The Cauchy problem of a fluid‐particle interaction model with external forces

In this paper, we consider the Cauchy problem of a fluid‐particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro‐micro decomposition.     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.     

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

Blow‐up of solution of wave equation with internal and boundary source term and non‐porous viscoelastic acoustic boundary conditions

In this paper we study the blow‐up of solution of a mixed problem associated to a nonlinear wave equation with dissipative and source term in a bounded domain of . On a boundary portion of the domain we consider a non‐porous viscoelastic acoustic boundary conditions to a non‐locally reacting boundary.     

Exponential convergence for a periodically driven semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a periodic source at the origin. We study the solution, which describes the equilibrium of this system and we prove that, as the space variable tends to infinity, the solution becomes, exponentially fast, asymptotic to a steady state. The key to the proof of this result is a Harnack type inequality, which we obtain using probabilistic ideas.     

Time‐dependent scattering of generalized plane waves by a wedge

We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet‐Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave. As an application, we establish (i) stability of long‐time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.     

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.     

Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth‐order integro‐differential equation

This paper presents general framework for solving the nth‐order integro‐differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth‐order integro‐differential equation. Copyright © 2011 John Wiley & Sons, Ltd.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

Nonradial positive solutions of the ‐Laplace Emden–Fowler equation with sign‐changing weight

In this paper we study the p‐Laplace Emden–Fowler equation with a radial and sign‐changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Therefore the equation has both a positive radial solution and a positive nonradial solution. Moreover, we prove in the one dimensional case that if the neighborhood is large, then a positive solution is unique.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

Edge asymptotics for the radiosity equation over polyhedral boundaries

In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double‐layer potential equation, the solution of the second kind integral equation with non‐compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. Copyright © 1999 John Wiley & Sons, Ltd.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Undercompressive shock waves in Hall‐magnetohydrodynamics

We study the propagation of nonlinear waves in a Hall‐magnetohydrodynamic model. An asymptotic method is used to derive the Gardner‐Burgers equation for fast magnetosonic waves; here, the flux function is nonconvex with both quadratic and cubic nonlinearities, and the evolution equation involves both second‐ and third‐order derivatives representing diffusion and dispersion terms, respectively. Effects of Hall parameter are discussed on the evolution of waves and their interaction by solving a pair of Riemann problems both analytically and numerically. It is shown that the Hall parameter is responsible for shock splitting—a phenomenon that is completely absent in ideal magnetohydrodynamic; indeed, the Hall parameter plays a significant role in deciding about the structure of the solution that involves undercompressive shocks and their interaction with refracted waves and the Lax shocks. It is found that increasing Hall parameter means increasing dispersion that triggers the physical mechanism causing speed and strength of an undercompressive shock to increase and the wave‐fan width to decrease; numerical solutions substantiate these features predicted by the analytical solution.