Fourier Method Research Articles

An adaptive stratification algorithm based on gradient fitting deviation and its application to acoustic ray-tracing algorithm

Accurate underwater positioning is a challenging task for complex marine environments where the study of sound velocity profiles (SVPs) is essential. To overcome the problem that the original SVP reduces the computational efficiency of the ray-tracing algorithm due to great capacity of data, this paper simplifies the SVP. The nature of the SVP and its influence on accurate underwater positioning are discussed. An adaptive stratification algorithm applicable to the ray-tracing algorithm is proposed, based on the gradient fitting deviation and characterized by the fitting treatment of the sound velocity gradient by the Fourier method. Measured SVP data and acoustic positioning data were used to validate this algorithm in the context of acoustic ray-tracing algorithm, to analyze and compare the simplification rates before and after SVP simplification, and the effect on the positioning accuracy. The experimental results in the shallow sea (water depth of 60 m) show that the simplification rate of the SVP is more than 95.56%, and the RMS of the deviation of the ray-tracing algorithm is better than 0.0112 m when determining the optimal number of unfolding levels and threshold; the experimental results in the deep sea (water depth of 1900 m) show that the simplification rate of the SVP is more than 99%, and the RMS of the deviation of the ray-tracing algorithm is better than 0.5 m. This indicates that, the algorithm can reconstruct the original SVP well and implement the automatic hierarchical simplification of the SVP while ensuring the positioning accuracy.

A comparison between Hankel and Fourier methods for photothermal radiometry analysis

AbstractPhotothermal radiometry has recently been investigated for use in the multidimensional thermal characterization of anisotropic samples. In application, there are two principal thermal models available for such characterization: a Cartesian model for the heat equation, which requires the application of three Fourier transforms to arrive at a solution (dubbed the Fourier technique), and a cylindrical model for the heat equation, which requires the application of a Hankel transform and a single Fourier transform (dubbed the Hankel technique). The Fourier technique allows for three‐dimensional characterization, while the Hankel technique is expected to greatly reduce the computational time required. As these models can be very computationally expensive, the potential to reduce this cost is of great interest. In this work, these multidimensional models are presented after which they are compared for accuracy, computational time, and assumption limitations. It was found that both the Fourier and Hankel techniques could accurately arrive at desired thermal properties, but that the Hankel Technique reduced the computational time by between 100× and 250× depending upon mesh spacings. Accuracy limitations were found as the eccentricity of the heating laser was increased with a less than 13% error being induced from a beam with a 3–1 axis ratio. The Hankel technique shows ideal application in computationally expensive models which employ a relatively circular beam shape.

Mid-Infrared Supercontinuum Generation in Highly Nonlinear Chalcogenide Fibers

Interest in mid-infrared broadband laser light sources has surged due to applications in trace gas detection, free-space communications, and countermeasures. Progress in supercontinuum generation leverages fiber-based near-infrared and bulk-optic mid-infrared pump sources. In this paper, the Generalized Nonlinear Schrödinger Equation has been solved, using the Split Step Fourier Method, to simulate the pulse propagation and mid-infrared supercontinuum generation, inside a fiber composed of highly nonlinear As2Se3/As2S3 chalcogenide glass. The effect of various parameters, including fiber nonlinearity, Group Velocity Dispersion (GVD), input power and pulse-width, anomalous and normal dispersion pumping regime, etc. on the output supercontinuum bandwidth has been extensively studied. A tapered chalcogenide fiber is modeled to facilitate continuous simultaneous modification of the GVD and the Kerr nonlinearity parameter. Pumping the waveguides with 230-fs secant pulses at a peak power of 4.2-kW yields a mid-IR supercontinuum extending from [Formula: see text] to [Formula: see text] micrometers.

