First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Nonlinear operator equations with a functional perturbation of the argument of neutral type

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties of the Jordan structure of the operator coefficients of the equation.     

The evolution of travelling waves in generalized Fisher equations via matched asymptotic expansions: Exponential corrections

In this paper we address an initial boundary value problem for a generalized Fisher equation. In particular we develop the matched asymptotic theory presented in Leach and Needham (2000) to consider the correction terms to the asymptotic approach to the wave-front of speed v = v*( 2) as t (time) . We establish the precise form of these corrections, and demonstrate that the rate of approach to the wave-front is algebraic in t when v* = 2 (there being two cases), but exponential in t when v* > 2.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

A new comparison theorem for solutions of stochastic differential equations

Let Z ₁(t) and Z ₂(t) be solutions of two stochastic differential equations. Then Z ₁(t)≦Z ₂(t) for all t?0 a.s. provided certain relations involving the coefficients and intial conditions of the equations hold. the diffusion coefficients are not required toi be the same for both equtions     

Envelope viscosity solutions of first- and second-order PDEs with u-dependence

General envelope methods are introduced which may be used to embed equations with u-dependence into equations without solution dependence. Furthermore, these methods present a rigorous way to consider so-called nodal solutions. That is, if w(t,x,z) is the viscosity solution of some pde, the nodal solution of an associated pde is a function u(t,x) so that w(t,x,u(t,x)) = 0. Examples are given to first- and second-order pdes arising in optimal control, differential games, minimal time problems, scalar conservation laws, geometric-type equations, and forward backward stochastic control.     

Symmetries and solutions for a Fisher equation with a proliferation term involving tumor development

We consider a generalized Fisher equation involving tumor development from the point of view of the theory of symmetry reductions in partial differential equations. The study of this equation is relevant as it includes generalizations within the proliferation rate, being interpreted in terms of the total mass of the tumor. Classical Lie point symmetries admitted by the equation are determined. Finally, we obtain some biologically meaningful solutions in terms of a hyperbolic tangent function, which describes the tumor dynamics.     

Boundary Conditions for Multidimensional Integrable Equations

We suggest an efficient method for finding boundary conditions compatible with integrability for multidimensional integrable equations of Kadomtsev-Petviashvili type. It is observed in all known examples that imposing an integrable boundary condition at a point results in an additional involution for the t-operator of the Lax pair. The converse is also likely to be true: if constraints imposed on the coefficients of the t-operator of the L-A pair result in a broader group of involutions of the t-operator, then these constraints determine integrable boundary conditions.New examples of boundary conditions are found for the Kadomtsev-Petviashvili and modified Kadomtsev-Petviashvili equations.     

Numerical Solution of the Bagley-Torvik Equation

We consider the numerical solution of the Bagley-Torvik equation Ay(t) + BD _* ^3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.     

A method for indefinite integration of oscillatory and singular functions

We propose a general method for computing indefinite integrals of the form

A Converse Comparison Theorem for <Emphasis Type="Italic">g</Emphasis>-Expectations

With the help of a limit property of solutions of backward stochastic differential equations(BSDEs),this paper establishes a converse comparison theorem for deterministic generators g of BSDEs under the assumption g(t, y, 0) ≡0.     

Diffusion approximation for the convection-diffusion equation with random drift

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ _t u _ɛ (t, x) = κΔ _x (t, x) + 1/ɛV(t/ɛ²,xɛ) ·∇ _x u _ɛ (t, x) with the initial condition u _ɛ(0,x) = u ₀(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R ^d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u _ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation. Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001     

Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes

We consider different types of processes obtained by composing Brownian motion B(t), fractional Brownian motion B _H (t) and Cauchy processes C(t) in different manners. We study also multidimensional iterated processes in ? ^d , like, for example, (B ₁(|C(t)|),…, B _d (|C(t)|)) and (C ₁(|C(t)|),…, C _d (|C(t)|)), deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly, we prove that some processes like C(|B ₁(|B ₂(…|B _n+1(t)|…)|)|) are governed by fractional diffusion equations.     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Exact elliptic compactons in generalized Korteweg–De Vries equations

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K*(l,p). Specifically we find that , where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006     

On the asymptotic stability for impulsive functional differential equations

We consider the asymptotic stability problems by Lyapunov functionals V for a class of functional differential equations with impulses of the form x′(t)=f(t,x _t ), x∈R ⁿ , t≧t ₀, t≠t _k ; △x=I _k (t,x(t ⁻)), t=t _k , k∈Z ⁺ . Some new asymptotic stability results are presented by using an idea originated by Burton and Makay [6] and developed by Zhang [1]. We generalize some known results about impulsive functional differential equations in the literature in which we only require the derivative of V to be negative definite on a sequence of intervals I _n =[s _n ,ξ _n ] which may or may not be contained in the sequence of impulsive time intervals [t _n ,t _n+1).     

Unbounded Solutions of Almost Periodically Forced Pendulum-Type Equations

In this paper, we prove that the forced pendulum-type equation , where with 1-periodicity in x satisfies the conditions: and , possesses infinitely many unbounded solutions on a cylinder S ¹×R for any almost periodic function p(t) with nonvanishing mean value. Received October 7, 1997, Accepted June 9, 1999     

Hopf Algebra Approaches to Nonlinear Algebraic Equations

Abstract

Let M be a k-vector space and R ∈ Hom(M ^?p , M ^?q ), we present a general version of the FRT-construction, we provide a method for examining whether an FRT-bialgebra A(R) has a pre-braided structure and whether M can be regarded as an A(R)-dimodule. We show that the FRT-relation plays a fundamental role in determining the algebra structure on the FRT-bialgebra and the compatibility condition of relevant dimodule. As an example, we give a Hopf algebra approach for solving both homogeneous and non-homogeneous nonlinear (algebraic) equations.