On Some Inequalities and Their Application to the Cauchy Problem
Derivation of inequalities and application to Cauchy problem for infinite system of equations
PROBLEM SOLVING↗
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On the Cauchy problem of the relativistic Boltzmann equation.
Cauchy problem for relativistic Boltzmann equation, discussing initial distribution function and scattering cross section
Bichteler, K.↗
Uniqueness of the Cauchy problem when the initial surface contains characteristic points.
Uniqueness of Cauchy problem when initial surface contains characteristic points
UNIQUENESS THEOREM↗
Cauchy problem in the scalar-tensor gravitational theory.
Cauchy problem for scalar-tensor gravitational theory, splitting field equations into initial value equations and evolution equations
Cocke, W. J.↗
On the Right Weak Solution of the Cauchy Problem for a Quasilinear Equation of First Order Final Report
Solution of Cauchy problem for quasilinear equation of first order
Hopf, E.↗
On the right weak solution of the Cauchy problem for a quasilinear equation of first order
Right weak solution of Cauchy problem for first order quasi-linear equation
Hopf, E.↗
Parameter identification for an abstract Cauchy problem by quasilinearization
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
Brewer, Dennis W.↗
Parameter identification for an abstract Cauchy problem by quasilinearization
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations.
Brewer, Dennis W.↗
On a general theory of characteristics and the method of invariant imbedding.
Invariant imbedding equations of transport theory, noting imbedding of boundary value problem into initial value problems leads to Cauchy problem
Meyer, G. H.↗
The reduction of the solution of a game of pursuit for survival payoff to the solution of a couchy problem for a first order partial differential equation
Reduction of solution of game of pursuit for survival payoff to solution of Cauchy problem for first order partial differential equation
GAME THEORY↗
On bilateral approximation in the numerical integration of Cauchy's problem for ordinary differential equations
Runge-Kutta formulas for ordinary differential equations
Zarubych, S. Z.↗
Remarks on nonlinear parabolic equations.
Nonlinear parabolic equations, considering existence, uniqueness theorems, abstract Cauchy problem, etc
BANACH SPACE↗
Postinstability models in elasticity
It is demonstrated that the instability caused by the failure of hyperbolicity in elasticity and associated with the problem of unpredictability in classical mechanics expresses the incompleteness of the original model of an elastic medium. The instability as well as the ill-posedness of the Cauchy problem are eliminated by reformulating the original model.
Zak, M.↗
Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems
Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.
Skollermo, G.↗
Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: Stationary modes
Spatially discrete difference approximations for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline approximation can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.
Warming, Robert F.↗
The transonic aerofoil problem with embedded shocks.
The integral equation approach to the mixed flow problem of infinite wings at high subsonic speeds is adopted for non-circulatory and circulatory (lifting) flows. The solutions are determined from a system of non-linear algebraic equations and, to ensure always unique solutions, the method of differentiation with respect to a parameter has been applied. The resulting Cauchy problem is then solved with the linearised flow solution as the initial value vector. For the case of embedded shocks in the flow field, the method of steepest descent has been added to the calculation scheme. Results for subcritical and supercritical flows past aerofoils are given and compared with solutions obtained by finite-difference techniques.
Norstrud, H.↗
Solitary waves in concentrated vortices.
A nonlinear integrodifferential equation governing finite amplitude wave propagation on concentrated vortices is solved numerically. The solution to the Cauchy problem shows a solitary wave development qualitatively similar to solutions of the Korteweg-de Vries equation. A perturbation solution of the stationary form of the evolution equation confirms the unsteady calculation.
Leibovich, S.↗
High speed flow past wings
The analytical solution to the transonic small perturbation equation which describes steady compressible flow past finite wings at subsonic speeds can be expressed as a nonlinear integral equation with the perturbation velocity potential as the unknown function. This known formulation is substituted by a system of nonlinear algebraic equations to which various methods are applicable for its solution. Due to the presence of mathematical discontinuities in the flow solutions, however, a main computational difficulty was to ensure uniqueness of the solutions when local velocities on the wing exceeded the speed of sound. For continuous solutions this was achieved by embedding the algebraic system in an one-parameter operator homotopy in order to apply the method of parametric differentiation. The solution to the initial system of equations appears then as a solution to a Cauchy problem where the initial condition is related to the accompanying incompressible flow solution. In using this technique, however, a continuous dependence of the solution development on the initial data is lost when the solution reaches the minimum bifurcation point. A steepest descent iteration technique was therefore, added to the computational scheme for the calculation of discontinuous flow solutions. Results for purely subsonic flows and supersonic flows with and without compression shocks are given and compared with other available theoretical solutions.
Norstrud, H.↗