E.C Young

BOUNDS FOR SOLUTIONS OF REACTION-DIFFUSION EQUATIONS

Publisher Summary This chapter describes bounds for the solutions of reaction–diffusion equations. Reaction–diffusion equations have been employed as models for problems in chemical morphogenesis. Reaction processes are usually nonlinear in character; hence, approximate solutions to the dynamical equations representing these phenomena are important for understanding behavior and can often be profitably exploited in the quantitative and qualitative study of such processes. The chapter describes a general method for constructing upper and lower bounds for the solutions of initial-boundary value problems associated with nonlinear reaction–diffusion equations. The nonlinear analysis approximation technique employs Lagranges theory of first-order characteristics, a transformation that is required to be monotone in the solution argument, and differential inequalities.

On a Cauchy problem for the damped wave equation

where m > 1, c a real parameter, and x = (x1 , x2 ,..., xsm-r), in space of any odd number of spatial dimensions. The solution is obtained by a method used by Copson in solving a Cauchy problem for the Euler-Poisson-Darboux equation [Z]. For the present problem, this method which depends on properties of the wave operator developed in [I] consists in finding a function e, for which L’(u) is a constant and such that v is regular and vanishes on the characteristic cone. The restriction to odd spatial dimensions is inherent in the method. But, as pointed out by Copson, this involves no loss of generality since the solution in space of an even number of spatial dimensions can be deduced by Hadamard’s method of descent. For brevity we shall denote a point in the 2m-dimensional space with coordinates (x1, x2 ,..., ~s,,-~, t) by (x, t), and a function of this point by U(X, t). Let (t, T) be the coordinates of a fixed point and let (x, t) be the coordinates of a variable point. The equation

Ocean acoustic models for navy applications: A model of underwater acoustic propagation

A model of underwater acoustic propagation is considered using a combination of normal mode expansion and truncation of the waveguide by introducing an artificial boundary. An outgoing cylindrical wave is assumed on the boundary on which a generalized radiation condition is prescribed. The resulting boundary value problem is then solved numerically by a finite difference scheme.

Uniqueness of solutions of improperly posed problems for singular ultrahyperbolic equations

for all values of the parameter a, oo < a < oo. The symbol A denotes the Laplace operator in the variables xl!---,xm, Dj indicates partial differentiation with respect to the variable ^ (1 ^ j | n), and the summation convention is adopted for repeated indices including (dtu) , where dt denotes differentiation with respect to the variable xf. The boundary value problems will be considered in the domain Q = X* x Y where X* is the parallelepiped denned by 0 < t < T, 0 < JC; < at{\ g i ^ m), and Y is a bounded domain in the space ylt•••,)/„. The parallelepiped defined by 0 < xt < at (1 ^ i ^ m) will be denoted by X. For brevity, we write x = (*!,•• •,*„,), y = (yi,---,yn)> a r | d denote a point in Q by (t,x,y). Throughout this paper, we assume that the coefficients aJk and c depend only on the variables yt, • • •, yn, and are continuous functions of these variables with