In this paper, we study Pontryagin's maximum principle for some optimal control problems governed by a non-well-posed parabolic differential equation. A new penalty functional is applied to derive Pontryagin's maximum principle and an application for this system is given.

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations. Under the framework of Constantin and Fefferman, a more relaxed regularity criterion in terms of the direction of vorticity is established.

We consider a nonlinear second-order elliptic boundary value problem in a bounded domain Ω ⊂ RN with mixed boundary conditions. The solution is found via linearisation. We design a robust and efficient approximation scheme. Error estimates for the linearisation algorithm are derived in L2(Ω), H1(Ω) and L∞(Ω) spaces under the minimal regularity assumptions of the exact solution.

This paper addresses the effect of general Lewis number and heat losses on the calculation of combustion wave speeds using an asymptotic technique based on the ratio of activation energy to heat release being considered large. As heat loss is increased twin flame speeds emerge (as in the classical large activation energy analysis) with an extinction heat loss. Formulae for the non-adiabatic wave speed and extinction heat loss are found which apply over a wider range of activation energies (because of the nature of the asymptotics) and these are explored for moderate and large Lewis number cases—the latter representing the combustion wave progress in a solid. Some of the oscillatory instabilities are investigated numerically for the case of a reactive solid.

We investigate oscillatory properties of a perturbed symplectic dynamic system on a time scale that is unbounded above. The unperturbed system is supposed to be nonoscillatory, and we give conditions on the perturbation matrix, which guarantee that the perturbed system becomes oscillatory. Examples illustrating the general results are given as well.

Multiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, a ∈ Z). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

The time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases α = 1 with arbitrary n, or n = 1 with arbitrary α. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

In this paper, we study the problem of robust H∞ stabilisation with definite attenuance for a class of impulsive switched systems with time-varying uncertainty. A norm-bounded uncertainty is assumed to appear in all the matrices of the state model. An LMI-based method for robust· H∞ stabilisation with definite attenuance via a state feedback control law is developed. A simulation example is presented to demonstrate the effectiveness of the proposed method.

In this paper we consider two spectral refinement schemes, elementary and double iteration, for the approximation of eigenelements of a compact operator using a new approximating operator. We show that the new method performs better than the Galerkin, projection and Sloan methods. We obtain precise orders of convergence for the approximation of eigenelements of an integral operator with a smooth kernel using either the orthogonal projection onto a spline space or the interpolatory projection at Gauss points onto a discontinuous piecewise polynomial space. We show that in the double iteration scheme the error for the eigenvalue iterates using the new method is of the order of , where h is the mesh of the partition and k = 0, 1, 2,… denotes the step of the iteration. This order of convergence is to be compared with the orders in the Galerkin and projection methods and in the Sloan method. The error in eigenvector iterates is shown to be of the order of in the new method, in the Galerkin and projection methods and in the Sloan method. Similar improvement is observed in the case of the elementary iteration. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. We illustrate this improvement in the order of convergence by numerical examples.

We show that Algorithm H* for the determination of the rate matrix of a block-GI/M/1 Markov chain is related by duality to Algorithm H for the determination of the fundamental matrix of a block-M/G/1 Markov chain. Duality is used to generate some efficient algorithms for finding the rate matrix in a quasi-birth-and-death process.

Over the past two decades considerable effort has been devoted to problems of stochastic stability, stabilisation, filtering and control for linear and nonlinear systems with Markovian jump parameters, and a number of results have been achieved. However, due to the exponential distribution of the Markovian chain, there are many restrictions on existing results for practical applications. In the present paper, we study systems whose jump parameters are semi-Markovian rather than fully Markovian. We consider only linear systems with semi-Markovian jump parameters, and also study systems with phase-type semi-Markovian jump parameters, because the family of phase-type distributions is dense in the families of all probability distributions on [0, +∞). Some stochastic stability results are obtained. An example is given to show the potential of the proposed techniques.

In this paper some existence results for third-order differential equations with nonlinear boundary value conditions are derived. Functional dependence in the data is allowed. In the proofs we use the method of upper and lower solutions, Schauder's fixed point theorem and results from Cabada and Heikkilä on third-order differential equations with linear and nonfunctional initial-boundary value conditions.

In mathematical programming, an important tool is the use of active set strategies to update the current solution of a linear system after a rank one change in the constraint matrix. We show how to update the general solution of a linear system obtained by use of the scaled ABS method when the matrix coefficient is subjected to a rank one change.

