Solution Of Equation Research Articles

Convergence of solutions of the porous medium equation with reactions

Consider the Cauchy problem of one-dimensional porous medium equation (PME) with reactions. We first prove a general convergence result, that is, any bounded global solution starting at a nonnegative compactly supported initial data converges as t\to \infty to a nonnegative zero of the reaction term or a ground state stationary solution. Based on it, we give out a complete classification on the asymptotic behavior of the solutions for PME with monostable, bistable and combustion types of nonlinearities.

Zero distribution of solutions of some linear differential equations

Assuming that A ( z ) = B 1 ( z ) e P ( z ) + B 2 ( z ) e ϱp ( z ) + Q ( z ) , where B 1 ( z ) , B 2 ( z ) , p ( z ) are nonzero polynomials, ϱ ∈ ( 0 , 1 ) , and Q ( z ) is an entire function of order less than deg ⁡ p , we study the oscillation of solutions of the linear differential equation f ( k ) + A ( z ) f = 0 . Some conditions on A ( z ) are given to guarantee that any non-trivial solution f satisfies λ ( f ) = ∞ or λ ( f ) ≥ σ ( A ) , where σ ( f ) and λ ( f ) denote respectively the order of f and the exponent of convergence of zeros of f. We also find some types of linear differential equations with zero-free solutions.

Lattice sums of I-Bessel functions, theta functions, linear codes and heat equations

We extend a certain type of identities on sums of I-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of theta functions such as the Dedekind eta function can be given by I-Bessel lattice sum identities with characters. We consider analogues of theta functions of lattices coming from linear codes and show that sums of I-Bessel functions defined by linear codes can be expressed by complete weight enumerators. We also prove that I-Bessel lattice sums appear as solutions of heat equations on general lattices. As a further application, we obtain an explicit solution of the heat equation on Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{Z}}^n$$\\end{document} whose initial condition is given by a linear code.

Description of the emission of cumulative secondary particles in collisions of heavy ions of intermediate energies based on the non-equilibrium hydrodynamic approach

We have analyzed the double-differential cross sections for the emission of cumulative protons, pions, kaons, and antiprotons in collisions of carbon nuclei on a fixed target at an energy of 19.6 GeV/nucleon obtained in the IHEP experiment at the U-70 accelerator. When describing these spectra, the nonequilibrium approach was taken into account as a result of the joint solution of the kinetic equation with the equations of hydrodynamics. Comparisons with other approaches are made.

CGC and saturation approach: Impact-parameter dependent model of perturbative QCD and combined HERA data

In this paper we confront the color glass condensate and saturation approach of Contreras [Non-linear equation in the re-summed next-to-leading order of perturbative QCD: The leading twist approximation, ] with the experimental combined HERA data and obtain its parameters. The model includes two features that are in accordance with our theoretical knowledge of deep inelastic scattering. These consist of: (i) the use of analytical solution for the nonlinear Balitsky-Kovchegov evolution equation and () the exponential behavior of the saturation momentum on the impact parameter b-dependence, characterized by Qs∝exp(−mb) which reproduces the correct behavior of the scattering amplitude at large b in accord with Froissart theorem. The model results are then compared to data at small-x for the structure function of the proton F2, the longitudinal structure function FL, the charm structure function F2cc¯, the exclusive vector meson (J/ψ,ϕ,ρ) production and deeply virtual Compton scattering. We obtain a good agreement for the processes in a wide kinematic range of Q2 at small-x. Our results provide a strong guide for finding an approach, based on color glass condensate and saturation effective theory for high-energy QCD, to make reliable predictions from first principles and for forthcoming experiments like the Electron Ion Collider and the LHeC. Published by the American Physical Society 2024

Local and Global Uniqueness Theorems of the N-th Order Partial Differential Equations

In this paper, we consider inequalities in which the function is an element of n-th partially order space. Local and Global uniqueness theorem of solutions of the n-the order Partial differential equation Obtained which are applications of Gronwall's inequalities.

Axisymmetric Cylindrical Green’s Function for the Steady-State Advection-Diffusion Operator with Uniform Velocity

The advection-diffusion equation has a wide range of applications as it governs energy and mass transport in a moving medium. Analytical solutions of the advection-diffusion equation for classes of boundary conditions amounts to finding exact expressions of the Green’s function of the advection-diffusion operator. In this work, the Green’s function of the steady-state advection-diffusion operator is obtained for an axisymmetric cylindrical problem with uniform velocity along the axis of the cylinder. The solution is exact, as axial diffusion (or conduction) is not neglected. The method of solution utilizes a formalism relating the Green’s function of the diffusion operator to that of the steady-state advection-diffusion operator with uniform velocity. The Green’s function obtained is applied to a Dirichlet boundary value problem. The distribution, (either particle concentration, or temperature), is plotted for various values of Péclet numbers. This work demonstrates how to invert the steady-state advection-diffusion operator with uniform velocity in a non-trivial geometry, namely the cylinder.

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

Existence and Uniqueness of Homogeneous Linear Equation Solutions in Supertropical Algebra

This research investigates the existence and uniqueness of solutions to homogeneous linear equations in supertropical algebra. We analyze the structure of supertropical matrices to identify the conditions in which nontrivial solutions exist for the system of equations A⊗𝒙 ⊨ 𝜀, where A is a matrix over a supertropical semiring and x is a vector. By applying determinant-based criteria, we demonstrate how tropical and supertropical values influence the solution space. The research applies theorems that determine the presence of trivial and nontrivial solutions and uses examples to illustrate practical methods for solving homogeneous matrix systems. This highlights the distinct characteristics of supertropical algebra compared to classical linear algebra. Our findings provide a deeper insight into solution behaviors in supertropical systems, paving the way for further research in tropical mathematics.

On Bäcklund transformations to stationary equations in hierarchy of the second Painlevé equation

The analytical properties of solutions to stationary equations of the second and fourth orders in the hierarchy of the second Painlevé equation are considered. For the second-order equation, it is shown that the Bäcklund transformation in the general case determines the formula of the addition theorem for the Weierstrass elliptic function. For the fourth and sixth-order equations, Bäcklund transformations and special classes of solutions are constructed. It has been established that, for a certain relationship between the parameters, the set of solutions to the first term of the stationary hierarchy is a subset of solutions to the second term and the set of solutions to the second term of the hierarchy is a subset of solutions of the six-order equation of the stationary hierarchy of the second Painlevé equation.

Stochastic soliton and rogue wave solutions of the nonlinear Schrödinger equation with white Gaussian noise

In this letter, we propose a novel integrable nonlinear Schrödinger equation and its Lax pair, influenced by Brownian motion and white Gaussian noise. We aim to construct and solve new integrable systems affected by the white Gaussian noise. Utilising the classical and generalised Darboux transformations, the stochastic soliton solutions and the stochastic rogue wave solutions of this novel integrable nonlinear Schrödinger equation are obtained and expressed in determinant form. Studies of stochastic soliton and rogue wave solutions of the NLS equation are essential for complex physical and mathematical phenomena where nonlinear interactions and randomness play crucial roles.

Existence and Uniqueness of Solution of Fractional Differential Equation Using Non-local Operator

Objectives : To investigate the existence and uniqueness of a solution to the non-local Cauchy problem of order using the Atangana Baleanu (AB) fractional derivative operator. Methods : Using Schauder's fixed point theorem and the Arzela-Ascoli theorem, the study proved the existence of a solution to the given problem. Further, it obtains results for the uniqueness of the solution. Findings : This study proves the existence of a solution to the non-local Cauchy problem under the given conditions. Results are also provided for the uniqueness of the solution. Novelty : A novel approach to fractional differential equations is represented by applying Atangana-Baleanu fractional derivative operator to the non-local Cauchy problem. Schauder's fixed point theorem and Arzela-Ascoli's theorem, are used to show existence and uniqueness. Further, one detailed example has been solved. Keywords: Existence, Uniqueness, Non-local operator, Schauder fixed point theorem, Arzela-Ascoli theorem, Cauchy problem

Existence of Positive Solutions for Singular Difference Equations with Nonlinear Boundary Conditions

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

Classification of the non-null electrovacuum solution of Einstein–Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions. Thus the classification of the electrovacuum stackel spaces of all types is completed and complete list of these spaces is constructed.

Numerical investigation of serration angle on trailing edge noise reduction

The influence of serration angle on the three-dimensional flow characteristics and noise attenuation mechanisms of a NACA6512-63 airfoil is explored in this research using large eddy simulation (LES) and spectral proper orthogonal decomposition (SPOD) methods. Analysis of the mean flow field reveals the formation of counter-rotating vortex pairs, resulting from the outward flow towards the serration edges and the downwash flow in the valleys. This flow modification alters the effective angle of the serration, which in turn impacts its noise reduction performance. Furthermore, the noise reduction mechanism is investigated by comparing the general solution of Lyu's semi-analytical equation with the SPOD mode results, and the frequency range associated with noise attenuation is analyzed. The findings indicate that the case with λ/h=0.15 demonstrates the most significant noise reduction effect. This research offers valuable insights into the interplay between serration geometry and the resulting flow characteristics and noise reduction mechanisms.

Acoustic pattern of a full scale simplified ship

Underwater radiated noise from ship propellers is one of the major contributors to the increase in overall noise levels in marine environments. The development of numerical tools for the computation of noise propagation is of paramount importance for the assessment of the noise impact on the environment. Hybrid approaches, based on the decoupling of the computation of the fluid flow and the successive evolution of the noise using an acoustic analogy, are typically conducted in open-water conditions, which do not correspond to a realistic environment. The Full Acoustic Analogy, which couples a propagation model based on the solution of the acoustic wave equation to the hybrid approach, enables the propagation of the noise generated by a marine propeller in a confined heterogeneous domain. The near-field scattering characteristics of a simplified hull positioned in front of a ship's propeller are analyzed in shallow and deep water, in the presence of the free surface, and compared to the open water case. The analysis determines the directivity of the propeller noise and how this may affect how the noise propagates in the marine environment.

Application of Analytical Solutions of the Reactive Transport Equation for Underground Methanation Reactors

AbstractFluctuations in the production of renewable-based electricity have to be compensated by converting and storing the energy for later use. Underground methanation reactors (UMR) are a promising technology to address this issue. The idea is to create a controlled bio-reactor system in a porous underground formation, where hydrogen obtained from renewable energy sources by electrolysis and carbon dioxide from industrial sources are fed into the reactor and converted into methane. Microorganisms, known as methanogenic archaea, catalyze the chemical reaction by using the two non-organic substrates as nutrients for their growth and for their respiratory metabolism. The generated synthetic methane is renewable and capable to compete with the fossil methane. Mathematical models play an important role in the design and planning of such systems. Usually, a numerical solution of the model is required since complex initial-boundary problems cannot be solved analytically. In this paper, an existing bio-reactive transport model for UMR is simplified to such an extent that an analytical solution of the advection-dispersion-reaction equation can be applied. A second analytical solution is used for the case without dispersion. The analytical solutions are shown for both the educt (hydrogen) and the reaction product (methane). In order to examine the applicability of the analytical models, they are compared with the significantly more complex numerical model for a 1D case and a 3D case. It was shown that there is an acceptable agreement between the two analytical solutions and the numerical solution in different spatial plots of hydrogen and methane concentration and in the methane concentration in the withdrawn gas. The mean absolute error in the mole fraction is well below 0.015 in most cases. The spatial distribution of the hydrogen concentration in the comparison to the 3D case shows a higher deviation with a mean absolute error of approx. 0.023. As expected, the model with dispersion shows a slightly lower error in all cases, as only here the gas mixing resulting in smeared displacement fronts can be represented. It is shown that analytical modeling is a good tool to get a first estimation of the behavior of an UMR. It allows to help in the design of well spacing in combination with the injection rate and injected gas composition. Nevertheless, it is recommended to use more complex models for the later detailed analysis, which require a numerical solution.

Prediction of the acoustic particle velocity in aerodynamic flows and its applications to the boundary element method

The acoustic variables are essential for understanding sound energy transmission and redistribution. The previous methods focused on acoustic pressure without the acoustic particle velocity, which is not enough for acoustic field evaluation. In this paper, an analytical formula for acoustic particle velocity prediction is developed based on the permeable surface integral solution of a vector wave equation. The validation examples are presented with stationary dipole and rotational monopole. The acoustic quantities, pressure, and velocity are solved at the same time and will be applied as input for the boundary element method (BEM) to evaluate the sound scattering problem in the frequency domain. Furthermore, the method is applied to study the acoustic field of flow passing a cylinder. The acoustic pressure shows good agreement with the flow observer. The acoustic particle velocity is also investigated.

Development and solution of the kinetics equation and adsorption isotherm for gold adsorption from cyanide solutions onto activated carbon

Development and solution of the kinetics equation and adsorption isotherm for gold adsorption from cyanide solutions onto activated carbon

Discretization of Continuous Spaces using Barycentric Subdivision Method with Metric Space Constraints on the Nearest Neighbour Edges

The Rao-Wilton-Glisson (RWG) is a commonly used basis function in the numerical solution of the electric field integral equation (EFIE) using the Method of Moments (MoM) and Galerkin approach. This method relies on triangular patches to approximate the surface current. Traditionally, barycentric subdivision of a primary triangle into n-sub-triangles has been used with RWG basis function to solve the EFIE using MoM. This paper presents a method of approximating a surface using triangular patches by sub-dividing a primary triangle into (2n-1) subtriangles. which creates a denser mesh than the widely used nine (9) points quadrature method. The structure is approximated by small square patches, which are further sub-divided into two primary triangles. By applying barycentric sub-division, the primary triangles are decomposed into sub-triangles to create a mesh over the surface of the structure. Using graph theory, the triangular meshes are defined by the function, G (V, E), where V and E are the vertices and edges of the triangles in the mesh space, respectively. The connectivity matrix of shared edges is found by imposing a constraint on the edge length using metric spaces. This method approximates a square patch with 32 scalene triangles and shows that it can be used to reconstruct equilateral patches and doubly split double ring into mesh structures.