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The purpose of this study is an application of the multi-wavelets Galerkin method to delay differential equations with vanishing delay known as Pantograph equation.

The method consists of expanding the required approximate solution at the elements of the Alpert multi-wavelets. Using the operational matrices of integration and wavelet transform matrix, the authors reduce the problem to a set of algebraic equations.

Because of the large size of the system, thresholding is used to obtain a new sparse system, and then this new system is solved to reduce the computational effort and related computer run time. The authors demonstrate that the solutions may be efficiently represented in a multi-wavelets basis because of flexible vanishing moments property of this type of multi-wavelets.

The L₂ convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.

The purpose of this paper is to explain the choice of Alpert multi‐wavelet as basis functions to discrete Fredholm integral equation of the second kind by using Petrov‐Galerkin method.

In this process, two kinds of matrices are obtained from inner product between basis of test space and trial space; some of them are diagonal with positive elements and some others are invertible. These matrices depend on type of selection of test and trial space basis.

In this process, solution of Fredholm integral equation of the second kind is found by solving the generated system of linear equations.

In previous work, convergence of Petrov‐Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree. In point of computation, because of appearance of diagonal and invertible matrices, a small dimension system with a more accurate solution is obtained. The numerical examples illustrate these facts.

The brushless direct current motor (BLDCM) is widely accepted and adopted by many industries instead of direct current motors due to high reliability during operation. Brushless direct current (BLDC) has outstanding efficiency as losses that arise out of voltage drops at brushes and friction losses are eliminated. The main factor that affects the performance is temperature introduced in the internal copper core windings. The control of motor speed generates high temperature in BLDC operation. The high temperature is due to presence of ripples in the operational current. The purpose is to present an effective controlling mechanism for speed management and to improve the performance of BLDCM to activate effective management of speed.

The purpose is to present an optimal algorithm based on modified moth-flame optimization algorithm over recurrent neural network (MMFO-RNN) for speed management to improve the performance. The core objective of the presented work is to achieve improvement in performance without affecting the design of the system with no additional circuitry. The management of speed in BLDCM has been achieved through reduction or minimization of ripples encircled with torque of the motor. The implementation ends in two stages, namely, controlling the loop of torque and controlling the loop of speed. The MMFO-RNN starts with error optimization, which arises from both the loops, and most effective values have been achieved through MMFO-RNN protocol.

The parameters are enriched with Multi Resolution Proportional Integral and Derivative (MRPID) controller operation to achieve minimal ripples for the torque of BLDC and manage the speed of the motor. The performance is increased by adopting this technique approximately 12% in comparison with the existing methodology, which is the main contributions of the presented work. The outcomes are analyzed with the existing methodologies through MATLAB Simulink tool, and the comparative analyses suggest that better performance of the proposed system produces over existing techniques, and proto type model is developed and cross verifies the proposed system.

The MMFO-RNN starts with error optimization, which arises from both the loops, and most effective values have been achieved through MMFO-RNN protocol. The parameters are enriched with MRPID controller operation to achieve nil or minimal ripples and to encircle the torque of Brushless Direct Current and manage the speed.

The principal intention behind the activity is to regulate the speed, current and commutation of the brushless DC (BLDC) motor. Thereby, the authors can control the torque.

In order to regulate the current and speed of the motor, the Multi-resolution PID (MRPID) controller is proposed. The altered Landsman converter is utilized in this proposed suppression circuit, and the obligation cycle is acclimated to acquire the ideal DC-bus voltage dependent on the speed of the BLDC motor. The adaptive neuro-fuzzy inference system-elephant herding optimization (ANFIS-EHO) calculation mirrors the conduct of the procreant framework in families.

Brushless DC motor's dynamic properties are created, noticed and examined by MATLAB/Simulink model. The performance will be compared with existing genetic algorithms.

The presented approach and performance will be compared with existing genetic algorithms and optimization of different structure of BLDC motor.

The purpose of this paper is to explore the challenges faced by the automatic recognition systems over the conventional systems by implementing a novel approach for detecting and recognizing the vehicle license plates in order to increase the security of the vehicles. This will also increase the societal discipline among vehicle users.

From a methodological point of view, the proposed system works in three phases which includes the pre-processing of the input image from the database, applying segmentation to the processed image, and finally extracting and recognizing the image of the license plate.

The proposed paper provides an analysis that demonstrates the correctness of the algorithm to correctly capture the license plate using performance metrics such as detection rate and false positive rate. The obtained results demonstrate that the proposed algorithm detects vehicle license plates and provides detection rate of 93.34 percent with false positive rate of 6.65 percent.

The proposed license plate detection system eliminates the need of manually used systems for managing the traffic by installing the toll-booths on freeways and bridges. The design implemented in this paper attempts to capture the license plate by using three phase detection process that helps to increase the level of security and contribute in making a sustainable city.

This paper presents a distinctive approach to detect the license plate of the vehicles using the various image processing techniques such as dilation, grey-scale conversion, edge processing, etc. and finding the region of interest of the segmented image to capture the license plate of the vehicles.

The purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.

L²[0, 1] was considered as solution space and the solution was projected to the subspaces of L²[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal.

In this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations.

Comparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.

The purpose of this paper is to reduce the destructive effects of existing unavoidable noises contaminating temperature data in inverse heat conduction problems (IHCP) utilizing the wavelets.

For noise reduction, sensor data were treated as input to the filter bank used for signal decomposition and implementation of discrete wavelet transform. This is followed by the application of wavelet denoising algorithm that is applied on the wavelet coefficients of signal components at different resolution levels. Both noisy and de‐noised measurement temperatures are then used as input data to a numerical experiment of IHCP. The inverse problem deals with an estimation of unknown surface heat flux in a 2D slab and is solved by the variable metric method.

Comparison of estimated heat fluxes obtained using denoised data with those using original sensor data indicates that noise reduction by wavelet has a potential to be a powerful tool for improvement of IHCP results.

Noise reduction using wavelets, while it can be implemented very easily, may also significantly relegate (or even eliminate) conventional regularization schemes commonly used in IHCP.

The real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.

To generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.

Adaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.

The proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.

Although many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations.

Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed.

Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method.

To the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.

Second‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.

For the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.

Similar techniques may be applied to scale‐decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDEs.

By decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes.

The paper explains second‐generation wavelet construction in a Lagrangian FE context. For 1D higher‐order and 2D first‐order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.