Non-uniform Haar Wavelet Method for Solving Singularly Perturbed Differential Difference Equations of Neuronal Variability

A non-uniform Haar wavelet method is proposed on specially designed non-uniform grid for the numerical treatment of singularly perturbed differential-difference equations arising in neuronal variability.We convert the delay and shift terms using Taylor series up to second order and then the problem with delay and shift is converted into a new problem without the delay and shift terms. Then it is solved by using non-uniform Haar wavelet. Two test examples have been demonstrated to show the accuracy of the non-uniform Haar wavelet method. The performance of the present method yield more accurate results on increasing the resolution level and converges fast in comparison to uniform Haar wavelet

A numerical solution for nonlinear heat transfer of fin problems using the Haar wavelet quasilinearization method

The aim of this paper is to study the new application of Haar wavelet quasilinearization method (HWQM) to solve one-dimensional nonlinear heat transfer of fin problems. Three different types of nonlinear problems are numerically treated and the HWQM solutions are compared with those of the other method. The effects of temperature distribution of a straight fin with temperature-dependent thermal conductivity in the presence of various parameters related to nonlinear boundary value problems are analyzed and discussed. Numerical results of HWQM gives excellent numerical results in terms of competitiveness and accuracy compared to other numerical methods. This method was proven to be stable, convergent and, easily coded

Stiffness parameter prediction for elastic supports of non-uniform rods

The present research focuses on establishing the stiffness parameter of elastic springs placed at the ends of non-uniform rods. The governing equation for the longitudinal vibrations of the rod was solved using the Haar wavelet integration method. The calculated natural frequency parameters closely aligned with those available in the literature. The normalised values of the first ten natural frequency parameters were used in the feature vector to predict the stiffness parameter of the springs. A feedforward neural network with two hidden layers made accurate predictions when the range of each natural frequency parameterwithin its domain exceeded one. The insights garnered from this study contribute to the design, optimisation and assessment of diverse engineering applications

Numerical solution of the inverse Gardner equation

In this paper, the numerical solution of the inverse Gardner equation will be considered. The Haar wavelet collocation method (HWCM) will be used to determine the unknown boundary condition which is estimated from an over-specified condition at a boundary. In this regard, we apply the HWCM for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. It is proved that the proposed method has the order of convergence O(∆x). The efficiency and robustness of the proposed approach for solving the inverse Gardner equation are demonstrated by one numerical example.Publisher's Versio

Solving nonlinear PDEs using the higher order Haar wavelet method on nonuniform and adaptive grids

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points

A haar wavelet series solution of heat equation with involution

It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations

Haari lainikute meetod omavõnkumiste analüüsiks ja parameetrite määramiseks

Tala on konstruktsioonielement, mille ülesandeks on vastu pidada erinevatele koormustele. Projekteerimisel alahinnatud koormused, ebatäpsused tootmisel, söövitav keskkond, konstruktsiooni vananemine ekspluatatsiooni käigus võivad talasid kahjustada ning põhjustada kogu konstruktsiooni purunemist. Seetõttu talade dünaamilise käitumise modelleerimine ja ekspluatatsiooni jälgimine on jätkuvalt aktuaalne teema konstruktsioonide mehaanikas. Käesolev väitekiri on suunatud süstemaatilisele lähenemisele võnkumiste analüüsimiseks ja purunemise parameetrite määramiseks Euler-Bernoulli tüüpi talades. Töös pakutakse välja Haari lainikute meetod sageduste arvutamiseks ja andmete töötlemiseks. Nimelt, väitekirja esimeses osas on Haari lainikuid ja nende integreerimist rakendatud vabavõnkumise ülesannete korral, kus lahendatavaks võrrandiks on muutuvate kordajatega diferentsiaalvõrrand, millel puudub analüütiline lahend (näiteks ebaühtlase ristlõikega tala, materjali funktsionaalse gradientjaotusega tala). Arvutused kinnitasid, et pakutud lähenemisviis on kiire ja täpne vabavõnkumiste sageduste arvutamisel. Väitekirja teine osa käsitleb vabavõnkumisega seotud pöördülesandeid: pragude, delaminatsioonide, elastsete tugede jäikuse, massipunktide parameetrite määramist modaalsete omaduste kaudu. Kuna purunemise asukoha ja ulatuse arvutamine võnkumise diferentsiaalvõrrandist ei ole analüütiliselt võimalik, kasutatakse antud töös tehisnärvivõrke ja juhumetsi. Andmekogumite genereerimiseks lahendati võnkumise võrrand ning tulemusi töödeldi Haari lainikute abil. Arvutused näitasid, et Haari lainikute abil genereeritud andmekogumite arvutamiseks kuluv aeg oli üle kümne korra väiksem kui vabavõnkumiste sagedustele põhinevate andmekogumite arvutusaeg; Haari lainikute abil genereeritud andmekogumid ennustasid paremini purunemise asukohta, samas vabavõnkumiste sagedused olid tundlikumad purunemise ulatuse suhtes; enamikel juhtudel andsid tehisnärvivõrgud sama täpseid ennustusi kui juhumetsad. Töös pakutud meetodeid ja mudeleid saab kasutada teistes teoreetilistes ülesannetes vabavõnkumiste ja purunemiste uurimiseks või rakendada talade purunemise diagnostikas.A beam is a common structural element designed to resist loading. Underestimated loads during the design stage, looseness during the manufacturing stage, corrosive environment, collisions, fatigue may introduce some damage to beams. If no action is taken, the damage can turn into a fault or a breakdown of the whole system. Hereof, the entirety of beams is a crucial issue. This dissertation proposes a systematic approach to vibration analysis and damage quantification in the Euler-Bernoulli type beams. The solution is sought on the modal properties such as natural frequencies and mode shapes. The forward problem of the vibration analysis is solved using the Haar wavelets and their integration since the corresponding differential equations do not have an analytical solution. Multiple numerical examples indicate that the proposed approach is fast and accurate. Damage quantification (location and severity) of a crack, a delamination, a point mass or changes in the stiffness coefficients of elastic supports on the bases of the modal properties is an inverse problem. Since it is not analytically possible to calculate the damage parameters from the vibration differential equation, the task is solved with the aid of artificial neural networks or random forests. The datasets are generated solving the vibration equations and decomposing the mode shapes into the Haar wavelet coefficients. Multiple numerical examples indicate that the Haar wavelet based dataset is calculated more than ten times faster than the frequency based dataset; the Haar wavelets are more sensitive to the damage location, while the frequencies are more sensitive to the damage severity; in most cases, the neural networks produce as precise predictions as the random forests. The results presented in this dissertation can help in understanding the behaviour of more complex structures under similar conditions, provide apparent influence on the design concepts of structures as well as enable new possibilities for operational and maintenance concepts.https://www.ester.ee/record=b539883

Higher Order Haar Wavelet Method for Solving Differential Equations

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the mesh and the method parameter values used

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior