Invariant Imbedding and Sequential Interpolating Filters for Nonlinear Processes

1969 ◽  
Vol 91 (2) ◽  
pp. 195-199 ◽  
Author(s):  
H. H. Kagiwada ◽  
R. E. Kalaba ◽  
A. Schumitzky ◽  
R. Sridhar
Keyword(s):  
Boundary Value Problem ◽  
Initial Value Problem ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Point Boundary ◽  
Invariant Imbedding ◽  
Nonlinear Processes ◽  
Initial Value ◽  
Imprecise Observations ◽  
Sequential Filter

Suppose imprecise observations are made on imprecisely defined nonlinear processes, and one wishes to estimate the state of the process at certain fixed instants of time lying within the interval of observation. Furthermore, assume that it is required to update these estimates as additional observations become available. This is precisely the problem of sequential interpolation. The equations of the sequential interpolating filter, when a least-squares estimation criterion is used, are obtained in this paper. The interpolation problem is first shown to be equivalent to a two-point boundary-value problem. The two-point boundary-value problem is converted to an initial-value problem using invariant imbedding. The initial-value problem leads directly to a sequential filter.

Dual extremum principles for the heat equation

1977 ◽  
Vol 77 (3-4) ◽  
pp. 273-292 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Initial Value Problem ◽  
General Class ