Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms

This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces.     

Inverse solutions for a second-grade fluid for porous medium channel and Hall current effects

Assuming certain forms of the stream function inverse solutions of an incompressible viscoelastic fluid for a porous medium channel in the presence of Hall currents are obtained. Expressions for streamlines, velocity components and pressure fields are described in each case and are compared with the known viscous and second-grade cases.     

Fourier Stability Analysis of a Numerical Method for Time Domain Electromagnetic Scattering from a Thin Wire

We present stability results for a numerical scheme for computing the current induced on the surface of a thin wire by an incident time-dependent electromagnetic field, using the exact kernel formulation. The results show that the algorithm is stable for arbitrarily small values of the mesh size.     

Exact Penalty Function and Asymptotic Strong Nonlinear Duality in Integer Programming

In this paper,a logarithmic-exponential penalty function with two parameters for integer program-ming is discussed.We obtain the exact penalty properties and then establish the asymptotic strong nonlinearduality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penaltyproperties.The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].     

Some High-Frequency Gravitational Waves Related to Exact Radiative Spacetimes

A formalism is introduced which may describe both standard linearized waves and gravitational waves in Isaacson's high-frequency limit. After emphasizing main differences between the two approximation techniques we generalize the Isaacson method to non-vacuum spacetimes. Then we present three large explicit classes of solutions for high-frequency gravitational waves in particular backgrounds. These involve non-expanding (plane, spherical or hyperbolical), cylindrical, and expanding (spherical) waves propagating in various universes which may contain a cosmological constant and electromagnetic field. Relations of high-frequency gravitational perturbations of these types to corresponding exact radiative spacetimes are described.     

Observability inequalities for thin shells

We consider the exact controllability problem from boundary for thin shells. Under some checkable geometric assumptions on the middle surface, we establish the observability inequalities via the Bochner technique for the Dirichlet control and the Neumann control problems. We also give several examples to verify the geometric assumptions.     

Arbitrary /-state approximate solutions of the Hulthén potential through the exact quantization rule

In this paper, using the Exact Quantization Rule, we present approximate analytical solutions of the radial Schrödinger equation with non-zero l values for the Hulthén potential in the frame of an approximation to the centrifugal potential for any l states. The energy levels of all bound states can be easily calculated from the Exact Quantization Rule. Specifically, the normalized analytical wave functions are also obtained. Some energy eigenvalues are numerically calculated and compared with those obtained by other methods such as asymptotic iteration, supersymmetry, numerical integration methods, and the schroedinger Mathematica package.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

Two interacting spins in external fields. Four‐level systems

In the present article, we consider the so‐called two‐spin equation that describes four‐level quantum systems. Recently, these systems attract attention due to their relation to the problem of quantum computation. We study general properties of the two‐spin equation and show that the problem for certain external backgrounds can be identified with the problem of one spin in an appropriate background. This allows one to generate a number of exact solutions for two‐spin equations on the basis of already known exact solutions of the one‐spin equation. Besides, we present some exact solutions for the two‐spin equation with an external background different for each spin but having the same direction. We study the eigenvalue problem for a time‐independent spin interaction and a time‐independent external background. A possible analogue of the Rabi problem for the two‐spin equation is defined. We present its exact solution and demonstrate the existence of magnetic resonances in two specific frequencies, one of them coinciding with the Rabi frequency, and the other depending on the rotating field magnitude. The resonance that corresponds to the second frequency is suppressed with respect to the first one.     

An economic interpretation of exact penalty functions using sensitivity analysis

It has already been demonstrated that under some assumptions, a local minimum of a constrained problem is also a local unconstrained minimum of a function which is called an exact penalty function. Here, we present the same result with a new demonstration. By using sensitivity analysis, we give an economic interpretation for exact penalty functions.