Solution Of Equation Research Articles

Accelerating wavepacket propagation with machine learning.

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.

Various optimized artificial neural network simulations of advection-diffusion processes

Abstract The aim of this research is to describe an artificial neural network (ANN) based method to approximate the solutions of the natural advection-diffusion equations. Although the solutions of these equations can be obtained by various effective numerical methods, feed forward neural network (FFNN) techniques combined with different optimization techniques offer a more practicable and flexible alternative than the traditional approaches to solve those equations. However, the ability of FFNN techniques to solve partial differential equations is a questionable issue and has not yet been fully concluded in the existing literature. The reliability and accuracy of computational results can be advanced by the choice of optimization techniques. Therefore, this study aims to take an effective step towards presenting the ability to solve the advection-diffusion equations by leveraging the inherent benefits of ANN methods while avoiding some of the limitations of traditional approaches. In this technique, the solution process requires minimizing the error generated by using a differential equation whose solution is considered as a trial solution. More specifically, this study uses a FFNN and backpropagation technique, one of the variants of the ANN method, to minimize the error and the adjustment of parameters. In the solution process, the loss function (error) needs to be minimized; this is accomplished by fitting the trial function into the differential equation using appropriate optimization techniques and obtaining the network output. Therefore, in this study, the commonly used techniques in the literature, namely gradient descent (GD), particle swarm optimization (PSO) and artificial bee colony (ABC), are selected to compare the effectiveness of gradient and gradient-free optimization techniques in solving the advection-diffusion equation. The calculations with all three optimization techniques for linear and nonlinear advection-diffusion equations have been run several times to obtain the optimum accuracy of the results. The computed results are seen to be very promising and in good agreement with the effective numerical methods and the physics-informed neural network (PINN) method in the literature. It is also concluded that the PSO-based algorithm outperforms other methods in terms of accuracy.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Infinitely many positive solutions for p-Laplacian equations with singular and critical growth terms

In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p-Laplacian type involving a singularity and a critical Sobolev exponent {−Δpu=up∗−1+λ|u|γ−1u,inΩ,u=0,on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta _{p}u=u^{p^{*}-1}+\\frac{\\lambda}{|u|^{\\gamma -1}u}, & \ ext{in} ~\\Omega , \\\\ u=0, &\ ext{on}~\\partial \\Omega , \\end{cases} $$\\end{document} where Ω is a bounded domain, p∗=NpN−p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p^{\\ast}=\\frac{Np}{N-p}$\\end{document} (N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N \\geq 3$\\end{document}) is the critical Sobolev exponent and λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda > 0$\\end{document}. Based on the cutoff technique, we prove that the above problem possesses infinitely many positive solutions with 0<γ<1,1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0 < \\gamma <1, 1< p < N$\\end{document}, and λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda > 0$\\end{document} small enough.

A Finite Integral Transform-Based Generalized Eigenvalue Formulation for Buckling of Orthotropic Rectangular Plates with Rotational Restraints

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

Empowering Discovery Learning and Direct Instruction with Computer-Assisted to Enhance Mathematical Performance

In the current educational context, the low academic performance of students in mathematics indicates the need for the implementation of more effective teaching methods to support the understanding of complex concepts. The effectiveness of learning can be enhanced through computer-assisted instruction, particularly in discovery learning (DL-CA) and direct instruction (DI-CA) utilizing web-based media. This study employs a quantitative approach with statistical analysis to examine the significant impact of these teaching methods on students' mathematical performance. Discovery learning and direct instruction were applied to two groups studying numerical methods, focusing on the topic of nonlinear equation solutions using the bisection method, with each group consisting of 33 students. The statistical test results indicated that there was no significant difference in mathematical performance between the DL-CA and DI-CA groups; both methods showed comparable effectiveness. However, both methods had a significant effect on the N-Gain of the learning outcomes in both groups. The improvement in mathematical performance for the DL-CA group was better than that for the DI-CA group, with more students showing high improvement (11 students) compared to the DI-CA group (1 student). Additionally, in the low improvement category, the DL-CA group had fewer students (2 students) than the DI-CA group (8 students). This suggests that both discovery learning and direct instruction have almost equivalent reliability in achieving students' mathematical abilities, but DL-CA is more effective in enhancing performance compared to DI-CA.

