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In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L∞. In addition, we provide numerical results to support our theoretical analysis.

The current investigation focuses on presenting an efficient and reliable computational technique for solving a general class of one‐ and two‐dimensional multi‐term time‐fractional diffusion and diffusion‐wave equation (2D‐MT‐TFD‐DWE) with constant coefficients using the Galerkin scheme based on two distinct choices of basis functions. The major idea for obtaining the proposed numerical algorithms is based on constructing trial functions in the Galerkin scheme as compact combinations of shifted Jacobi polynomials. The existence and uniqueness of the considered problem are investigated using the Lax–Milgram theorem. Five numerical examples are considered with comparisons with analytical solutions and with numerical solutions that are given in the literature using some other existing techniques to confirm the effectiveness and accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of a new augmented formulation in terms of vorticity, velocity and pressure for the Stokes equations with variable viscosity and mixed boundary conditions. The well-posedness of the continuous problem holds under assumptions on the viscosity. When the domain is a parallelepiped, the spectral discretization is proposed using the Galerkin method with numerical integration. Then, we prove the well-posedness of the obtained discrete problem under the same type of conditions on the viscosity. A priori error estimates is then derived for the three unknowns. Finally, numerical experiments are presented that confirm the interest of the discretization. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the numerical approximation of a three‐parameter family of Boussinesq systems. The systems have been proposed as models of the propagation of long internal waves along the interface of a two‐layer system of fluids with rigid‐lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of the systems on well‐posedness, conservation laws, Hamiltonian structure, and solitary‐wave solutions, using the results for analogous models for surface wave propagation. Then the corresponding periodic initial‐value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the semidiscrete approximation are proved. The spectral semidiscretizations are numerically integrated in time by a fourth‐order Runge–Kutta‐composition method based on the implicit midpoint rule. Numerical experiments illustrate the accuracy of the fully discrete scheme, in particular its ability to simulate accurately solitary‐wave solutions of the systems. [ABSTRACT FROM AUTHOR]

In this work, we present a numerical study of the wave stability of steady solitary waves over a localised topographic obstacle through the full Euler equations. There are two branches of the solutions: one from the perturbed uniform flow and the other from the perturbed solitary-wave flow. We find that steady waves from the perturbed uniform flow are always stable with respect to perturbations of its amplitude. Regarding the perturbed solitary wave, when the perturbed initial condition has smaller amplitude than the steady solution, we notice a certain type of stability. Yet, when the perturbed initial condition has larger amplitude than the steady solution an onset of wave-breaking seem to occur. [ABSTRACT FROM AUTHOR]

The electrochemical behavior of disordered systems, such as high-entropy alloys, is a stochastic random process. To accurately predict and analyze the behavior of such systems under operating conditions, it is necessary to use new computational and experimental methods along with classical electrochemical methods. Using equimolar rare-earth alloys GdTbDyHoSc and GdTbDyHoY as an example, we demonstrate the efficiency of using fast Fourier transform and wavelet analysis to estimate the electrochemical behavior of stochastic systems. The time series of potential fluctuations of alloy samples are measured in 0.01 M NaCl solution within 12 h at a current density of 0.2–0.5 mA/cm2. Fast Fourier transform analysis of the obtained time series shows that the slope of the logarithm of spectral power density to the logarithm of frequency increases with the current density. In particular, coefficient β changes from –1.93 to –1.77 for a GdTbDyHoY sample and from –1.46 to –1.35 for a GdTbDyHoSc sample. In addition, wavelet analysis is used to process the time series obtained for both alloys at current densities from 0.2 to 0.5 mA/cm2. To illustrate the intensity of the electrochemical dissolution of the alloy surface, we construct scalograms for the obtained time series. The scalograms are used to calculate the global energy spectra distributed over frequency ranges and the total energies of the systems under study. The GdTbDyHoY alloy exhibits higher total energies as compared to the GdTbDyHoSc alloy. The total energy for the GdTbDyHoY alloy increases from 0.97 to 2.03 kV2 when the current density increases from 0.2 to 0.5 mA/cm2. For the GdTbDyHoSc alloy, the total energy increases from 0.50 to 0.84 kV2. Fast Fourier transform and wavelet analysis are found to be effective tools for understanding the electrochemical behavior of locally disordered chemical systems, such as high-entropy GdTbDyHoSc and GdTbDyHoY alloys, in addition to classical electrochemical methods. [ABSTRACT FROM AUTHOR]

