Maximum Absolute Error Research Articles

Determination of CO2 diffusion coefficient in sodium carbonate solutions and implications for decarbonization kinetics

Sodium carbonate solution is a promising carbon capture absorber due to its low cost, rich content, environmental friendliness, and low heat requirement for regeneration, but its large-scale application is limited by its low diffusion coefficient and thus unsatisfied chemical absorption kinetics. Most importantly, the diffusion coefficient parameters of CO2 absorption into sodium carbonate solutions at different concentrations and temperatures are lacked in the existing literature, which leads to a certain bias in the understanding of its absorption kinetics. Based on this, in this paper, a molecular dynamics simulation method was used to calculate the diffusion behavior of CO2 in sodium carbonate solution at different parameters and compared with the existing fitting equations in the literature. The data calculated from literature presented a severe overestimation of diffusion coefficient in low temperature solvent, underestimation of diffusion coefficient in high temperature solvent. The maximum deviation between our simulated data and calculated data of fitted formula in literature exceeded 30%. Based on this, a new fitting formula was proposed in this paper using the temperature adjusted method. The results showed that the maximum error between simulated data and fitted new formula is 24.82%, AARD, SD, RMSD are only 9.21%, 11.13%, and 11.15% respectively. The maximum absolute error was 11.73%, AARD, SD, RMSD are only 4.55%, 5.53%, and 5.53% for CO2 mass transfer kinetics calculation, which significantly contributes to the accuracy of carbon capture process calculations.

An I–V characteristic reconstruction-based partial shading diagnosis and quantitative evaluation for photovoltaic strings

The partial shading condition (PSC) is the most common abnormality occurred in photovoltaic (PV) systems. Accurate quantitative evaluation of the shaded area and the severity of the shading is of potential importance in optimizing the maintenance strategy for PV systems. In this paper, we propose a PSC diagnosis and quantitative evaluation method by analyzing the measured string current–voltage (I–V) characteristic obtained from the PV inverter with the I–V scanning function, which includes pre-diagnosis of the system abnormality based on the operational power deviation, the diagnosis of PSCs based on the derivatives characteristics of the PV string, and the quantitative evaluation based on the I–V characteristic reconstruction. The quantitative evaluation of PSCs is the main focus in this paper, where the I–V characteristics of the unshaded PV modules in the partially shaded PV string are reconstructed according to different mismatch levels, respectively. The number of shaded PV modules and the corresponding severity of the partial shadings are estimated according to the reconstructed I–V characteristics. The simulation and experimental results verify that both the proposed diagnosis and quantitative evaluation method is effective with decent accuracy, especially for severe mismatch conditions. Experimental results show that the maximal mean absolute error of the quantified shaded area and quantified shaded rate are approximately 1.4446 units of 1/3 PV modules and 0.026, respectively.

A novel hybrid optimized incremental relevance vector machine and filtering technique for state of charge estimation of lithium-ion batteries

Despite not requiring a comprehensive understanding of the intricate electrochemical reactions within the battery, the data-driven approach faces challenges regarding state of charge (SOC) fluctuations during abrupt changes in battery current. Currently, research on data-driven filtering methods predominantly focuses on neural network-based algorithms, with limited exploration of the relevance vector machine (RVM) algorithm that possesses advantages in handling small sample sizes, high sparsity, and probabilistic distributions. Drawing inspiration from this premise, the present manuscript introduces an innovative fusion algorithm, founded upon the framework of the adaptive Kalman filtering (AKF) algorithm, which intricately melds the optimized incremental RVM (OIRVM) with the Coulomb counting methodology, called Coulomb_AKF_OIRVM (CFR) algorithm. Experimental findings suggest that the CFR algorithm displays exceptional generalization capacity across different driving cycles and temperatures, exhibiting high estimation accuracy and robustness. The average root mean square error, determination coefficient, mean absolute error and maximum absolute error of the public datasets and experimental datasets are 0.25 %, 0.02 %, 0.20 %, and 0.85 %, respectively.

