In this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.

For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.

In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

Two boundary value problems are solved for potential steady-state 2D Darcian seepage flows towards a line sink in a homogeneous isotropic soil from a ponded land surface, which is not flat but profiled. The aim of this shaping is ‘uniformisation’ of the velocity and travel time between this surface and a horizontal drain modelled by a line sink. The complex potential domain is a half-strip, which is mapped onto a reference plane. Either the velocity magnitude or a vertical coordinate along the land surface are control variables. Either a complexified velocity or complex physical coordinate is reconstructed by solving mixed boundary-value problems with the help of the Keldysh-Sedov formula via singular integrals, the kernel of which are the control functions. The flow nets, isotachs and breakthrough curves are found by computer algebra routines. A designed soil hump above the drain ameliorates an unwanted ‘preferential flow’ (shortcut) and improves leaching of salinised soil of a cropfield during a pre-cultivation season.

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.

In this paper, we consider a two-point boundary value problem with Caputo fractional derivative, where the second order derivative of the exact solution is unbounded. Based on the equivalent form of the main equation, a finite difference scheme is derived. The L∞ convergence of the difference system is discussed rigorously. The convergence rate in general improves previous results. Numerical examples are provided to demonstrate the theoretical results.

Kernel-based methods are popular in computer graphics, machine learning, and statistics, among other fields; because they do not require meshing of the domain under consideration, higher dimensions and complicated domains can be managed with reasonable effort. Traditionally, the high order of accuracy associated with these methods has been tempered by ill-conditioning, which arises when highly smooth kernels are used to conduct the approximation. Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems. This paper will extend these techniques to the solution of boundary value problems using collocation. The method of particular solutions will also be considered for elliptic problems, using Gaussian eigenfunctions to stably produce an approximate particular solution.

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight aproblem that can arise in the numerical approximation of nonlinear dynamical systems oncomputers with a finite precision floating point number system. We describe the dynamicalsystem generated by Burgers’ equation with mixed boundary conditions, summarize some ofits properties and analyze the equilibrium states for finite dimensional dynamical systemsthat are generated by numerical approximations of this system. It is important to notethat there are two fundamental differences between Burgers’ equation with mixedNeumann–Dirichlet boundary conditions and Burgers’ equation with both Dirichlet boundaryconditions. First, Burgers’ equation with homogenous mixed boundary conditions on a finiteinterval cannot be linearized by the Cole–Hopf transformation. Thus, on finite intervalsBurgers’ equation with a homogenous Neumann boundary condition is truly nonlinear. Second,the nonlinear term in Burgers’ equation with a homogenous Neumann boundary condition isnot conservative. This structure plays a key role in understanding the complex dynamicsgenerated by Burgers’ equation with a Neumann boundary condition and how this structureimpacts numerical approximations. The key point is that, regardless of the particularnumerical scheme, finite precision arithmetic will always lead to numerically generatedequilibrium states that do not correspond to equilibrium states of the Burgers’ equation.In this paper we establish the existence and stability properties of these numericalstationary solutions and employ a bifurcation analysis to provide a detailed mathematicalexplanation of why numerical schemes fail to capture the correct asymptotic dynamics. Weextend the results in [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov,Math. Comput. Modelling 35 (2002) 1165–1195] and provethat the effect of finite precision arithmetic persists in generating a nonzero numericalfalse solution to the stationary Burgers’ problem. Thus, we show that the results obtainedin [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput.Modelling 35 (2002) 1165–1195] are not dependent on a specifictime marching scheme, but are generic to all convergent numerical approximations ofBurgers’ equation.

For the $n$-th order nonlinear differential equation, ${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k\,+\,j)$-point boundary conditions for $1\,\le \,j\,\le \,n\,-\,1$ and $1\,\le \,k\,\le \,n\,-\,j$. We define $(k;\,j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n\,-\,{{j}_{0}};\,{{j}_{0}})$-point unique solvability implies $(k;\,j)$-point unique solvability for $1\,\le \,j\,\le \,{{j}_{0}}$, and $1\,\le \,k\,\le \,n\,-\,j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\,\le \,k\,\le \,n\,-\,1$.

Existence results of positive solutions for a two point boundary value problem are established. No asymptotic condition on the nonlinear term either at zero or at infinity is required. A classical result of Erbe and Wang is improved. The approach is based on variational methods.

In this paper, we deal with the multiplicity of solutions for a fourth-order impulsive differential equation with a parameter. Using variational methods and a ‘three critical points’ theorem, we give some new criteria to guarantee that the impulsive problem has at least three classical solutions. An example is also given in order to illustrate the main results.

The aim of this paper is to determine the optimal trajectory and maximum payload of flexible link manipulators in point-to-point motion. The method starts with deriving the dynamic equations of flexible manipulators using combined Euler–Lagrange formulation and assumed modes method. Then the trajectory planning problem is defined as a general form of optimal control problem. The computational methods to solve this problem are classified as indirect and direct techniques. This work is based on the indirect solution of open-loop optimal control problem. Because of the offline nature of the method, many difficulties like system nonlinearities and all types of constraints can be catered for and implemented easily. By using the Pontryagin's minimum principle, the obtained optimality conditions lead to a standard form of a two-point boundary value problem solved by the available command in MATLAB®. In order to determine the optimal trajectory a computational algorithm is presented for a known payload and the other one is then developed to find the maximum payload trajectory. The optimal trajectory and corresponding input control obtained from this method can be used as a reference signal and feedforward command in control structure of flexible manipulators. In order to clarify the method, derivation of the equations for a planar two-link manipulator is presented in detail. A number of simulation tests are performed and optimal paths with minimum effort, minimum effort-speed, maximum payload, and minimum vibration are obtained. The obtained results illustrate the power and efficiency of the method to solve the different path planning problems and overcome the high nonlinearity nature of the problems.

In this paper, by using the Leggett–Williams fixed point theorem, we prove the existence of three nonnegative solutions to second-order nonlinear impulsive differential equations with a three-point boundary value problem.

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problemssuffers under a coarse boundary approximation. To overcome this drawback the methodis furnished withan a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme.In contrast to similar, but rather theoretical, concepts already described in theliterature our approach combines a high generality with an easy-to-implement algorithm.

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.