A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

A Simple Semi-Implicit Scheme for Partial Differential Equations with Obstacle Constraints

We develop a simple and efficient numerical scheme to solve a class of obstacle problems encountered in various applications. Mathematically, obstacle problems are usually formulated using nonlinear partial differential equations (PDE). To construct a computationally efficient scheme, we introduce a time derivative term and convert the PDE into a time-dependent problem. But due to its nonlinearity, the time step is in general chosen to satisfy a very restrictive stability condition. To relax such a time step constraint when solving a time dependent evolution equation, we decompose the nonlinear obstacle constraint in the PDE into a linear part and a nonlinear part and apply the semi-implicit technique. We take the linear part implicitly while treating the nonlinear part explicitly. Our method can be easily applied to solve the fractional obstacle problem and min curvature flow problem. The article will analyze the convergence of our proposed algorithm. Numerical experiments are given to demonstrate the efficiency of our algorithm.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

   Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

Mathematical analysis of a nonlinear PDE model for European options with counterparty risk

In this work, we analyze a nonlinear partial differential equation (PDE) model for the total value adjustment on European options in the presence of a counterparty risk. We transform the nonlinear PDE into an equivalent one, involving a sectorial operator, and prove the existence and uniqueness of a solution.     

Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension

We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.     

A uniqueness result for a nonlinear hyperbolic equation

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function.     

Numerical Solutions for Option Pricing Models Including Transaction Costs and Stochastic Volatility

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.     

Spatial dynamical behavior of narrow composite beams

In this article, we study the nonlinear dynamical behavior of composite beams with nonlinear forced term. We obtain a fourth order partial differential equation (PDE) from the motion law of composite beams and discretize the PDE system. We analyze the corresponding discrete system and obtain the nonlinear spatial behavior such as chaotic and bifurcation behavior of interlaminar displacement.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

Symmetry reductions and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1+1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solutions with arbitrary functions for the PKPp.     

Threshold of Effective Degree SIR Model

The effective degree SIR model is a precise model for the SIR disease dynamics on a network. The original ODE model is only applicable for a network with finite degree distributions. The new generating function approach rewrites with model as a PDE and allows infinite degree distributions. In this paper, we first prove the existence of a global solution. Then we analyze the linear and nonlinear stability of the disease-free steady state of the PDE effective degree model, and show that the basic reproduction number still determines both the linear and the nonlinear stability. Our method also provides a new tool to study the effective degree SIS model, whose basic reproduction number has been elusive so far.     

The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification

We consider a spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification. We are interested in solutions bifurcating from the spatially homogeneous ground state in the case when pairs of imaginary eigenvalues simultaneously cross the imaginary axis. For this system we prove the bifurcation of a family of invariant tori which may contain quasiperiodic solutions. There is a serious difficulty in obtaining this result, because the linearization at the ground state possesses an essential spectrum up to the imaginary axis for all values of the bifurcation parameter. To construct the invariant tori, we use their invariance under the flow which manifests in a condition in PDE form. The nonlinear terms of this resulting PDE exhibit a loss of regularity. Since the linear part of this PDE is not smoothing, an adaption of the hard implicit function theorem (or Nash-Moser scheme) and energy estimates will be used to prove our result.     

The Strong Stability and Instability of a Fluid-Structure Semigroup

The strong stability problem for a fluid-structure interactive partial differential equation (PDE) is considered. The PDE comprises a coupling of the linearized Stokes equations to the classical system of elasticity, with the coupling occurring on the boundary interface between the fluid and solid media. Because of the nature of the unbounded coupling between fluid and structure, the resolvent of the associated semigroup generator will not be a compact operator. In consequence, the classical solution to the stability problem, by means of the Nagy-Foias decomposition, will not avail here. Moreover, it is not practicable to write down explicitly the resolvent of the fluid-structure generator; this situation thus makes it problematic to use the well-known semigroup stability result of Arendt-Batty and Lyubich-Phong. When a locally supported boundary dissipative mechanism is in place, we derive here a result of strong decay for this fluid-structure PDE. In the absence of said dissipative mechanism, we show the lack of asymptotic decay for solutions corresponding to arbitary initial data of finite energy.     

Damage processes in thermoviscoelastic materials with damage‐dependent thermal expansion coefficients

In this paper, we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage‐dependent thermal expansion coefficient. This term implies the presence of nonlinear coupling terms in the PDE system, which make the analysis more challenging. Copyright © 2015 John Wiley & Sons, Ltd.     

Hilbert空间中的二阶微分方程问题

运用多值分析、单调算子理论和Schauder不动点定理讨论了一类具有多点边值条件的二阶微分包含问题．作为一个预备性的结果，给出了一类二阶发展方程的解的存在唯一性和对初值的连续依赖性．最后，给出了以上结论在最优化和偏微分方程方面的两个应用．     

Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets

In the present work, unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.     

A New Jacobi Elliptic Function Expansion Method for Solvinga Nonlinear PDE Describing Pulse Narrowing Nonlinear Transmission Lines

In this article, we apply the first elliptic function equation to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing pulse narrowing nonlinear transmission lines are given with the aid of computer program, e.g. Maple or Mathematica. Based on Kirchhoff's current law and Kirchhoff's voltage law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and can be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.     

Multiple shooting methods for parabolic optimal control problems with control constraints

In the context of ordinary differential equations, shooting techniques are a state-of-the-art solver component, whereas their application in the framework of partial differential equations (PDE) is still at an early stage. We present two multiple shooting approaches for optimal control problems (OCP) governed by parabolic PDE. Direct and indirect shooting for PDE optimal control stem from the same extended problem formulation. Our approach reveals that they are structurally similar but show major differences in their algorithmic realizations. In the presented numerical examples we cover a nonlinear parabolic optimal control problem with additional control constraints. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)