Nonlinear First-order Partial Differential Equations Research Articles

Double-null coordinates for the Vaidya metric.

Einstein's equations with spherical symmetry are formulated in double-null coordinates, and the high-frequency approximation to a unidirectional radial flow of unpolarized radiation (the Vaidya metric) is studied in detail. For this case the Einstein equations reduce to a single first-order nonlinear partial differential equation. Integration of this equation introduces an arbitrary function (of one null variable) which must be chosen so as to regularize the metric across horizons. Although the problem is, in general, not analytically solvable, we are able to extend the class of known analytic solutions from the constant-mass case (Kruskal-Szekeres metric) to linear and exponential mass functions. In the linear case we give the first explicit regular covering of a spacetime with a naked shell-focusing singularity.

The variational approach to shape from shading

We develop a systematic approach to the discovery of parallel iterative schemes for solving the shape-from-shading problem on a grid. A standard procedure for finding such schemes is outlined, and subsequently used to derive several new ones. The shape-from-shading problem is known to be mathematically equivalent to a nonlinear first-order partial differential equation in surface elevation. To avoid the problems inherent in methods used to solve such equations, we follow previous work in reformulating the problem as one of finding a surface orientation field that minimizes the integral of the brightness error. The calculus of variations is then employed to derive the appropriate Euler equations on which iterative schemes can be based. The problem of minimizing the integral of the brightness error term is ill posed, since it has an infinite number of solutions in terms of surface orientation fields. A previous method used a regularization technique to overcome this difficulty. An extra term was added to the integral to obtain an approximation to a solution that was as smooth as possible. We point out here that surface orientation has to obey an integrability constraint if it is to correspond to an underlying smooth surface. Regularization methods do not guarantee that the surface orientation recovered satisfies this constraint. Consequently, we attempt to develop a method that enforces integrability, but fail to find a convergent iterative scheme based on the resulting Euler equations. We show, however, that such a scheme can be derived if, instead of strictly enforcing the constraint, a penalty term derived from the constraint is adopted. This new scheme, while it can be expressed simply and elegantly using the surface gradient, unfortunately cannot deal with constraints imposed by occluding boundaries. These constraints are crucial if ambiguities in the solution of the shape-from-shading problem are to be avoided. Differrent schemes result if one uses different parameters to describe surface orientation. We derive two new schemes, using unit surface normals, that facilitate the incorporation of the occluding boundary information. These schemes, while more complex, have several advantages over previous ones.

A simple approximation for a moving interface in a coastal aquifer

An approximate analytical solution for a moving interface in a confined coastal aquifer is found by solving a nonlinear first-order partial differential equation. The solution, which assumes that flow approaching the interface changes discontinuously with time from one constant value to another, coincides with exact steady-flow solutions obtained from the Dupuit approximation at both t = 0 and t = ∞. A comparison with numerical solutions suggests that maximum errors of about ten percent occur in the toe location for intermediate values of time and that the approximate solution tends to overestimate both the rate of advance and retreat of the toe. The approximation has a very simple form and, in contrast to numerical solutions, is easy to evaluate. Additional comparisons with numerical solutions also suggest that the analytical approximation can be applied with confidence to an unconfined flow only when the sudden change in approaching flow rate occurs at distances from the interface toe that are not too great. This is because free-surface storage effects slow the movement of the flow rate change before its effects are eventually felt downstream at the interface toe.

The variational approach to shape from shading

Abstract We develop a systematic approach to the discovery of parallel iterative schemes for solving the shape-from-shading problem on a grid. A standard procedure for finding such schemes is outlined, and subsequently used to derive several new ones. The shape-from-shading problem is known to be mathematically equivalent to a nonlinear first-order partial differential equation in surface elevation. To avoid the problems inherent in methods used to solve such equations, we follow previous work in reformulating the problem as one of finding a surface orientation field that minimizes the integral of the brightness error. The calculus of variations is then employed to derive the appropriate Euler equations on which iterative schemes can be based. The problem of minimizing the integral of the brightness error term is ill posed, since it has an infinite number of solutions in terms of surface orientation fields. A previous method used a regularization technique to overcome this difficulty. An extra term was added to the integral to obtain an approximation to a solution that was as smooth as possible. We point out here that surface orientation has to obey an integrability constraint if it is to correspond to an underlying smooth surface. Regularization methods do not guarantee that the surface orientation recovered satisfies this constraint. Consequently, we attempt to develop a method that enforces integrability, but fail to find a convergent iterative scheme based on the resulting Euler equations. We show, however, that such a scheme can be derived if, instead of strictly enforcing the constraint, a penalty term derived from the constraint is adopted. This new scheme, while it can be expressed simply and elegantly using the surface gradient, unfortunately cannot deal with constraints imposed by occluding boundaries. These constraints are crucial if ambiguities in the solution of the shape-from-shading problem are to be avoided. Differrent schemes result if one uses different parameters to describe surface orientation. We derive two new schemes, using unit surface normals, that facilitate the incorporation of the occluding boundary information. These schemes, while more complex, have several advantages over previous ones.

Viscosity Solutions of Hamilton-Jacobi Equations at the Boundary

When considering classical solutions of boundary value problems for nonlinear first-order scalar partial differential equations, one knows that there are parts of the boundary of the region under consideration where one cannot specify data and would not expect to require data in order to prove uniqueness. Of course, classical solutions of such problems rarely exist in the large owing to the crossing of characteristics. The theory of a sort of generalized solution—called "viscosity solutions"—for which good existence and uniqueness theorems are valid has been developed over the last few years. In this note we give some results concerning parts of the boundary on which one need not know (prescribe) viscosity solutions to be able to prove comparison (and hence uniqueness) results. In this context, this amounts to identifying boundary points with the property that solutions in the interior which are continuous up to the boundary are also viscosity solutions at the boundary point. Examples indicating the sharpness of the results are given.

Viscosity solutions of Hamilton-Jacobi equations at the boundary

When considering classical solutions of boundary value problems for nonlinear first-order scalar partial differential equations, one knows that there are parts of the boundary of the region under consideration where one cannot specify data and would not expect to require data in order to prove uniqueness. Of course, classical solutions of such problems rarely exist in the large owing to the crossing of characteristics. The theory of a sort of generalized solution—called "viscosity solutions"—for which good existence and uniqueness theorems are valid has been developed over the last few years. In this note we give some results concerning parts of the boundary on which one need not know (prescribe) viscosity solutions to be able to prove comparison (and hence uniqueness) results. In this context, this amounts to identifying boundary points with the property that solutions in the interior which are continuous up to the boundary are also viscosity solutions at the boundary point. Examples indicating the sharpness of the results are given.

Large axisymmetric deformation of a non-linear viscoelastic circular membrane

The problem of a viscoelastic circular membrane deforming under its own weight is considered. The membrane, clamped in a horizontal plane at its boundary, is formed from a non-linear viscoelastic material whose behaviour is modelled by a non-linear single integral constitutive equation. The sag of the membrane is considered quasi-static and provision is made in the model for making the relaxation time stretch-dependent. The analysis leads to a system of three equations, two of which are first-order non-linear partial-differential integral equations while the third is an algebraic relation. Time descretization enables the differential equations to be solved as ordinary differential equations which may be dealt with by a familiar step-wise integration process. Numerical solutions are obtained for membranes with both stretch-dependent and stretch-independent relaxation times.

Solution for Free-Surface Flow in Porous Media Using Rayleigh-Ritz Expansions

Free-surface flow can be modelled by the Laplace equation for the velocity potential and a nonlinear first-order partial differential equation for the free surface. The potential problem is reformulated as a variational problem and then solved approximately by a Rayleigh-Ritz expansion. The free-surface equation is solved using finite differences. The procedure is applied to a particular problem, and excellent results are obtained.

Stability of a feedback controlled distributed system by modal representation

In this work a linear distributed parameter system controlled by a finite number of lumped linear feedback loops is considered. The spatial portion of the system is decomposed using generalized Fourier expansions, and a complex frequency domain transcendental characteristic determinant results. The theory developed is then applied to the automatic regulation of water flow in a ‘ reach ’ of an open irrigation canal. The dynamics of water flow in the transient state are described by a set of two first-order non-linear partial differential equations. The boundary conditions, governed by the water-flow dynamics under the gate, are also non-linear equations. For this particular system the feedback controller signal originates at the downstream boundary and controls the gates at the ‘ head ’ of the canal. The stability analysis of the complete closed-loop system is based on perturbation about some steady-state flow conditions which allows for the linearization of the system's non-linear partial differential equations and boundary conditions. Subsequent application of the Nyquist criterion to a modal representation of the distributed parameter portion of the control loop in cascade with the lumped parameter controller yields the stability conditions.

Large Axisymmetric Deformation of a Nonlinear Viscoelastic Membrane Due to Spinning

The problem of a viscoelastic membrane undergoing large planar deformation due to spinning is solved. The membrane, rigidly bonded at its inner boundary and traction-free at the outer, consists of a nonlinear viscoelastic solid whose behavior is modeled by a nonlinear single integral constitutive equation. The model includes the possibility that the relaxation time is influenced by the amount of stretching. Spinning is considered to be sufficiently slow so that shear effects can be neglected. The formulation uses radial and circumferential stretch ratios as dependent variables. These satisfy a system of first-order nonlinear partial-differential integral equations. The numerical procedure at each time step obtains the current spatial derivatives of the stretch ratios in terms of the current and previously determined stretch ratios. This gives essentially a first-order nonlinear system of differential equations, of the same structure as that obtained in the elastic version of the problem [1], which is integrated numerically. Solutions are obtained for both strain-dependent and strain-independent relaxation times.