO alone), low pressure (PPO+Vacuum), and an atmosphere enriched with carbon dioxide (CO2) (PPO+CO2). For all PPO treatments, PPO was directly introduced into the fumigation chamber at seven or more concentrations using different micro-syringes of different volume ranges: 2.5-25 µl l-1, 1.5-45 µl l-1, 1.5-30 µl l-1, and 0.5-15 µl l-1 for eggs, larvae, pupae, and adults, respectively. The results indicate that 100% mortality was observed in eggs at concentrations of 25 µl l-1, 20 µl l-1 and 20 µl l-1; in larvae at concentrations of 45 µl l-1, 30 µl l-1 and 40 µl l-1; in pupae at concentrations of 30 µl l-1, 15 µl l-1 and 25 µl l-1; and in adults at concentrations of 15 µl l-1, 10 µl l-1 and 10 µl l-1 for the PPO alone, PPO+Vacuum, and PPO+CO2 treatments, respectively. The results obtained from the developed regression models for insect mortality reveal that these models generally exhibited a better fit when described by exponential and third-order polynomial functions. In summary, this study indicates that PPO treatments hold significant promise for rapid insect control, particularly in the case of the Indian meal moth, rendering them invaluable for quarantine purposes.

Comparison of Ridge and Kernel Ridge Models in Predicting Thermal Stability of Zn-MOF Catalysts

This study investigates machine learning-based quantitative structure-property relationship (QSPR) models for predicting the thermal stability of zinc metal-organic frameworks (Zn-MOF). Utilizing a dataset comprising 151 Zn-MOF compounds with relevant molecular descriptors, ridge (R) and kernel ridge (KR) regression models were developed and evaluated. The results demonstrate that the R model outperforms the KR model in terms of prediction accuracy, with the R model exhibiting exceptional performance (R² = 0.999, RMSE = 0.0022). While achieving high accuracy, opportunities for further improvement exist through hyperparameter optimization and exploration of polynomial functions. This research underscores the potential of ML-based QSPR models in predicting the thermal stability of Zn-MOF compounds and highlights avenues for future investigation to enhance model accuracy and applicability in materials science.

A framework for performance enhancement of classifiers in detection of prostate cancer from microarray gene

Prostate cancer is a major world health problem for men. This shows how important early detection and accurate diagnosis are for better treatment and patient outcomes. This study compares different ways to find Prostate Cancer (PCa) and label tumors as normal or abnormal, with the goal of speeding up current work in microarray gene data analysis. The study looks at how well several feature extraction methods work with three feature selection strategies: Harmonic Search (HS), Firefly Algorithm (FA), and Elephant Herding Optimization (EHO). The techniques tested are Expectation Maximization (EM), Nonlinear Regression (NLR), K-means, Principal Component Analysis (PCA), and Discrete Cosine Transform (DCT). Eight classifiers are used for the task of classification. These are Random Forest, Decision Tree, Adaboost, XGBoost, and Support Vector Machine (SVM) with linear, polynomial, and radial basis function kernels. This study looks at how well these classifiers work with and without feature selection methods. It finds that the SVM with radial basis function kernel, using DCT for feature extraction and EHO for feature selection, does the best of all of them, with an accuracy of 94.8 % and an error rate of 5.15 %.

Development and Application of an IoT-Based System for Soil Water Status Monitoring in a Soil Profile.

Soil water content (θ), matric potential (h) and hydraulic conductivity (K) are key parameters for hydrological and environmental processes. Several sensors have been developed for measuring soil θ-h-K relationships. The cost of such commercially available sensors may vary over several orders of magnitude. In recent years, some sensors have been designed in the framework of Internet of Things (i.e., IoT) systems to make remote real-time soil data acquisition more straightforward, enabling low-cost field-scale monitoring at high spatio-temporal scales. In this paper, we introduce a new multi-parameter sensor designed for the simultaneous estimation of θ and h at different soil depths and, due to the sensor's specific layout, the soil hydraulic conductivity function via the instantaneous profile method (IPM). Our findings indicate that a second-order polynomial function is the most suitable model (R2 = 0.99) for capturing the behavior of the capacitive-based sensor in estimating θ in the examined soil, which has a silty-loam texture. The effectiveness of low-cost capacitive sensors, coupled with the IPM method, was confirmed as a viable alternative to time domain reflectometry (TDR) probes. Notably, the layout of the sensor makes the IPM method less labor-intensive to implement. The proposed monitoring system consistently demonstrated robust performance throughout extended periods of data acquisition and is highly suitable for ongoing monitoring of soil water status.

Computational Construction of Weierstrass Sections for Semi-Invariant Polynomial Functions on Lie Algebras

In this paper, we present a computational approach to construct Weierstrass sections for semi-invariant polynomial functions on Lie algebras, extending the foundational work of Bourbaki and Popov. We focus on simple Lie algebras of type B, C, or D, and their associated parabolic subalgebras, particularly those with Levi factors composed of successive blocks of size two. Our method extends the notion of Weierstrass sections introduced by Popov, enabling us to explicitly construct these sections and establish their polynomiality. Furthermore, we demonstrate how these sections facilitate the linearization of semi-invariant generators. Central to our approach is the construction of an adapted pair, akin to a principal sl_2-triple in the non-reductive case. We provide computational algorithms and implementations for constructing these Weierstrass sections, offering a novel avenue for research in Lie algebra theory and algebraic geometry.