Compressible and anelastic governing-equation solution methods for thermospheric gravity waves with realistic background parameters

Abstract abstract A previously developed numerical-multilayer modeling approach for systems of governing equations is extended so that unwanted terms, resulting from vertical variations in certain background parameters, can be removed from the dispersion-relation polynomial associated with the system. The new approach is applied to linearized anelastic and compressible systems of governing equations for gravity waves including molecular viscosity and thermal diffusion. The ability to remove unwanted terms from the dispersion-relation polynomial is crucial for solving the governing equations when realistic background parameters, such as horizontal velocity and temperature, with strong vertical gradients, are included. With the unwanted terms removed, previously studied dispersion-relation polynomials, for which methods for defining upgoing and downgoing vertical wavenumber roots already exist, are obtained. The new methods are applied to a comprehensive set of medium-scale time-wavepacket examples, with realistic background parameters, lower boundary conditions at 30 km altitude, and modeled wavefields extending up to 500 km altitude. Results from the compressible and anelastic model versions are compared, with compressible governing-equation solutions understood as the more physically accurate of the two. The new methods provide significantly less computationally expensive alternatives to nonlinear time-step methods, which makes them useful for comprehensive studies of the behavior of viscous/diffusive gravity waves and also for large studies of cases based on observational data. Additionally, they generalize previously existing Fourier methods that have been applied to inviscid problems while providing a theoretical framework for the study of viscous/diffusive gravity waves. Graphic abstract

Computational and Implementation Problems in Option Pricing with Long Maturity

In this paper we explore the significance of maturity parameter when the Fourier method is applied in option pricing by using two different analytical solutions of the Black-Scholes equation, that are expressed as infinite series of reflections and Fourier series. Even for fixed barriers contracts different solutions give the same answer when all the terms have been added up but the rate of convergence of the sum to the solution can be quite different, depending on the time to expiry (maturity), i.e. the duration of the contract. Thus, we demonstrate the significance of the maturity parameter in option contracts when analytical methods like Fourier one is applied not only in jump-diffusion models.

Separation and Extraction of Compound-Fault Signal Based on Multi-Constraint Non-Negative Matrix Factorization.

To solve the separation of multi-source signals and detect their features from a single channel, a signal separation method using multi-constraint non-negative matrix factorization (NMF) is proposed. In view of the existing NMF algorithm not performing well in the underdetermined blind source separation, the β-divergence constraints and determinant constraints are introduced in the NMF algorithm, which can enhance local feature information and reduce redundant components by constraining the objective function. In addition, the Sine-bell window function is selected as the processing method for short-time Fourier transform (STFT), and it can preserve the overall feature distribution of the original signal. The original vibration signal is first transformed into time-frequency domain with the STFT, which describes the local characteristic of the signal from the time-frequency distribution. Then, the multi-constraint NMF is applied to reduce the dimensionality of the data and separate feature components in the low dimensional space. Meanwhile, the parameter WK is constructed to filter the reconstructed signal that recombined with the feature component in the time domain. Ultimately, the separated signals will be subjected to envelope spectrum analysis to detect fault features. The simulated and experimental results indicate the effectiveness of the proposed approach, which can realize the separation of multi-source signals and their fault diagnosis of bearings. In addition, it is also confirmed that the proposed method, juxtaposed with the NMF algorithm of the traditional objective function, is more applicable for compound fault diagnosis of the rotating machinery.

Non-Intrusive Frequency Measurements With Laser Shadowgraphy And Event-Based Schlieren Imaging Of Bluff Body Wakes In Compressible Flow

Event-based imaging is an emerging technology in thermodynamics and fluid dynamics. In this work, we are leveraging an event-based camera with a conventional schlieren set-up to investigate the vortex shedding dynamics behind a blunt central injector in a transonic nozzle flow. We offer different methods to evaluate the data stream of the event-based camera. Standard Fourier methods are not directly applicable to this data due to a non-constant time step. Therefore, the data needs to be treated with special care. Clusters of on- and off-events alternate and show the periodic nature of the wake. Using a clustering algorithm on the event groups also provides an estimation of vortex shedding frequency. These results are compared with laser shadowgraphy, a second novel method to measure the vortex shedding frequency in the wake. Both approaches to measuring the vortex shedding frequency are agreeing with each other within 1%. They each offer a simple set-up with comparatively low cost for their high temporal resolution.