We investigate in the article the L2‐norm state constrained control problem with first bi‐harmonic equation. First, the optimality conditions of the control problem are derived carefully, and spectral discretization of the problem is established. Based on the property of projection operator, we establish a priori error analysis for control variable, state, and adjoint state variable. Furthermore, the efficient projected gradient algorithm is proposed, and the numerical results we obtained verify the analytical results that it can provide high order accuracy and fast convergence rate. [ABSTRACT FROM AUTHOR]

In this study, a steady magnetohydrodynamic (MHD) flow due to stretchable rotating disk in the presence of gyrotactic microorganisms is investigated. The governing equations modeling the flow are solved numerically using the newly introduced simple iteration method (SIM) that seeks to linearize a system using relaxation technique that effectively decouples the system. To verify the convergence and accuracy of the method, solution error and residual error analysis are carried out, respectively. The obtained results suggest that the SIM is a highly efficient method that produces convergent and highly accurate solutions. The effects of various parameters as well as combined parameter effects on the solution profiles are also investigated. An increase in the Hall and permeability parameters leads to a corresponding rise in the microorganism’s density and nanoparticle volume fraction.

Purpose. The development of transient processes calculation method in electric circuits with thyristors based on the use of functions approximation by orthogonal polynomials. Methodology. Functions approximation by orthogonal polynomials, numerical methods of differential equations integration, matrix methods, programming, theory of electric circuits. Obtained results. The method of solution function polynomial approximation of integro-differential equations of state, which describes the transient processes of an electric circuit with thyristors, is used in this paper. The used method showed the advantages over other known methods in increasing the accuracy and reducing the simulation time of transient electrical processes by more than 6 times. Findings. The solution is approximated by a series of Chebyshev polynomials. The integro-differential equations of state are transformed into linear algebraic equations for special depiction of the solution functions. The depiction of functions of true currents in the equivalent circuit is interpreted as direct currents. Such a schematic model creates visibility for a researcher performing simulation of transient electrical processes. Practical value. The proposed methods discover the possibility of using the apparatus of direct current electric circuits’ theory for transient processes in complex schemes modeling with thyristors.

In this work, we synthesized and characterized the properties of a series of new fluorescent DB3(n) narrow-groove ligands. DB3(n) compounds based on dimeric trisbenzimidazoles have the ability to bind to the AT regions of DNA. The synthesis of DB3(n), whose trisbenzimidazole fragments are linked by oligomethylene linkers of different lengths (n = 1, 5, 9), is based on the condensation of the MB3 monomeric trisbenzimidazole with α,ω-alkyldicarboxylic acids. DB3(n) proved to be effective inhibitors of the catalytic activity of HIV-1 integrase at submicromolar concentrations (0.20–0.30 µM). DB3(n) was found to inhibit the catalytic activity of DNA topoisomerase I at low micromolar concentrations. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral properties of the periodized Radial Basis Function interpolation matrix as well as the related harmonic operators discretized using Radial Basis Functions. For Gaussian RBF, this procedure could be easily extended to an arbitrarily high dimensional space on a tensor-product grid as presented in the later parts of the paper. The experimental result of Boyd's condition number [1] is analytically well predicted in the context of periodized RBF. [ABSTRACT FROM AUTHOR]

This paper is a short retrospective review of the predictive methods of turbulent flows in Computational Fluid Dynamics over the last 50 years since the first development of computers. The different schools of turbulence modeling are presented with the aim to guide both users and researchers involved in numerical simulation of turbulent flows.

In this study, discontinuities and major tectonic boundaries are interpreted in and around Türkiye by Bouguer gravity anomaly. The World Gravity Map 2012 is used for the interpretation of major tectonic features in the Anatolia Region. Radial average power spectrum (RAPS) and band-pass filter are used for long and short wavelength separation. For the whole study area, four depth segments are detected. Moreover, the radial average depths of these depth segments are 54.9 km, 32.2 km, 21.9 km and 8.0 km. In order to conduct better interpretation, the study area was divided into three subareas from the west to the east (area 1 to area 3). In area 1 (41.4 km, 21.2 km and 7.8 km) and area 2 (48.1 km, 20.0 km and 6.6 km), three depth sources are detected. Furthermore, four various depth segments are analysed in area 3 (54.3 km, 29.8 km, 20.8 km and 8.6 km). The interpretation of the whole study area, area 1, area 2 and area 3 showed that depth of the sediment accumulation in the Western Anatolia is estimated as 7.8 km.

Introduction: Clustering is usually the first exploratory analysis step in empirical data. When the data set comprises graphs, the most common approaches focus on clustering its vertices. In this work, we are interested in grouping networks with similar connectivity structures together instead of grouping vertices of the graph. We could apply this approach to functional brain networks (FBNs) for identifying subgroups of people presenting similar functional connectivity, such as studying a mental disorder. The main problem is that real-world networks present natural fluctuations, which we should consider. Methods: In this context, spectral density is an exciting feature because graphs generated by different models present distinct spectral densities, thus presenting different connectivity structures. We introduce two clustering methods: k-means for graphs of the same size and gCEM, a model-based approach for graphs of different sizes. We evaluated their performance in toy models. Finally, we applied themto FBNs of monkeys under anesthesia and a dataset of chemical compounds. Results: We show that our methods work well in both toy models and real-world data. They present good results for clustering graphs presenting different connectivity structures even when they present the same number of edges, vertices, and degree of centrality. Discussion: We recommend using k-means-based clustering for graphs when graphs present the same number of vertices and the gCEM method when graphs present a different number of vertices. [ABSTRACT FROM AUTHOR]

The spectral Tau method to compute eigenpairs of ordinary differential equations is implemented as part of the Tau Toolbox—a numerical library for the solution of integro-differential problems. This mathematical software enables a symbolic syntax to be applied to objects to manipulate and solve differential problems with ease and accuracy. The library is explained in detail and its application to various problems is illustrated: numerical approximations for linear, quadratic, and nonlinear differential eigenvalue problems. [ABSTRACT FROM AUTHOR]

We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be transferred between different graphs. It is hence important to study the GCNN transferability : the capacity of the network to have approximately the same repercussion on different graphs that represent the same phenomenon. Transferability ensures that GCNNs trained on certain graphs generalize if the graphs in the test set represent the same phenomena as the graphs in the training set. In this paper, we consider a model of transferability based on graphon analysis. Graphons are limit objects of graphs, and, in the graph paradigm, two graphs represent the same phenomenon if both approximate the same graphon. Our main contributions can be summarized as follows: 1) we prove that any fixed GCNN with continuous filters is transferable under graphs that approximate the same graphon, 2) we prove transferability for graphs that approximate unbounded graphon shift operators, which are defined in this paper, and 3) we obtain non-asymptotic approximation results, proving linear stability of GCNNs. This extends current state-of-the-art results which show asymptotic transferability for polynomial filters under graphs that approximate bounded graphons. • We study the generalization error of graph convolutional neural networks. • Graphs representing the same phenomenon are modeled via graphon analysis. • We show that networks can be transferred between graphs sampling the same phenomenon. • Our analysis allows working with generic continuous filters. • By introducing unbounded graphons, Euclidean CNNs are a special case of our analysis. [ABSTRACT FROM AUTHOR]