The Effect of Air–Water Mixture on the Dynamic Response of a Hydrodynamic Journal Bearing With Inclined Grooves

Abstract Water-lubricated hydrodynamic journal bearings with deep inclined grooves are often used to support radial loads and dampen pump vibrations in nuclear power plants. The inclined grooves increase the pumping effect of the lubricant thus reducing wear and friction during rotor startup and coast-down. The deep grooves also facilitate the removal of debris without damaging the bearing. The load capacity and the stability of this type of bearing are studied in the present work. A sensitivity study of the impact of the number of grooves on bearing performance underlined the superiority of the journal bearing with two inclined grooves. Moreover, these journal bearings may operate with a mixture of water and air. This compressible two-phase lubricant will modify the load capacity and the stability of the journal bearing. The impact of the ingested volume fraction of air is therefore investigated. A physical model of the homogeneous air–water mixture linking the local air volume fraction with the pressure was used in conjunction with the numerical solution of the compressible Reynolds equation on an unstructured grid. It was found that the air ingestion in the two-inclined grooves journal bearing decreases the load capacity but improves the stability for mild and low loads.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Review of physics-informed machine-learning inversion of geophysical data

We review five types of physics-informed machine-learning (PIML) algorithms for inversion and modeling of geophysical data. Such algorithms use the combination of a data-driven machine-learning (ML) method and the equations of physics to model or invert geophysical data (or both). By incorporating the constraints of physics, PIML algorithms can effectively reduce the size of the solution space for ML models, enabling them to be trained on smaller data sets. This is especially advantageous in scenarios in which data availability may be limited or expensive to obtain. In this review, we restrict the physics to be that from the governing wave equation, either as a constraint that must be satisfied or by using numerical solutions of the wave equation for modeling and inversion. This approach ensures that the resulting models adhere to physical principles while leveraging the power of ML to analyze and interpret complex geophysical data. There are several potential benefits of PIML compared to standard numerical modeling or inversion of seismic data computed by, for example, finite-difference solutions to the wave equation. 1) Empirical tests suggest that PIML algorithms constrained by the physics of wave propagation can sometimes resist getting stuck in a local minima compared with standard full-waveform inversion (FWI). 2) After the weights of the neural network are found by training, then the forward and inverse operations by PIML can be more than several orders of magnitude more efficient than FWI. However, the computational cost for general training can be enormous. 3) If the ML inversion operator [Formula: see text] is locally trained on a small portion of the recorded data [Formula: see text], then there is sometimes no need for millions of training examples that aim for global generalization of [Formula: see text]. The benefit is that the locally trained [Formula: see text] can be used to economically invert the remaining test data [Formula: see text] for the true velocity [Formula: see text], where [Formula: see text] can comprise more than 90% of the recorded data.

Numerical solution of nonlinear partial differential equations using shifted Legendre collocation method

In life important applications are modeled mathematically by nonlinear partial differential equations. Primarily, the objective is to transform such equations into a system of algebraic equations to get their solutions. The Legendre collocation method is demonstrated by the differentiation operational matrix via shifted Legendre polynomials in such a transformation. The validity and effectiveness of the prospective method are manifested by illustrative examples, an error analysis, residual analysis, and comparison with others.

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.

κ-Fractional Derivative Operator and Solution of Fractional Kinetic Equations Involving (κ,𝔍)- Second Appell Hypergeometric Matrix Functions with the two Variables

κ-Fractional Derivative Operator and Solution of Fractional Kinetic Equations Involving (κ,𝔍)- Second Appell Hypergeometric Matrix Functions with the two Variables

On quadrirational pentagon maps

Abstract We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps R : ( x , y ) ↦ ( u ( x , y ) , v ( x , y ) ) , where u , v are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang–Baxter maps from entwining pentagon maps.

Least Squares Approximation Method for Estimation of Volterra Fractional Integro-differential equation using Hermite Polynomial as the basis functions

This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate solution of degree N, which is then substituted into the fractional-order integro-differential equation under investigation. After evaluating the integral, the equation is rearranged to isolate one side, allowing the application of the least squares method. Three examples were solved using this approach. In Example 1, the numerical results for α=0.9 and α=0.8 were compared to the exact solution for α=1. In Examples 2 and 3, the results for α=1.9 and α=1.8 were compared to the exact solution for α=2. These comparisons showed favorable alignment with the exact solutions. The numerical results and graphical illustrations demonstrate the validity, competence, and accuracy of the proposed method.

