Hermite Wavelet Research Articles

Hermite Wavelet Method for Fractional Delay Differential Equations

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

ADAPTIVE TRIGONOMETRIC HERMITE WAVELET FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS

Owing to its good approximation characteristics of trigonometric functions and the multi-resolution local characteristics of wavelet, the trigonometric Hermite wavelet function is used as the element interpolation function. The corresponding trigonometric wavelet beam element is formulated based on the principle of minimum potential energy. As the order of wavelet can be enhanced easily and the multi-resolution can be achieved by the multi-scale of wavelet, the hierarchical and multi-resolution trigonometric wavelet beam element methods are proposed for the adaptive analysis. Numerical examples have demonstrated that the aforementioned two methods are effective in improving the computational accuracy. The trigonometric wavelet finite element method (WFEM) proposed herein provides an alternative approach for improving the computational accuracy, which can be tailored for the problem considered.

Trigonometric wavelet-based method for elastic thin plate analysis

Dealing with the boundary conditions is one of the difficult problems when using wavelet function as trial function to carry out structural analysis. In this paper, the two-dimensional tensor product trigonometric Hermite wavelet that has both good approximation characteristics of trigonometric function and multi-resolution, local characteristics of wavelet is proposed as trial function, and the united formulation of elastic bending, vibration and buckling of rectangle thin plate (on elastic foundation) with different boundary conditions is derived based on the principle of minimum potential energy. Two approaches, hierarchical and multi-resolution approach, are presented to improve calculation accuracy. The impact of proposed method is discussed by different numerical examples. Due to the Hermite interpolation properties, the proposed trigonometric wavelet method can process all kinds of boundary conditions conveniently. The solution accuracy of hierarchical method can be increased steadily with raising the order of wavelet, while the solution accuracy of multi-resolution method can be improved along with increasing the scale of wavelet.

Finite element analysis of beam structures based on trigonometric wavelet

Taking advantages of trigonometric Hermite wavelet that has both good approximation characteristics of trigonometric function and multi-resolution, local characteristics of wavelet as interpolating...

Finite element analysis of beam structures based on trigonometric wavelet

Taking advantages of trigonometric Hermite wavelet that has both good approximation characteristics of trigonometric function and multi-resolution, local characteristics of wavelet as interpolating function, the trigonometric wavelet finite beam element is formulated in the paper to carry out the bending, free vibration and buckling of beam structures. Due to the Hermite interpolation properties of trigonometric wavelet, the proposed trigonometric wavelet finite beam element formulation can deal with the boundary conditions and connection between adjacent elements as the traditional finite element method does. Several numerical examples on the bending, free vibration and buckling analysis of beam structures have demonstrated that the trigonometric wavelet finite element method can achieve a good accuracy with less element adopted, especially for free vibration analysis.

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cln,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \({\mathbb{R}^{n}}\) . We express the admissibility condition in terms of the Cln,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cln,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.

Modelling cardiovascular physiological signals using adaptive Hermite and wavelet basis functions

This study presented a unified perspective of adaptive basis functions to compare Hermite decomposition and wavelet transform for the analysis of cardiovascular physiological signals. Three different algorithms were presented to carry out physiological signal modelling with adaptive Hermite basis functions (HBFs), orthonormal wavelet basis functions (OWBFs) and adaptive wavelet basis functions (AWBFs). The modelling with OWBFs is computationally efficient. However, the concomitant restrictions in mathematics make OWBFs not optimal for compact modelling. In contrast, the optimised AWBFs can model cardiovascular physiological signals compactly with the cost of losing orthonormality. It not only sacrifices the fast implementation but also degrades AWBFs in discriminant analysis. In summary, merely HBFs achieve a balanced performance in compact modelling and discriminant analysis.

Hermitian Clifford-Hermite wavelets: an alternative approach

Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing a higher dimensional continuous wavelet transform theory. In this setting a very specific construction of the wavelets has been established, encompassing all dimensions at once as opposed to the usual tensorial approaches, and being based on generalizations to higher dimension of classical orthogonal polynomials on the real line, such as the radial Clifford--Hermite polynomials, leading to Clifford--Hermite wavelets. More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the orthogonal case. In this new setting a Clifford--Hermite continuous wavelet transform has already been introduced in earlier work, its norm preserving character however being expressed in terms of suitably adapted scalar valued inner products on the respective $L_2$--spaces of signals and of transforms involved. In this contribution we present an alternative Hermitian Clifford--Hermite wavelet theory with Clifford algebra valued inner products, based on an orthogonal decomposition of the space of square integral functions, which is obtained by introducing a new Hilbert transform in the Hermitian setting.

Two-index Clifford–Hermite polynomials with applications in wavelet analysis

Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on specific types of multi-dimensional orthogonal polynomials, such as the Clifford–Hermite polynomials, which form the building blocks for so-called Clifford–Hermite wavelets, offering a refinement of the traditional Marr wavelets. In this paper, a generalization of the Clifford–Hermite polynomials to a two-parameter family is obtained by taking the double monogenic extension of a modulated Gaussian, i.e. the classical Morlet wavelet. The eventual goal being the construction of new Clifford wavelets refining the Morlet wavelet, we first investigate the properties of the underlying polynomials.

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O ( N ) entries for the weakly singular kernel representation matrix with dimensions N 2 which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.

Closed-form correlation functions of generalized Hermite wavelets

A closed-form expression is given for the correlation functions of generalized Hermite wavelets, constructed from an also-generalized definition of Hermite polynomials. Due to their Gaussianity, these wavelets can be used as a tool in the analysis or design of systems involving nonsinusoidal wavelets as well as to model impulsive waveforms found in real-world applications and signal processing problems. As such, the formula is potentially applicable to various areas of science.