Boundary Correction Research Articles

A Strongly Mass Conservative Method for the Coupled Brinkman-Darcy Flow and Transport

In this paper, a strongly mass conservative and stabilizer-free scheme is designed and analyzed for the coupled Brinkman–Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered discontinuous Galerkin scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method.

Intercomparison and Assessment of Stand-Alone and Wavelet-Coupled Machine Learning Models for Simulating Rainfall-Runoff Process in Four Basins of Pothohar Region, Pakistan

The science of hydrological modeling has continuously evolved under the influence of rapid advancements in software and hardware technologies. Starting from simple rational formulae for estimating peak discharge and developing into sophisticated univariate predictive models, accurate conversion of rainfall into runoff and the assessment of inherent uncertainty has been a prime focus for researchers. Therefore, alternative data-driven methods have gained widespread attention in hydrology. Moreover, scientists often couple conventional machine learning models with data pre-processing techniques, i.e., wavelet transformation (WT), to enhance modelling accuracy. In this context, this research work attempts to explore the latent linkage between rainfall and runoff in Pothohar region of Pakistan by developing a novel linkage of five streamline techniques of machine learning, including single decision tree (SDT), decision tree forest (DTF), tree boost (TB), multilayer perceptron (MLP), and gene expression modeling (GEP), with a more sophisticated variant of WT, i.e., maximal overlap discrete wavelet transformation (MODWT), for boundary correction of the transformed components of timeseries data. This study also implements these machine learning models in a stand-alone mode for a more comprehensive comparative analysis of performances. Furthermore, the study uses a combined-basin approach that divides Pothohar region into two basins to compensate for the complex topographic division of the study area. The results indicate that MODWT-based DTF outperformed other stand-alone and hybrid models in terms of modeling accuracy. In the first scenario, considering the Bunha-Kahan River basin, MODWT-DTF yielded the highest NSE (0.86) and the lowest RMSE (220.45 mm) and R2 (0.92 at lag order 3 (Lo3)) when transformed with daubechies4 (db4) at level three. While in the Soan-Haro River basin, MODWT-DTF produced the highest accuracy modeling at lag order 4 (Lo4) (NSE = 0.88, RMSE = 21.72 m3/s, and R2 = 0.91). The highly accurate performance of 3- and 4-days lagged models reflects the temporal consistency in hydrological response of the study area. The comparison of simple and hybrid model performance indicates up to a 55% increase in modeling accuracy due to data pre-processing with wavelet transformation.

A Lake Extraction Method Combining the Object-Oriented Method with Boundary Recognition

The China–Pakistan Economic Corridor is the pilot area of the Belt and Road, where glaciers and lakes are widely distributed. Recent years, global warming has accelerated the expansion of glacier lakes, which increased the risk of natural disasters such as glacier lake outburst. It is important to monitor the glacier lakes in this region. In this paper, we propose a method combining the object-oriented image analysis with boundary recognition (OOBR) to extract lakes in several study areas of China–Pakistan Economic Corridor (CPEC). This method recognized the lake boundary with the symmetrical characteristic according to the principle of seed growth of watershed algorithm, which can correct the boundary extracted by the object-oriented method. The overall accuracy of the proposed method is up to 98.5% with Landsat series images. The experiments also show that the overall accuracy of our method is always higher than that of the object-oriented method with different segmentation scales mentioned in this paper. The proposed method improved the overall accuracy on the basis of the results obtained by the object-oriented method, and the results with the proposed method are more robust to the seeds than that with the boundary correction method of the watershed algorithm. Therefore, the proposed method can obtain a high extraction accuracy while reducing the complexity of the object-oriented extraction.

커널 밀도 추정 기법을 이용한 풍력발전량에 대한 확률론적 모델

As the proportion of renewable energy sources that are intermittent and variability increases, system uncertainty is increasing, and a probabilistic analysis that can reflect these characteristics is required. In this paper, we propose a method for generating a probablistic wind power generation model using kernel density estimation. The boundary bias problem caused by using kernel density estimation for wind power generation data is eliminated through the boundary correction method, and a bandwidth suitable for the data can be selected through the bandwidth selector. Scenarios were created from probabilistic models made in various ways, and the accuracy of the models was validated by comparing them with real data. As a result, the accuracy was improved when the scenario was created by generating the model directly from the wind power generation data by estimating the kernel density estimation.

A New Surface Node Method to Accurately Model the Mechanical Behavior of the Boundary in 3D State-Based Peridynamics

Peridynamics is a non-local continuum theory capable of modeling crack initiation and propagation in solid bodies. However, the layer near the boundary of the body exhibits a stiffness fluctuation due to the so-called surface effect and the inaccurate way of imposing the boundary conditions. Moreover, in numerical models discretized using the meshfree method with uniform grid spacing, there are no nodes on the external surface of the body where the boundary conditions should be applied. Inspired by the method of the fictitious nodes with the Taylor-based extrapolation, we propose an innovative method that introduces a new type of nodes lying on the external surface of the body, i.e., the surface nodes. These nodes represent the interactions between the nodes within the body and the fictitious nodes surrounding the body, and they are used to mitigate the surface effect and properly impose the boundary conditions via the concept of force flux. Moreover, a procedure to compute the analytical solution of peridynamic problems is developed: a manufactured displacement field is prescribed and the volume and surface forces, to obtain that displacement field, are computed. The benefits of the surface node method are shown by means of several 2D and 3D quasi-static examples by comparing the numerical results with other methods with or without boundary corrections.

DM-MOGA: a multi-objective optimization genetic algorithm for identifying disease modules of non-small cell lung cancer

BackgroundConstructing molecular interaction networks from microarray data and then identifying disease module biomarkers can provide insight into the underlying pathogenic mechanisms of non-small cell lung cancer. A promising approach for identifying disease modules in the network is community detection.ResultsIn order to identify disease modules from gene co-expression networks, a community detection method is proposed based on multi-objective optimization genetic algorithm with decomposition. The method is named DM-MOGA and possesses two highlights. First, the boundary correction strategy is designed for the modules obtained in the process of local module detection and pre-simplification. Second, during the evolution, we introduce Davies–Bouldin index and clustering coefficient as fitness functions which are improved and migrated to weighted networks. In order to identify modules that are more relevant to diseases, the above strategies are designed to consider the network topology of genes and the strength of connections with other genes at the same time. Experimental results of different gene expression datasets of non-small cell lung cancer demonstrate that the core modules obtained by DM-MOGA are more effective than those obtained by several other advanced module identification methods.ConclusionsThe proposed method identifies disease-relevant modules by optimizing two novel fitness functions to simultaneously consider the local topology of each gene and its connection strength with other genes. The association of the identified core modules with lung cancer has been confirmed by pathway and gene ontology enrichment analysis.

Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

We consider a class of non-integrable 2D Ising models whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature via a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest neighbor Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green’s function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

Computer-Aided Detection of Human Lung Nodules on Computer Tomography Images via Novel Optimized Techniques.

As the mortality rate of lung cancer is enormously high, its impact is also extremely higher than the other types of cancer. Lung malignancy is thus considered one of the deadliest diseases with a high death rate in the world. It is reported that nearly 1.2 million people are diagnosed with this disease and about 1.1 million individuals are died due to this type of cancer every year. The early detection of this disease is the only solution for minimizing the death rate or maximizing the survival rate. However, the timely identification of lung malignant growth is a complex process and hence various imaging algorithms are employed in the process of detecting lung cancer on time. The Computer-Aided Diagnosis (CAD) is highly beneficial for the radiologist to rapidly detect and diagnose the irregularities in advance. The CAD systems usually focus on identifying and detecting the lung nodules. As the treatment of this disease is provided on the basis of its stages, the early detection of cancer has to be given much importance. The major drawbacks of existing CAD systems are less accuracy in segmenting the nodule and staging the lung cancer. The major aim of this work is to categorize the lung nodules from the CT image and classify the tumorous cells for identifying the exact position of cancer with higher sensitivity, precision, and accuracy than other strategies. The methods employed in this study are listed as follows: (i) For the process of de-noising and edge sharpening of lung image, the curvelet transform was used. (ii) The Fuzzy thresholding technique was used to perform lung image binarization and lung boundary corrections. (iii) Segmentation was performed by implementing the K-means algorithm. (iv) By using Convolutional Neural Network (CNN), different stages of lung nodules, like benign and malignant, were identified. The proposed classifier achieves optimal accuracy of 97.3%, a sensitivity of 98.6% and a specificity of 96.1% which are significantly better than the other approaches. Thus, the proposed approach is highly helpful in detecting lung cancer in its early stages. The results validate that the proposed algorithms are highly capable of classifying the lung images into various stages, which effectively helps the radiologist in the decision-making process.

Stepwise decomposition-integration-prediction framework for runoff forecasting considering boundary correction

Predicting river runoff accurately is of substantial significance for flood control, water resource allocation, and basin ecological dispatching. To explore the reasonable and effective application of time series decomposition in runoff forecasting, this study proposed a novel stepwise decomposition-integration-prediction considering boundary correction (SDIPBC) framework by using the stepwise decomposition sampling method and multi-input neural network. On this basis, we implemented a hybrid forecasting model combining seasonal-trend decomposition procedures based on loess (STL) with the long short-term memory (LSTM) network called STL-LSTM (SDIPBC) to estimate mid-long term river runoff. The reliability of the method was assessed using the historical runoff series of the Lianghekou and Jinping I Reservoirs in the Yalong River Basin, China, and developed several single models and hybrid models for comparative experiments. The results show that the existing decomposition-based hybrid forecasting frameworks are not suitable for practical runoff forecasting. The proposed SDIPBC framework can avoid using future information and improve the prediction accuracy of the single prediction model. For the Nash-Sutcliffe efficiency coefficient (NSE), the ten-day runoff forecasting accuracy of STL-LSTM (SDIPBC) in Lianghekou reservoir and Jinping I Reservoirs reached 0.845 and 0.862 respectively, which improved 1.81 % and 2.38 % than the single LSTM model, indicating that this is a practical and reliable decomposition-based hybrid runoff forecasting method.

Subcortical and hippocampal brain segmentation in 5-year-old children: Validation of FSL-FIRST and FreeSurfer against manual segmentation.

Developing accurate subcortical volumetric quantification tools is crucial for neurodevelopmental studies, as they could reduce the need for challenging and time‐consuming manual segmentation. In this study, the accuracy of two automated segmentation tools, FSL‐FIRST (with three different boundary correction settings) and FreeSurfer, were compared against manual segmentation of the hippocampus and subcortical nuclei, including the amygdala, thalamus, putamen, globus pallidus, caudate and nucleus accumbens, using volumetric and correlation analyses in 80 5‐year‐olds.Both FSL‐FIRST and FreeSurfer overestimated the volume on all structures except the caudate, and the accuracy varied depending on the structure. Small structures such as the amygdala and nucleus accumbens, which are visually difficult to distinguish, produced significant overestimations and weaker correlations with all automated methods. Larger and more readily distinguishable structures such as the caudate and putamen produced notably lower overestimations and stronger correlations. Overall, the segmentations performed by FSL‐FIRST's default pipeline were the most accurate, whereas FreeSurfer's results were weaker across the structures.In line with prior studies, the accuracy of automated segmentation tools was imperfect with respect to manually defined structures. However, apart from amygdala and nucleus accumbens, FSL‐FIRST's agreement could be considered satisfactory (Pearson correlation > 0.74, intraclass correlation coefficient (ICC) > 0.68 and Dice score coefficient (DSC) > 0.87) with highest values for the striatal structures (putamen, globus pallidus, caudate) (Pearson correlation > 0.77, ICC > 0.87 and DSC > 0.88, respectively). Overall, automated segmentation tools do not always provide satisfactory results, and careful visual inspection of the automated segmentations is strongly advised.

Transient heat transfer analysis of an orthotropic composite plate with oblique cracks using dual-phase-lagging model

Cracks have a significant effect on heat transfer in composites, yet the influence of oblique cracks on the temperature response of composites has not been fully investigated under strong transient thermal shock, in which the heat propagation velocity cannot be assumed to be infinite. In this work, the dual-phase-lagging heat transfer model is employed to capture the transient temperature response. By imposing different boundary conditions on the crack surface, an extended boundary correction algorithm has been derived to avoid the problem of non-convergence when dealing with large and small angle cracks. The effects of crack orientation angle and interaction between cracks on dual-phase-lagging heat conduction in orthotropic materials are analyzed at the macroscale. For the two-dimensional representative volume element (RVE) of fiber reinforced composite material, the influence of the multi-crack distribution on dual-phase-lagging heat conduction at the mesoscale is further discussed in detail.

A Weak Galerkin Mixed Finite Element Method for Second Order Elliptic Equations on 2D Curved Domains

This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $\alpha>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which exhibits an $O(h^{1/2})$ convergence for all $\alpha\ge 1$. It is a little surprising to see that even the lowest-order WG-MFEM ($\alpha=1$) experiences a loss of accuracy. This is different from known results for the finite element method (FEM) or the mixed FEM, and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain. Second, we propose a remedy to bring the approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes, which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements. Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all $\alpha$. Numerical results further confirm this conclusion.

MCI-Net: Multi-scale context integrated network for liver CT image segmentation

Owing to the various object scales and high similarity with the surrounding organs (e.g., kidney, stomach, and spleen), it is difficult to accurately segment the liver region from the abdominal computed tomography images. In this study, we propose a multi-scale context integration network called MCI-Net for liver image segmentation. Specifically, we first design a simplified residual module to prevent network degradation. Given the scale variability of objects, we propose a multi-scale context extraction module by combining four cascaded branches of hybrid dilated convolutions to capture broader and deeper features. In addition, we introduce an external attention mechanism based on two external, learnable and shared memory units, which helps to perceive the most discriminative information and suppress redundant features. Finally, we provide a boundary correction block to further improve the localization ability of boundary information. Extensive experiments on two liver CT image benchmark datasets qualitatively and quantitatively illustrate that our method is effective in improving liver segmentation accuracy and outperforms several state-of-the-art methods.

A divergence-free finite element method for the Stokes problem with boundary correction

Abstract This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

Trailing-Edge Noise Comparability in Open, Closed, and Hybrid Wind Tunnel Test Sections

Open-jet, hard-wall, and hybrid test section configurations are typically used in wind tunnel tests, aiming to represent the ideal free-flight condition. Each test section type requires special adaptations of the experimental hardware and specific correction methodologies that account for systematic measurement errors. Differently from previous research, this paper investigates the comparability of measurements performed in three test section configurations in the same wind tunnel. With this approach, systematic errors related to the facility or boundary-layer tripping methodology are minimized. Basic 2D aerodynamic boundary corrections are evaluated using a DU97W300 airfoil. The corrected lift curve collapsed well while differences in stall behavior and distribution at larger angles of attack are nonnegligible. The trailing-edge noise produced by a NACA-0012, NACA-0018, and NACA-63018 served as reference aeroacoustic sources in this comparability analysis. Moreover, the noise reduction from trailing-edge serrations served as an additional reference for the comparability of relative noise levels. The chord-based Reynolds number of the experiments ranged from 260,000 to 660,000. Unsteady wall pressure measurements performed near the trailing-edge provided a reference for the aeroacoustic noise source terms, demonstrating a negligible change between the different test section types. The applied experimental hardware and acoustic corrections yielded aeroacoustic sources of comparable absolute far-field noise level in the three test section types within . The relative noise levels are comparable within . These results show that aeroacoustic measurements of trailing-edge noise performed in different test section configurations are equivalent, provided that adequate experimental and postprocessing methodologies are systematically implemented.

Variational Optimization of Continuous Matrix Product States.

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

BOMSC-Net: Boundary Optimization and Multi-Scale Context Awareness Based Building Extraction From High-Resolution Remote Sensing Imagery

Automatic building extraction from high-resolution remote sensing imagery has various applications, such as urban planning and land use management. However, the existing methods for automatic building extraction focus on regional accuracy but not on boundary quality. As a result, these building extraction methods yield discontinuous results due to tree and shadow occlusions and complex building roof materials. In this article, BOMSC-Net, which can consider boundary optimization and multi-scale context awareness, is proposed to address these two problems. BOMSC-Net is an encoder-to-decoder network composed of a basic network, a multi-scale context-aware module (MSCAM), and a direction feature optimization module (DFOM). The basic network learns to predict building segmentation maps through attention gates (AGs) that connect low-level feature jumps to advanced semantics. MSCAM is used as a bridge, which uses graph reasoning blocks (GRBs) to solve intra-class discontinuities. Finally, the DFOM constructs a direction field (DF) and then uses direction information for boundary correction to achieve fine building segmentations. Experiments were conducted on the open Massachusetts building dataset, WHU aerial building dataset, and Inria aerial image labeling dataset. The quantitative and qualitative analyses demonstrate that the proposed BOMSC-Net can help building extraction overcome interruptions caused by shadows and occlusions, extract buildings with different scales and materials, and realize boundary optimization and complete segmentation of buildings.

Kernel density smoothing of composite spatial data on administrative area level

Composite spatial data on administrative area level are often presented by maps. The aim is to detect regional differences in the concentration of subpopulations, like elderly persons, ethnic minorities, low-educated persons, voters of a political party or persons with a certain disease. Thematic collections of such maps are presented in different atlases. The standard presentation is by Choropleth maps where each administrative unit is represented by a single value. These maps can be criticized under three aspects: the implicit assumption of a uniform distribution within the area, the instability of the resulting map with respect to a change of the reference area and the discontinuities of the maps at the borderlines of the reference areas which inhibit the detection of regional clusters.In order to address these problems we use a density approach in the construction of maps. This approach does not enforce a local uniform distribution. It does not depend on a specific choice of area reference system and there are no discontinuities in the displayed maps. A standard estimation procedure of densities are Kernel density estimates. However, these estimates need the geo-coordinates of the single units which are not at disposal as we have only access to the aggregates of some area system. To overcome this hurdle, we use a statistical simulation concept. This can be interpreted as a Simulated Expectation Maximisation (SEM) algorithm of Celeux et al (1996). We simulate observations from the current density estimates which are consistent with the aggregation information (S-step). Then we apply the Kernel density estimator to the simulated sample which gives the next density estimate (E-Step).This concept has been first applied for grid data with rectangular areas, see Groß et al (2017), for the display of ethnic minorities. In a second application we demonstrated the use of this approach for the so-called “change of support” (Bradley et al 2016) problem. Here Groß et al (2020) used the SEM algorithm to recalculate case numbers between non-hierarchical administrative area systems. Recently Rendtel et al (2021) applied the SEM algorithm to display spatial-temporal clusters of Corona infections in Germany.Here we present three modifications of the basic SEM algorithm: 1) We introduce a boundary correction which removes the underestimation of kernel density estimates at the borders of the population area. 2) We recognize unsettled areas, like lakes, parks and industrial areas, in the computation of the kernel density. 3) We adapt the SEM algorithm for the computation of local percentages which are important especially in voting analysis.We evaluate our approach against several standard maps by means of the local voting register with known addresses. In the empirical part we apply our approach for the display of voting results for the 2016 election of the Berlin parliament. We contrast our results against Choropleth maps and show new possibilities for reporting spatial voting results.

NONPARAMETRIC KERNEL DISTRIBUTION FUNCTION ESTIMATION NEAR ENDPOINTS

In this paper, two kernel cumulative distribution function estimators are introduced and investigated in order to improve the boundary effects, we will restrict our attention to the right boundary. The first estimator uses a self-elimination between modify theoretical Bias term and the classical kernel estimator itself. The basic technique of construction the second estimator is kind of a generalized reflection method involving reflection a transformation of the observed data. The theoretical properties of our estimators turned out that the Bias has been reduced to the second power of the bandwidth, simulation studies and two real data applications were carried out to check these phenomena and are conducted that the proposed estimators are better than the existing boundary correction methods.