We consider the mathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction. For steady-state solutions, the model can be reduced to the ordinary differential equation (22.1) $$ u'' - \lambda u' + F(\lambda ,u,\beta ,u) = 0 $$ with boundary conditions 22.2 $$ u'(0) = \lambda u(0), u'(1) = 0 $$ where $$ F(\lambda ,\mu ,\beta ,u) = \lambda \mu (\beta - u)exp(u) $$ (see [1]). The unknown u represents the steady-state temperature of the reaction, and the parameters λ, µ and β represent the Peclet number, the Damkohler number and the dimensionless adiabatic temperature rise respectively. This problem has been studied by various Authors (e.g. [2,3,4]) who have demonstrated numerically the existence of solutions (sometimes multiple solutions) for particular parameter ranges.

An ordinary differential equation with a parameter in the boundary conditions describes the steady state in an adiabatic tubular chemical reactor. In this paper, the problem is considered as a Hammerstein integral equation and solutions are obtained using Adomian's decomposition method.

SynopsisA radiation condition is obtained, and is then used together with weighted Sobolev spaces and the limiting absorption method to establish the unique existence of solutions to the diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.

The problem of scattering by an obstacle inR3, the index of refraction of which differs from the index of refraction of free space, is examined. The problem reduces to an integral equation in the region defined by the obstacle. A scheme is proposed for regularizing the derived integral equation which ensures convergence of the iteration procedure.

In this paper we prove a decomposition formula for solutions of mixed boundary-value problems for elliptic and parabolic equations into regular and singular parts, which provides an explicit description of the behaviour of the solution near the corners of the boundary and also, in parabolic cases, shows how the singularity due to the corners varies in time. We also give some sufficient conditions for the regularity of the solution of time-dependent problems.

A simple iterative method for solving many of the integral equations arising in scattering problems is presented. By introducing a relaxation parameter the equation is changed to one which may be solved as a Neumann series. An explicit choice of the relaxation parameter is proposed which does not require detailed knowledge of the spectrum nor does the method require the symmetrization of the, in general, non-selfadjoint integral operators that occur. Convergence of the method is demonstrated in examples where the Neumann series for the original equation either diverges or converges at a much lower rate.

It is shown that a generalized overrelaxation method, when applied to the domain integral equation that arises in scattering by penetrable objects, results in a convergent iterative solution. This modified Born series converges when the original Born series diverges for a wide range of indices of refraction and scatterer size. Moreover the modified series converges more rapidly than the original Born series in those cases for which the original series converges. Numerical examples demonstrating the effectiveness of this method are presented in one, two, and three dimensions.

In this paper we prove a decomposition formula of Grisvard type, but for mixed boundary value problems of Dirichlet-oblique derivative type for general elliptic equations of second order in a plane domain with piecewise differentiable boundary. The main result extends previous results in this area by showing that, if the problem is posed in the Sobolev space H2(Ω), then a decomposition of the solution into a regular and an explicitly determined singular part holds for all combinations of the measure of the angle at the corner and the type of boundary conditions imposed in its neighbourhood.

This paper presents a constructive algorithm for solving the problem of reconstructing the shape of a scattering object from measurements of the scattered far field when the unknown object is illuminated by a known incident wave. The problem is recast as an optimization problem with a penalty term. The cost functional consists of a term which assesses the difference between the measured far field and the far field of the solution of the field equation to a particular surface and the penalty term which measures the error in satisfying the boundary conditions on that surface. A complete family of radiating solutions of the Helmholtz equation is employed to construct approximate solutions by solving finite dimensional minimization problems. Existence of solutions of the original problem as well as the finite dimensional approximations is established. Moreover convergence of the approximate solutions to a solution of the original problem is proven. Some preliminary numerical results are presented to indicate the viability of the method.

Le problème consiste a trouver la forme d’un cylindre parfaitement conducteur éclairé par une onde electromagnétique a polarisation E. Il est ramené á un problème d’ optimisation à pénalité avec recherche, dans une classe à contrainte appropriée, d’une surface qui minimise une expression fonctionnelle à deux termes. Le premier terme mesure l’écart quadratique moyen du champ lointain de la solution du problème direct de diffraction pour une surface de la classe admissible à partir de données de champ lointain mesurées, tandis que le deuxième terme dit de pénalité, mesure l’erreur faite en satisfaisant les conditions aux limites imposées. On utilise un algorithme d’optimisation qui ne requiert pas la resolution d’un probléme direct dans un processus de mise à jour. Des reconstructions explicites illustrent l’efficacité de la méthode. Quelques resultats indiquant la sensibilité de l’algorithme au choix du paramètre de pénalité et au nombre d’ondes incidentes sont donnés.

The scattering problem is formulated as a boundary integral equation over the scattering surface that, through the use of a modified Green’s function, is uniquely solvable for all wavenumbers of interest. This integral equation is further transformed, by operating with the adjoint, into a self-adjoint equation into which a parameter has been introduced by adding and subtracting multiples of the identity. This equation is discretized and is solved in a Neumann series that has been shown to converge, with a suitable choice of the parameter, with no restriction on the wavenumber. Numerical results from several examples—spheres, spheroids, and finite cylinders—are presented and shown to be in complete agreement with those obtained using T-matrix methods. Results of a numerical experiment are presented that show how the parameter choice affects the rate of convergence.

The spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error estimates in the operator norm. In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively.

The problem considered is that of determining the shape of a three-dimensional scattering object, illuminated by an acoustic field, from a knowledge of scattered far-field data. The far-field data are the asymptotic form of the solution of an exterior transmission problem for the Helmholtz equation. The problem is reformulated as an optimisation problem, specifically, finding that surface, in a suitably restricted class, which minimises an appropriate functional of the far field generated by the surface through the solution of the direct problem. Through the use of complete families of solutions, the problem is further reduced to finding a surface which minimises error in satisfying the transmission conditions.

SynopsisA scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

On etablit des conditions pour qu'un operateur T sur un espace norme reticule X soit contractif dans un sous-espace de X par rapport a une norme bien construite