Fractional Spline Research Articles

Norm inequalities with fractional integrals

Fractional spline wavelet systems are considered. Together with the Battle–Lemarié scaling and wavelet functions of natural orders, they are used to find conditions that ensure certain inequalities between the norms of images and pre-images of fractional integration operators in Besov spaces with Muckenhoupt weights on R \mathbb {R} .

High order numerical method for a subdiffusion problem

We consider a subdiffusive fractional differential problem characterized by an equation that incorporates a time Riemann-Liouville fractional derivative of order 1−α, α∈(0,1), on its right-hand side, while the diffusive coefficient is allowed to vary with both space and time. An high order numerical method for the subdiffusion problem is derived based on the fractional splines of degree β∈(1,2]. The main purpose of this work is to apply fractional splines for approximating the fractional integral in the definition of the Riemann-Liouville fractional derivative, and hence explain how to solve the subdiffusion problem using this approach. It is discussed how the rate of convergence of the numerical method depends on the solution, the degree of the spline and the order of the fractional derivative.

A maximum-likelihood-based approach to estimate the long memory parameter using fractional spline wavelets

Fractional spline wavelets act as fractional differentiators for essentially low-pass signals, which is a useful property to analyze time series with fractal behavior. Moreover, under suitable scaling of wavelet coefficients, one can whiten a fractional Gaussian noise inputted in the wavelet channels. In this article, we demonstrate the two former statements and, by employing them, propose a method based on maximum likelihood to estimate the long-memory parameter using fractional spline wavelets. The proposed estimator exhibits competitive behavior in terms of mean squared error in the simulation studies we conducted, dominating all other methods as the sample size increases. In empirical applications, the proposed method outperformed other approaches in 10 of 12 evaluations for three different time series. Moreover, the proposed method allows for estimation in a continuous search grid, even when the Hurst coefficient lies outside of stationarity boundaries, which is a clear advantage over many algorithms: in such cases, the proposed method displays a stable behavior and an increasing outperformance margin. We also propose an identification procedure for the persistence or antipersistence of a given signal, based on the energy concentration of the coefficients of wavelet decomposition and its theoretical properties.

Explicit representations for Banach subspaces of Lizorkin distributions

The Lizorkin space is well suited to the study of operators like fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. In return, we show that it is dense in [Formula: see text], a property that is shared by the larger Schwartz space and that turns out to be useful for applications. Based on this result, we investigate subspaces of Lizorkin distributions that are Banach spaces and for which a continuous representation operator exists. Then, we introduce a variational framework that involves these spaces and that makes use of the constructed operator. By investigating two particular cases of this framework, we are able to strengthen existing results for fractional splines and 2-layer ReLU networks.

Comparative Numerical Solution of Fractional Spline with Continuity Equations

In this paper, constructed a fractional polynomial spline to compute the solution of FDEs; the spline interpolation with fractional polynomial coefficients must be constructed using the Caputo fractional derivative. For the provided spline function, error bounds were studied and a stability analysis was completed. To consider the numerical explanation for the provided method and compared, three examples were studied. The fractional spline function, which interpolates data, appears to be useful and accurate in solving unique problems, according to the research.

On the numerical solution of Volterra and Fredholm integral equations using the fractional spline function method

In this article, the researchers develop a new type of spline function with fractional order which constructs two distinct formulas for the proposed method by using fractional boundary conditions and fractional continuity conditions. These methods are used to solve linear Volterra and Fredholm-integral equations of the second kind. The convergence analysis is studied. Moreover, some numerical examples are provided and compared to illustrate the efficiency and applicability of the proposed methods.

Non-polynomial fractional spline method for solving Fredholm integral equations

A new type of non-polynomial fractional spline function for approximating solutions of Fredholm-integral equations has been presented. For this purpose, we used a new idea of fractional continuity conditions by using the Caputo fractional derivative and the Riemann Liouville fractional integration to generate fractional spline derivatives. Moreover, the convergence analysis is studied with proven theorems. The approach is also well-explained and supported by four computational numerical findings, which show that it is both accurate and simple to apply.

Optimized Trigonometric Spline Function with Conjugate GradientMethod for Solving FDEs

We investigate the non-polynomial spline function to solve the fractional differential equations with the conformable conjugate gradient method. The fractional derivative was described using the Caputo fractional derivative to construct the spline scheme with polynomial fractional order. Therefore, transform the problem to an equivalent iterative linear system that can be solved by Gauss-Seidel and conjugate gradient methods. For the given spline function, error bounds were studied and a stability analysis was completed, the error estimation is also calculated as different values of (n) depend on the step size oh (h). Numerical examples with known analytical solutions are shown to verify the method's accuracy. The outcomes are in satisfactory correlation with the exact answers according to the numerical experiments. Moreover, the convergence analysis was investigated with the drive some theorems. Also, the procedure is explained in depth and supported by computational examples and the results show that the fractional spline function which interpolates data is productive and profitable in solving unique problems and compare with the exact solutions.

Long-memory parameter estimation based on fractional spline wavelets

Fractional splines extend Schoenberg's B-splines to fractional orders, which were proven to fulfill all requirements to form wavelet bases. Moreover, some of these fractional splines wavelets act approximately as fractional difference operators for signals that are essentially lowpass and concentrated around the origin, which makes them helpful in analyzing series with fractal behavior. In this article, we propose a novel estimator for the long memory parameter of a time series based on the fractional spline discrete wavelet transform (FrDWT), using the fact that it works approximately as a fractional differentiator. We perform simulations and examples to illustrate the proposed method. The proposed estimator outperforms traditional and widely used estimators in simulated data. We also show that, as the sample size increases, the bias in the simulation tends to decrease, as well as the variance, which might indicate that the estimator is consistent.

Wavelet sparsity enhancement for extracting transient vibration signatures of bearing structural damages

Wavelet methods are widely used in mechanical transient vibration signature detection and fault diagnosis. Undesirable artifacts (e.g., spurious noise spikes and pseudo-Gibbs components) are serious issues for the mainstream methods, probably resulting in inaccurate analysis results. For this reason, a new wavelet sparsity enhancement methodology is proposed to achieve artifact-free extraction of bearing transient vibration signatures. The wavelet sparsity is enhanced through two aspects—that is, wavelet basis design and wavelet coefficient processing. Specifically, we use a new family of wavelets called fractional B-spline wavelets on bearing fault signal analysis and propose the adaptive undecimated fractional spline wavelet transform which addresses the shift-invariant problem and also permits to customize an optimal wavelet basis according to the signal itself adaptively. Meanwhile, we introduce a two-step wavelet processing method including a nonlinear operator and a generalized hard-thresholding (with symmetric or asymmetric thresholds determined automatically by wavelet coefficients at each level). Moreover, unlike the most existing wavelet methods, the approximation coefficients are also processed along with the detailed coefficients to remove the possible low-frequency noise. The final transient vibration signatures are reconstructed with the processed coefficients and would be sparser, more accurate, and almost free of artifacts. The validity of the methodology is verified with the analysis results of vibration signals measured from fault-injection experiment and industrial wind turbine transmission system. The comparisons highlight the advantages of the methodology over several common methods in suppressing artifacts and extracting the sparse transient vibration signatures of bearing structural damages.

Spline Fractional Polynomial for Computing Fractional Differential Equations

We propose a fractional spline method for solving fractional differential equations subject to initial conditions. Using the Caputo fractional integral and derivative have to construct the spline interpolation with polynomial coefficients. For the given spline function, error bounds were studied and a stability analysis was completed. The numerical explanation for the provided method was considered using three examples. The results show that the fractional spline function which interpolates data is productive and profitable in solving unique problems.

Shape Preserving Piecewise KNR Fractional Order Biquadratic C 2 Spline

In a recent article, a piecewise cubic fractional spline function is developed which produces C 1 continuity to given data points. In the present paper, an interpolant continuity class C 2 is preserved which gives visually pleasing piecewise curves. The behavior of the resulting representations is analyzed intrinsically with respect to variation of the shape control parameters t and s. The data points are restricted to be strictly monotonic along real line.

A comparison of statistical methods for age-specific reference values of discrete scales

Age-specific reference values are important in medical science to evaluate the normal ranges of subjects and to help physicians signal potential disorders as early as possible. They are applied to many types of measurements, including discrete measures obtained from questionnaires and clinical tests. These discrete measures are typically skewed to the left and bounded by a maximum score of one (or 100%). This article investigates the performances of various statistical methods, including quantile regression, the Lambda-Mu-Sigma (LMS) method and its extensions, and the generalized additive models for location, scale, and shape with zero and one-inflated distributions implemented with either fractional polynomials or splines, for age-specific reference values on discrete measures. Their large-sample performances were investigated using Monte-Carlo simulations, and the consistency of splines and fractional polynomials age profiles with quantile regression had been demonstrated as well. The advantages and disadvantages of these methods were illustrated with data on the Infant Motor Profile, a test score on motor behavior in children of 3–18 months. We concluded that quantile regression with fractional polynomials approach is a robust and computationally efficient method for setting age-specific reference values for discrete measures.

Fractional spline wavelets within the surface texture analysis

The article describes the possibilities of applying fractional spline wavelets for diagnosing the surface texture of machine parts. The research was conducted on the surface profiles of samples prepared using additive technology and face milling. The surface profiles were analysed using fractional spline wavelets for various scaling function order values. The obtained results were compared with filtration results acquired using a one-dimensional wavelet transform, for various groups of mother wavelets. In order to better illustrate the spline wavelet application-wise possibilities, the analysis was conducted at several decomposition levels. The executed tests indicated the advantages and disadvantages of an approach towards surface texture using fractional spline wavelets, which can be directly translated to better surface diagnostics of manufactured machine parts.

Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

In this article, we begin by introducing two classes of lacunary fractional spline functions by using the Liouville–Caputo fractional Taylor expansion. We then introduce a new higher-order lacunary fractional spline method. We not only derive the existence and uniqueness of the method, but we also provide the error bounds for approximating the unique positive solution. As applications of our fundamental findings, we offer some Liouville–Caputo fractional differential equations (FDEs) to illustrate the practicability and effectiveness of the proposed method. Several recent developments on the the theory and applications of FDEs in (for example) real-life situations are also indicated.

Sparse representations for fault signatures via hybrid regularization in adaptive undecimated fractional spline wavelet transform domain

Recent studies on vibration-based machine diagnostics have highlighted the role played by the wavelet transform (WT). However, common WT-based denoising methods (e.g. wavelet thresholding and non-penalty regularization) are often challenging in attaining accurate sparse representations of fault signatures in practice due to artifacts. In this paper, a method via hybrid regularization in the adaptive undecimated fractional spline WT (AUFrSWT) domain is introduced to achieve the accurate sparse representations of fault signatures from strong noise environments. The method promotes wavelet sparsity by two aspects: wavelet basis design and wavelet coefficient processing. For the former, a new wavelet family, i.e. a fractional spline wavelet, is used, and the AUFrSWT is originally proposed to customize the optimal wavelet basis and meanwhile address the translation-invariant issue. For the latter, the wavelet coefficients are estimated by minimizing a single convex model function with hybrid regularization, where nonconvex arctangent penalty is employed for wavelet sparsity promoting and total variation is used to improve the reconstruction performance of the shrunken coefficients. The method is validated by the analysis of actual bearing and gear vibration data from fault-injection experiments. Analysis and comparison results show great potential for signal sparse representations and machinery fault diagnostics.

Numerical method with fractional splines for a subdiffusion problem

We consider a subdiffusion problem described by a time fractional Riemann–Liouville derivative of order $$0<\alpha <1$$ . The main purpose of this work is to show how we can apply fractional splines of order $$0<\beta \le 1$$ to approximate a fractional integral and hence how to solve the subdiffusion problem using this approach. To discuss the convergence of the numerical method we present the error bounds for the fractional splines and the fractional integral approximations and study the von Neumann stability analysis. We observe that, depending on the smoothness of the solution, the order of convergence will be affected by the values of $$\alpha $$ and $$\beta $$ . Numerical tests are presented along the work to highlight several properties of the fractional splines and the numerical tests in the end illustrate the performance of the numerical method.

Sparsity-based fractional spline wavelet denoising via overlapping group shrinkage with non-convex regularization and convex optimization for bearing fault diagnosis

Vibration monitoring has become a relatively common method for the fault diagnosis of rolling bearings in recent years. Vibration-based fault diagnosis technologies essentially rely upon accurate fault signature extraction from noisy vibration signals. To this end, this paper presents a sparsity-assisted denoising method using the fractional spline wavelet transform (FrSWT) and overlapping group shrinkage (OGS) with non-convex regularization and convex optimization (OGSNCRCO) to diagnose bearing faults. The FrSWT is used to promote sparsity of the wavelet coefficients for bearing fault signals. The OGSNCRCO balances data consistency and sparsity by using non-convex regularization and convex optimization, and is applied to shrink the wavelet coefficients of noisy signals in the fractional spline wavelet domain to extract bearing fault signals. Simulations and experiments validate the efficiency of the proposed method for fault diagnosis of rolling bearings. The results of analysis suggest that the proposed method is superior to other state-of-art techniques also commonly used for extracting fault signatures of rolling bearings, including the L1-norm denoising method (FrSWT with L1-norm regularization) and spectral kurtosis method.

Analysis on Wavelet Family based Channel Estimation for MIMO-OFDM System

Channel estimation for (MIMO-OFDM) is an important part for present and future generation broadband wireless communications. OFDM, which uses for the spaced subcarriers to improving the performance. The channel estimation schemes based on pilot reduces the transmission rate and spectral efficiency. Many conventional schemes of channel estimation are not effective in reducing noise. It leads to poor quality signals at receiver at final stage. To rectify this problem, in this paper a wavelet family based channel estimation technique, is proposed and analyzed. The proposed research aims to estimate channels and reconstruct signal via wavelet transform, dyadic wavelet transform and fractional spline wavelet transform which enhances the spectrum efficiency and transmission rate. Simulation results shows that the fractional spline wavelet transform performs well for channel estimation and data signal reconstruction.

Influence of the edge effect on the wavelet analysis process

The article describes the research results in which the size of the edge effect was evaluated. There were analysed results of the filtration process of the surface profiles, obtained by different manufacturing methods and selected wavelet transformations. Measurements were made using contact profilometer over an evaluation length of 17.5 mm. Comparative tests were carried out on the profiles obtained as a result of filtration whole measuring section, with the tests taking into account only the central part profile and filtration carried out only on the central part of the profile. The research was carried out using discrete wavelet transformation, complex frequency B-Spline wavelet and fractional spline wavelet. For each method it was evaluated the size of the edge effect on a particular level of decomposition and based on results it was concluded that a fractional spline wavelet is a suitable tool for analysing the profiles of the surface texture.