A model for thermal ignition by intense light is studied. The governing non-linear parabolic equation is linearized in a two-step manner with the aid of a non-linear ordinary differential equation which captures the salient features of the non-linear parabolic equation. The critical parameters are computed from the steady-state solution of the ordinary differential equation, which can be obtained without actually solving the equation. Comparison with available data shows that the present method yields good results.

In earlier papers (Parts I and II) existence and uniqueness of the solutions to a coupled pair of nonlinear elliptic partial differential equations with linear boundary conditions was considered. These equations arise when material is undergoing an exothermic chemical reaction which is sustained by the diffusion of a reactant. In this paper we establish the existence of multiple solutions for many different values of the parameters not considered in the earlier parts. It is shown that the case, also omitted in earlier parts, with perfect thermal and mass transfer on the boundary (the double-Dirichlet case) does have a unique solution for sufficiently large values of the exothermicity or an equivalent parameter. The methods of solution provide specific bounds on the region of existence of multiple solutions.

We give here some extensions of inequalities of Popoviciu and Rado. The idea is to use an inequality [C. P. Niculescu and L. E. Persson, Convex functions. Basic theory and applications (Universitaria Press, Craiova, 2003), Page 4] which gives an approximation of the arithmetic mean of n values of a given convex function in terms of the value at the arithmetic mean of the arguments. We also give more general forms of this inequality by replacing the arithmetic mean with others. Finally we use these inequalities to establish similar inequalities of Popoviciu and Rado type.

The reduction of an important class of triple integral equations to a pair of simultaneous Fredholm equations has been carried out by Cooke [1]. In this paper, Cooke's equations are transformed to new uncoupled Fredholm equations which, for certain important cases, are shown to be simpler than Cooke's and also superior for the purposes of solution by iteration.

A local existence and uniqueness result is proved for the three-dimensional Euler-Poisson system without a pressure term which arises in plasma physics.

This paper examines the predictions of shallow water theory for steady and unsteady withdrawal flows through an extended sink from fluid of finite depth. Two-dimensional plane flows and three-dimensional axi-symmetric flow through a circular drain are examined. Shallow water theory indicates the presence of limiting configurations, where the surface of the fluid collapses directly into the sink. In addition, this theory suggests that some previously computed steady solutions may be unstable.

This paper is concerned with the oscillation of first-order delay differential equations

where p(t) and τ(t) are piecewise continuous and nonnegative functions and τ(t) is non-decreasing. A new oscillation criterion is obtained.

We revisit the singular eigensolution to the steady state one-speed transport equation for an isotropically scattering and multiplying heterogeneous slab. It is proved that this solution is a sum of Stieltjes integrals over the resolvent set of only the operator of multiplication by the angular variable.

We consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are from Lp spaces and Bullen-type inequalities.

Designing a reasonable M/G/1 retrial queue system that enhances service efficiency and reduces energy consumption is a challenging issue in Information and Communication Technology systems. This paper presents an M/G/1 retrial queue system incorporating random working vacation (RWV) and improved service efficiency during vacation (ISEV) policies, and examines its optimal queuing strategies. The RWV policy suggests that the server takes random working vacations during reserved idle periods, effectively reducing energy consumption. In contrast, the ISEV policy strives to augment service efficiency during regular working periods by updating, inspecting or maintaining the server on vacations. The system is transformed into a Cauchy problem to investigate its well-posedness and stability, employing operator semigroup theory. Based on the system’s stability, steady-state performance measures, such as service efficiency, energy consumption and expected costs, are quantified using the steady-state solution. The paper subsequently demonstrates the existence of optimal queuing strategies that achieve maximum efficiency and minimum expected costs. Finally, two numerical experiments are provided to illustrate the effectiveness of the system.

In this paper we have proved that the generalized incomplete gamma functions and their extensions are mutually related through integral and differential representations.

Consideration of functions whose second difference along the trajectories of a difference equation is positive gives a stability theorem for autonomous discrete-time systems. Such functions can be used to estimate domains of nonglobal stability.

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic equations. The unknown coefficient of the elliptic equations depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic equations are uniquely solvable for the given class of coefficients. Proof of the existence of a quasisolution of the inverse problems is obtained.

Facility location problems are one of the most common applications of optimization methods. Continuous formulations are usually more accurate, but often result in complex problems that cannot be solved using traditional optimization methods. This paper examines theuse of a global optimization method—AGOP—for solving location problems where the objective function is discontinuous. This approach is motivated by a real-world application in wireless networks design.

Using geometric quantization, and accepting the quantum Hamiltonian of previous authors, we propose some candidate formulae for the quantum operator of an observable which is a quadratic form in the momenta.

Recently in several papers the boundary element method has been applied to non-linear problems. In this paper we extend the analysis to strongly nonlinear boundary value problems. We shall prove the convergence and the stability of the Galerkin method in Lp spaces. Optimal order error estimates in Lp space then follow. We use the theory of A-proper mappings and monotone operators to prove convergence of the method. We note that the analysis includes the u4 -nonlinearity, which is encountered in heat radiation problems.

A method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

We consider fully three-dimensional time-dependent outflow from a source into a surrounding fluid of different density. The source is distributed over a sphere of finite radius. The nonlinear problem is formulated using a spectral approach in which two streamfunctions and the density are represented as a Fourier-type series with time-dependent coefficients that must be calculated. Linearized theories are also discussed and an approximate stability condition for early stages in the outflow is derived. Nonlinear solutions are presented and different outflow shapes adopted by the fluid interface are investigated.

In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account and a stock whose price is a log normal diffusion. The bank pays at an interest rate r(t) for any deposit, and vice takes at a larger rate r′(t) for any loan. Optimal strategies are obtained via Hamilton-Jacobi-Bellman (HJB) equation which is derived from dynamic programming principle. For the specific HARA case, we get the optimal consumption and optimal investment explicitly, which coincides with the classical one under the condition r′(t) ≡ r(t)

A monoenergetic point-source solution of the steady-state cosmic-ray equation of transport for cosmic-rays in the interplanetary region in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by Lie Theory. A spherically-symmetric model of the propagation region is assumed incorporating anisotropic diffusion with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speed V. Because of the point release the solution is non-spherically-symmetric.