System Of Equations Research Articles

Small data non-linear wave equation numerology: The role of asymptotics

Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to our classification. Furthermore, we gave two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [13].

Anisotropic resistivity estimation and uncertainty quantification from borehole triaxial electromagnetic induction measurements: Gradient-based inversion and physics-informed neural network

Rapid and accurate petrophysical reservoir description and quantification is important for subsurface energy resource modeling and engineering. Triaxial borehole resistivity measurements enable the estimation of key in-situ rock properties, such as the volumetric concentration of shale and sandstone resistivity. However, traditional approaches for estimating these properties from triaxial, or anisotropic, borehole resistivity measurements rely on analytical solutions that solve a system of equations simultaneously, which can lead to numerical instability and inefficiency. By reformulating the system of anisotropic resistivity equations as an inverse problem, we can achieve a more stable, accurate, and efficient estimation of key petrophysical properties. We propose two methods, namely nonlinear gradient-based and physics-informed neural network (PINN) inversion, to estimate the volumetric concentration of shale and sandstone resistivity from the parallel- and perpendicular-to-bedding-plane resistivity logs, posed as the solution of an inverse problem. Furthermore, we compare the PINN, nonlinear gradient-based inversion and analytical solution methods in terms of accuracy, computational efficiency, and uncertainty quantification using four different datasets of anisotropic resistivity logs, i.e., two synthetic and two field cases. The PINN inversion technique estimates the petrophysical properties within 0.5 CPU milliseconds with an accuracy between 91% and 99%, while nonlinear gradient-based inversion can estimate the petrophysical properties with an accuracy between 98% and 99% but requires several CPU minutes depending on the size of the dataset. The proposed PINN technique is therefore capable of providing fast and accurate estimation and uncertainty quantification of key petrophysical properties from triaxial resistivity logs, with comparable accuracy and approximately 106 speedup compared to nonlinear gradient-based inversion, and can be used for real-time applications such as automatic shale properties estimation, logging-while-drilling measurement interpretation, automated well geosteering, and time-lapse reservoir monitoring.

Method of undetermined coefficients for circuits and filters using Legendre functions

This article presents a new way to implement matching networks and filters using the method of undetermined coefficients. A method is proposed for approximating the transmission coefficient of the synthesized filter, taking into account the required amplitude-frequency characteristics. To synthesize the filter, an approximating function (AF) was used using orthogonal Legendre polynomials, which is a mathematical description using a system of equations. Filter properties whose implementation is based on modified Legendre approximating functions usually depend on the interval on which they are defined and have the property that they are orthogonal on this interval. An example of seventh order filter synthesis using modified Legendre approximating functions is given. The filter circuit is implemented, the elements of the filter circuit are calculated based on the selected approximating modified function. The criteria used were minimization of the unevenness of the group delay time (GDT) and minimization of the complex approximation error for given values of the AF parameters. As a result, the number of filter elements, the group delay value and the complex approximation error are significantly reduced.

Modelling soil - Vegetation - Atmospheric interactions of radon products in a Belgian Scots pine forest site.

A soil-vegetation-atmospheric transfer (SVAT) model for radon and its progeny is presented to improve process-level understanding of the role of forests in taking-up radionuclides from soil radon outgassing. A dynamic system of differential equations couples soil, tree (Scots pine) and atmospheric processes, treating the trees as sources, sinks and conduits between the atmosphere and the soil. The model's compartments include a dual-layer soil column undergoing hydrological and solute transport, the tree system (comprising roots, wood, litter, and foliage) and the atmosphere, with physical processes governing the transfers of water and radon products between these compartments. A dose post-processor calculates dose rates to the trees from internal, external, and surface radiation exposures. The model parameterisation is based on measurement data from the Grote Nete Valley in the Belgian Campine region. The model results suggest that the tree intake of radon progeny is principally from the atmosphere, whereas radium is mainly taken-up from soil by the root uptake process, leading to an additional fraction of ingrown radon progeny in the tree by this route. It is also suggested that atmospheric uptake of radon is an essential mechanism when evaluating the tree uptake of 214Po, 214Pb and 218Po and subsequent decay products. The model also indicates a slow uptake of radionuclides by the tree roots, with timescales in the order of years, leading to different dose rates for young and mature trees. The importance of foliar surface deposition, leading to a dominance of surface doses to the tree needles, is also highlighted. These mechanisms, ignored in most assessment models, are necessary improvements for assessment tools dealing with the impact of radon and its progeny in forests, with regard to legacy sites with 226Ra contamination.