Renormalized Solutions for the Non-local Equations in Fractional Musielak–Sobolev Spaces

Renormalized Solutions for the Non-local Equations in Fractional Musielak–Sobolev Spaces

Effect of the dissipative states of non-ideal second-type superconductors on their current-carrying capacity

The influence of the dissipative states of non-ideal second-type superconductors on their current-carrying capacity is investigated. The study performed is based on the numerical solution of non-stationary Maxwell and Fourier equations with different features of the non-linear rise of their I-V characteristics. First, the power equation of the I-V characteristic with various n-values was used to examine the states of superconductors that occur when the I-V characteristic continuously increases. Second, the obtained results are compared with the results of computer experiments simulating the thermo-electrodynamic states of superconductors when the electric field is not present in their I-V characteristics in the subcritical current range. A piecewise linear I-V equation describes such idealized modes. The results of the simulations indicate that a superconductor's ability to carry the transport current drops as the n-value decreases. Accordingly, the maximum transport current (quench current) flowing stably through a superconductor with an idealized I-V characteristic are always higher than the corresponding value calculated for superconductor with the same critical current but with a continuously increasing I-V characteristic. Moreover, a deterioration in cooling conditions or an increase in the current charging rate will also lead to a reduction in the current-carrying capacity of non-ideal second-type superconductors. The non-trivial temperature change of superconductors during formation of stable modes should necessarily be taken into account in experiments to investigate the voltage-current characteristics of superconductors, their current-carrying capacity, and loss theory.

NUMA-aware parallel sparse LU factorization for SPICE-based circuit simulators on ARM multi-core processors

In circuit simulators that resemble the Simulation Program with Integrated Circuit Emphasis (SPICE), one of the most crucial steps is the solution of numerous sparse linear equations generated by frequency domain analysis or time domain analysis. The sparse direct solvers based on lower-upper (LU) factorization are extremely time-consuming, so their performance has become a significant bottleneck. Despite the existence of some parallel sparse direct solvers for circuit simulation problems, they remain challenging to adapt in terms of performance and scalability in the face of rapidly evolving parallel computers with multiple NUMA hardware based on ARM architecture. In this paper, we introduce a parallel sparse direct solver named HLU, which re-examines the performance of the parallel algorithm from the viewpoint of parallelism in pipeline mode and the computing efficiency of each task. To maximize task-level parallelism and further minimize the thread waiting time, HLU devises a fine-grained scheduling method based on an elimination tree in pipeline mode, which employs depth-first search (DFS-like) to iteratively search for parent tasks and then place dependent tasks in the same task queue. HLU also suggests two NUMA node affinity strategies: thread affinity optimization based on NUMA nodes topology to guarantee computational load balancing and data affinity optimization to enable effective memory placement when threads access data. The rationality and effectiveness of the sparse solver HLU are validated by the SuiteSparse Matrix Collection. In comparison with KLU and NICSLU, the experimental results and analysis show that HLU attains a speedup of up to 9.14× and 1.26x (geometric mean) on a Huawei Kunpeng 920 Server, respectively.

Parametric identification of coefficients for a model of fatigue stiffness degradation of a composite material

The problem of finding the fatigue characteristics of a composite material based on test results is considered. The results of endurance tests of unidirectional polymer composite materials with different initial stiffness, breaking stress and working cycle stress were used as the initial data. As a mathematical model of stiffness degradation, a nonlinear ordinary differential equation with five unknown parameters is used, reflecting characteristic changes in the properties of the material. It is required to find such parameter values that the solution of the differential equation should describe the available test results with sufficient accuracy. The solution procedure is reduced to the problem of optimizing the objective function, the value of which characterizes the achieved accuracy. As optimization methods, a method simulating the behavior of a flock of moths and a method of sequential reduction of the search set were used. A step-by-step algorithm for finding unknown model parameters is proposed, and numerical results of processing input data containing information on changing the elasticity modulus of the composite material in the course of applying load cycles are presented.

Whitham modulation theory and the classification of solutions to the Riemann problem of the Fokas–Lenells equation

AbstractIn this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time.

Consequences of the Pb-S Bond Formation for Lead Halide Perovskites.

Lead halide perovskites are structurally not stable due to their ionic bonds. Using sulfur agents in the crystal growth improves the stability and performance of the photovoltaic and light-emitting devices. In this theoretical work, we use a small toy S-radical in place of A cation in the bulk of lead iodide perovskite, and highlight the significance of the Pb-S covalent-double-bond formation for: the charge redistribution on the neighboring bonds that also turn to be covalent, phase transformation to a stable non-perovskite structure, and superior optoelectronic properties. The chemical analysis was performed with the Quantum Theory of Atoms In Molecules (QTAIM) and Non-Covalent Interactions (NCI) index. Excitonic properties were obtained from the solution of ab initio Bethe-Salpeter equation. Presence of the spin-orbit coupling triggers an interplay between the Frenkel and charge-transfer multiexcitons, switching between the photovoltaic and laser applications. Multiexcitons obey the exciton-fission preconditions.