An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.     

On the solution of Dirichlet problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the second type

Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.     

反射天线设计的一个数学理论

本文讨论反射天线面设计中出现的一个偏微分方程, 这是一个完全非线性Monge-Ampère型方程. 我们先给出一个广义解的定义, 然后介绍如何得到广义解的存在性、唯一性和正则性, 最后我们给出求数值解的一个方法.     

Existence of distributional solutions of a closed Dirichlet problem for an elliptic-hyperbolic equation

The closed Dirichlet problem for an elliptic-hyperbolic equation arising in linearizing the Monge-Ampère type equation is shown to admit a distributional solution by using an a-b-c integral method.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian

For a class of elliptic Hessian operators raised by Caffarelli-Nirenberg-Spruck, the corresponding parabolic Monge-Ampère equation was studied, the existence and uniqueness of the admissible solution to the first initial-boundary value problem for the equation were established, which extended a result of Ivochkina-Ladyzhenskaya.     

Solution of the Monge-Ampère equation on Wiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge-Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400-428], w.r.t the Wiener measure is also the strong solution of the Monge-Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.     

Weak Solutions of Monge-Ampére Type Equations in Optimal Transportation

This paper concerns the weak solutions of some Monge-Ampére type equations in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampére type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.     

The Sinc-collocation method for solving the Thomas–Fermi equation

A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results.     

On the intermediate integral for Monge-Ampère equations

Goursat showed that in the presence of an intermediate integral, the problem of solving a second-order Monge-Ampère equation can be reduced to solving a first-order equation, in the sense that the generic solution of the first-order equation will also be a solution of the original equation. An attempt by Hermann to give a rigorous proof of this fact contains an error; we show that there exists an essentially unique counterexample to Hermann's assertion and state and prove a correct theorem.

     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.

     

Approximations of the nonlinear Volterra's population model by an efficient numerical method

In this paper, we apply the new homotopy perturbation method to solve the Volterra's model for population growth of a species in a closed system. This technique is extended to give solution for nonlinear integro‐differential equation in which the integral term represents the total metabolism accumulated fromtime zero. The approximate analytical procedure only depends on two components. The newhomotopy perturbationmethodwas applied to nonlinear integro‐differential equations directly and by converting the problem into nonlinear ordinary differential equation. We also compare this method with some other numerical results and show that the present approach is less computational and is applicable for solving nonlinear integro‐differential equations and ordinary differential equations as well. Copyright © 2011 John Wiley & Sons, Ltd.     

Nets of lines depending on a parameter and sufficient conditions for their convergence

We study criteria for the global regularity of a net of lines defined on the plane or a part of it by an ordinary differential equation of first order and second degree, nets depending on a parameter, and questions of convergence on the parameter. We use an analytic technique connected with hyperbolic systems of equations of a special form. We give applications to surfaces of negative Gaussian curvature and to hyperbolic Monge-Ampère equations.Translated fromItogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 22, 1990, pp. 3–36.     

Solution of a laminar boundary layer flow via a numerical method

In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.     

Boundedness of maximal function related to the Monge-Ampère equation

In this paper, we establish the boundedness of maximal function on Morrey spaces related to the Monge-Ampère equation.     

Numerical method for determining the inhomogeneity boundary in the Dirichlet problem for Laplace’s equation in a piecewise homogeneous medium

The Dirichlet problem for Laplace’s equation in a two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary from additional information on the solution of the Dirichlet problem is considered. A numerical method based on the linearization of the nonlinear operator equation for the unknown boundary is proposed for solving the inverse problem. The results of numerical experiments are presented.     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.