Alternative Transient and Steady State Analysis of Some Gaver’s Parallel Systems

This paper presents one of the most interesting generalizations of Gaver’s basic two-unit parallel system sustained by a cold standby unit and attended by a repairman with multiple vacations. The system was studied using the supplementary variables technique, like many other similar semi-Markov systems. D.P. Gaver, Jr. was the first to apply this method for constructing and studying reliability models, and since then it has been then widely used by other researchers to study various reliability problems. As a result, non-classical boundary value problem of mathematical physics with nonlocal boundary conditions has been obtained. Until now, a solution to this problem was obtained in terms of Laplace transforms. Naturally, the most significant part of the problem is a system of partial differential equations (Kolmogorov forward equations). In this study, we demonstrate that Kolmogorov equations are redundant and we can solve the problem by avoiding the necessity of using them. We present here a novel, purely probabilistic approach. The results are formulated as rigorous mathematical statements, offering a significant simplification in the reliability analysis of stochastic systems. Our findings show that this novel approach can be applied to study both semi-Markov and some non semi-Markov models where the supplementary variables technique is used.

Diminishing spectral bias in physics-informed neural networks using spatially-adaptive Fourier feature encoding.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for solving partial differential equation (PDE) systems in computer mechanics. However, PINNs still struggle in simulating systems whose solution functions exhibit high-frequency patterns, especially in cases with wide frequency spectrums. Current methods apply Fourier feature mappings to the input to improve the learning ability of model on high-frequency components. However, they are largely problem-dependent which require proper selection of hyperparameters and introduces additional training difficulty into the optimization. To this end, we present a spatially adaptive Fourier feature encoding method accompanied by a tree-based sampling strategy in this work. Specifically, we propose to guide the Fourier feature mappings of input by gradually exposing the input coordinate from low to higher encoding frequencies during training through the feedback loop of loss. Meanwhile, we also propose to refine the sampling of residual points by presenting a novel tree-based sampling strategy. This method represents the input domain by a tree and formulates the sampling of residual points as a resource allocation problem which optimizes the sampling of residual points during training and assigns more computational capacity to the underfit region. The effectiveness of our proposed method is demonstrated in several challenging PDE problems, including Poisson equation, heat equation, Navier-Stokes equations, Reynolds-Averaged Navier-Stokes equations, and Maxwell equation. The results indicate that our method can better allocate the computational resources during training and enable the model to fit the local frequencies of target function adaptively.

On Mathematical Analysis of a Micro-Scale Boltzmann-Maxwell’s PDEs Model for a Plasma Flow Influence by Non-linear Sinusoidal External Electric Field: Novel Irreversibility Analysis of Plasma Kinetic Theory

Unfortunately, the scientific research concerned with the micro-scale mathematical models in plasma has very few numbers compared with the other mathematical models. However, the microscopic fields are related to all modern technology like nano and micro technology, quantum computers, and many other essential applications. This study focuses on the mathematical analysis of a micro-scale Boltzmann-Maxwell partial differential equations (PDE) system for a gaseous plasma flow influenced by a non-linear, non-uniform external electric field, specifically in the context of a novel irreversibility micro-scale analysis of the kinetic theory of plasma. We did that using the analytical solution of the PDE system. The governing equations were developed using the moment and traveling-wave techniques in a new irreversible non-equilibrium thermodynamics (INT) methodology. To our knowledge, this was applied for the first time. We are investigating the distinct behavior of the electron velocity distribution functions (VDF) and the non-equilibrium velocity functions, representing an essential INT novel study. As a result, critical micro-scale INT variables would employ the generated non-equilibrium VDF to get the equilibrium time for each species. We aim to show how the impacts of various thermodynamic forces on internal energy change (IEC) are maintained. The research generates visual representations of physical variables in 3D. Extensive physics, electric manufacturing, and nano-electro-mechanical systems applications in various manufacturing contexts attest to the research’s standing. We aim to calculate the percentages between the numerous contributions of IEC in diamagnetic and paramagnetic plasma based on the total derivatives of the extensive parameters. We apply the results to a typical model of laboratory helium plasma because of the helium’s various excellent applications.

A new finite element formulation unifying fluid-structure and fluid-fluid interaction problems

When the accurate simulation of two materials that interact through their common and deformable interface is of interest, the efficient treatment of the interface determines the success or failure of a numerical method. In this work, we propose a new, robust and easy-to-code finite element formulation for such interaction problems. The remedy of the interface constraints, namely the continuity of velocities and stresses, is accomplished using a single-node approach and the same continuous basis functions for the velocities in both materials. Given that only Newtonian fluids will be examined, we do not have to introduce basis functions for the stress components. The XFEM method, which enriches locally the continuous basis function of a variable that presents a discontinuity, is employed to tackle the discontinuous behavior of the pressure across the interface. The incorporation of Petrov-Galerkin stabilization schemes enhances further our formulation and allows the usage of equal order interpolants for velocities and pressure. We solve the coupled system of equations in a monolithic manner to alleviate the convergence problems of the segregated approach. The novel aspect of our method is that its ingredients do not differentiate based on the constituent materials of the problem, and it can be used interchangeably for either a fluid-structure or a fluid-fluid interaction problem. The accuracy of the new finite element formulation is assessed by comparing its numerical results to those of the literature in three problems: i) the flow through a partially collapsible channel, ii) the induced motion of a flexible elastic plate, iii) the filament stretching of a Newtonian thread surrounded by another immiscible viscous fluid. In all cases, we are in agreement with the results of the literature. Furthermore, we conduct a challenging, 3D simulation for a setup that resembles the motion of a three-leaflet stented aortic heart valve.

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with L∞ coefficients

Let Ω⊂Rn+1, n≥1, be an open set with s-Ahlfors regular boundary ∂Ω, for some s∈(0,n], such that either s=n and Ω is a corkscrew domain with the pointwise John condition, or s<n and Ω=Rn+1∖E, for some s-Ahlfors regular set E⊂Rn+1. In this paper we provide a unifying method to construct Varopoulos type extensions of BMO and Lp boundary functions. In particular, we show that a) if f∈BMO(∂Ω), there exists F∈C∞(Ω) such that dist(x,Ωc)|∇F(x)| is uniformly bounded in Ω and the Carleson functional of dist(x,Ωc)s−n|∇F(x)| as well the sharp non-tangential maximal function of F are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of f, and F→f in a certain non-tangential sense Hs|∂Ω-almost everywhere; b) if f¯∈Lp(∂Ω), 1<p≤∞, there exists F¯∈C∞(Ω) such that the non-tangential maximal functions of F¯ and dist(⋅,Ωc)|∇F¯| as well as the Carleson functional of dist(⋅,Ωc)s−n|∇F¯| are in Lp(∂Ω) with norms controlled by the Lp-norm of f¯, and F¯→f¯ in some non-tangential sense Hs|∂Ω-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support, then both F and F¯ can be constructed so that they are also Lipschitz on Ω‾ and converge to the boundary data continuously. The latter results hold without the additional assumption of the pointwise John condition. Finally, for elliptic systems of equations in divergence form with merely bounded complex-valued coefficients, we show some connections between the solvability of Poisson problems with interior data in the appropriate Carleson or tent spaces and the solvability of Dirichlet problem with Lp and BMO boundary data.

Learning dynamical systems from noisy data with inverse-explicit integrators

We introduce the mean inverse integrator (MII), a novel approach that improves accuracy when training neural networks to approximate vector fields of dynamical systems using noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge–Kutta methods. We show that the class of mono-implicit Runge–Kutta methods (MIRK) offers significant advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained by inserting the training data in the MIRK formulae. This allows the use of symmetric and high-order integrators without requiring the solution of non-linear systems of equations. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with extensive numerical experiments using data from chaotic and high-dimensional Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, providing theoretical support for the favorable performance of MII and symmetric MIRK methods.

Understanding key interactions between NOx and C2-C5 alkanes and alkenes: The ab initio kinetics and influences of H-atom abstractions by NO2

This study aims to reveal the important role and the respective rate rules of H-atom abstractions by NO2 for better understanding NOX/hydrocarbon interactions. To this end, H-atom abstractions from C2-C5 alkanes and alkenes (15 species) by NO2, leading to the formation of three HNO2 isomers (trans-HONO, HNO2, and cis-HONO) and their respective products (45 reactions), are first characterized through quantum chemistry computation, where electronic structures, single point energies, C-H bond dissociation energies and 1-D hindered rotor potentials are determined at DLPNO-CCSD(T)/cc-pVDZ//M06–2X/6−311++g(d,p). The rate coefficients for all studied reactions, over a temperature range from 298.15 to 2000 K, are computed using transition state theory with the Master Equation System Solver program. Comprehensive analysis of branching ratios elucidates the diversity and similarities between different species, HNO2 isomers, and abstraction sites, from which accurate rate rules are determined. With the rate rules, the rate coefficients at various reaction sites on heavier hydrocarbons (e.g., > C5) can be reliably estimated by analogy. Incorporating the updated rate parameters into a detailed chemical kinetic model reveals the significant influences of this type of reaction on model prediction results, where the simulated ignition delay times are either prolonged or reduced, depending on the original rate parameters presented in the selected model. Sensitivity and flux analysis further highlight the critical role of this type of reaction in affecting system reactivity and reaction pathways, emphasizing the need for adequately representing these kinetics in existing chemistry models. This can now be sufficiently achieved for alkanes and alkenes based on the results from this study.

Equivalent system frequency response model with energy storage

Providing Frequency Response (FR) using energy storage system (ESS) has been adopted in power systems worldwide to reduce the maximum frequency deviation. This paper presents a new equivalent system frequency response model with ESS. The model can be conveniently used to assess the system frequency nadir and calculate the capacity and equivalent droop of storage considering the maximum frequency deviation in a synchronous generator (SG) dominated system. Unlike existing calculation methods based on control strategies such as fuzzy-logic, robust control, data-driving control and predictive algorithm, the proposed method (i) uses the equivalent capacity and equivalent droop to describe the FR of SG and ESS, thereby avoids the difficulty in solving high-order FR model and (ii) does not need the control strategy of SGs in the network. Therefore, the proposed method provides a simple yet systematic method in calculating the capacity and equivalent droop of ESS. The effectiveness of the proposed method is demonstrated using the four-generator two-area (4G2A) system with 0 %, 25 % and 50 % penetration of renewable resources and different types of SGs (hydraulic and steam SG) based on the given maximum frequency deviation..

Dynamical boundary value problem for a viscoelastic half-space with cut

The dynamical boundary value problem for viscoelastic half-space with cut in the form of a strip is considered. The problem is reduced to the singular integral equation of first kind. Using the method of orthogonal polynomials, the integral equation is reduced to an infinite system of linear algebraic equations. The quasi-completely regularity of the obtained system is proved and the reduction method for approximate solution is developed.

Quantum algorithm for partial differential equations of nonconservative systems with spatially varying parameters

Quantum algorithm for partial differential equations of nonconservative systems with spatially varying parameters

A Robust Algorithm for Solving Heterogeneous Systems of Equations arising in Kinetostatics of Compliant Mechanisms

A Robust Algorithm for Solving Heterogeneous Systems of Equations arising in Kinetostatics of Compliant Mechanisms

Two-dimensional steady-state thermal analytical model of permanent magnet linear motor in Cartesian coordinates