In this paper we propose a new affine scaling interior trust region algorithm with a nonmonotonic backtracking technique for nonlinear equality constrained optimisation with nonnegative constraints on the variables. In order to deal with large problems, the general full trust region subproblem is decomposed into a pair of trust region subproblems in horizontal and vertical subspaces. The horizontal trust region subproblem in the algorithm is defined by minimising a quadratic function subject only to an ellipsoidal constraint in a null tangential subspace and the vertical trust region subproblem is defined by the least squares subproblem subject only to an ellipsoidal constraint. By adopting Fletcher's penalty function as the merit function, combining a trust region strategy and a nonmonotone line search, the mixing technique will switch to a backtracking step generated by the two trust region subproblems to obtain an acceptable step. The global convergence of the proposed algorithm is proved while maintaining a fast local superlinear convergence rate, which is established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some highly nonlinear cases.

The present work deals with the problem of recovering a local image from localised projections using the concept of approximation identity. It is based on the observation that the Hilbert transform of an approximation identity taken from a certain class of compactly supported functions with sufficiently many zero moments has no significant spread of support. The associated algorithm uses data pertaining to the local region along with a small amount of data from its vicinity. The main features of the algorithm are simplicity and similarity with standard filtered back projection (FBP) along with the economic use of data.

Humoral immunity is that aspect of specific immunity that is mediated by B lymphocytes and involves the neutralising of disease-producing microorganisms, called pathogens, by means of antibodies attaching to the pathogen's binding sites. This inhibits the pathogen's entry into target cells. We present a master equation in both discrete and in continuous form for a ligand bound at n sites becoming a ligand bound at m sites in a given interaction time. To track the time-evolution of the antibody-ligand interaction, it is shown that the process is most easily treated classically and that in this case the master equation can be reduced to an equivalent one-dimensional diffusion equation. Thus well-known diffusion theory can be applied to antibody-ligand interactions. We consider three distinct cases depending on whether the probability of antibody binding compared to the probability of dissociation is relatively large, small or comparable, and numerical solutions are given.

We consider the stability of high Reynolds number flow past a heated, curved wall. The influence of both buoyancy and curvature, with the appropriate sense, can render a flow unstable to longitudinal vortices. However, conversely each mechanism can make a flow more stable; as with a stable stratification or a convex curvature. This is partially due to their influence on the basic flow and also due to additional terms in the stability equations. In fact the presence of buoyancy in combination with an appropriate local wall gradient can actually increase the wall shear and these effects can lead to supervelocities and the promotion of a wall jet. This leads to the interesting discovery that the flow can be unstable for both concave and convex curvatures. Furthermore, it is possible to observe sustained vortex growth in stably stratified boundary layers over convexly curved walls. The evolution of the modes is considered in both the linear and nonlinear régimes.

The steady flow of an incompressible viscous non-Newtonian electrically conducting fluid and heat transfer due to the rotation of an infinite disk are studied considering the Hall effect. The effects of an externally applied uniform magnetic field, the Hall current, and the non-Newtonian fluid characteristics on the velocity and temperature distributions as well as the heat transfer are considered. Numerical solutions of the nonlinear equations which govern the magnetohydrodynamics (MHD) and energy transfer are obtained over the entire range of the physical parameters.

A multiple fractional Brownian motion (FBM)-based traffic model is considered. Various lower bounds for the overflow probability of the associated queueing system are obtained. Based on a probabilistic bound for the busy period of an ATM queueing system associated with a multiple FBM-based input traffic, a minimal dynamic buffer allocation function (DBAF) is obtained and a DBAF-allocation algorithm is designed. The purpose is to create an upper bound for the queueing system associated with the traffic. This upper bound, called a DBAF, is a function of time, dynamically bouncing with the traffic. An envelope process associated with the multiple FBM-based traffic model is introduced and used to estimate the queue size of the queueing system associated with that traffic model.

A stable linear time-invariant classical digital control system with several widely different small coefficients multiplying the lowest functions is considered. It is formulated as a multi-parameter singularly perturbed system. Perturbation methods are developed for both initial and boundary value problems based on asymptotic expansions of the perturbation parameters. The approximate solution consists of an outer solution and a number of boundary layer correction solutions equal to the number of initial conditions lost in the process of degeneration. An example is provided for illustration.