Kernel Of Equation Research Articles

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

The role of conjugacy in the dynamics of time of arrival operators

The construction of time of arrival (TOA) operators canonically conjugate to the system Hamiltonian entails finding the solution of a specific second-order partial differential equation called the time kernel equation (TKE). In this paper, we provide an exact analytic solution of the TKE for a special class of potentials satisfying a specific separability condition. The solution enables us to investigate the time evolution of the eigenfunctions of the conjugacy-preserving TOA operators (CPTOA) and show that they exhibit unitary arrival at the intended arrival point at a time equal to their corresponding eigenvalues. We also compare the dynamics between the TOA operators constructed by quantization and those independent of quantization for specific interaction potentials. We find that the CPTOA operator possesses smoother and sharper unitary dynamics over the Weyl-quantized one within numerical accuracy.

Neural operators for robust output regulation of hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of first-order hyperbolic and parabolic partial differential equation (PDE) systems. Due to the global approximation ability of the DeepONet, the NO provides approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, the NO theory is leveraged to solve the robust output regulation problem for a class of uncertain hyperbolic PDE systems under the design framework of backstepping-based regulator. The NO is trained offline on a dataset containing a sufficient number of system parameters and corresponding prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Once the NO is trained, the kernel equation does not need to be solved ever again, for any new system parameters that do not exceed the range of the training set. Based on the internal model principle, the regulator is inherently robust to a degree of parameter uncertainty and error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equations are solvable and the parameter uncertainty leads to an asymptotically stable origin. We provide a series of theory proofs and a numerical test under the approximate control and observation gains to demonstrate the robust regulation problem. It is shown that the NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10−4 in the test. This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying system parameters.

Short-time expansion of one-dimensional Fokker-Planck equationswith heterogeneous diffusion.

We formulate a short-time expansion for one-dimensional Fokker-Planck equationswith spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation(the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equationswhich can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

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

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

BVP with a Load in the Form of a Fractional Integral

A boundary value problem for a nonhomogeneous heat equation with a load in the form of a fractional Riemann–Liouville integral of an order β∈0,1 is considered. By inverting the differential part, the problem is reduced to an integral equation with a kernel with a special function. The special function is presented as a generalized hypergeometric function. The limiting cases of the order β of the fractional derivative are studied: it is shown that the interval for changing the order of the fractional derivative can be expanded to integer values β∈0,1. The results of the study remain unchanged. The kernel of the integral equation is estimated. Conditions for the solvability of the integral equation are obtained.

Predicting resting-state brain functional connectivity from the structural connectome using the heat diffusion model: a multiple-timescale fusion method

Objective. Understanding the intricate relationship between structural connectivity (SC) and functional connectivity (FC) is pivotal for understanding the complexities of the human brain. To explore this relationship, the heat diffusion model (HDM) was utilized to predict FC from SC. However, previous studies using the HDM have typically predicted FC at a critical time scale in the heat kernel equation, overlooking the dynamic nature of the diffusion process and providing an incomplete representation of the predicted FC. Approach. In this study, we propose an alternative approach based on the HDM. First, we introduced a multiple-timescale fusion method to capture the dynamic features of the diffusion process. Additionally, to enhance the smoothness of the predicted FC values, we employed the Wavelet reconstruction method to maintain local consistency and remove noise. Moreover, to provide a more accurate representation of the relationship between SC and FC, we calculated the linear transformation between the smoothed FC and the empirical FC. Main results. We conducted extensive experiments in two independent datasets. By fusing different time scales in the diffusion process for predicting FC, the proposed method demonstrated higher predictive correlation compared with method considering only critical time points (Singlescale). Furthermore, compared with other existing methods, the proposed method achieved the highest predictive correlations of 0.6939 ± 0.0079 and 0.7302 ± 0.0117 on the two datasets respectively. We observed that the visual network at the network level and the parietal lobe at the lobe level exhibited the highest predictive correlations, indicating that the functional activity in these regions may be closely related to the direct diffusion of information between brain regions. Significance. The multiple-timescale fusion method proposed in this study provides insights into the dynamic aspects of the diffusion process, contributing to a deeper understanding of how brain structure gives rise to brain function.

Developing predictions for pion fragmentation functions

Exploiting crossing symmetry, the hadron scale pion valence quark distribution function is used to predict the kindred elementary valence quark fragmentation function (FF). This function defines the kernel of a quark jet fragmentation equation, which is solved to obtain the full pion FFs. After evolution to a scale typical of FF fits to data, the results for quark FFs are seen to compare favourably with such fits. However, the gluon FF is markedly different. Notably, although FF evolution equations do not themselves guarantee momentum conservation, inclusion of a gluon FF which, for four quark flavours, distributes roughly 11% of the total light-front momentum fraction, is sufficient to restore momentum conservation under evolution. Overall, significant uncertainty is attached to FFs determined via fits to data; hence, the features of the predictions described herein could potentially provide useful guidance for future such studies.

Multi-agent deployment in 3-D via reaction–diffusion system with radially-varying reaction

This paper considers the problem of the deployment of a set of agents distributed on a disk-shaped grid onto three-dimensional (3-D) profiles, by using a continuum approximation (valid in the limit for a large number of agents) and then a control methodology for partial differential equations (PDEs). The agents’ collective behavior is modeled by a pair of radially-varying diffusion–reaction PDEs in polar coordinates, whose state determines the agents’ position. Having a radially-varying reaction coefficient not only increases the challenge of kernel equations becoming singular in radius, but also brings more potential deployment manifolds. To stabilize and increase the convergence of the deployment, a boundary controller and a boundary observer are designed by combining an infinite-dimensional backstepping approach with a Fourier series decomposition technique, thus driving all agents to the desired profile. A key feature of the presented result is that the desired profile only needs to be known by the leaders, with the followers only needing to follow a simple control strategy which requires only the measurement of its current position and communication with its neighbors as defined by the multi-agent system topology. The method provides closed-loop exponential stability with any prescribed convergence rate in the L2 norm. Simulation tests are shown to prove the effectiveness of the proposed algorithm.

SOLUTIONS OF SOME INTEGRAL EQUATIONS OF THE SECOND KIND

The article examines the solutions of some integral equations of the second kind. Such equations arise when using Levi's method to construct a fundamental solution of the Cauchy problem for a degenerate equation of the Kolmogorov type. The equation may also contain a degeneracy on the initial hyperplane. The coefficients of this equation are bounded in the group of principal terms and ones are increasing functions in the group of lowest terms. The considered classes of kernels of integral equations make it possible to preserve the function that determines the growth of the coefficients of the parabolic equation when evaluating the resolvent. In the evalutions of the kernels of integral equations, there are evaluation functions that arise when constructing the corresponding fundamental solution, and fundamental solution of the Cauchy problem of the model equation with constant coefficients.

“Spectral Method” for Determining a Kernel of the Fredholm Integral Equation of the First Kind of Convolution Type and Suppressing the Gibbs Effect

A set of one-dimensional (as well as one two-dimensional) Fredholm integral equations (IEs) of the first kind of convolution type is solved. The task for solving these equations is ill-posed (first of all, unstable); therefore, the Wiener parametric filtering method (WPFM) and the Tikhonov regularization method (TRM) are used to solve them. The variant is considered when a kernel of the integral equation (IE) is unknown or known inaccurately, which generates a significant error in the solution of IE. The so-called “spectral method” is being developed to determine the kernel of an integral equation based on the Fourier spectrum, which leads to a decrease of the error in solving the IE and image improvement. Moreover, the authors also propose a method for diffusing the solution edges to suppress the possible Gibbs effect (ringing-type distortions). As applications, the problems for processing distorted (smeared, defocused, noisy, and with the Gibbs effect) images are considered. Numerical examples are given to illustrate the use of the “spectral method” to enhance the accuracy and stability of processing distorted images through their mathematical and computer processing.

Location, separation and approximation of solutions of nonlinear Hammerstein-type integral equations

From Newton's method, we construct a Newton-type iterative method that allows studying a class of nonlinear Hammerstein-type integral equations. This method is reduced to Newton's method if the kernel of the integral equation is separable and, unlike Newton's method, can be applied to approximate a solution if the kernel is nonseparable. In addition, from an analysis of the global convergence of the method, we can locate and separate solutions of the nonlinear Hammerstein-type integral equations involved. For this study of the global convergence, we use auxiliary functions and obtain restricted global convergence domains that are usually balls.

On the dynamical kernels of fermionic equations of motion in strongly-correlated media

On the dynamical kernels of fermionic equations of motion in strongly-correlated media

Regularity for parabolic equations with singular non-zero divergence vector fields

We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.

Inverse problem of determining the kernel of integro-differential fractional diffusion equation in bounded domain

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven.

TUMME 2023: Tsinghua University Minnesota Master Equation program. New version announcement

This is an updated version of TUMME (Tsinghua University Minnesota Master Equation program), which is a computer program for setting up and solving master equations for chemical kinetics of unimolecular and bimolecular reactions. The master equation is discretized in terms of energy bins, and rate constants are calculated from chemically significant eigenmodes. TUMME has interfaces to Gaussian 16, Polyrate 2023, and MSTor 2023 output files that allow the master equation code to obtain the microcanonical flux coefficients needed for the kernel of the master equation as calculated by conventional transition state theory (TST), variational transition state theory (VTST) with various tunneling methods, or multi-structural or multi-path VTST (MS-VTST or MP-VTST) with various tunneling methods. The tunneling methods supported include zero-curvature tunneling (ZCT), small-curvature tunneling (SCT), large-curvature tunneling (LCT), and microcanonically optimized multidimensional tunneling (μOMT). For mechanisms involving only unimolecular isomerization (no bimolecular pairs), TUMME can set up and solve a conservative master equation for both rate constants and time-dependent energy-bin populations. For mechanisms involving bimolecular pairs, TUMME 2023 can set up and solve two kinds of master equation: (i) a nonconservative master equation for calculating rate constants of bimolecular reactions and (ii) a conservative master equation that includes bimolecular association in the transition matrix and that can be used for calculating the time evolution of the concentration of a pseudo-first-order bimolecular reactant. TUMME is written in double precision with Python 3; quadruple and octuple precision are also available for some subtasks in C++. The Python code can run in serial or parallel (MP or MPI), and C++ code can run on a single processor or on multiple processors with OpenMP. The program includes a manual and a tutorial. New version program summaryProgram Title: TUMME 2023CPC Library link to program files:https://doi.org/10.17632/whcnvm2mc9.2Developer's repository link:https://doi.org/10.5281/zenodo.7943283Licensing Provisions: Apache-2.0 for the program and CC-BY-4.0 for the manualProgramming languages: Python 3 and C++Does the new version supersede the previous version?: YesJournal reference of previous version: Comput. Phys. Commun. 270 (2022) 108140 [1]External libraries: Numpy, Scipy, Numba, mpi4py (optional), modified mpack (optional), qd (optional), omp (optional)Interfaces to other programs: Gaussian 16 [2], Polyrate 2023 [3], and/or MSTor 2023 [4]Nature of problem: Characterize and calculate rate constants for a temperature-dependent and pressure-dependent complex reaction systemSolution method: Solve the energy-grained master equation based on the chemically significant eigenmodes to get phenomenological rate constants and time evolutions of the populations [1,5–7].Reason for new version: The new version increases the functionality and accuracy, and it offers an easier way to run the program. In addition, some bugs have been corrected.Summary of revisions:1.Added an option (available in high precision) to treat pseudo-first-order bimolecular reactions to obtain the time evolution of the reactants and intermediates. [8]2.Added an inverse-Laplace-transform option in the REACTION block.3.Abandoned rounding the sizes of energy bins to an integer wavenumber (ESOT).4.Changed the way to run TUMME by providing an executable bash script.5.Slightly adjusted the strategy for eigenmode mergers to deal with some unexpected cases.6.Adjusted the output for the partition functions.7.Added options for large-curvature tunneling (LCT) and microcanonically optimized multidimensional tunneling (μOMT).8.Added the keyword pyrfile to support a Polyrate 2023 [3] file that ends with a suffix .fu100.9.Fixed some bugs.10.Updated the manual and added a tutorial to the program distribution, so the distribution now has both a manual and a tutorial.

Bilateral boundary control of an input delayed 2-D reaction–diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction–diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy–Schwarz inequality, the 2-D Fourier series, and the Parseval’s theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.

Robust stabilization of 2 × 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of 2 × 2 first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.

Kernel well-posedness and computation by power series in backstepping output feedback for radially-dependent reaction–diffusion PDEs on multidimensional balls

Recently, the problem of boundary stabilization and estimation for unstable linear constant-coefficient reaction–diffusion equation on n-balls (in particular, disks and spheres) has been solved by means of the backstepping method. However, the extension of this result to spatially-varying coefficients is far from trivial. Some early success has been achieved under simplifying conditions, such as radially-varying reaction coefficients under revolution symmetry, on a disk or a sphere. These particular cases notwithstanding, the problem remains open. The main issue is that the equations become singular in the radius; when applying the backstepping method, the same type of singularity appears in the kernel equations. Traditionally, well-posedness of these equations has been proved by transforming them into integral equations and then applying the method of successive approximations. In this case, with the resulting integral equation becoming singular, successive approximations do not easily apply. This paper takes a different route and directly addresses the kernel equations via a power series approach (in the spirit of the method of Frobenius for ordinary differential equations), finding in the process the required conditions for the radially-varying reaction (namely, analyticity and evenness) and showing the existence and convergence of the series solution. This approach provides a direct numerical method that can be readily applied, despite singularities, to both control and observer boundary design problems.

Control Problem Concerned With the Process of Heating a Thin Plate

Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution.Ранее была рассмотрена математическая модель следующей задачи. На части границы правого прямоугольника расположен нагреватель с регулируемой температурой. Требуется найти такой режим его работы, чтобы средняя температура в каком-либо районе достигала некоторого заданного значения. В данной работе рассматривается задача граничного управления, связанная с параболическим уравнением на правом прямоугольнике. На части границы рассматриваемой области указано значение решения с управляющим параметром. Ограничения на управление задаются таким образом, чтобы среднее значение решения в некоторой части рассматриваемой области принимало заданное значение. Вспомогательная задача решается методом разделения переменных, а рассматриваемая задача сводится к интегральному уравнению Вольтерра. Кроме того, в статье дается определение обобщенного решения данной начально-краевой задачи и доказывается существование такого решения. Методом преобразования Лапласа найдено решение интегрального уравнения Вольтерра и доказана теорема существования допустимых управляющих функций. Также показано, что начальное значение допустимой функции управления равно нулю с помощью замены переменной в интегральном уравнении. Доказательство этого исходит из того, что ядра интегральных уравнений положительны и конечны, а система имеет однозначное решение.