Exact Solutions to a Finite-difference Model of a Nonlinear Reaction-advection Equation: Implications for Numerical Analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

Perturbations of Dynamic Equations

We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Small time and semiclassical asymptotics for stochastic heat equations driven by Lévy noise

The paper is devoted to the study of stochastic heat equations driven by Lévy noise. Applying the WKB method, we obtain multiplicative small time and semiclassical asymptotics for the Green functions and for solutions of the Cauchy problem for the heat equation under some natural additional assumptions on their coefficients. The first step in this construction consists in solving the corresponding stochastic Hamilton-Jacobi equations which constitute the "classical part" of the semiclassical approximation. In its turn, the corresponding Hamilton-Jacobi equations can be solved via solutions of the corresponding Hamiltonian systems, which gives rise to the method of stochastic characteristics. The relevant theory of stochastic Hamiltonian systems and stochastic Hamilton-Jacobi equations was developed in our previous papers. Here we put the final rung on the ladder: stochastic Hamiltonian systems, stochastic Hamilton-Jacobi equations, stochastic heat equations.     

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.     

Remarks on the Notion of Order of Difference Equations

In the paper, the notion of order of a difference equation is introduced in such a way that this order is invariant with respect to the change of the independent variable. For the general case, a formula for the general solution of linear difference equation of k -th order is given. It is shown that, in contrast to differential equations, the dimension of the linear space of solutions of linear homogenous difference equation can be lowered if their domain of definition is restricted appropriately.     

Difference Equations∶ Detecting Oscillations via Higher Order Characteristic Equations

In these notes we introduce an alternative procedure to detect the presence of a wide range of oscillatory solutions-the exponential oscillatory ones-in both continuous-time and discrete linear difference equations. The technique is addressed to the scalar case of the linear two-delay difference equation, but it can be extended to other types of evolution equations including the nonlinear ones, in higher dimensions. In particular, we give an illustration of its applicability to the case of a simple two dimensional difference equation. Oscillatory solutions are presented in their closed formulae.     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

On Periodic Solutions of Second Order Linear Difference Equations

In this paper, we apply a new procedure initially developed in Refs. [H. El-Owaidy and H.Y. Mohamed. "On the periodic solutions for nth order difference equations". Journal of Applied Mathematics and Computation , (to appear); "The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations". Journal of Applied Mathematics and Computation , (to appear)] to simplify the use of Carvalho's method to the case of discrete difference equations, in order to find the periodic solutions of second order linear difference equations. We can also find the complex periodic solutions.     

The Bismut-Elworthy formula for backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.     

Linear Generalized Analytic Functions

Complex-valued continuous functions are approximated using solutions of complex partial differential equations of Vekua type.     

Risk-sensitive portfolio optimization on infinite time horizon

We consider a continuous time portfolio optimization problems on an infinite time horizon for a factor model, recently treated by Bielecki and Pliska ["Risk-sensitive dynamic asset management", Appl. Math. Optim. , 39 (1990) 337-360], where the mean returns of individual securities or asset categories are explicitly affected by economic factors. The factors are assumed to be Gaussian processes. We see new features in constructing optimal strategies for risk-sensitive criteria of the portfolio optimization on an infinite time horizon, which are obtained from the solutions of matrix Riccati equations.     

Difference Equations and Divisibility Properties of Sequences

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of the form y n = y n ( p , a )= ~ k =0 X n kp a k where p is an arbitrary prime. Let 𝜌 p ( m ) denote the exponent of the highest power of a prime p which divides m . Recently, the author obtained lower bounds for 𝜌 p ( y n ) based on recurrence relations of order p and p m 1. The cases with tight bounds have also been characterized. In this paper, we prove that 𝜌 p ( y np ( p , a ))= n for 𝜌 p ( a +1)=1, p S 3. We obtain alternative difference equations of order p 2 for y n and order p for the p -sected sequence y np by a generating function based method. We also extend general divisibility results relying on the arithmetic properties of the coefficients and initial values.     

On Detecting Solutions of Polynomial Nonlinear Difference Equations

In this paper a method for discovering solutions of nonlinear polynomial difference equations is presented. It is based on the concepts of i -operator and star-product. These notions create a proper algebraic background by means of which we can find linear equations "included" into the original nonlinear one and to seek for solutions among them. A corresponding algorithm and some examples are also provided.     

Nonstandard Finite Difference Schemes for Differential Equations

This paper gives an introduction to nonstandard finite difference methods useful for the construction of discrete models of differential equations when numerical solutions are required. While the general rules for such schemes are not precisely known at the present time, several important criterion have been found. We provide an explanation of their significance and apply them to several model ordinary and partial differential equations. The paper ends with a discussion of several outstanding problems in this area and other related issues.     

Discontinuous Oblique Derivative Problem for Second Order Equations of Mixed Type in General Domains

In [L. Bers (1958). Mathematical Aspects of Subsonic and Transonic Gas Dynamics . Wiley, New York; A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; J.M. Rassias (1990). Lecture Notes on Mixed Type Partial Differential Equations. World Scientific, Singapore; H.S. Sun (1992). Tricomi problem for nonlinear equation of mixed type. Sci. in China ( Series A ), 35 , 14-20], the authors proposed and discussed the Tricomi problem of second order equations of mixed type in a special domain, and in [G.C. Wen (1998). Oblique derivative problems for linear mixed equations of second order. Sci. in China ( Series A ), 41 , 346-356], the author discussed the oblique derivative problem of second order equations of mixed type in a special domain. The present article deals with the discontinuous oblique derivative problem for quasilinear second order equations of mixed (elliptic-hyperbolic) type in general domains. Firstly, we give the formulation of the above boundary value problem, and then prove the existence of solutions for the above problem in general domains, in which the complex analytic method is used.     

Global Behaviour of Solutions of Nonlinear Delay Difference Equations*

In this paper global asymptotic stability and asymptotic behaviour of solutions of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global asymptotic stability are derived.     

On the Global Character of the Difference Equation X n + 1 =

We investigate the global stability, the periodic character, and the boundedness nature of solutions of the difference equation x n +1 = f + n x n m (2 k +1) + i x n m 2 l A + x n m 2 l , n =0,1,… where k and l are non-negative integers, the parameters f , n , i , A are non-negative real numbers with f + n + i >0, and the initial conditions are non-negative real numbers. We show that the solutions exhibit a trichotomy character depending upon the parameters n , i and A .     

Obstacle problem for nonlinear integro-differential equations arising in option pricing

Abstract  We study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate, which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle. We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity. Keywords: Integro-differential equations, Obstacle problem, Viscosity solutions, American options Mathematics Subject Classification (2000): 45K05, 35K85, 49L25, 91B24