Hybrid Hamilton–Jacobi–Poisson wall distance function model

Expensive to compute wall distances are used in key turbulence models and also for the modeling of peripheral physics. A potentially economical, robust, readily parallel processed, accuracy improving, differential equation based distance algorithm is described. It is hybrid, partly utilising an approximate Poisson equation. This also allows auxiliary front propagation direction/velocity information to be estimated, effectively giving wall normals. The Poisson normal can be used fully, in an approximate solution of the eikonal equation (the exact differential equation for wall distance). Alternatively, a weighted fraction of this Poisson front direction (effectively, front velocity, in terms of the eikonal equation input) information and that implied by the eikonal equation can be used. Either results in a hybrid Poisson–eikonal wall distance algorithm. To improve compatibility of wall distance functions with turbulence physics a Laplacian is added to the eikonal equation. This gives what is termed a Hamilton–Jacobi equation. This hybrid Poisson–Hamilton–Jacobi approach is found to be robust on poor quality grids. The robustness largely results from the elliptic background presence of the Poisson equation. This elliptic component prevents fronts propagated from solid surfaces, by the hyperbolic eikonal equation element, reflecting off zones of rapidly changing grid density. Where this reflection (due to poor grid quality) is extreme, the transition of front velocity information from the Poisson to Hamilton–Jacobi equation can be done more gradually. Consistent with turbulence modeling physics, under user control, the hybrid equation can overestimate the distance function strongly around convex surfaces and underestimate it around concave. If the former trait is not desired the current approach is amenable to zonalisation. With this, the Poisson element is automatically removed around convex geometry zones.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

A class of collocation methods for numerical integration of initial value problems

For the numerical solution of initial value problems a general procedure to determine global integration methods is derived and studied. They are collocation methods which can be easily implemented and provide a high order accuracy. They further provide globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients are generated by a recurrence formula and no integrals are involved in the calculation. Numerical experiments provide favorable comparisons with other existing methods.     

An iterative multistep formula for solving initial value problems

The iterative multistep method (IMS) introduced by Hyman (1978) for solving initial value problems in ordinary differential equations has the advantage of being able to offer a higher degree of accuracy than the Runge-Kutta formulas by continuing the iteration process. In this article, another IMS formula is developed based on the geometric means predictor-corrector formulas introduced by Sanugi and Evans (1989). A numerical example is provided that shows that this formula can be used as a competitive alternative to Hyman's IMS formula.     

A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ. A stability analysis is performed, showing that even for the explicit scheme that θ=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.     

Solution of initial value problems by the differential quadrature method with Hermite bases

The differential quadrature method (DQM) is used to solve the first-order initial value problem. The initial condition is given at the beginning of the interval. The derivative of a space-independent variable at a sampling grid point within the interval can be defined as a weighted linear sum of the given initial conditions and the function values at the sampling grid points within the defined interval. Hermite polynomials have advantages compared with Lagrange and Chebyshev polynomials, and so, unlike other work, they are chosen as weight functions in the DQM. The proposed method is applied to a numerical example and it is shown that the accuracy of the quadrature solution obtained using the proposed sampling grid points is better than solutions obtained with the commonly used Chebyshev–Gauss–Lobatto sampling grid points.     

Minimum time control problems for non-autonomous differential equations

In this paper, we investigate a minimum time problem for controlled non-autonomous differential systems, with dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we will prove that it is related to a standard front propagation problem via the reachability function. Two simple numerical examples are given to illustrate our approach.     

Observability functions for linear and nonlinear systems

A generalization of the zero-state observability function is considered for nonlinear systems. The linear time-invariant case is considered as an application in model reduction problems.     

Variational iteration method for singular perturbation initial value problems

In this paper, the variational iteration method (VIM) is applied to solve singular perturbation initial value problems (SPIVPs). The obtained sequence of iterates is based on the use of Lagrange multipliers. Some convergence results of VIM for solving SPIVPs are given. Moreover, the illustrative examples show the efficiency of the method.     

Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians

In a series of papers, we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The results applied to problems which satisfy a positivity condition on the integral of the Lagrangian. This positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length. In this note, we prove a new theorem which characterizes the exit time value function as the unique bounded-from-below viscosity solution of the Bellman equation that vanishes on the target. The theorem applies to problems satisfying an asymptotic condition on the trajectories, including cases where the positive integral condition is not satisfied. Our results are based on an extended version of “Barb lat's lemma”. We apply the theorem to variants of the Fuller problem and other examples where the Lagrangian is degenerate.     