Challenges by latent heat calculation – Competition among analytical and computational methods

Determining the latent heat of multi-component cast aluminum alloys is complex. These alloys solidify within the mushy zone, with their latent heat release influenced by factors like composition, cooling rate, and microstructure. Techniques such as DSC and DTA, along with software tools like JMatPro, FactSage, and ThermoCalc, can determine this value, understand solidification parameters, and calculate latent heat. This paper introduces a novel approach using Newtonian, Fourier, and Energy Balance methods simultaneously, for the first time, to calculate the latent heat of pure aluminum and its AlSi7Cu1 alloy. These methods offer advantages in foundry conditions and require no specialized operator for data interpretation. Commercial software typically provides parameters for standard alloys only, necessitating alternative sources for accurate data. Thermal analysis techniques offer a reliable method to acquire missing parameters and calculate latent heat with high precision. Comparing the accuracy of these methods for pure aluminum and its AlSi7Cu1 alloy, using DSC measurement data and commercial software values, indicates successful application on the foundry floor to determine thermophysical properties accurately.

Denoising of Dense Spatial Array Data Using the Curvelet Transform

ABSTRACT To remove background noise from seismic data recorded by spatially dense arrays, we have developed a space-based denoising procedure using the discrete curvelet transform. Based on a detailed statistical characterization of noise coefficients through the empirical cumulative distribution function method within a pre-event time window, signal and noise can be separated effectively by nonlinear thresholding. After synthetic test, we applied this method on data from an industry 3D seismic experiment recorded at an array deployed near Utica, Ohio. The denoising results show good waveform consistency with a significantly enhanced signal-to-noise ratio. Our curvelet approach allows a more computationally efficient spatial–temporal localization analysis of seismic data than conventional curvelet techniques by avoiding the assumption of stationary Gaussian-distributed noise and can be implemented as a complement of time-domain wavelet methods with fewer frequency losses after denoising. This new method provides a fast and convenient way to recover signals from noisy recordings with dense 2D arrays, leading to a considerable improvement in data quality compared with conventional Fourier, wavelet, and curvelet methods. The partitioned seismic signals and noise would yield advanced earth structure imaging, small-event detection, ambient noise tomography, and others.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

A theoretical study on deep ultraviolet supercontinuum generation in subwavelength optical fiber

In order to obtain supercontinuum (SC) spectra in the deep ultraviolet (UV) region, a generation method of UV SC is proposed based on subwavelength fiber (SWF) composed of background material MgF2. The nonlinear Schrödinger equation is solved by the split-step Fourier method, the optical fiber transmission characteristics are analyzed by the full-vector finite element, and the evolution of the pumping light into SC is numerically simulated in detail. The results of the numerical analysis show that the SC spectra have a spectral width extending from 229.11 nm to 2366.68 nm when 350 fs input pulses of 6 kW peak power, centered about λ0 = 400 nm, are launched into a 4 cm long MgF2 SWF. Such a deep UV SC spectrum has many applications, especially in telecommunication and military fields.

The Digital Twin Framework for the Physical Wideband and Long‐Haul Optical Fiber Communication Systems

AbstractDigital twin (DT) modeling is essential to optical fiber communication systems, particularly for enhancing system performance, controlling the system in real time, and understanding signal nonlinearity. Conventional split‐step Fourier method ‐based simulations, however, struggle with wide‐band transmissions, plagued by increasing complexity and inherent biases due to inconsistent link parameter availability. Addressing these challenges, a hybrid data‐driven and model‐driven DT approach for the wide‐band and long‐haul physical systems with various system effects is developed. The approach utilizes a neural network (NN) to capture fiber nonlinear features as well as biased perturbations as “lumped” stochastic noises while offloading the linear effects to modules described by physical models of link elements. The model, tested in a 30.5‐Tbps 1200 km fiber transmission link with 40 channels, achieves a mean Q factor error of less than 0.1 dB and a maximum runtime of 1.3 s for NN processing under various launch powers, transmission lengths, and optical signal‐to‐noise ratios. Furthermore, the study has implemented a nonlinear compensation algorithm on the DT model, yielding a consistent enhancement in experimental data. The accuracy and adaptability of the DT model underline its suitability for planning, design, and optimization within the physical optical fiber communication systems.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

