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This paper deals with solutions $u(t,x)$ of parabolic differential inequalities (a) $u_1 \leqq Lu$, or (b) $u_1 \geqq Lu$, respectively, where L is a linear, weakly elliptic differential operator of second order. The behavior of u for large t is studied under the assumption, that on the lateral boundary a nonlinear boundary condition of the form (a) ${{\partial u} / {\partial v}} \leqq f(u)$, or (b) ${{\partial u} / {\partial v}} \geqq f(u)$, is imposed, where $f(z) \to \infty $ as $z \to \infty $. It is shown that the value of the integral $\int^\infty {{dz} / {f(z)f'(z)}}$ is crucial for the growth properties of u. If this integral is infinite, then we have the case of global existence, i.e., any solution u of (a) is bounded in bounded sets. If, on the other hand, the integral is finite, then all solutions u of (b) with large initial values become infinite in finite time.

A basic theoretical model is presented for the study of tropical malaria epidemics. A predatory-prey type of formalism results in two pairs of partial differential equations which describe the latent and infectious states and two pairs of integro-differential equations which describe recovery and dying rates for humans or mosquitoes. The complete system of equations is solved numerically for a practical case, since coupled nonlinear boundary conditions destroy the linearity of the system.

In this paper we prove the convergence of projection methods for the equationLx=Gx with a linear operatorL and a Frechet-differentiable OperatorG. This theorem is used to establish the convergence of Galerkin's method and certain projection methods with splines for boundary value problems with nonlinear boundary conditions.

An essentially elementary proof of the existence of a solution to the boundary value problem $x'' = f(t,x),(x(0),x'(0)) \in \Lambda _0 ,(x(1), - x'(1)) \in \Lambda _1 $, provided there exist functions $\alpha $, $\beta $ such that $\alpha ''(t) \geqq f(t,\alpha (t))$ and $\beta ''(t) \leqq f(t,\beta (t))$, where $\Lambda _0 $ and $\Lambda _1 $ are appropriate sets in $R^2$ depending upon $\alpha $ and $\beta $, is given. The boundary conditions in this case are sufficiently general to include all the previous linear and nonlinear boundary conditions known to the authors (cf. the references). A similar theorem is established for the periodic boundary value problem.

A variety of Galerkin methods are studied for the parabolic equationu t =??(a(x)? u),x?Ω?? n ,t? (O,T], subject to the nonlinear boundary conditionu v =g(x,t,u),x??Ω,t? (O,T] and the usual initial condition. Optimal order error estimates are derived both inL 2 (Ω) andH 1 (Ω) norms for all methods treated, including several that produce linear computational procedures.

Nonlinear ordinary differential equations in the forms $x^{(n)} (t) = f(t,x,x', \cdots ,x^{(n - 1)} )$, where $n \geqq 2$ and $x'(t) = f(t,x)$ and $x(t)$ is a vector, are considered when If i is bounded by a linear function in the space variables. Existence of solutions of boundary value problems is proved by analysis of the funnel of solutions. For linear boundary conditions, our proof is a more elementary and geometrically more intuitive proof of a known result due to Z. Opial [J. Differential Equations, 3 (1967), pp. 580–594]. For nonlinear boundary conditions, the results appear to be new. A geometric explanation of the above result of Opial is given.

The transient, one-dimensional temperature distribution is determined for bodies with internal heat generation and nonlinear boundary condition in the form: k·e·gradθ+ε(θn−T0n)+ε1(θ−T0)=0 Approximate analytical solutions are derived with the aid of Biot’s variational method. The additional boundary condition introduced by Lardner is modified, and this modification makes it possible to solve the problem. The solution has been obtained assuming a parabolic profile of temperature distribution. Formulas are given for plates, cylinders, and spheres. Some results are illustrated with the graphs, and compared with the exact solution for the case of convective heat transfer.

BOUNDARY layer methods have been used in a number of recent papers [1] for solving problems on the equilibrium and motion of a fluid with surface tension in a strong gravitational field (more precisely, with large Bond numbers). The present note proves the suitability of the method for Investigating the equilibrium of a fluid with surface tension, enclosed in a cylindrical vessel of arbitrary cross-section, in a gravitational field. Mathematically, the problem amounts to solving asymptotically an equation with a small parameter in a quasilinear elliptic operator under nonlinear boundary conditions. The method of obtaining the asymptotic solution is similar to that developed in [2, 3], while the proof of it follows the lines of [4].

Constrained minimization problems are formulated from a quasilinear parabolic boundary value problem (possibly with nonlinear boundary conditions), making use of the latter’s (conditional) inverse-positive property. Approximate solutions and three error bounds can be obtained by solving these minimization problems by linear programming and discretization techniques. Numerical results are obtained using splines as basis functions.

Periodic solutions of heat conduction equations with boundary conditions of the relay kind are found for a finite interval, and the behavior of such solutions at unlimited time increase is analyzed. Periodic solutions of heat conduction equations with nonlinear boundary conditions were considered in [1–4, 10], while in [5, 6] periodic solutions of nonhomogeneous heat conduction equations with their right-hand sides nonlinear with respect to the unknown functions are presented, and the asymptotic behavior of related initial problems is analyzed. Solutions of this kind define self-oscillating processes occurring in various branches of hydrodynamics (theory of filtration and diffusion [3–6]).

The special features of the method of successive approximations as applied to the simulation of nonlinear boundary conditions by electrical networks are examined. The dependence of the convergence of the method on the initial data is analyzed. Results of an experimental test of the conclusions are presented.

Concerning the heat conduction problem with a nonlinear boundary condition, as in the case of a boiling process, the effect of the time interval in the numerical solution scheme on the error of the solution is analyzed here.

We propose the use of electron tubes in the modeling of nonlinear boundary conditions in the solution of problems of steady heat conduction by an electric-model method, thus making it possible to solve the most complex of problems on the simplest of analog computers.

Two solutions are presented to the problem of calculating the velocity field within a solitary wave and its celerity and profile, given its height and the depth of water in which it propagates. The first is analytic and approximate and may be regarded as a continuation of the calculations by J. Boussinesq and Lord Rayleigh. The second solution is numerical and is a type of iterative procedure which is necessary to take into account the nonlinear boundary condition which must be imposed on the solution (Bernoulli's theorem). No attempt is made to establish convergence of the iteration procedure, but if convergence is assumed, the method seems to be as accurate as the calculator wishes it to be. The results of the two solutions are compared with experimental results, and it is found that the second method of solution yields results for the velocity components which agree, on the whole, better with the experimental results.