Purpose Compared with the time-consuming numerical method and the complex lumped parameter thermal network method to solve the steady-state heat distribution of the permanent magnet (PM) linear motor, there is no analytical method based on the thermal partial differential equations. This paper aims to propose a two-dimensional (2-D) analytical model for predicting the steady-state temperature distribution of PM linear motors to improve the prediction accuracy and speed up the calculation. Design/methodology/approach Based on the complex Fourier series theory and Cauchy’s product theorem, this paper presents for the first time a general analytical solution for 2-D temperature field in Cartesian coordinates. Then, by combining the electromagnetic field finite element model (FEM), the copper loss, iron loss and PM eddy current loss are used as the heat sources of the thermal analytical model. Finally, the solution to the temperature field is obtained by solving the system equations under boundary and interface conditions. Findings The analytical results are in good agreement with those from the thermal FEM, and the calculation speed is significantly faster than that of the thermal FEM. Originality/value The multilayer model proposed in this paper can consider heat conduction, convection and radiation. It is not only suitable for PM linear motors but also has significant application value for the thermal analysis of electromagnetic devices modeled in 2-D Cartesian coordinates.

Terminal value problem for the system of fractional differential equations with additional restrictions

This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ–tempered fractional derivatives of Caputo type is considered.

Thermal inspection on MHD Prandtl fluid flow past a stretching sheet in the presence of heat generation/absorption and suction/injection

Abstract For the past several years, researchers have been investigating the thermal boundary layer behavior under various assumptions due to contemporary requirements. From the existing literature, it has been noticed that no investigation has been conducted to examine the thermal and mass transport of a magnetohydrodynamic (MHD) flow of a Prandtl fluid past a stretching sheet with heat generation/absorption and suction/injection. To address this gap, this study looks at how the changed thermal flux, heat generation/absorption, and suction/injection affect the convective flow of MHD Prandtl fluid over a stretching surface. The governing system of nonlinear partial differential equations (PDEs) has been reduced to a system of ordinary differential equations (ODEs) through appropriate transformations. The desired numerical solutions of the resulting ODEs have been obtained using the bvp5c MATLAB package. We have analyzed and presented the effects of the physical parameters on the velocity, temperature, and concentration fields using graphs and tables. Also, the drag coefficient, heat, and mass transport rates were examined. Additionally, an entropy analysis has been conducted to explore the reusability of heat production. It has been observed that intensifying the magnetic and porosity parameters reduces the flow velocity while increasing the Eckert number, heat generation parameter, and the Biot number significantly increases the temperature. The concentration drops with an increase in the Schmidt number and the chemical reaction parameter. This study holds significant implications for industries that rely on drug delivery, heat exchanger technology, polymerization, and refining processes.

Spectral Properties of The Finite System of Klein-Gordon S-Wave Equations with Condition Depends on Spectral Parameter

The spectral characteristics of the operator L is studied where L is defined within the Hilbert space L2(R+,CV ) given by a finite system of Klein-Gordon type differential equations and boundary condition depends on spectral parameter. The research of the Klein-Gordon type operator continues to be an important topic for researchers due to the range of applicability of them in numerous branches of mathematics and quantum physics. Contrary to the previous works, we take the potential as complex valued and generalize the problem to the matrix Klein-Gordon operator case. The spectrum is derived by determining the Jost function and resolvent operator of the prescribed operator. Further, we provide the conditions that must be met for the certain quantitative properties of the spectrum.

Estimation of stationary and non-stationary moving average processes in the correlation domain.

This paper introduces a novel approach for the offline estimation of stationary moving average processes, further extending it to efficient online estimation of non-stationary processes. The novelty lies in a unique technique to solve the autocorrelation function matching problem leveraging that the autocorrelation function of a colored noise is equal to the autocorrelation function of the coefficients of the moving average process. This enables the derivation of a system of nonlinear equations to be solved for estimating the model parameters. Unlike conventional methods, this approach uses the Newton-Raphson and Levenberg-Marquardt algorithms to efficiently find the solution. A key finding is the demonstration of multiple symmetrical solutions and the provision of necessary conditions for solution feasibility. In the non-stationary case, the estimation complexity is notably reduced, resulting in a triangular system of linear equations solvable by backward substitution. For online parameter estimation of non-stationary processes, a new recursive formula is introduced to update the sample autocorrelation function, integrating exponential forgetting of older samples to enable parameter adaptation. Numerical experiments confirm the method's effectiveness and evaluate its performance compared to existing techniques.