A tensor network is a diagram that specifies a way to ``multiply"" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case. [ABSTRACT FROM AUTHOR]

Streptomycetes produce a huge variety of bioactive metabolites, including antibiotics, enzyme inhibitors, pesticides and herbicides, which offer promise for applications in agriculture as plant protection and plant growth-promoting products. The aim of this report was to characterize the biological activities of strain Streptomyces sp. P-56, previously isolated from soil as an insecticidal bacterium. The metabolic complex was obtained from liquid culture of Streptomyces sp. P-56 as dried ethanol extract (DEE) and possessed insecticidal activity against vetch aphid (Medoura viciae Buckt.), cotton aphid (Aphis gossypii Glov.), green peach aphid (Myzus persicae Sulz.), pea aphid (Acyrthosiphon pisum Harr.) and crescent-marked lily aphid (Neomyzus circumflexus Buckt.), as well as two-spotted spider mite (Tetranychus urticae). Insecticidal activity was associated with production of nonactin, which was purified and identified using HPLC-MS and crystallographic techniques. Strain Streptomyces sp. P-56 also showed antibacterial and antifungal activity against various phytopathogenic bacteria and fungi (mostly for Clavibacfer michiganense, Alternaria solani and Sclerotinia libertiana), and possessed a set of plant growth-promoting traits, such as auxin production, ACC deaminase and phosphate solubilization. The possibilities for using this strain as a biopesticide producer and/or biocontrol and a plant growth-promoting microorganism are discussed. [ABSTRACT FROM AUTHOR]

In this paper, the spectral method is developed as a reduced-order model for the solution of parametric problems within the building refurbishment framework. We propose to use the spectral reduced-order method to solve parametric problems in an innovative way, integrating the unknown parameter as one of the coordinates of the decomposition. The residual is minimized combining the Tau–Galerkin method with the Collocation approach. The developed method is evaluated in terms of accuracy and reduction of the computational time in three different cases. The dynamic behaviour of unidimensional moisture diffusion is investigated. The cases focus on solving parametric problems in which the solution depends on space, time, diffusivity and material thickness. Results highlight that the parametric spectral reduced-order method provides accurate solutions and can reduce 10 times the degree of freedom of the solution. It allows efficient computation of the physical phenomena with a lower error when compared to traditional approaches. [ABSTRACT FROM AUTHOR]

This paper examines the movement of waves that occur in a fuel tank—both with and without a wave breaker—when a car is travelling at a constant speed and then suddenly brakes. This phenomenon is known as slosh noise, and the paper presents an analysis of the movement of free surfaces in relation to the level of noise generated. The paper focuses on mathematical models of the fluid flow for both tanks—one without any technical solutions for breaking waves, and the other with a solution for breaking waves. The model is constructed based on a set of initial hypotheses about the fluid flow within the tank, by developing the speed potential in a series of fundamental solutions and considering the main variables that affect the phenomenon of sloshing, such as the depth of the liquid, the tank's geometry, and the frequency and amplitude of the initial external force acting on the tank. The analysis of free surface movement is used to find the correlation with the sound generated in the tank. Nonlinearities that arise from the sudden braking are also modelled and numerically studied using MATLAB software. Following the mathematical model, a technical wave-breaking solution was implemented and tested, and it was shown that the amplitude of the movement of the free surface is reduced by half. Further research on the correspondence between the free surface movement based on the behaviour of potential energies in the two cases may be developed. [ABSTRACT FROM AUTHOR]

• Physicochemical interaction mechanisms between SMP and TMP were explored. • Fluorescence enhancement effect in the SMP-TMP interaction was elucidated. • Carboxyl, carbonyl and hydroxyl groups were mainly responsible for the interaction. • π-π and electrostatic interactions were involved in the SMP-TMP binding. Soluble microbial products (SMPs), dissolved organic matter excreted by activated sludge, can interact with antibiotics in wastewater and natural water bodies. Interactions between SMPs and antibiotics can influence antibiotic migration, transformation, and toxicity but the mechanisms involved in such interactions are not fully understood. In this study, integrated spectroscopy approaches were used to investigate the mechanisms involved in interactions between SMPs and a representative antibiotic, trimethoprim (TMP), which has a low biodegradation rate and has been detected in wastewater. The results of liquid chromatography-organic carbon detection-organic nitrogen detection indicated that the SMPs used in the study contained 15% biopolymers and 28% humic-like substances (based on the total dissolved organic carbon concentration) so would have contained sites that could interact with TMP. A linear relationship of fluorescent intensities of tryptophan protein-like substances in SMP was observed (R 2>0.99), indicating that the fluorescence enhancement between SMP and TMP occurred. Fourier-transform infrared spectroscopy and X-ray photoelectron spectroscopy indicated that carboxyl, carbonyl, and hydroxyl groups were the main functional groups involved in the interactions. The electrostatic and π-π interactions were discovered by the UV-vis spectra and 1H nuclear magnetic resonance spectra. Structural representations of the interactions between representative SMP subcomponents and TMP were calculated using density functional theory, and the results confirmed the conclusions drawn from the 1H nuclear magnetic resonance spectra. The results help characterize SMP–TMP complexes and will help understand antibiotic transformations in wastewater treatment plants and aquatic environments. [Display omitted]. [ABSTRACT FROM AUTHOR]

We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew‐Hermitian differentiation matrix. While a theory of such L2 ‐orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterize such systems completely as appropriately weighted Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev‐orthogonal system whose leading N coefficients can be computed in O(NlogN)$ \mathcal{O} (N\log N)$ operations. [ABSTRACT FROM AUTHOR]

A small spherical source discharges a fluid into a porous medium that is already fully saturated with another fluid. The injected fluid has higher density than the ambient fluid, and so it forms a plume that moves downward under the effects of gravity. We present a simple asymptotic analysis assuming the two fluids do not mix that gives the width of the plume far from the source as a function of the injected volume flux. A spectral method is then developed for solving the full nonlinear problem in Boussinesq theory. Accurate numerical solutions are presented, which show in detail the evolution of the plume of heavier injected fluid over time. Close agreement with the asymptotic plume shape far from the source is demonstrated at later times. [ABSTRACT FROM AUTHOR]

In this work, we propose the use of non-homogeneous grids in 1D and 2D for the study of various nonlinear physical equations using spectral methods. As is well known, the use of spectral methods allow a faster resolution of the problem via the application of the ubiquitous Fast Fourier Transform (FFT) algorithm. We will center our investigation on the search of fast and accurate schemes to solve the spectral operators in the Fourier space. In particular, we will use the Conjugate Gradient (CG) iterative method, with a preconditioning matrix to accelerate the inversion process of the non-uniform Fast Fourier Transform (NFFT). As it will be shown, the results obtained are in good agreement with the expected values. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms. [ABSTRACT FROM AUTHOR]

IntroductionClustering is usually the first exploratory analysis step in empirical data. When the data set comprises graphs, the most common approaches focus on clustering its vertices. In this work, we are interested in grouping networks with similar connectivity structures together instead of grouping vertices of the graph. We could apply this approach to functional brain networks (FBNs) for identifying subgroups of people presenting similar functional connectivity, such as studying a mental disorder. The main problem is that real-world networks present natural fluctuations, which we should consider.MethodsIn this context, spectral density is an exciting feature because graphs generated by different models present distinct spectral densities, thus presenting different connectivity structures. We introduce two clustering methods: k-means for graphs of the same size and gCEM, a model-based approach for graphs of different sizes. We evaluated their performance in toy models. Finally, we applied them to FBNs of monkeys under anesthesia and a dataset of chemical compounds.ResultsWe show that our methods work well in both toy models and real-world data. They present good results for clustering graphs presenting different connectivity structures even when they present the same number of edges, vertices, and degree of centrality.DiscussionWe recommend using k-means-based clustering for graphs when graphs present the same number of vertices and the gCEM method when graphs present a different number of vertices.

In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the effciency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed. [ABSTRACT FROM AUTHOR]

Active–dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterized by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative) and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth-order dissipation and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular, we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalizing the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilizing a criterion involving the rate of growth of suitable norms of the |$n$| th spatial derivative of the solution, with respect to the spatial variable, as |$n$| tends to infinity. An estimate of the rate of growth of the |$n$| th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if |$\gamma $|⁠ , the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e. if |$\gamma $| is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active–dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active–dissipative–dispersive pseudo-differential equations. [ABSTRACT FROM AUTHOR]

Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “ m = ∞ ” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature. [ABSTRACT FROM AUTHOR]

We study the problem of ranking regression, in which a dataset of rankings is used to learn Plackett-Luce scores as functions of sample features. We propose a novel spectral algorithm to accelerate learning in ranking regression. Our main technical contribution is to show that the Plackett-Luce negative log-likelihood augmented with a proximal penalty has stationary points that satisfy the balance equations of a Markov Chain. This allows us to tackle the ranking regression problem via an efficient spectral algorithm by using the Alternating Directions Method of Multipliers (ADMM). ADMM separates the learning of scores and model parameters, and in turn, enables us to devise fast spectral algorithms for ranking regression via both shallow and deep neural network (DNN) models. For shallow models, our algorithms are up to 579 times faster than the Newton's method. For DNN models, we extend the standard ADMM via a Kullback-Leibler proximal penalty and show that this is still amenable to fast inference via a spectral approach. Compared to a state-of-the-art siamese network, our resulting algorithms are up to 175 times faster and attain better predictions by up to 26% Top-1 Accuracy and 6% Kendall-Tau correlation over five real-life ranking datasets. [ABSTRACT FROM AUTHOR]

Long electromagnetic transients occur in electrical systems because of switching and impulse actions As a result, the simulation time of such processes can be long, which is undesirable. Simulation time is significantly increased if the circuit in the study is complex, and also if this circuit is described by a rigid system of state equations. Modern requests of design engineers require an increase in the speed of calculations for realizing a real-time simulation. This work is devoted to the development of a unified spectral method for calculating electromagnetic transients in electrical circuits based on the representation of solution functions by series in algebraic and orthogonal polynomials. The purpose of the work is to offer electrical engineers a method that can significantly reduce the time for modeling transients in electrical circuits. Research methods. Approximation of functions by orthogonal polynomials, numerical methods for integrating differential equations, matrix methods, programming and theory of electrical circuits. Obtained results. Methods for calculating transients in electrical circuits based on the approximation of solution functions by series in algebraic polynomials as well as in the Chebyshev, Hermite and Legendre polynomials, have been developed and investigated. The proposed method made it possible to convert integro-differential equations of state into linear algebraic equations for images of time-dependent functions. The developed circuit model simplifies the calculation method. The images of true current functions are interpreted as direct currents in the proposed equivalent circuit. A computer program for simulating the transient process in an electrical circuit was developed on the basis of the described methods. The performed comparison of methods made it possible to choose the best method and a way to use it. The advantages of the presented method over other known methods are to reduce the simulation time of electromagnetic transients (for the considered examples by more than 6 times) while ensuring the required accuracy. The calculation of the process in the circuit over a long time interval showed a decrease and stabilization of errors, which indicates the prospects for using research methods for calculating complex electrical circuits. [ABSTRACT FROM AUTHOR]

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along a suitable orthonormal basis. Interestingly, this approach can be extended to cope with more general differential problems. In this paper we sketch this fact, by considering some relevant examples. [ABSTRACT FROM AUTHOR]

In this paper, the collocation spectral method (CSM) combined with domain decomposition method is developed to solve the radiative transfer equation in two-dimensional irregular domains. Three benchmark problems consist of the square enclosure, the L-shaped enclosure and the square enclosure with a centered obstruction are solved and compared with the published data to validate the ability of present developed method. The comparison shows good agreements and indicates that the method has a good accuracy for all problems. Then, the performances of influence matrix technique and iterative substructuring technique to exchange the radiative information between subdomains are compared. It is found that, in the serial computations, the computational cost of influence matrix technique is hundreds of times more expensive than that of iterative substructuring technique. The high cost of influence matrix technique is due to that the radiative intensity is a high dimensional variable with angular dependence, and tremendous subproblems have to be solved to construct the influence matrix. Finally, the modified CSM with domain decomposition, in which the radiative intensity is decomposed into three components, the real wall-related one, the virtual shared interface-related one, and the medium-related one, is proposed. The first two components are solved analytically, and the last one is still solved by the CSM. With such treatments, the computational cost is slightly increased, but the ray effect originated from step-change temperature of medium or step-change optical parameters can be effectively mitigated. Besides, the modified CSM with domain decomposition can also avoid the ray effect due to shadowing singularities. But it should be noted that, in such case, the ray effect due to inhomogeneous spatial distribution of source term may emerge. In conclusion, the CSM combined with domain decomposition is a good alternative method for thermal radiation calculation in complex geometry which can be decomposed into regular subdomains. [ABSTRACT FROM AUTHOR]

In this paper, we study numerical methods for simulating acoustic scattering by multiple three-dimensional objects of different nature (penetrable, sound-soft, sound-hard and absorbing targets) simultaneously present in the background media. We derive and analyze a boundary integral system of equations that arises when the solution of the problem is represented via single-layer potentials. We give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations and show that spectral methods satisfy these conditions. Superalgebraic convergence order of the discrete method for smooth objects is illustrated in some test cases. After that, we tackle the inverse problem of finding the shape of objects of different unknown nature from measurements of the total field at a set of receptors. We propose a numerical algorithm based on the computation of the topological energy of a weighted multifrequency least squares cost functional and present some numerical examples to illustrate its capabilities. [ABSTRACT FROM AUTHOR]

This study aims at extending and implementing an iterative spectral scheme for fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel. In this scheme, the unknown function u(x, t) is estimated by using shifted Gegenbauer polynomials vector Λ(x, t), and Picard iterative scheme is used to handle underlying nonlinearities. Some novel operational matrices are developed for the first time in order to approximate the singular integral like, $ \int_0^x {\int_0^y {u(p{a_1} + {b_1}, q{a_2} + {b_2}, t)/{{({x^{{\rho _1}}} - {p^{{\rho _1}}})}^{{\alpha _1}}}{{({y^{{\rho _2}}} - {q^{{\rho _2}}})}^{{\alpha _2}}}{\text{d}}q{\text{d}}p} } $ \end{document} and $ \int_0^t {{u^\gamma }({\bf{x}}, \xi)/{{({t^{{\rho _3}}} - {\xi ^{{\rho _3}}})}^{{\alpha _3}}}{\text{d}}\xi } $, where ρ's > 1, 0 < α's < 1 by means of shifted Gegenbauer polynomials vector. The advantage of this extended method is its ability to convert nonlinear problems into systems of linear algebraic equations. A computer program in Maple for the proposed scheme is developed for a sample problem, and we validate it to compare the results with existing results. Six new problems are also solved to illustrate the effectiveness of this extended computational method. A number of simulations are performed for different ranges of the nonlinearity n, α, fractional-order, ρ, and convergence control M, parameters. Our results demonstrate that the extended scheme is stable, accurate, and appropriate to find solutions of complex problems with inherent nonlinearities.

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios. [ABSTRACT FROM AUTHOR]