Integrals,infinite series and closed forms

A number of examples of infinite series are given whose sum can be expressed in closed form, sometimes as an integral. This leads naturally to the determination of a closed form for the sum of a less well-known series in terms of an interesting definite integral. Other examples are included, as well as suggestions for further work.     

A closed form solution to one dimensional robin boundary problems

Many cellular and subcellular biological processes can be described in terms of diffusing and chem- ically reacting species (e.g. enzymes). In this paper, we will use reflected and absorbed Brownian motion and stochastic differential equations to construct a closed form solution to one dimensional Robin boundary prob- lems. Meanwhile, we will give a reasonable explanation to the closed form solution from a stochastic point of view. Finally, we will extend the problem to Robin boundary problem with two boundary conditions and give a specific solution by resorting to a stopping time.     

Solvable Three-Body Problem and Painlevé Conjectures

For a special choice of the three interparticle coupling constants in the three-body version of a many-body problem in the plane that was recently investigated, the general solution of the equations of motion can be written in closed form (and is remarkably simple). We also discuss another analogous three-body problem and obtain two third-order highly nonlinear autonomous ODEs whose general solutions, we conjecture, are entire. In other words, we conjecture that these ODEs feature (a strong version of) the Painlevé property.     

Differential algebraic equations with after-effect

In this paper, we are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.     

Existence of a fundamental solution of partial differential equations associated to Asian options

We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.

In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.     

Parallel Multi-Stage & Multi-Step Method in ODEs

In this paper, the theory of parallel multi-stage and multi-step method is discussed, which is a form of combining Runge-Kutta method with linear multi-step method that can be used for parallel computation.     

A modified Markov model for the estimation of computer software performance

The model proposed by Trivedo and Shooman [8] is extended and modified by assuming that (1) the error occurrence rate when the machine is running is proportional to the number of errors in the system; (2) the error correction rate has two components, either an error is corrected with correction rate μ₀ or an error is corrected but a new error is created with ineffective correction rate μ₁. The solution of the differential equations corresponding to the model is obtained in closed form.     

Closed paths of light trapped in a closed Fermat curve

Geometric constructions have previously been shown that can be interpreted as rays of light trapped either in polygons or in conics, by successive reflections. The same question, trapping light in closed Fermat curves, is addressed here. Numerical methods are used to study the behaviour of the reflection points of a triangle when the degree of the curve varies, including a generalization to non integer powers.     

On-line solution of a stochastic pursuit-evasion game

In this paper, the concepts of prior-commitment and delayed-commitment strategies for zero-sum, linear-quadratic differential games with noise-corrupted measurements are applied to a detailed example of a pursuit-evasion game. Quasilinearization is used to solve the nonlinear two-point boundary-value problem of the prior-commitment game. A closed form solution is obtained for the delayed-commitment strategies. Comparison of the payoff functionals confirms the relationships discussed in Ref. 1.     

Forward-backward stochastic differential equation games with delay and noisy memory

Abstract

The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of equation has not been studied before. The maximum principles give conditions for determining the Nash equilibrium of the game. We use this to derive a closed form Nash equilibrium for an economic model where the players maximize their consumption with respect to recursive utility.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

A numerical scheme based on mean value solutions for the helmholtz equation on triangular grids

A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.

     

Solvability of partial differential equations by meshless kernel methods

This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well-posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101205). Robert Schaback’s research in Hong Kong was sponsored by DFG and City University of Hong Kong.     

Analytic solutions of initial-boundary-value problems of transient conduction using symmetries

Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface temperature or constant heat flux condition. The solutions are obtained in a manner highlighting the systematic procedure of extending the symmetry method for a PDE to investigate BVPs of the PDE. A comparative analysis of numerical and closed form solutions is carried out for a physical problem of heat conduction in a semi-infinite solid bar made of AISI 304 stainless steel.     

Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross sections

We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross‐sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle and localized, cross‐wise uniform initial distributions, short‐time skewness is calculated exactly to be positive, while long‐time asymptotics shows it to be negative. Monte Carlo simulations confirm these predictions, and document the timescale for sign change. The equilateral triangle appears to be the only regular polygon with this property—all others possess positive skewness at all times. Alternatively, closed‐form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how both nonzero short‐time skewness and the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (more mass to the right of the mean) or back (more mass to the left of the mean) “loading” properties of the evolving tracer distribution along the pipe. Short‐ and long‐time asymptotics confirm this condition, and Monte Carlo simulations verify this at all times. The simulations are also used to examine the role of corners and boundaries on the distribution for short‐time evolution of point source , as opposed to cross‐wise uniform, initial data.     

New anisotropic models from isotropic solutions

We establish an algorithm that produces a new solution to the Einstein field equations, with an anisotropic matter distribution, from a given seed isotropic solution. The new solution is expressed in terms of integrals of known functions, and the integration can be completed in principle. The applicability of this technique is demonstrated by generating anisotropic isothermal spheres and anisotropic constant density Schwarzschild spheres. Both of these solutions are expressed in closed form in terms of elementary functions, and this facilitates physical analysis. Copyright © 2005 John Wiley & Sons, Ltd.     

Some analysis of a stochastic logistic growth model

We derive several new results on a well-known stochastic logistic equation. For the martingale case, we compute the distribution of the solution, mean passage times, and the distribution of hitting times, all in closed form. For the case of constant coefficients, we also find mean passage times and for the general equation we give the weak solution expressed in terms of stochastic quadratures. We also show how these quadratures may be considerably simplified using the results for the martingale case. As it turns out, the martingale case has a particularly elegant weak solution, and to a large degree its structure carries over to the general case.