Right-hand Side Research Articles

Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization

Abstract We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent of temperature, we propose an a posteriori error estimator and study its reliability and efficiency properties. We illustrate the theory with numerical examples.

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Abstract We consider the semi-direct products $G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$, and ${\mathbb{Z}}^{2}\rtimes \Gamma (2)$ (where $\Gamma (2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^{*}$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in $K_{0}$ provide explicit generators that are matched by the Baum–Connes assembly map.

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ2. The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side

Difference schemes that approximate dynamic systems are considered discrete models of the same phenomena that are described by continuous dynamic systems. Difference schemes with t-symmetry and midpoint and trapezoid schemes are considered. It is shown that these schemes are dual to each other, and, from this fact, we derive theorems on the inheritance of quadratic integrals by these schemes (Cooper’s theorem and its dual theorem on the trapezoidal scheme). Using examples of nonlinear oscillators, it is shown that these schemes poses challenges for theoretical research and practical application due to the problem of extra roots: these schemes do not allow one to unambiguously determine the final values from the initial values and vice versa. Therefore, we consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible schemes with t-symmetry can be easily constructed for any dynamical system with a quadratic right-hand side. As an example of such a dynamic system, a top fixed at its center of gravity is considered in detail. In this case, the discrete theory repeats the continuous theory completely: (1) the points of the approximate solution lie on some elliptic curve, which at Δt→0 turns into an integral curve; (2) the difference scheme can be represented using quadrature; and (3) the approximate solution can be represented using an elliptic function of a discrete argument. The last section considers the general case. The integral curves are replaced with closures of the orbits of the corresponding Cremona transformation as sets in the projective space over R. The problem of the dimension of this set is discussed.

A mathematical model for the formation of the mineral composition of rocks removed to the surface during a volcano eruption


 
 
 The article examines the origin of chemicals obtained as a result of cooling on the Earth’s surface during a volcanic eruption. For this, we propose a mathematical model of the formation of the mineral composition of rocks brought to the surface during a volcanic eruption, depending on the crystallization temperature. To do this, we consider a certain inverse problem for the heat equation and prove its solvability under certain conditions. The proposed method can also be used in case of an earthquake. Thus, by studying the composition of rocks on the surface, we can determine at what depth the corresponding layer of rocks is located. Mathematically, this means that you need to find the unknown functions f (x) and u (x, t), where the function f (x) characterizes the source of the formation of rocks, and the function u (x, t) characterizes the crystallization of rocks depending on tem- perature because each rock has its crystallization temperature depending on temperature. We assume that the functions h (x, t), g (x, t) are known. Unknown functions f (x) and u (x, t). Using the inverse problem, we determine the function u (x, t) – the temperature in the lower layers of the earth, and knowing the temperature of crystallization of rocks, we can determine what depth the required rock is. The main purpose of the inverse problem is to determine the composition of rocks below the earth’s surface using data obtained at the earth’s surface. Experimentally, this mathematical model means that with the help of certain sensors on the surface, it is possible to give information about the subterranean depths. This information is determined by the nature of the object under study and the experimental complex used in this study. In such situations, for the diagnosis of objects and their internal structure, mathematical processing, and interpretation of observation results are required .We are talking about those tasks in which it is required to determine the causes if the consequences obtained as a result of observation are known. For example, determining the location and power of an earthquake from the vibrations measured on the earth’s surface. Or, by the structure of minerals on the surface of the earth, determining the depth of their location underground, etc. When processing the experimental data, a conclusion is made about the internal connections of the phenomenon or process. When the mathematical model of the process under study is known, the problem of identifying the mathematical model is posed. Namely, the determination of the coefficients of differential equations, their right-hand side, the boundary of the region, boundary, and initial conditions – these problems are related to inverse problems of mathematical physics. In general, inverse problems.
 
 

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.</p></abstract>

Математическая теория ускоренного расширения Вселенной на основе принципа наименьшего действия и модели Фридмана и Милна-Маккри

In classical works, equations for the fields of gravity and electromagnetism are proposed without deriving the right-hand sides. Here we give the derivation of the right-hand sides and analysis of the stress–energy tensor within the framework of the Vlasov–Maxwell–Einstein equations and models of the Milne–McCrea and Friedman type. New models of the accelerated expansion of the Universe without Einstein's lambda are proposed.

To boundary value problems for a generalized hyperbolic equation with two fractional differentiation operators

Boundary value problems for a one-space-dimensional hyperbolic equation with two fractional differentiation operators and variable coefficients are considered. The method of finite differences is used for the approximate solution to the problems posed. Difference schemes for each of the problems are constructed, and by the method of energy inequalities for various relations between the orders of the fractional derivative (α, β) a priori estimates in differential and difference interpretations are obtained. The obtained estimates imply the uniqueness and stability of the solution with respect to the right-hand side and initial data, as well as the convergence of the solution of the difference problem to the solution of the original differential problem. Algorithms for the numerical solution of the problems posed are constructed.