Experimental investigation of brain contusion characteristics and dynamic response in low-age children using an animal model

IntroductionBrain contusion is a prevalent traumatic brain injury (TBI) in low-age children, bearing the potential for coma and fatality. Hence, it is imperative to undertake comprehensive research in this field. MethodsThis study employed 4-week-old piglets as surrogates for children and introduced self-designed devices for both free-fall drop impact tests and drop-hammer impact tests. The study explored the characteristics of brain contusion and dynamic responses of brain under these distinct testing conditions. ResultsBrain contusions induced by free-fall and drop-hammer conditions both were categorized as the coup injury, except that slight difference in the contusion location was observed, with contusion occurring mainly in the surrounding regions beneath the impact location under free-fall condition and the region just right beneath the impact location under drop-hammer condition. Analysis of impact force and intracranial pressure (ICP) curves indicated similar trends in impact forces under both conditions, yet different trends in ICPs. Further examination of the peak impact forces and ICPs elucidated that, with increasing impact energy, the former followed a combined power and first-order polynomial function, while the latter adhered to a power function. The brain contusion was induced at the height (energy) of 2 m (17.2 J), but not at the heights of 0.4, 0.7, 1, 1.35 and 1.7 m, when the vertex of the piglet head collided with a rigid plate. In the case of a cylindrical rigid hammer (cross-sectional area constituting 40 % of the parietal bone) striking the head, the brain contusion was observed under the energy of 21.9 J, but not under energies of 8.1 J, 12.7 J and 20.3 J. Notably, the incidence of brain contusion was more pronounced under the free-fall condition. ConclusionsThese findings not only facilitate a comprehensive understanding of brain contusion dynamics in pediatric TBIs, but also contribute to the validation of theories and finite element models for piglet heads, which are commonly employed as surrogates for children.

Mechanical Characteristics and Macro–Microscopic Response Mechanisms of Cemented Paste Backfill under Different Curing Temperatures

Macroscopic and microscopic properties of cemented paste backfill (CPB) were studied through uniaxial compressive testing, acoustic emission (AE) monitoring, and microscopic feature analysis. The research shows that the uniaxial compressive strength (UCS) and elastic modulus have an exponential function type positive correlation with the increase in curing time and a polynomial function type with the rise of curing temperature; the mechanical parameters reach the maximum when the curing temperature is 40 °C. Increasing the curing time and curing temperature can promote the transition from shear crack to tensile crack. Increasing the curing time and raising the curing temperature both promote the transition of shear crack to tensile crack in the CPB. Overall, the crack mode is a combination of tensile and shear crack. At room temperature, the shear cracks dominates in the initial stage, but the proportion of the shear cracks decreases as the pressure increases in uniaxial compression test. At a curing temperature of 60 °C, the crack mode transitions to a tensile-shear mixed crack, with tension becoming the dominant crack mode. Microscopic analysis suggests an excellent linear correlation between the pore fractal dimension, UCS, and elastic modulus. When the pore fractal dimension decreases, the mechanical parameters also decrease. The pore fractal dimension can effectively characterize the macroscopic mechanical properties. Finally, the curing temperature is divided into two stages, with 40 °C as the dividing line for analysis. In the first stage, the increase in curing temperature effectively improved the mechanical parameters; in the second stage, the excessively high hydration reaction rate weakened the mechanical parameters.

New experimental measurements of X-ray emissions induced by protons of several tens of MeV on elements with Z=22 to Z=79: Toward quantitative high-energy PIXE

K-shell X-ray production cross-sections measurement campaigns were carried out at the ARRONAX cyclotron with titanium, chromium, copper, molybdenum, silver and gold targets for different energies from 15 to 68 MeV. These measurements, carried out using three detectors (SDD, CdTe and HPGe), complete the database with 66 new values. Our different measurements agree with previously published data and are consistent for all three detectors, confirming the absence of experimental bias. The universal ionisation cross-section representation allows an optimisation of the data using a 3rd-order polynomial function. This function can be used for HE-PIXE quantification for elements with atomic numbers between 22 (Ti) and 79 (Au) in the studied energy region.

Physical mechanism-corrected degradation trend prediction network under data missing

Accurate degradation trend prediction (DTP) is crucial for optimizing equipment operation and maintenance, thereby boosting production efficiency. This study introduces a novel Data Repair and Dual-data-stream LSTM (DR-DLSTM) network to tackle the challenge of missing data in equipment DTP. The proposed DR-DLSTM framework employs convex optimization to consider both the trend and periodic variations in the data, incorporating polynomial and trigonometric functions into the implicit feature matrix to construct latent vectors for missing data rectification. The network features a Dual-LSTM block with dual data streams to enhance feature extraction, with two gating update units correlating time series components and redistributing feature weights. The Dual-LSTM enables separate and accurate prediction of trend and periodic components, thereby enhancing the feature extraction capability of the prediction model. Additionally, the integration of physical rule information through Fourier and wavelet transform frequency correction modules allows for dynamic adjustments in prediction outcomes, from global trends to localized details. The DR-DLSTM's effectiveness is demonstrated through comprehensive comparisons with state-of-the-art models across multiple datasets, highlighting its superior performance. The results demonstrate the superiority of the proposed model. These algorithms were implemented in Python using Torch on a 2.9 GHz Intel I7 CPU and TITAN Xp GPU.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.