APPLICATION OF DIFFERENTIAL EQUATIONS IN VARIOUS FIELDS OF SCIENCE

The article, "Application of Differential Equations in Various Fields of Science," explores the use of differential equations for modeling economic and natural phenomena. It examines two main models of economic dynamics: the Evans model for the market of a single product, and the Solow model for economic growth.The author emphasizes the importance of proving the existence of solutions to differential equations in order to verify the accuracy of mathematical models. They also discuss the role of electronic computers in developing the theory of differential equations and its connection with other branches of mathematics such as functional analysis, algebra, and probability theory.Furthermore, the article highlights the significance of various solution methods for differential equations, including the Fourier method, Ritz method, Galerkin method, and perturbation theory.Special attention is paid to the theory of partial differential equations, the theory of differential operators, and problems arising in physics, mechanics, and technology. Differential equations are the theoretical foundation of almost all scientific and technological models and a key tool for understanding various processes in science, such as in physics, chemistry, and biology.Examples of processes described by differential equations include normalreproduction, explosive growth, and the logistic curve. Cases of using differential equations to model deterministic, finite-dimensional, and differentiable phenomena, as well as the impact of catch quotas on population dynamics, are discussed.In conclusion, the significance of differential equations for research and their role in stimulating the development of new mathematical areas is emphasized.

GLOBAL SOLVABILITY OF AN INVERSE PROBLEM FOR A MOORE-GIBSON-THOMPSON EQUATION WITH PERIODIC BOUNDARY AND INTEGRAL OVERDETERMINATION CONDITIONS

This article studies the inverse problem of determining the pressure and convolution kernel in an integro-differential Moore-Gibson-Thompson equation with initial, periodic boundary and integral overdetermination conditions on the rectangular domain. By Fourier method this problem is reduced to an equivalent integral equation and on based of Banach’s f ixed point argument in a suitably chosen function space, the local solvability of the problem is proven. Then, the found solutions are continued throughout the entire domain of definition of the unknowns.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

Advancing computational models for wave dynamics applications with quartic trigonometric tension b-spline techniques

This paper presents a computational model using Quartic Trigonometric Tension B-spline (QT-TB) function with collocation approach for nonlinear dispersive wave equation. Shallow water wave modeling plays a crucial role in various fields, including coastal engineering, oceanography, and natural hazard assessment. The QT-TB function is employed for spatial derivatives, integrating the nonlinear term through the linearization technique. While maintaining most characteristics of traditional polynomial B-splines, this method improves numerical solutions, enhancing accuracy in modeling. The performance of the method is rigorously assessed across three benchmark problems, with results compared to those of prior studies employing identical parameters. Detailed numerical illustrations are presented. Graphical representations are utilized to illustrate the single solitary wave motion, dynamics of coupled solitary waves and dynamics of triplet solitary waves in this study. Numerical experiments validate the method's accuracy and efficiency in capturing RLW-driven wave dynamics, establishing it as a reliable tool for various applications. The presented work includes an analysis of the stability of the proposed scheme, employing the Fourier method and shows that its unconditional stable.

Nonlinear impairment compensation in multi-channel communication systems based on correlated digital backpropagation with separation of walk-off effect

Digital backpropagation (DBP) is a commonly used method to compensate for the nonlinear impairments of optical fiber channels in the electrical domain, offering a new approach to enhancing the data capacity without affecting the system performance. However, due to the repeated Fourier transform and inverse transform of the split-step Fourier method (SSFM), the complexity of DBP algorithm has increased significantly. In this paper, a correlated backpropagation algorithm with the separation of the walk-off effect is developed, which improves the computational step and significantly reduces the complexity of the algorithm. Simulation results demonstrate that in a 6-channel PDM-16QAM system, the proposed hybrid algorithm reduces the computational complexity by 73.96 % compared to the traditional DBP algorithm for an 800 km transmission.

Calculation of RF pulse evolution due to dispersion in travelling wave linacs using Fourier methods

An rf pulse propagating through a travelling wave linac can be seen to evolve due to dispersion. To accurately predict the field amplitude experienced by electron bunches travelling through the cavity, this pulse evolution must be well known. Here we present an method to predict the pulse evolution using Fourier methods. The method requires two inputs to calculate the dispersion: the group velocity as a function of cell number, and the phase advance. Attenuation is included in the model with the addition of a third input: the cavity Q0. The method is faster and more simple than 3D modelling, and allows subtle details of the pulse evolution to be revealed, without the need to know the exact dimensions of the structure. A synchronism condition can be added to predict the voltage experienced by a particle beam. The model is tested on the first CLARA linac, and shows good agreement both with measurements of the rf pulse and with measurements of the beam momentum as the time of beam injection is varied.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions