Asymptotic structure for solutions of the Cauchy problem for Burgers type equations

Large time asymptotic structure for solutions of the Cauchy problem for a generalized Burgers equation is determined. In particular, Gelfand’s question about location of viscous shock waves for such equations is answered.     

Classical solutions to relativistic Burgers equations in FLRW space-times

We are interested in the classical solutions to the Cauchy problem of relativistic Burgers equations evolving in Friedmann-Lemat?tre-Robertson-Walker(FLRW)space-times,which are spatially homogeneous,isotropic expanding or contracting universes.In such kind of space-times,we first derive the relativistic Burgers equations from the relativistic Euler equations by letting the pressure be zero.Then we can show the global existence of the classical solution to the derived equation in the accelerated expanding space-times with small initial data by the method of characteristics when the spacial dimension n=1 and the energy estimate when n 2,respectively.Furthermore,we can also show the lifespan of the classical solution by similar methods when the expansion rate of the space-times is not so fast.     

Large‐time behaviour of solutions to non‐linear wave equations: higher‐order asymptotics

The Cauchy problems for the Korteweg–de Vries–Burgers equation and the Benjamin–Bona– Mahony–Burgers equation are studied. Using subtle estimates of solutions to the linearized equations, the higher‐order terms of the asymptotic expansion as of solutions are derived. Copyright © 1999 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations

We show that the solutions of the initial value problems for a large class of Burgers type equations approach with time to the sum of appropriately shifted wave-trains and of diffusion waves.

Résumé

Nous montrons que les solutions du problème de Cauchy pour une grande classe d'équations de type de Burgers sont approchées en temps grand vers des sommes d'ondes de diffusion et d'ondes progressives adéquatement translatées.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

On the Cauchy problem for linear hyperbolic functional-differential equations

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.     

Parabolic Type Equations and Markov Stochastic Processes on Adeles

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.     

Cauchy problem of semi-linear parabolic equations with weak data in homogeneous space and application to the Navier-Stokes equations

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data n the homogeneous spaces.We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in C(σ,s,p)and L~q([O,T);H~(s,p)by introducing the concept of both admissiblequintuptet and compatible space and establishing estblishing time-space estimates for solutions to the linear parabolic typeequations For the small data,we prove that these results can be extended globally in time. We also study the     

A Note on the Linear Stability of Burgers Vortex

A two-parameter family of analytical solutions of the linearized equations for axially dependent disturbances to the three-dimensional base strain field associated with the well-known axisymmetric Burgers vortex is presented. The solutions are valid asymptotically at large axial distances from the stagnation point. By a formal perturbation analysis, perturbative solutions are also found for disturbances to the Burgers vortex for small Reynolds numbers. The solutions are believed to provide important insights into the nature of the as-yet-unsolved problem of the linear stability of Burgers vortex to axially varying disturbances.     

Convergence estimates of Galerkin-wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential equations

A class of Cauchy problems for interesting complicated periodic pseudodifferential equations is considered. By the Galerkin-wavelet method and with weak solutions one can find sufficient conditions to establish convergence estimates of weak Galerkin-wavelet solutions to a Cauchy problem for this class of equations.

     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

Space-time Estimates for Parabolic Type Operator and Application to Nonlinear Parabolic Equations

In this present paper we establish space-time estimates of solutions for linear parabolic type equations based on classical multipliers theory or operator semigroup theory. According to space-time estimates we first construct suitable work space L^q(0, T; L^P), moreover we study the Cauchy problem and initial boundary value problem for semilinear parabolic equation in L^q(0, T; L^P) type space.     

Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

     

A Probability Approach to the Method of Vanishing Viscosity

We clarify conditions under which solutions to the Cauchy problem for a general (fully nondiagonal) system of linear and nonlinear parabolic equations admit probability representations. Such representations are also used for constructing and studying solutions to the Cauchy problem for nonlinear hyperbolic systems. Bibliography: 26 titles.     

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.     

一般非线性扩散方程Cauchy问题广义解的渐近性质

本文研究了比较一般的非线性扩散方程的Cauchy问题广义解的渐近性质.利用试验函数与积分估计的方法,获得了广义解当t→∞时的渐近性质以及其广义整体解u(x,t),推广了该类方程Cauchy问题广义解的性质.     

Equations containing locally Henstock-Kurzweil integrable functions

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.     

On the dichotomy of linear autonomous systems of functional-differential equations

We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form x _n = Γx _n?1 with a compact operator Γ, in Banach space. The spectrum of Γ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.     

Numerical study of homotopy‐perturbation method applied to Burgers equation in fluid

In this article, the problem of Burgers equation is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. Comparison is made between the HPM and Exact solutions. The obtained solutions, in comparison with the exact solutions, admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010