On the value function for nonautonomous optimal control problems with infinite horizon

In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by L¹-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton–Jacobi–Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function is a Dini solution of the HJB equation. We obtain a verification result for the class of Dini sub-solutions of the HJB equation and also prove a minimax property of the value function with respect to the sets of Dini semi-solutions of the HJB equation. We introduce the concept of viscosity solutions of the HJB equation in infinite horizon and prove the equivalence between this and the concept of Dini solutions. In the Appendix we provide an existence theorem.     

Sixth order non-dissipative C 1 spline collocation method for oscillatory ordinary initial value problems

In this paper we construct a global method, based on quintic C ¹-spline, for the integration of first order ordinary initial value problems (IVPs) including stiff equations and those possessing oscillatory solutions as well. The method will be shown to be of order six and in particular is A-stable. Attention is also paid for the phase error (or dispersion) and it is proved that the method is dispersive and has dispersion order six with small phase-lag (compared with the extant methods having the same order (cf. [7])). Moreover, the method may be regarded as a continuous extension of the closed four-panel Newton–Cotes formula (NC4) (typically it is a continuous extension of an implicit Runge–Kutta method). In additiona priori error estimates, in the uniform norm, together with illustrative test examples will also be presented.     

A continuous analysis framework for the solution of location–allocation problems with dense demand

Location–allocation problems arise in several contexts, including supply chain and data mining. In its most common interpretation, the basic problem consists of optimally locating facilities and allocating customers to facilities so as to minimize the total cost. The standard approach to solving location–allocation problems is to model alternative location sites and customers as discrete entities. Many problem instances in practice involve dense demand data and uncertainties about the cost and locations of the potential sites. The use of discrete models is often inappropriate in such cases. This paper presents an alternative methodology where the market demand is modeled as a continuous density function and the resulting formulation is solved by means of calculus techniques. The methodology prioritizes the allocation decisions rather than location decisions, which is the common practice in the location literature. The solution algorithm proposed in this framework is a local search heuristic (steepest-descent algorithm) and is applicable to problems where the allocation decisions are in the form of polygons, e.g., with Euclidean distances. Extensive computational experiments confirm the efficiency of the proposed methodology.     

Solving linear and nonlinear initial value problems by the projected differential transform method

In this work, we propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM). The PDTM can be easily applied to the initial value problems with less computational work. For the several illustrative examples, the computational results are compared with those obtained by many other methods; the Adomian decomposition, the variational iteration and the spline method. For all examples, the PDTM provides exact solutions. It has been shown that the PDTM is a reliable algorithm in obtaining analytic as well as approximate solution for the initial value problems.     

Boundary value problems for a class of impulsive functional equations

This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.     

Singular H∞ control in nonlinear systems: positive semidefinite storage functions and separation principle

A framework is developed for the general nonlinear H_∞ output feedback control problem, in which two major restrictions are relaxed, i.e., the non-singular penalty in H_∞ cost and the positive definite solution of Hamilton–Jacobi inequality at present state space nonlinear H_∞ control literatures. As illustrated in an example, positive semidefinite solution simplifies the structure of the H_∞ controller. Based on this framework, some sufficient conditions are derived. While specialized to linear systems, the controller reduces to the so-called central controller. © 1997 by John Wiley & Sons, Ltd.     

Solutions of the Cahn–Hilliard equation

In this paper, we found some exact solutions of the Cahn–Hilliard equation and the system of the equations by considering a modified extended tanh function method. A numerical solution to a Cahn–Hilliard equation is obtained using a homotopy perturbation method (HPM) combined with the Adomian decomposition method (ADM). The comparisons are given in the tables.     

On fuzzy boundary value problems

The statement that a two-point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by Lakshmikantham et al. and O’Regan et al. Recently Bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two-point boundary value problems have no solutions. Under a new structure and certain conditions we show that a two-point boundary value problem is equivalent to a fuzzy integral equation. We also prove the existence of solutions to the two-point boundary value problem. In some sense, this is an amendment to results of Lakshmikantham et al. and O’Regan et al., and it is an answer to one of Bede’s problems.     

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.     

On the numerical solution of initial value problems for nonlinear trapezoidal formulas with different types

In this paper, the local truncation errors of the trapezoidal formulas such as arithmetic mean (AM), geometric mean (GM), heronian mean (HeM), harmonic mean (HaM), contraharmonic mean (CoM), root mean square (RMS), logarithmic mean (LM) and centroidal mean (CeM) are investigated and the stability analysis of these formulas are found. Finally, it is applied to various initial value problems.