Abstract PO1-07-12: Inverse modeling with surface temperature accurately detects the presence of breast cancer

Abstract Breast cancer affects over 250,000 women in the United States every year. Patient outcomes including overall survival significantly decline with large sized tumors and nodal involvement on initial presentation. The introduction of mammogram screening has been instrumental in the early detection of cancer. However, it has lower sensitivity in dense breast tissue and limited specificity which leads to unnecessary additional invasive testing. Several methods such as magnetic resonance imaging (MRI), computed tomography (CT) and ultrasound imaging are used as adjunctive methods. However, there is still a need for cost-effective techniques without additional radiation exposure for breast cancer screening for use with mammography. We propose an adjunctive screening method using surface temperature measurements and inverse breast modeling for early detection of breath cancer. Methods: We enrolled patients who presented with mammogram detected breast tumors that were confirmed to be cancerous lesions by biopsy. We used prone position steady state infrared imaging of bilateral breasts to measure surface temperatures and multi-view MRI images to create patient specific 3D breast models. The inverse technique applies Levenberg-Marquard algorithms (LMA) and utilizes commercial software for thermal transport modeling. The inverse technique predicts a heat generation map normalized for baseline surface temperatures to localize the presence of an underlying tumor. Results: A total of 25 breast tumors (diameter range from 5 to 27 mm) from 24 patients were included in the analysis with a median patient age of 67 years. One patient had bilateral breast tumors. Mammogram data showed inclusion of mixed breast tissue composition including dense and extremely dense breast tissue. LMA accurately detected the tumor in all 25 patients with maximum absolute errors of 7 mm in location of the tumor in the breast and 1.78 mm in diameter. The accuracy of detection was not affected by tumor histology which included invasive ductal and lobular carcinomas, ductal and lobular carcinoma in-situ and atypical ductal hyperplasia. No heat sensitive tumors were detected in the 23 contralateral breasts which acted as internal negative controls. Conclusion: Infrared temperature profiles and inverse modeling successfully detected malignant breast tumors with no missed tumors or false positive results and could be used as adjunctive screening along with mammography, especially in patients with dense breast tissue. Citation Format: Nithya Sritharan, Carlos Gutierrez, Isaac Perez-Raya, Jose-Luis Gonzalez-Hernandez, Alyssa Owens, Donnette Dabydeen, Lori Medeiros, Satish Kandlikar, Pradyumna Phatak. Inverse modeling with surface temperature accurately detects the presence of breast cancer [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO1-07-12.

Optimal transonic buffet aerodynamic noise PSD predictions with Random Forest: Modeling methods and feature selection

To address challenges such as long computational cycles and high experimental measurement costs in obtaining Power Spectral Density (PSD) of aerodynamic noise, this study aims to enable a rapid assessment of buffet frequency and aerodynamic noise levels. The paper conducts a comparative analysis of the impact of different modeling methods and input features on prediction accuracy, proposing the time domain model, the full frequency domain model, and the single frequency model. The research reveals that the frequency domain model has an advantage over the time domain model in predicting aerodynamic noise, emphasizing the importance of selecting appropriate modeling methods. Additionally, based on whether frequency information is used as input features, the study introduces the full frequency domain model and the single frequency model. Results indicate that the single frequency model can significantly reduce the maximum relative error and Root Mean Square Error of the full frequency domain model by approximately three orders of magnitude, lowering the reconstruction error of the Proper Orthogonal Decomposition method by 2–3 orders of magnitude. Furthermore, this model demonstrates generalization across Mach numbers, angles of attack, and spatial positions, ensuring that the maximum absolute error of discrete peak frequencies is controlled within 1 Hz, and the relative error of discrete narrowband peaks is maintained at around 1 %.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.

Development of a mathematical model of stabilisation of device for small-sized cargo transportation

The relevance of the study is conditioned by the need to improve the efficiency and safety of transportation of small-sized cargo. The purpose of this study was to build a mathematical model of the dynamics of stabilisation of device for small-sized cargo transportation. For this, the equations of motion of the system were formulated in the form of a system of second order Lagrange differential equations of the second kind. A grey box approach was used to determine the unknown coefficients of the equations of motion. To implement the approach, an optimisation criterion was constructed that reflected the parameters of the root-mean-square and maximum absolute errors of the differences between theoretical and experimental data of the tilt angle and angular velocity of the device. A modified Ring-Rot-PSO particle swarm method was used to minimise the criterion. The unknown parameters of the device model were found, and the adequacy of the obtained mathematical model was assessed by individual components of the criterion, which proved the adequacy of the obtained mathematical model. To find the unknown parameters, namely the coefficients of the equation of motion of the device, a grey box approach was applied. For this, experimental studies of the device stabilisation were performed, and the difference function was formed as an objective function of theoretical, obtained based on analytical equations of motion and experimental data. The objective function was minimised using the modified particle swarm method Ring-Rot-PSO. As a result of the optimisation, the unknown parameters of the system were obtained: the moments of inertia of the frame I1c = 5.52·10-4 kg·m² and the wheel Iwc = 2.75·10-3 kg·m², the wheel mass mw = 3.31·10-1 kg. These data allowed obtaining an adequate mathematical model model of the stabilisation of the device, which underlies further solving of the problem of synthesising optimal control of its motion

Computational Analysis of Predicting Stress Intensity Factor Using Hdmr Technique

The present research focused to develop meta-model (MM) as a low-cost substitute for the Finite Element Method (FEM), to forecast the Stress Intensity Factor (SIF) of a semi-elliptic crack propagating. Multi-linear regression (MLR) based MM is developed for predicting the crack propagation. Three parameters—Root Mean Square Error (RMSE), Maximum Absolute Error (AAE), and Average Absolute Error—are examined to compare the MM's precision (AAE).

Comprehensive thermal performance analysis of an air-cooled staggered configured Li-ion battery pack- A numerical and experimental approach

The discharge rate, ambient temperature, and airflow cooling velocity of a Li-ion battery pack BTMS have a direct impact on the battery's temperature, performance, and safety. Monitoring these is essential for avoiding overheating, ensuring optimal performance, and extending the battery's lifespan. This study comprehensively assesses the thermal performance of staggered structured cells of a 24.00V, 10.00Ah Li-ion battery pack cooled by ambient air. Both experimental and numerical methods are employed to achieve this evaluation. The primary objective is to investigate how varying discharge rates (0.50, 1.00, 1.50, 2.00, and 2.50; C), the seasonal temperatures in Prayagraj, India (282.00, 296.00, 305.00, and 315.00; K), and airflow velocities (0.50, 1.00, 1.50, and 2.00; m/s) influence the battery pack's thermal behaviour. A comparative analysis is also conducted between the present study and the author's prior research, focusing on the aligned configuration. The study observed a temperature increase in the airflow direction, from the inlet to the outlet, with the highest temperature recorded in the second-to-last column of the battery pack. Both numerical analysis and experimental testing confirmed that decreasing the airflow velocity from 2.00 m/s to 0.50 m/s reduced thermal performance by enhancing temperature in the airflow path, as well as transverse and circumferential temperature variations. Conversely, as the load on the battery (discharge rate) decreased from 2.50 C to 0.50 C, the thermal performance estimating parameters; Tmax, Tavg., ΔTmax, and σT, exhibited a declined trend. This is due to the greater heat generation at higher discharge rates. Furthermore, it was observed that as the ambient temperature increased from 282.00 K to 315.00 K, the temperature of the battery cells also increased. Nonetheless, the cooling patterns remained similar across these different temperature range conditions. Moreover, experimental results compete with the numerical outcomes with a maximum absolute percentage error of 1.21%. The staggered configuration gives lower and uniform temperature distributions within the battery packs than the aligned arrangements.

