Solutions Of Evolution Equations Research Articles

Compacton–anticompacton collisions in the Rosenau–Hyman [formula omitted] equation by numerical simulations with hyperviscosity

Compactons are robust solitary waves with compact support, since they almost retain their shape after mutual interactions. They are solutions of evolution equations with either degenerate nonlinear dispersion or a sublinearity. The most extensively studied case is the Rosenau–Hyman K(p,p) equation, whose compactons are strong solutions for 13. This equation also has anticompactons, but compacton–anticompacton collisions has only been studied for p=2 in symmetric head-on collisions of a compacton and an anticompacton with the same absolute amplitude. Here, asymmetric compacton–anticompacton collisions for integer p≥2 are studied by using computational simulations with small hyperviscosity. For odd p=3, 5, and 7, robust compacton–anticompacton collisions are observed, even with very small hyperviscosity parameter. But for even p=2, 4, and 6, the solution blows up unless the quotient of the absolute amplitudes of the compacton and the anticompacton are below a given threshold that depends on p and the hyperviscosity parameter. Our results stress that, for physical applications, the K(p,p) equation requires the addition of a small amount of viscosity for the robustness of the compacton and anticompacton interactions.

Calculating heat and wave propagation from lateral Cauchy data

UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Systems of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy with an economical computational cost.

CHARACTERISTIC FUNCTION OF THE SYSTEM D–EQUATIO

This paper is devoted to the problems of studying the multiperiodic solution of some evolutionary equations. The article also discusses the existence and uniqueness of a multiperiodic solution with respect to vector functions for an evolutionary reduced equation. Studies have been conducted on the characteristic function of a certain system of the evolutionary equation. Some properties of the vector function are proved. They can be used in the further study of oscillatory bounded solutions of evolutionary equations. Based on the argumentation of the theorem on the existence and uniqueness of an almost multiperiodic solution of the specified system, considered using the method of shortening the characteristic function. All estimates of the characteristic function are based on the enhanced Lipschitz condition, first introduced by academician K. P. Persidskiy. The results will also be useful in the study of periodic solutions of evolutionary equations of mathematical physics

Spectral density for the Schrödinger operator with magnetic field in the unit complex ball: solutions of evolutionary equations and applications to special functions

The Schwartz kernel of the spectral density for the Schrodinger operator with magnetic field in the n-dimensional complex ball is given. As applications, we compute the heat, resolvent and the wave kernels. Moreover, the resolvent and wave kernels are used to establish two new formulas for the Gauss-hypergeometric function.

Forward scattering for non-linear wave propagation in (3 + 1)-dimensional Jimbo-Miwa equation using singular manifold and group transformation methods

Forward scattering of (3 + 1)-dimensional Jimbo-Miwa (JM) equation is investigated by singular manifold method (SMM). The detected Lax pair is reduced through three parameters group method into a compatible couple of ordinary differential equations (ODEs). Three forms of symmetry variables have been discussed leading to integrable system of ODEs. Some new closed form solutions have been obtained. The three parameters group method, which is rarely used in solving PDEs, has emphasized its efficiency in creating symmetry solutions of evolution equations.

Existence and regularity of periodic solutions for neutral evolution equations with delays

The aim of this paper is to study the periodic problem for neutral evolution equation \t\t\t(u(t)−G(t,u(t−ξ)))′+Au(t)=F(t,u(t),u(t−τ)),t∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(u(t)-G\\bigl(t,u(t-\\xi )\\bigr)\\bigr)'+Au(t)=F \\bigl(t,u(t),u(t-\\tau )\\bigr), \\quad t\\in \\mathbb{R}, $$\\end{document} in a Banach space X, where A:D(A)subset Xrightarrow X is a closed linear operator, and −A generates a compact analytic operator semigroup T(t) (tgeq 0). With the aid of the analytic operator semigroup theories and some fixed point theorems, we obtain the existence and uniqueness of periodic mild solution for the neutral evolution equation. The regularity of periodic mild solutions for the evolution equation with delay is studied, and some existence results of the classical and strong solutions are obtained. In the end, we give an example to illustrate the applicability of abstract results. Our works greatly improve and generalize the relevant results of existing literature.

Solutions of evolutionary equation based on the anisotropic variable exponent Sobolev space

In this paper, we are concerned with the equation $$\begin{aligned} u_t= \sum _{i=1}^N\frac{\partial }{\partial x_i} \left( a_i(x)\left| u_{x_i} \right| ^{p_i(x) - 2} u_{x_i} \right) +\sum _{i=1}^N\frac{\partial b_i(u)}{\partial x_i},\ \ (x,t) \in \Omega \times (0,T), \end{aligned}$$ where $$a_i(x)|_{x\in \partial \Omega }=0$$ and $$a_i(x)|_{x\in \Omega }>0$$ . By the theory of anisotropic variable exponent Sobolev spaces, we study the well-posedness of weak solutions of this equation. Since $$a_i(x)$$ is degenerate on the boundary, the stability of weak solutions may be established without any boundary value condition. The main feature which distinguishes this paper from other related works lies in the fact that we propose a novel analytical method to deal with the stability of weak solutions.

Steady plastic wave fronts and scale universality of strain localization in metals and ceramics

Mechanisms of structural relaxation are linked with the metastability of nonequilibrium potential of solid with defects and the generation of collective modes of defects responsible for the plastic strain and damage localization. It is shown that spatial-temporal dynamics of collective modes (auto-solitary and blow-up dissipative structures) provide the anomalous relaxation ability of nonlinear system “solid with defects” in the conditions of the specific type of criticality – structural-scaling transition. These modes have the nature of self-similar solutions of evolution equations for damage parameter (defect-induced strain) and represent the “universality class” providing the four power law for a steady plastic front, splitting of an elastoplastic shock wave front, and elastic precursor decay kinetics. Wide-range constitutive equations reflecting the linkage between defect-induced mechanisms and structural relaxation are used in the numerical simulation for shock wave loading of metals and ceramics in the comparison with experiments.

Solutions of evolution equations for medium-induced QCD cascades

In this paper we present solutions of evolution equations for inclusive distribution of gluons as produced by jet traversing quark–gluon plasma. We reformulate the original equations in such a form that virtual and unresolved-real emissions as well as unresolved collisions with medium are resummed in a Sudakov-type form factor. The resulting integral equations are then solved most efficiently with use of newly developed Markov Chain Monte Carlo algorithms implemented in a dedicated program called MINCAS. Their results for a gluon energy density are compared with an analytical solution and a differential numerical method. Some results for gluon transverse-momentum distributions are also presented. They exhibit interesting patterns not discussed so far in the literature, in particular a departure from the Gaussian behaviour – which does not happen in approximate analytical solutions.

Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

Exact traveling (solitary) wave solutions of nonlinear partial differential equations (NLPDEs) are analyzed for third-order nonlinear evolution equations. These equations have indeterminant homogenous balance and therefore cannot be solved by the Power Index Method (PIM). Some evolution equations are linearizable where solutions are transferred from those of a linear PDE. For other evolution equations transforming to a NLPDE which has a homogenous balance gives rise to possible solutions by the PIM. The solutions for evolution equations that are not linearizable are developed here.

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

Abstract It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

Asymptotic behavior of $$C^0$$ C 0 -solutions of evolution equations with uncertainties

In this paper, we study Cauchy problem for evolution equations with uncertainties in semilinear metric space $$C(J_\infty ,\mathcal {T}_{\mathcal {F}})$$ of symmetric triangular fuzzy-valued functions. We use fixed point approach and measure of noncompactness to investigate the solvability, the existence of decay $$C^0$$ -solutions and the asymptotic stability of equilibrium point of the problem. We present several examples to illustrate the theoretical results.

Stationary and non-stationary solutions of the evolution equation for neutrino in matter

We study solutions of the equation which describes the evolution of a neutrino propagating in dense homogeneous medium in the framework of the quantum field theory. In the two-flavor model the explicit form of Green function is obtained, and as a consequence the dispersion law for a neutrino in matter is derived. Both the solutions describing the stationary states and the spin-flavor coherent states of the neutrino are found. It is shown that the stationary states of the neutrino are different from the mass states, and the wave function of a state with a definite flavor should be constructed as a linear combination of the wave functions of the stationary states with coefficients, which depend on the mixing angle in matter. In the ultra-relativistic limit the wave functions of the spin-flavor coherent states coincide with the solutions of the quasi-classical evolution equation. Quasi-classical approximation of the wave functions of spin-flavor coherent states is used to calculate the probabilities of transitions between neutrino states with definite flavor and helicity.

Estimates of Wiman–Valiron type for evolution equations

We establish estimates of Wiman–Valiron type for solutions of evolution equations with a pseudodifferential operator of the Hormander class in a Hilbert space. Estimates of this type characterize the behavior of the solution of the problem as t→∞ or t → 0 depending on the decay or growth rate of the Fourier coefficients of the initial data.

Almost periodic solutions of evolution equations

We state existence theorems for almost periodic solutions of evolution problems, namely, quasi-autonomous problems and more generally, time dependent evolution equations. We apply these theorems firstly, to a boundary value quasilinear hyperbolic equation of first order, and secondly, to a boundary value quasi-parabolic equation.

A numerical framework for computing steady states of structured population models and their stability.

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

Fractional approximation of solutions of evolution equations

AbstractWe show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order

Weighted pseudo asymptotically periodic mild solutions of evolution equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.

Feynman formulas for solutions of evolution equations on ramified surfaces

We obtain a representation, using a Feynman formula, for the operator semigroup generated by a second-order parabolic differential equation with respect to functions defined on the Cartesian product of the line ℝ and a graph consisting of n rays issuing from a common vertex.