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Partial differential equations finite difference approximation, applying convergence theorems with probabilistic method
Elliptic partial differential equations are used to generate a smooth grid that permits a one-to-one mapping in such a way that mesh lines of the same family do not cross. Problems that arise due to lack of clustering at crucial points or intersections of mesh lines at highly acute angles, are examined and various forcing or source terms are used (to correct the problems) that are either compatible with the maximum principle or are so locally controlled that mesh lines do not intersect. Attention is given to various schematics of unclustered grids and grid detail about (highly cambered) airfoils.
Identification in partial differential equations by Laplace equation
The parabolic partial differential equation considered is u sub t = u sub xx + f(u), where minus infinity x plus infinity and o t plus infinity. Under suitable hypotheses pertaining to f, a class of initial data is exhibited: phi(x), minus infinity x plus infinity, for which the corresponding solutions u(x,t) appraoch zero as t approaches the limit of plus infinity. This convergence is uniform with respect to x on any compact subinterval of the real axis.
Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
Hamilton-Jacobi partial differential equation integration method, using variational principle and transversality condition
Integration of Hamilton-Jacobi partial differential equation using Euler equations and Legendre transform - classical example of wave fronts or parallel surfaces
The hyperbolic system of partial differential equations with a real constant square coefficient matrix A is considered. The problem of finding an energy conserving norm for the solution of the system is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm. The conditions serve as criteria on a given system which determine whether or not the solution will have its energy conserved in some norm. Examples of specific systems and boundary conditions are also provided.
Singular perturbation problems for partial differential equations
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
The feasibility of using parabolic partial differential equations for grid generation is examined. Source terms in the form of a linear interpolation between the current grid and the outer boundary are used in generating a grid for a two-dimensional airfoil flow. Grid generation equations are derived by assuming that grid spacings are locally nonuniform on the computational domain. The grid spacing control method is described in detail, and the local orthogonality of the grid lines is discussed. An O-mesh and an H-mesh generated by the described method are shown.
Application of Lie series to solution of linear ordinary differential equations of second order
Parameter and domain dependence of eigenvalues of elliptic partial differential equations
Semigroups, groups and Liapunov stability of partial differential equation, obtaining operator differential equations in Hilbert spaces
Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.
Boundary value problem for elliptic-parabolic partial differential equations in theory of random processes, obtaining analytical solution in terms of hypergeometric functions
Lie series solutions for 33 partial differential equations resulting from separation of Helmholtz equation into 11 coordinate systems
Lie transformation group for solution of partial differential equations