Vector Partial Differential Equations Research Articles

Theory of high-frequency linear response of isotropic type-II superconductors in the mixed state.

A self-consistent approach to vortex dynamics, including the effects of nonlocal vortex interaction, pinning, and creep, is further clarified and unified by its restatement in terms of an initial-boundary value problem. We derive and solve a single vector partial differential equation describing the linear response of a type-II superconductor in the mixed state at frequencies well below the gap frequency. The solution of this equation, presented here for several sample geometries, provides the phenomenological superconductor dispersion relation, accompanying complex penetration depths, and complex response functions. The theory is expected to have applicability to a wide range of experiments involving vortex dynamics.

Coupled nonlinear electrodynamics of type-II superconductors in the mixed state.

A theory of the nonlinear electrodynamics of isotropic high-\ensuremath{\kappa} type-II superconductors containing an array of vortices is presented. The theory generalizes a self-consistent approach to vortex dynamics wherein the effects of nonlocality, vortex inertia, pinning, flux flow, and flux creep are treated in a unified fashion. We derive and solve a single vector partial differential equation describing the nonlinear response of a type-II superconductor at frequencies well below the gap frequency. The generation of nth-order harmonics due to bilinear field nonlinearity is discussed.

Solving Oseen's system of equations by boundary elements using multiple reciprocity method

The Multiple Reciprocity Method has been used successfully to solve scalar partial differential equations whose integral representation formulae possesses a relatively complex kernel, in terms of a sequence of simple higher order fundamental solutions of given related equations, and also to reduce domain integrals to surface integrals using the same basic idea. In this work, we will extend the use of the Multiple Reciprocity Method to solve Oseen's system of vector partial differential equations in an unbounded domain. In this case, Oseen's kernel is given in terms of Stokes' higher order fundamental solutions.

Oscillation of nonlinear vector differential equations

Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈Em, where G is an exterior domain in En, and b is a continuous nonnegative valued function in G × Em. A solution v: G→Em is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in En satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G.

The Three-Dimensional Hollow or Solid Truncated Cone Under Axisymmetric Torsionless End Loading

The Saint-Venant problems of solid or hollow truncated cone are investigated under axisymmetric torsionless end loading with the ruled sides being free from stress. Total-stress problems are formulated in terms of a vector partial differential equation whose component variables are stresses or of stress-type. A biorthogonality condition is derived which permits the numerical solution of boundary-value problems, and the results of a sample application of the method are presented.

A vector partial differential equation model for air pollution

A vector partial differential equation model for air pollution

A vector partial differential equation model for air pollution

A general system of non-linear partial differential equations for the concentrations of all the components of polluted air is given and rewritten in vector matrix form thus establishing a general mathematical model for air pollution.

Gronwall’s inequality for systems of partial differential equations in two independent variables

This paper presents a generalization for systems of partial differential equations of Gronwall’s classical integral inequality for ordinary differential equations. The proof is by reducing the vector integral inequality to a vector partial differential inequality and then using a vector generalization of Riemann’s method to obtain the final inequality. The final inequality involves a matrix function in the integrand which is a generalization of the scalar Riemann function. The proof includes a successive approximations argument to guarantee the existence and positivity property of this matrix function. The inequality is applied to prove a uniqueness theorem for a nonlinear vector hyperbolic partial differential equation, a comparison theorem for a linear hyperbolic vector partial differential equation, and a continuous dependence theorem for a nonlinear vector boundary value problem. The inequality also appears to have many applications in stability problems and in numerical solutions of partial differential equations. All of these results hold for the corresponding Volterra integral equations and the method of proof of the main result shows that the function on the right-hand side of the final inequality is the solution of the integral equation and hence is the maximal solution of the original inequality.

Determination of heart dipole vector P electrically' and magnetically'. Comparison.

Even though the electrical activity of the heart varies very slowly with time, a complete and thorough investigation of the heart activity in terms of time-varying potentials and fields has been studied. Starting from Maxwell's equations in their vector partial differential equation form, a vector partial differential equation of the electric Hertzian vector potential\(\bar \Pi \) is set up in terms of the assumed electric dipole moment per unit volume of the electrical heart activityP. The equation is solved in terms of an infinite series in\(\bar \Pi \) and then approximated only to its first term since it converges extremely fast. The potential distribution inside the body ϕ, the electric field intensity\(\bar E\), and the magnetic field intensity\(\bar H\), are derived from\(\bar \Pi \) in terms of\(\bar P\). Thus ϕ,\(\bar E\) and\(\bar H\) are known in terms of\(\bar P\), the source.

Distributed parameter differential games: A variational calculus approach†

This paper extends certain results known for lumped parameter differential games to differential games which involve distributed systems. The distributed system is described by a linear second-order vector partial differential equation with appropriate boundary conditions. Simultaneous distributed and boundary controls are considered and all performance criteria involved are quadratic. Three types of solutions are sought for the non-zero sum gaino: The Nash equilibrium, the non-inferior and the minimax. Open-loop strategies are found using techniques of variational calculus. It is noted that these solutions differ from closed-loop solutions obtained using dynamic programming. The solution of certain partial differential equations arising in this method is proposed using either the normal modes approach or Letov's analytical design approach extended to distributed systems.

Distributed parameter differential games: A dynamic programming approach†

This paper extends certain results known for lumped parameter differential games to differential games which involve distributed systems described by partial differentia] equations. In particular the distributed system is described by a linear second-order vector partial differential equation with appropriate boundary conditions. Simultaneous distributed and boundary controls are considered and all performance criteria involved are quadratic. Three types of solutions are sought for the non-zero sum game: The Nash equilibrium, the non-inferior and the minimax. Closed-loop strategies are found using techniques of dynamic programming. Finally, the solution of certain partial differential equations which arise in this method is considered using the normal modes approach.

EQUATIONS OF HEAT DISTRIBUTION WITHIN THE BODY.

A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.

Nonsteady Helical Flows of Second-Order Fluids

Some exact solutions of the third-order vector partial differential equation describing nonsteady flows of incompressible second-order fluids are presented. Although periodic simple shearing flow is also discussed, the main emphasis here is on helical flows between moving coaxial cylinders, particularly periodic flows of the Couette and Poiseuille type. The calculations suggest that, for such periodic helical flows, measurements of the radial thrusts on the bounding cylindrical walls supply practicable methods for the determination of normal-stress coefficients.