Furtivity and masking problems in time dependent electromagnetic obstacle scattering

In this paper, we consider furtivity and masking problems in time-dependent three-dimensional electromagnetic obstacle scattering. That is, we propose a criterion based on a merit function to minimize or to mask the electromagnetic field scattered by a bounded obstacle when hit by an incoming electromagnetic field and, with respect to this criterion, we derive the optimal strategy. These problems are natural generalizations to the context of electromagnetic scattering of the furtivity problem in time-dependent acoustic obstacle scattering presented in Ref. 1. We propose mathematical models of the furtivity and masking time-dependent three-dimensional electromagnetic scattering problems that consist in optimal control problems for systems of partial differential equations derived from the Maxwell equations. These control problems are approached using the Pontryagin maximum principle. We formulate the first-order optimality conditions for the control problems considered as exterior problems defined outside the obstacle for systems of partial differential equations. Moreover, the first-order optimality conditions derived are solved numerically with a highly parallelizable numerical method based on a perturbative series of the type considered in Refs. 2-3. Finally, we assess and validate the mathematical models and the numerical method proposed analyzing the numerical results obtained with a parallel implementation of the numerical method in several experiments on test problems. Impressive speedup factors are obtained executing the algorithms on a parallel machine when the number of processors used in the computation ranges between 1 and 100. Some virtual reality applications and some animations relative to the numerical experiments can be found in the website http://www.econ.unian.it,/recchioni/w10/.

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

On the Canonical Formulation of Electrodynamics and Wave Mechanics

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

The hypergeometric function 1F1 of a matrix argument Y is a symmetric entire function in the eigenvalues y1,...,ym of Y. It appears in the distribution function of the largest eigenvalue of a Wishart matrix and its numerical evaluation is important in multivariate distribution theory. Hashiguchi et al. (J. Multivariate Analysis, 2013) proposed an efficient algorithm for evaluating the matrix 1F1 by the holonomic gradient method (HGM). The algorithm is based on the system of partial differential equations (PDEs) satisfied by the matrix 1F1 given by Muirhead (Ann. Math. Statist., 1970) and it cannot be applied to the diagonal cases, i.e. the cases where several yi's are equal because the system of PDEs has singularities on the diagonal region. Hashiguchi et al. derived an ordinary differential equation (ODE) satisfied by 1F1(y,y) in the bivariate case from some relations which are obtained by applying l'opital rule to the system of PDEs for 1F1(y1,y2). In this paper we generalize this approach for computing systems of PDEs satisfied by the matrix 1F1 for various diagonalization patterns. We show that the existence of a system of PDEs for a diagonalized 1F1 is reduced to the non-singularity of the matrices systematically derived from the diagonalization pattern. By checking the non-singularity numerically, we show that there exists a system of PDEs for a diagonalized 1F1 if the size of each diagonal block ≤ 36. We have computed an ODE for 1F1(y,...,y) up to m=22. We made a test implementation of HGM for diagonal cases and we show some numerical results.

Many problems of science and technique lead to partial differential equations. The mathematical theories of such equations, especially of the nonlinear ones, are very intricate, such that their numerical study becomes inevitable. To classify the partial differential equations, this chapter uses various criteria, that is, considering the order of the derivatives (first order, second order, or nth order), the linearity character (linear, quasilinear, or nonlinear equations), the influence of the integration domain at a point (equations of elliptic, parabolic, or hyperbolic type), and the types of limit conditions (Dirichlet, Neumann, or mixed problems). It presents the numerical methods for the integration of partial differential equations and of systems of partial differential equations. The numerical methods are point matching method, Ritz's method, Galerkin's method, and method of the least squares. Finally, the chapter gives numerical examples and some applications.

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

In testing systems with distributed interfaces/ports, we may place a separate tester at each port. It is known that this approach can introduce <it>controllability problems</it> which have received much attention in testing from finite state machines. Message sequence charts (MSCs) form an alternative, commonly used, language for modelling distributed systems. However, controllability problems in testing from MSCs have not been thoroughly investigated. In this paper, controllability problems in MSC test cases are analysed with three notions of observability: local, tester and global. We identify two types of controllability problem in MSC-based testing. It transpires that each type of controllability problem is related to a type of MSC pathology. <it>Controllability problems of timing</it> are caused by races but not every race causes controllability problems; <it>controllability problems of choice</it> are caused by non-local choices and not every non-local choice causes controllability problems. We show that some controllability problems of timing are avoidable and some controllability problems of choice can be overcome when testers have better observational power. Algorithms are provided to tackle both types of controllability problems. Finally, we show how one can overcome controllability problems using a coordination service with status messages based on algorithms developed in this paper.

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

Singular perturbation problems for systems of partial differential equations of elliptic type

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

Kranc: a Mathematica package to generate numerical codes for tensorial evolution equations