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The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs.

Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods.

This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased.

This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

The formation of the well known element mass, geometric stiffness and stiffness matrices based on Hermite polynomials can be very easily automated using Computer Algebra systems. The algebraic language REDUCE has been used to obtain the algebraic expressions for such matrices in one and two dimensions and to automatically produce FORTRAN code. The program developed can derive the matrices for any number of nodes in the element. The aim is to demonstrate how the Computer Algebra system can greatly reduce the effort in forming these classical matrices. It can thus be used as a teaching aid.

Realistic composite vehicles with 2, 3, 5 and 9 axles, consisting of a truck with one or two trailers, are addressed in this paper by computational models for vehicle–bridge interaction analysis.

The vehicle–bridge interaction (VBI) models are formed by sets of 2-D rigid blocks interconnected by mass, damping and stiffness elements to simulate their suspension system. The passage of the vehicles is performed at different speeds. Several rolling surface profiles are admitted, considering the maintenance grade of the pavement. The spectral density functions are generated from an experimental database to form the longitudinal surface irregularity profiles. A computational code written in Phyton based on the finite element method was developed considering the Euler–Bernoulli beam model.

Several models of composite heavy vehicles are presented as manufactured and currently travel on major roads. Dynamic amplification factors are presented for each type of composite vehicle.

The VBI models for compound heavy vehicles are 2-D.

This work contributes to improving the safety and lifetime of the bridges, as well as the stability and comfort of the vehicles when passing over a bridge.

The structural response of the bridge is affected by the type and size of the compound vehicles, their speed and the conservative grade of the pavement. Moreover, one axle produces vibrations that can be superposed by the vibrations of the other axles. This effect can generate not usual dynamic responses.

This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.

The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.

The method is computationally very attractive and gives very accurate results.

Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).

The proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.

To demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.

The H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

The Hermite collocation method of discretization can be used to determine highly accurate solutions to the steady‐state one‐dimensional convection‐diffusion equation (which can be used to model the transport of contaminants dissolved in groundwater). This accuracy is dependent upon sufficient refinement of the finite‐element mesh as well as applying upstream or downstream weighting to the convective term through the determination of collocation locations which meet specified constraints. Owing to an increase in computational intensity of the application of the method of collocation associated with increases in the mesh refinement, minimal mesh refinement is sought. Very often this optimization problem is the one where the feasible region is not connected and as such requires a specialized optimization search technique. This paper aims to focus on this method.

An original hybrid method that utilizes a specialized adaptive genetic algorithm followed by a hill‐climbing approach is used to search for the optimal mesh refinement for a number of models differentiated by their velocity fields. The adaptive genetic algorithm is used to determine a mesh refinement that is close to a locally optimal mesh refinement. Following the adaptive genetic algorithm, a hill‐climbing approach is used to determine a local optimal feasible mesh refinement.

In all cases the optimal mesh refinements determined with this hybrid method are equally optimal to, or a significant improvement over, mesh refinements determined through direct search methods.

Further extensions of this work could include the application of the mesh refinement technique presented in this paper to non‐steady‐state problems with time‐dependent coefficients with multi‐dimensional velocity fields.

The present work applies an original hybrid optimization technique to obtain highly accurate solutions using the method of Hermite collocation with minimal mesh refinement.

The purpose of this paper is to present the analytical solution to the Hermite collocation discretization of a quadratically forced steady‐state convection‐diffusion equation in one spatial dimension with constant coefficients, defined on a uniform mesh, with Dirichlet boundary conditions. To improve the accuracy of the method “upstream weighting” of the convective term is used in an optimal way. The authors also provide a method to determine where the forcing function should be optimally sampled. Computational examples are given, which support and illustrate the theory of the optimal sampling of the convective and forcing term.

The authors: extend previously published results (which dealt only with the case of linear forcing) to the case of quadratic forcing; prove the theorem that governs the quadratic case; and then illustrate the results of the theorem using computational examples.

The algorithm developed for the quadratic case dramatically decreases the error (i.e. the difference between the continuous and numerical solutions).

Because the methodology successfully extends the linear case to the quadratic case, it is hoped that the method can, indeed, be extended further to more general cases. It is true, however, that the level of complexity rose significantly from the linear case to the quadratic case.

Hermite collocation can be used in an optimal way to solve differential equations, especially convection‐diffusion equations.

Since convection‐dominated convection‐diffusion equations are difficult to solve numerically, the results in this paper make a valuable contribution to research in this field.

The paper aims to compare and clarify the differences and between the two well-known decomposition spectral techniques; the Winer–Chaos expansion (WCE) and the Winer–Hermite expansion (WHE). The details of the two decompositions are outlined. The difficulties arise when using the two techniques are also mentioned along with the convergence orders. The reader can also find a collection of references to understand the two decompositions with their origins. The geometrical Brownian motion is considered as an example for an important process with exact solution for the sake of comparison. The two decompositions are found practical in analysing the SDEs. The WCE is, in general, simpler, while WHE is more efficient as it is the limit of WCE when using infinite number of random variables. The Burgers turbulence is considered as a nonlinear example and WHE is shown to be more efficient in detecting the turbulence. In general, WHE is more efficient especially in case of nonlinear and/or non-Gaussian processes.

The paper outlined the technical and literature review of the WCE and WHE techniques. Linear and nonlinear processes are compared to outline the comparison along with the convergence of both techniques.

The paper shows that both decompositions are practical in solving the stochastic differential equations. The WCE is found simpler and WHE is the limit when using infinite number of random variables in WCE. The WHE is more efficient especially in case of nonlinear problems.

Applicable for SDEs with square integrable processes and coefficients satisfying Lipschitz conditions.

This paper fulfils a comparison required by the researchers in the stochastic analysis area. It also introduces a simple efficient technique to model the flow turbulence in the physical domain.

This paper presents new approach to the integration of neural classifiers. Typically only the best trained network is chosen, while the rest is discarded. However, combining the trained networks helps to integrate the knowledge acquired by the component classifiers and in this way improves the accuracy of the final classification. The aim of the research is to develop and compare the methods of combining neural classifiers of the heart beat recognition.

Two methods of integration of the results of individual classifiers are proposed. One is based on the statistical reliability of post‐processing performance on the trained data and the second uses the least mean square method in adjusting the weights of the weighted voting integrating network.

The experimental results of the recognition of six types of arrhythmias and normal sinus rhythm have shown that the performance of individual classifiers could be improved significantly by the integration proposed in this paper.

The presented application should be regarded as the first step in the direction of automatic recognition of the heart rhythms on the basis of the registered ECG waveforms.

The results mean that instead of designing one high performance classifier one can build a number of classifiers, each of not superb performance. The appropriate combination of them may produce a performance of much higher quality.

The purpose of this paper is to design an intelligent ECG classifier using programmable IC technologies to implement many functional blocks of signal acquisition and processing in one compact device. The main microprocessor also simulates the TSK neuro-fuzzy classifier in testing mode to recognize the ECG beats. The design brings various theoretical solutions into practical applications.

The ECG signals are acquired and pre-processed using the Field-Programmable Analog Array (FPAA) IC due to the ability of precise configuration of analog parameters. The R peak of the QRS complexes and a window of 300 ms of ECG signals around the R peak are detected. In this paper we have proposed a method to extract the signal features using the Hermite decomposition algorithm, which requires only a multiplication of two matrices. Based on the features vectors, the ECG beats are classified using a TSK neuro-fuzzy network, whose parameters are trained earlier on PC and downloaded into the device. The device performance was tested with the ECG signals from the MIT-BIH database to prove the correctness of the hardware implementations.

The FPAA and Programmable System on Chip (PSoC) technologies allow us to integrate many signal processing blocks in a compact device. In this paper the device has the same performance in ECG signal processing and classifying as achieved on PC simulators. This confirms the correctness of the implementation.

The device was fully tested with the signals from the MIT-BIH databases. For new patients, we have tested the device in collecting the ECG signals and QRS detections. We have not created a new database of ECG signals, in which the beats are examined by doctors and annotated the type of the rhythm (normal or abnormal, which type of arrhythmia, etc.) so we have not tested the classification mode of the device on real ECG signals.

The compact design of an intelligent ECG classifier offers a portable solution for patients with heart diseases, which can help them to detect the arrhythmia on time when the doctors are not nearby. This type of device not only may help to improve the patients’ safety but also contribute to the smart, inter-networked life style.

The device integrate a number of solutions including software, hardware and algorithms into a single, compact device. Thank to the advance of programmable ICs such as FPAA and PSoC, the designed device can acquire one channel of ECG signals, extract the features and classify the arrhythmia type (if detected) using the neuro-fuzzy TSK network in online mode.