State of Health (SOH) Estimation of Lithium-Ion Batteries Based on ABC-BiGRU

As a core component of new energy vehicles, accurate estimation of the State of Health (SOH) of lithium-ion power batteries is essential. Correctly predicting battery SOH plays a crucial role in extending the lifespan of new energy vehicles, ensuring their safety, and promoting their sustainable development. Traditional physical or electrochemical models have low accuracy in measuring the SOH of lithium batteries and are not suitable for the complex driving conditions of real-world vehicles. This study utilized the black-box characteristics of deep learning models to explore the intrinsic correlations in the historical cycling data of lithium batteries, thereby eliminating the need to consider the internal chemical reactions of lithium batteries. Through Pearson correlation analysis, this study selects health indicators (HIs) from lithium battery cycling data that significantly impact SOH as input features. In the field of lithium batteries, this paper applies ABC-BiGRU for the first time to SOH prediction. Compared with other recursive neural network models, ABC-BiGRU demonstrates superior predictive performance, with maximum root mean square error and mean absolute error of only 0.016799317 and 0.012626847, respectively.

The capacity degradation path prediction for the prismatic lithium-ion batteries based on the multi-features extraction with SGPR

Accurate capacity degradation path estimation of lithium-ion batteries plays a crucial role in ensuring the safety and reliability of electric vehicles. In recent years, the evolution of the status of health (SOH) prediction with machine learning techniques has become a research hotspot because of its powerful computing power and robustness. Thus, this study employes the sparse gaussian process regression (SGPR) data-driven approach with multi-features extracted from the battery cycling processes to project the potential degradation patterns of the lithium-ion batteries. Firstly, a battery life test platform was built. The accelerated life aging tests of batteries at different temperatures (25 °C, or 60 °C) and different discharge rates (1C, or 2C) were conducted to establish the dataset for the multi-features extraction and training of the SGPR algorithm. Secondly, different battery characteristic features were extracted based on the battery cycle charge-discharge curves. Various order-reduction treatments, i.e., the filter-based, embedding-based, and fusion-based selection algorithms, were performed to suppress the over-fitting and improve the estimate's accuracy. Finally, using the extracted features as inputs, SGPR models are constructed to estimate the degradation path of the battery. With the 50 % training data, the SGPR has a higher estimation accuracy than the regular GPR, and the average maximum absolute errors for the batteries are 2.80 %, 1.22 %, 3.57 %, and 1.83 %, respectively.

Assembly deviation analysis of satellite flexible parts integrating rigid and flexible properties: Modeling and experiment

With the advancement of space technology, satellites are assuming an increasingly pivotal role in various fields. Currently, satellite structures incorporate a growing number of thin-walled parts. During the assembly process, the movement and connection of these parts are prone to generating assembly deviations, which seriously affects the overall assembly quality of the satellites. In order to accurately predict assembly deviations before assembly, this paper introduced an assembly deviation analysis model which took rigid and flexible properties into account. Starting from the attitude adjustment of satellite assembly, the transmission law of manufacturing error was established. Then, the assembly deviation analysis model integrating rigid and flexible properties was established. Finally, the assembly experiment was carried out to verify the effectiveness of the proposed model. The results showed that the average absolute error between the measurement results and calculation results was only 0.032 mm and the maximum absolute error was 0.074 mm. The average solving accuracy of the assembly deviation was 94%. The result fully proved the effectiveness of the proposed model.

A modified sliding mode controller based on fuzzy logic to control the longitudinal dynamics of the autonomous vehicle

This article delves into the intricate world of controlling the longitudinal dynamics of autonomous vehicles. In the first part, we studied two distinct controllers: the Super Twisting Sliding Mode Control and its modified version enriched with the integration of fuzzy logic, applied to the longitudinal dynamics of the autonomous automobile to follow a desired speed longitudinal profile, the two controllers are compared with a Neural Network-Based Non-singular Terminal Sliding-Mode Control, the system takes the throttle and brake as its inputs and delivers speed and acceleration as outputs. The overarching objective is to ensure that the controlled vehicle maintains a close and precise alignment with the desired speed profile. The second part of our research is dedicated to the development of adaptive cruise control systems and cruise control according to the safety conditions. This controller consists of two blocks low and upper controller, in upper controller the inputs are the speeds of the automobile in front and of the autonomous automobile itself, the safety distance, the measured distance. The output is the desired acceleration. The objective is to maintain a distance between the front vehicles, greater than or equal to the safety distance. For this, to achieve this task, we have implemented a control system known as the Proportional Integral Derivative (PID) controller in the adaptive cruise control system to control this system. In the low controller block, the same controllers used in the first part: the Super Twisting Sliding Mode Control and its modified version based on fuzzy logic, are applied, the system inputs are throttle and brake, and the outputs are speed and acceleration. This system is processed by MATLAB code, we obtained a better result with our proposed controller such that the maximum absolute speed error is equal to 0.0144 m/s in the first case of speed tracking, and to 0.006 m/s in the second case of the using adaptive cruise control, the illustrations below show the efficiency and robustness of these controllers.

State of Health Estimation and Remaining Useful Life Prediction of Lithium-Ion Batteries by Charging Feature Extraction and Ridge Regression

The state of health (SOH) and remaining useful life (RUL) of lithium-ion batteries are critical indicators for assessing battery reliability and safety management. However, these two indicators are difficult to measure directly, posing a challenge to ensure safe and stable battery operation. This paper proposes a method for estimating SOH and predicting RUL of lithium-ion batteries by charging feature extraction and ridge regression. First, three sets of health feature parameters are extracted from the charging voltage curve. The relationship between these health features and maximum battery capacity is quantitatively evaluated using the correlation analysis method. Then, the ridge regression method is employed to establish the battery aging model and estimate SOH. Meanwhile, a multiscale prediction model is developed to predict changes in health features as the number of charge-discharge cycles increases, combining with the battery aging model to perform multistep SOH estimation for predicting RUL. Finally, the accuracy and adaptability of the proposed method are confirmed by two battery datasets obtained from varying operating conditions. Experimental results demonstrate that the prediction curves can approximate the real values closely, the mean absolute error (MAE) and root mean square error (RMSE) calculations of SOH remain below 0.02, and the maximum absolute error (AE) of RUL is no more than two cycles.

A maximal overlap discrete wavelet packet transform coupled with an LSTM deep learning model for improving multilevel groundwater level forecasts

Developing precise groundwater level (GWL) forecast models is essential for the optimal usage of limited groundwater resources and sustainable planning and management of water resources. In this study, an improved forecasting accuracy for up to 3 weeks ahead of GWLs in Bangladesh was achieved by employing a coupled Long Short Term Memory (LSTM) network-based deep learning algorithm and Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) data preprocessing. The coupled LSTM-MODWPT model’s performance was compared with that of the LSTM model. For both standalone LSTM and LSTM-MODWPT models, the Random Forest feature selection approach was employed to select the ideal inputs from the candidate GWL lags. In the LSTM-MODWPT model, input GWL time series were decomposed using MODWPT. The ‘Fejér-Korovkin’ mother wavelet with a filter length of 18 was used to obtain a collection of scaling coefficients and wavelets for every single input time series. Model performance was assessed using five performance indices: Root Mean Squared Error; Scatter Index; Maximum Absolute Error; Median Absolute Deviation; and an a-20 index. The LSTM-MODWPT model outperformed standalone LSTM models for all time horizons in GWL forecasting. The percentage improvements in the forecasting accuracies were 36.28%, 32.97%, and 30.77%, respectively, for 1-, 2-, and 3-weeks ahead forecasts at the observation well GT3330001. Accordingly, the coupled LSTM-MODWPT model could potentially be used to enhance multiscale GWL forecasts. This research demonstrates that the coupled LSTM-MODWPT model could generate more precise GWL forecasts at the Bangladesh study site, with potential applications in other geographic locations globally.

A Fuzzy-PI Clock Servo with Window Filter for Compensating Queue-Induced Delay Asymmetry in IEEE 1588 Networks.

Clock synchronization is one of the popular research topics in Distributed Measurement and Control Systems (DMCSs). In most industrial fields, such as Smart Grid and Flight Test, the highest requirement for synchronization accuracy is 1 μs. IEEE 1588 Precision Time Protocol-2008 (PTPv2) can theoretically achieve sub-microsecond accuracy, but it relies on the assumption that the forward and backward delays of PTP packets are symmetrical. In practice, PTP packets will experience random queue delays in switches, making the above assumption challenging to satisfy and causing poor synchronization accuracy. Although using switches supporting the Transparent Clock (TC) can improve synchronization accuracy, these dedicated switches are generally expensive. This paper designs a PTP clock servo for compensating Queue-Induced Delay Asymmetry (QIDA), which can be implemented based on ordinary switches. Its main algorithm comprises a minimum window filter with drift compensation and a fuzzy proportional-integral (PI) controller. We construct a low-cost hardware platform (the cost of each node is within USD 10) to test the performance of the clock servo. In a 100 Mbps network with background (BG) traffic of less than 70 Mbps, the maximum absolute time error (max |TE|) does not exceed 0.35 μs, and the convergence time is about half a minute. The accuracy is improved hundreds of times compared with other existing clock servos.

Lossless and Near-Lossless Compression Algorithms for Remotely Sensed Hyperspectral Images.

Rapid and continuous advancements in remote sensing technology have resulted in finer resolutions and higher acquisition rates of hyperspectral images (HSIs). These developments have triggered a need for new processing techniques brought about by the confined power and constrained hardware resources aboard satellites. This article proposes two novel lossless and near-lossless compression methods, employing our recent seed generation and quadrature-based square rooting algorithms, respectively. The main advantage of the former method lies in its acceptable complexity utilizing simple arithmetic operations, making it suitable for real-time onboard compression. In addition, this near-lossless compressor could be incorporated for hard-to-compress images offering a stabilized reduction at nearly 40% with a maximum relative error of 0.33 and a maximum absolute error of 30. Our results also show that a lossless compression performance, in terms of compression ratio, of up to 2.6 is achieved when testing with hyperspectral images from the Corpus dataset. Further, an improvement in the compression rate over the state-of-the-art k2-raster technique is realized for most of these HSIs by all four variations of our proposed lossless compression method. In particular, a data reduction enhancement of up to 29.89% is realized when comparing their respective geometric mean values.

A method for estimating ship berthing angle based on 2D bird's eye view point cloud

The accurate estimation of berthing angle is essential for the parallel berthing of inshore ships. Herein, a new method for calculating the ship berthing angle using a shore-based LiDAR system is proposed. First, the LiDAR-scanned point cloud data is preprocessed, including coordinate conversion, noise filtering, and data downsampling. Then, the processed point cloud data is projected onto a two-dimensional (2D) plane, and the height value is used as the pixel value to generate bird's eye view (BEV) images. Finally, an improved Hough transform is proposed to detect the straight line representing the ship's nearshore profile in the BEV images and replace the ship midline with the detected line for calculating the berthing angle. The performance of the proposed method is verified by applying it on simulated and real ship data. Simulation experiment results show that the maximum absolute error of our method is only 0.151°. Real ship experiments were conducted at Dalian Port (China) to verify the reliability and stability of the method compared with other berthing angle estimation methods and the compass equipment. Overall, the proposed method can effectively obtain the berthing angle of ships, assist in parallel berthing, and improve berthing efficiency.

Calculation of ship floating state by quasi-Newton iteration method for onboard loading computer

ABSTRACT Calculating the floating state of a ship is the basic function of onboard loading computer, which is crucial for ship safety. Aiming at the deficiencies in the existing algorithms, we used the quasi-Newton iteration method to calculate the floating state of a ship. Firstly, the section model database of ship's hull was established by cutting the stereolithographic (STL) 3D model which is created by ship design software. Then, Sutherland–Hodgma polygon clipping algorithm and Shoelace Formula were used to calculate the Jacobian matrix. Finally, We selected a 108,000 DWT bulk carrier, ADMIRAL SCHMIDT, as an example to verify the correctness of the algorithm. We performed 100,000 random loading numerical tests, the average number of iterations was 2–3, and the number of convergence failures was zero. We calculated 24 typical standard loading conditions. We found the maximum average absolute error of the draught and trim were 0.0081 and −0.0283 m, respectively. The calculation error was far less than the requirements of the classification society. The results showed that the Quasi-Newton iteration method is simple to implement and easy to program, and has high robustness and accuracy and appropriate convergence. The method is suitable for the real-time calculation of a ship's floating state and has considerable engineering application value.

THEORETICAL ANALYSIS AND NUMERICAL SOLUTION OF LINEAR AND NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations.