Frequency formula for a class of fractal vibration system

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

Image design and interaction technology based on Fourier inverse transform

As one of the main directions of applied mathematics research, inverse Fourier transform (FT) has been widely used in image speech analysis and other fields in recent decades of development. FT is the basic content of digital image processing technology. In practical analysis, image design and interaction can be realised by using time-space domain and frequency domain, which can accurately obtain image information characteristics and achieve the expected application goals. In this paper, based on the understanding of FT and inverse transform, an improved algorithm is used to lay the foundation for the realisation of image design and interactive technology.

A boundary division guiding synchrosqueezed wave packet transform method for rolling bearing fault diagnosis

Synchrosqueezed wave packet transform (SSWPT) can effectively reconstruct the band-limited components of the signal by inputting the specific reconstructed boundaries and it provides an alternative bearing fault diagnosis method. However, the selection of reconstructed boundaries can significantly affect the fault feature extraction performance of SSWPT. Accordingly, this paper presents a boundary division guiding SSWPT (BD-SSWPT) method. In this method, an adaptive boundary division method is developed to effectively determine the reconstructed boundaries of SSWPT. Firstly, the marginal spectrum of SSWPT, more robust to noise than the Fourier spectrum, is defined for the scale-space division to obtain the initial boundaries. Secondly, the inverse transform of SSWPT is conducted based on the initial boundaries to obtain the initial reconstructed components. Thirdly, a boundary redefinition scheme, composed of clustering and combination, is conducted to redefine the boundaries. Finally, the potential components are extracted by the inverse transform of SSWPT based on the redefined boundaries. The validity of BD-SSWPT is verified by simulated and experimental analysis, and the superiority of BD-SSWPT is highlighted through comparison with singular spectrum decomposition (SSD) and an adaptive parameter optimized variational mode decomposition (AVMD). The results demonstrate that BD-SSWPT identifies more significant fault features and has higher computational efficiency than SSD and AVMD.

DCT based Fusion of Variable Exposure Images for HDRI

Combining images with different exposure settings are of prime importance in the field of computational photography. Both transform domain approach and filtering based approaches are possible for fusing multiple exposure images, to obtain the well-exposed image. We propose a Discrete Cosine Trans- form (DCT-based) approach for fusing multiple exposure images. The input image stack is processed in the transform domain by an averaging operation and the inverse transform is performed on the averaged image obtained to generate the fusion of multiple exposure image. The experimental observation leads us to the conjecture that the obtained DCT coefficients are indicators of parameters to measure well-exposedness, contrast and saturation as specified in the traditional exposure fusion based approach and the averaging performed indicates equal weights assigned to the DCT coefficients in this non- parametric and non pyramidal approach to fuse the multiple exposure stack.

Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models

This paper aims to value the cliquet-style equity-linked insurance product with death benefits. Whether the insured dies before the contract maturity or not, a benefit payment to the beneficiary is due. The premium is invested in a financial asset, whose dynamics are assumed to follow an exponential jump diffusion. In addition, the remaining lifetime of an insured is modelled by an independent random variable whose distribution can be approximated by a linear combination of exponential distributions. We found that the valuation problem reduced to calculating certain discounted expectations. The Laplace inverse transform and techniques from existing literature were implemented to obtain analytical valuation formulae.

Two-sided fractional quaternion Fourier transform and its application

In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

Nonstatinary deconvolutive Radon transform

Different versions of the Radon transform (RT) are widely used in seismic data processing tofocus the recorded seismic events. Multiple separation, data interpolation, and noise attenuationare some of RT applications in seismic processing work-flows. Unfortunately, the conventional RTmethods cannot focus the events perfectly in the RT domain. This problem arises due to theblurring effects of the source wavelet and the nonstationary nature of the seismic data. Sometimes,the distortion results in a big difference between the original data and its inverse transform. Wepropose a nonstationary deconvolutive RT to handle these two issues. Our proposed algorithm takesadvantage of a nonstationary convolution technique. that builds on the concept of block convolutionand the overlap method, where the convolution operation is defined separately for overlapping blocks.Therefore, it allows the Radon basis function to take arbitrary shapes in time and space directions. Inaddition, we introduce a nonstationary wavelet estimation method to determine time-space-varyingwavelets. The wavelets and the Radon panel are estimated simultaneously and in an alternative way.Numerical examples demonstrate that our nonstationary deconvolutive RT method can significantlyimprove the sparsity of Radon panels. Hence, the inverse RT does not suffer from the distortioncaused by the unfocused seismic events.

The Analysis of Fractional-Order Helmholtz Equations via a Novel Approach

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.