Nonlinear Mixed Boundary Value Problems Research Articles

Existence result for a nonlinear mixed boundary value problem for the heat equation

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set Ω⊆Rn we remove a cavity ω⊆Ω. On the exterior boundary of Ω∖ω‾ we prescribe a Neumann boundary condition, while on the interior boundary we set a nonlinear Robin-type condition. Under suitable assumptions on the data and by means of Leray Schauder Fixed-Point Theorem, we prove the existence of (at least) one solution u∈C01+α2;1+α([0,T]×(Ω‾∖ω)).

Multiplicity results for nonlinear mixed boundary value problem

The aim of this paper is to establish multiplicity results of nontrivial and nonnegative solutions for mixed boundary value problems with the Sturm-Liouville equation. The approach is based on variational methods.

Method of Adaptive-Gradient Elements for Computational Mechanics

For tackling high-gradient, localized, or singular boundary value problems, the concept of an adaptive-gradient (AG) element family is introduced to advance the utility of discretization methods. Capable of encompassing regular and singular elements as special cases, a basic but versatile family of AG elements for multidimensional applications is derived whose gradient and singularity can be controlled parametrically to handle a wide variety of functional behavior with standard mesh configurations. As illustrations, examples of usage and performance in a set of linear and nonlinear mixed-boundary value problems are presented.

On the galerkln boundary element method for a mixed non-linear boundary value problem

We consider a nonlinear mixed boundary value problem for the Lapla-cian in the plane. Using Green's formula the problem is converted into a system of boundary integral equations. For this nonlinear operator equation the existence and uniqueness is proved under the monotonicity assumption on the nonlinearity. Furthermore, we show that the nonlinear mapping is a-proper and when the nonlinear perturbation is Frechot-differentiable the linearized operator is a-proper and injective. Then we use weak coercivity of the linearized operator in order to derive quasiop-timality estimates for the Galerkin boundary element solutions.

Asymmetric boundary-value problems for a transversely isotropic elastic medium

This paper considers general boundary-value problems (displacement, traction, mixed) related to a transversely isotropic elastic half-space. The geometry of the problems considered is axisymmetric but the load is asymmetric. A cylindrical coordinate system is employed in the analysis and Fourier expansion and Hankel integral transforms are applied with respect to circumferential and radial coordinates, respectively. The boundary-value problems are formulated in terms of a system of non-singular integral equations for each Fourier harmonic. The kernels of the integral equations are displacement and traction Green's functions corresponding to buried ring loads with appropriate circumferential dependence. In general a total of 18 Green's functions are required for each Fourier harmonic. Explicit solutions for Green's functions corresponding to an arbitrary Fourier harmonic are presented in terms of infinite integrals of Lipschitz-Hankel type. The versatility of the solution scheme is illustrated by solving a traction boundary-value problem corresponding to a pressurized hemispherical cavity, displacement boundary-value problems related to an embedded rigid cylindrical body and a non-linear mixed boundary-value problem corresponding to an embedded rigid hemisphere with an elastic-perfectly plastic interface.

Non-linear mixed boundary value problems for parabolic partial differential equations

SynopsisA sufficient condition on the angles of a bounded open subset Ω of ℝn is given for the best possible regularity of a solution to a class of parabolic problems with non-linear mixed boundary conditions.

Non-linear mixed boundary value problems for elliptic partial differential equations

SynopsisA sufficient condition on the angles of a bounded open subset of ℝn is given, ensuring the best regularity of solutions of a class of elliptic problems with non-linear mixed boundary conditions.

On nonlinear mixed boundary value problems for second order elliptic differential equations on domains with corners

SynopsisWe investigate the existence, uniqueness and regularity of solutions to the linear differential equationLu=funder nonlinear mixed boundary conditions on domains with singular boundary points.

The integration of a nonlinear mixed boundary value problem in partial derivatives

The integration of a nonlinear mixed boundary value problem in partial derivatives

On the asymptotic integration of a nonlinear mixed boundary value problem in partial derivatives

On the asymptotic integration of a nonlinear mixed boundary value problem in partial derivatives

Application of differential sensitivity analysis to nonlinear mixed boundary value problems - I An algorithm for recycle reactor optimization

Differential sensitivity analysis is developed as a basis for boundary condition iteration methods. By including the interdependence of process and control variables in the derivation of the sensitivity equations, the analysis is extended to optimal control problems involving mixed boundary conditions. The technique is illustrated by a study of temperature optimization of a tubular reactor with free recycle. A two-level scheme for boundary and control iterations is formulated using differential sensitivity measures and is found to give rapid convergence under mild restrictions. The relationship to quasilinearization is explored.

Application of differential sensitivity analysis to nonlinear mixed boundary value problems—II. Computational results and convergence studies

The well-known consecutive reaction system A → B → C is chosen for a digital computer study of the optimization of a recycle reactor. The associated nonlinear mixed boundary value problem is solved by a weighted gradient-cum-Newton—Raphson iteration which uses differential sensitivity measures developed in Part I. The effectiveness of the method is borne out by a comparative numerical study of the quasilinearization scheme applied to the same problem. The global convergence properties of the algorithm are established by means of a convergence theorem, which also proves that the rate of approach to the optimum is superlinear. Das wohlbekannte Folgereaktionssystem A → B → C wurde gewählt für eine Digital Computer Studie der Optimisierung eines Rücklaufreaktors. Das damit verbundene Problem des nichtlinearen gemischten Grenzwertes wird mittels bewerterem Gradienten und Newton— Raphsonscher Iteration unter Verwendung der im 1. Teil entwickelten Differentialsensitivitätsanalyse gelöst. Eine vergleichende numerische Studie des Quasilinearisierungsschemas, auf das gleiche Problem angewendet, bewies die Wirksamkeit der Methode. Die globalen Konvergenziegenschaften des Algorithmus werden mit Hilfe eines Konvergenztheorems festgestellt, das ebenfalls beweist, dass die Annäherungsrate an das Optimum superlinear ist.