Stability analysis of an implicit fractional integro-differential equation via integral boundary conditions

The primary objective of this research study is to analyze a boundary problem involving Caputo fractional integro-differential equations. The focus is on a differential equation with a nonlinear right-hand side composed of two terms. The stability analysis of a fractional integro-differential equation is presented using Ulam's concept. Furthermore, this research study establishes the correlation between the stated problem and the Volterra integral equation. The investigation proceeds by utilizing the renowned Banach and Krasnoselskii's fixed point theorems to explore the existence and uniqueness of solutions for the problem. Additionally, to provide tangible evidence of the abstract findings, two illustrative examples are presented.

Enhancing neutrosophic fuzzy goal programming approach for solving multi-objective probabilistic linear programming problems

This article considers a stochastic multi-objective linear programming problem in a neutrosophic framework. The right-hand sides of the constraints are random variables associated with their means and variances, and the single-valued trapezoidal neutrosophic numbers are included in the coefficients of the objective functions to investigate the problem. The problem is successfully converted into the related deterministic programming model based on the formulation of the scoring function and several distributions, such as the Uniform, Exponential, and Gama distributions. Applying fuzzy goal programming, the deterministic problem is solved. An example is then solved to demonstrate the solution methodology.

Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling

In this study, we introduce a way to control the viscosity of the numerical approximation in the Godunov-like smoothed particle hydrodynamics (SPH) methods. This group of SPH methods includes momentum and energy fluxes in the right-hand sides of the equations, which are calculated by the solution of the Riemann problem between each pair of neighboring particles within the support radius of the smoothing kernel, which is similar to the procedure for the calculation of fluxes across cell boundaries in Godunov schemes. Such SPH methods do not require the use of artificial viscosity since the significant numerical viscosity is already introduced by a Riemann problem solution. We demonstrate that such a numerical viscosity may be measured and obtain the explicit expression for it depending on smoothed particle properties. In particular, we have found that Godunov-like SPH method with interparticle contact algorithms produces numerical viscosity several orders of magnitude higher than physical viscosity in materials. Modern approaches, such as SPH with monotonic upstream-centered scheme for conservation laws or weighted essentially non-oscillatory reconstruction techniques, have not only lower numerical viscosity but also too large for modeling real-world viscous flows. By constructing a correcting viscous stress tensor based on the analytical solution for discontinuous viscous flow, it is possible to reduce the viscous stresses of numerical origin. The use of such a correction makes it possible to improve the agreement with experiments in the simulation of viscous flows without using schemes of higher order reconstruction.

Anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with superlinear nonlinearities

<abstract><p>In this paper we consider a class of anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with nonlinear right-hand sides that are superlinear at $ \pm\infty $. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.</p></abstract>

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

On the stability of a nonlinear nonautonomous scalar equation with variable delay

The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

CSPlib: A performance portable parallel software toolkit for analyzing complex kinetic mechanisms

Computational singular perturbation (CSP) is a method to analyze dynamical systems. It targets the decoupling of fast and slow dynamics using an alternate linear expansion of the right-hand side of the governing equations based on eigenanalysis of the associated Jacobian matrix. This representation facilitates diagnostic analysis, detection and control of stiffness, and the development of simplified models. We have implemented CSP in a C++ open-source library CSPlib1 using the Kokkos2 parallel programming model to address portability across diverse heterogeneous computing platforms, i.e., multi/many-core CPUs and GPUs. We describe the CSPlib implementation and present its computational performance across different computing platforms using several test problems. Specifically, we test the CSPlib performance for a constant pressure ignition reactor model on different architectures, including IBM Power 9, Intel Xeon Skylake, and NVIDIA V100 GPU. The size of the chemical kinetic mechanism is varied in these tests. As expected, the Jacobian matrix evaluation, the eigensolution of the Jacobian matrix, and matrix inversion are the most expensive computational tasks. When considering the higher throughput characteristic of GPUs, GPUs performs better for small matrices with higher occupancy rate. CPUs gain more advantages from the higher performance of well-tuned and optimized linear algebra libraries such as OpenBLAS. Program summaryProgram Title: CSPlibCPC Library link to program files:https://doi.org/10.17632/p9gb7z54sp.1Developer's repository link:https://github.com/sandialabs/csplibLicensing provisions: BSD 2-clauseProgramming language: C++Nature of problem: Dynamical systems can involve coupled processes with a wide range of time scales. The computational singular perturbation (CSP) method offers a reformulation of these systems which enables the use of dynamically-based diagnostic tools to better comprehend the dynamics by decoupling fast and slow processes. CSPlib is an open-source software library for analyzing general ordinary differential equation (ODE) and differential algebraic equation (DAE) systems, with specialized implementations for detailed chemical kinetic ODE/DAE systems. It relies on CSP for the analysis of these systems. CSPlib has been used in gas kinetic and heterogeneous catalytic kinetic models.Solution method: CSP analysis seeks a set of basis vectors to linearly decompose the right-hand side (RHS) of a dynamical system in a manner that decouples fast and slow processes. The CSP basis vectors are often well approximated with the right eigenvectors of the RHS Jacobian. And the left basis vectors are found by the inversion of the matrix, whose columns are the CSP basis vectors. Accordingly, the right and left CSP basis vectors are orthonormal. CSP defines mode amplitudes as the projections of the left basis vectors on the RHS; the time scales as the reciprocals of the RHS Jacobian eigenvalue magnitudes; and the CSP pointers, which are the element-wise multiplication of the transpose of the right CSP basis vectors with the left CSP basis vectors. For kinetic models that can be cast as the product of a generalized stoichiometric matrix and a rate of progress vector, CSP defines the participation index, which represents the contribution of a chemical reaction to each mode. Further, it defines the slow and fast importance indices, which describe the contribution of a chemical reaction to the slow and fast dynamics of a state variable, respectively. These indices are useful in diagnostic studies of dynamical systems and the construction of simplified models.Additional comments including restrictions and unusual features: CSPlib is a portable library that carries out many CSP analyses in parallel and can be used in modern high-performance platforms.

CONVEXITY OF REACHABLE SETS OF QUASILINEAR SYSTEMS

This paper investigates convexity of reachable sets for quasilinear systems under integral quadratic constraints. Drawing inspiration from B.T. Polyak's work on small Hilbert ball image under nonlinear mappings, the study extends the analysis to scenarios where a small nonlinearity exists on the system's right-hand side. At zero value of a small parameter, the quasilinear system turns into a linear system and its reachable set is convex. The investigation reveals that to maintain convexity of reachable sets of these systems, the nonlinear mapping's derivative must be Lipschitz continuous. The proof methodology follows a Polyak's scheme. The paper's structure encompasses problem formulation, exploration of parameter linear mapping and image transformation, application to quasilinear control systems, and concludes with illustrative examples.

Разностная схема повышенного порядка аппроксимации для уравнения Аллера с переменными коэффициентами

The initial boundary value problem for the one-dimensional Aller’s equation with variable coefficients and boundary conditions of the first kind is studied. The problem under study describes the processes of heat transfer in a heterogeneous environment, moisture transfer in soils, and fluid filtration in fractured porous media. To numerically solve the problem posed, a difference scheme of high order of accuracy was constructed - fourth order of accuracy in h and second order of accuracy in τ. Using the method of energy inequalities, an a priori estimate of the solution in a difference treatment is obtained. From this estimate it follows that the solution is unique and stable with respect to the right-hand side and initial data. Under the assumption of the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies that the solution of the constructed difference problem converges to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. The goal and scientific novelty of the work is to obtain a new numerical scheme of a higher order of approximation when solving the Dirichlet problem for the Aller’s equation with variable coefficients.

EXISTENCE RESULTS AND NUMERICAL SOLUTION OF FULLY FOURTH ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a fully fourth-order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments.By the reduction of the problem to operator equation for the right-hand side nonlinear function, we establish the existence and uniqueness of the solution and construct iterative methods on both continuous and discrete levels for solving it. We obtain the total error estimate for the discrete iterative solution. Many examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

Interhemispheric asymmetrical change in gray matter volume in patients with unilateral hippocampal sclerosis.

To clarify the interhemispheric asymmetrical change in gray matter volume (GMV) in unilateral hippocampal sclerosis (HS), we compared changes in GMV relative to normal subjects between the HS and contralateral or non-HS sides. Forty-five patients with unilateral HS and 30 healthy subjects were enrolled. We quantified changes in GMV in the patients with HS as compared to GMV in the normal subjects by introducing the Z-score (Z-GMV) in each region or region of interest in unilateral HS. Then, we assessed the asymmetrically decreased regions, that is, regions with significantly higher Z-GMV on the HS side than the contralateral or non-HS side. Z-GMV was calculated according to the two templates of 58 regions per hemisphere covering the whole brain by anatomical automatic labeling (AAL) and 78 regions per cerebral hemisphere using the Anatomy Toolbox. Seven and four regions in AAL and 17 and 11 regions in Anatomy Toolbox were asymmetrically decreased in the Left Hand Side (LHS) and Right Hand Side (RHS), respectively. Hippocampus and Caudate in AAL, five subregions of the hippocampus (CA1-3, Dentate Gyrus and hippocampus-amygdala-transition-area and 4 extrahippocampal regions including two subregions in amygdala (CM: Centromedial, SF: Superficial), basal forebrain (BF) (Ch4), and thalamus (temporal) in anatomy toolbox were common among LHS and RHS concerning asymmetrically decreased regions. By introducing Z-GMV, we demonstrated the regions with asymmetrically decreased GMV in LHS and RHS, and found that the hippocampus and extrahippocampal regions, including the BF, were the common asymmetrically decreased regions among LHS and RHS.

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.