A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

再论弹性大挠度问题von Kármán方程与量子本征值问题Schrǒdinger方程的关系

本文是文[1]的继续和改善。利用本文的结果,还可以改善文[2～3]中有关弹性大挠度问题的讨论。在本文中,我们再次对弹性大挠度问题的von Kármán方程进行简化,使它最终成为非线性Schr?dinger方程。其次,在本文中我们对AKNS方程在多维条件下进行了更为对称的拓展。由于非线性Schr?dinger方程与AKNS方程即Dirac方程的可积性条件相联系,因此,弹性大挠度问题可以用逆散射方法求得其精确解,也就是说,它完全成了量子本征值问题。对于正交各向异性大挠度问题,本文也作了推论。     

Optimal control in coefficients for weak variational problems in Hilbert space

In this paper we give sufficient conditions for the existence of solutions of a problem of parametric optimization. We use continuity with respect to a functional parameter of weak solutions of a variational problem in a Hilbert space.We consider a problem of optimization with the control in coefficients of linear parabolic equation as an example. Using results of Spagnolo we characterize the closure of the reachable set. Finally, we construct an example of an optimization problem with the control in coefficients of a parabolic equation which does not have an optimal solution.     

On determining the source function in heat and mass transfer problems under integral overdetermination conditions

We examine an inverse problem of determining the right-hand side (the source function) in a parabolic equation from integral overdetermination data. By a solution to a parabolic equation we mean a weak solution, and the right-hand side in this equation can be a distribution of a certain class. Under some conditions on the data of the problem, we demonstrate that this inverse problem is well posed and, in particular, some stability estimates hold.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

Optimal control for a parabolic problem in a domain with highly oscillating boundary

In this article we study the homogenization of an optimal control problem for a parabolic equation in a domain with highly oscillating boundary. We identify the limit problem, which is an optimal control problem for the homogenized equation and with a different cost functional.     

Portfolio Optimization with Stochastic Volatilities and Constraints: An Application in High Dimension

In this paper we are interested in an investment problem with stochastic volatilities and portfolio constraints on amounts. We model the risky assets by jump diffusion processes and we consider an exponential utility function. The objective is to maximize the expected utility from the investor terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB in short) equation which could not be solved when the risky assets number exceeds three. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semilinear equation. We prove the existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We present an example that shows the importance of this reduction for numerical study of the optimal portfolio. We then compute the optimal strategy of investment by solving the associated optimization problem.     

On The Trace Problem For Solutions Of The Vlasov Equation

We study tlie trace problem for weak solutions of the Vlasov equation set in a domain. When the force field has Sobolev regularity, we prove the existence of a trace on the boundaries, which is defined thanks to a Green formula, and we show that the trace can be renormalized. We apply these results to prove existence and uniqueness of tlie Cauchy problem for the Vlasov equation witli specular reflection at the boundary. We also give optimal trace theorems and solve the Cauchy problem witli general Dirichlet conditions at the boundary     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Green's function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.     

Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations

We consider a multiparameter spectral problem for a weakly coupled system of ordinary differential equations in which every equation is Hamiltonian and contains two unknown functions. Using the notion of the number of an eigenvalue for a problem with one such equation, we give a statement of the problem of finding the desired eigentuple of values for the problem with several equations. We prove the existence and uniqueness of a solution of this problem and suggest and study a numerical solution method.     

A Black-Scholes option pricing model with transaction costs

We consider a boundary value problem for a nonlinear differential equation which arises in an option pricing model with transaction costs. We apply the method of upper and lower solutions in order to obtain solutions for the stationary problem. Moreover, we give conditions for the existence of solutions of the general evolution equation.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.     

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain. In many practice applications, we cannot know the initial temperature during heat process, therefore we consider a non-characteristic Cauchy problem for the heat equation. The method of fundamental solutions is applied to solve this problem. Due to ill-posedness of this problem, we first discretize the problem and then regularize it in the form of discrete equation. Numerical tests are conducted for showing the effectiveness of the proposed method.     

A Boundary Value Problem for Mixed-Type Equations in Domains with Multiply Connected Hyperbolicity Subdomains

Unlike the familiar problems for mixed-type equations, we pose a boundary value problem for the Lavrent'ev–Bitsadze equation in domains with unbounded multiply connected hyperbolicity subdomains. We prove unique solvability in explicit form. We also study this problem for the general Lavrent'ev–Bitsadze equation.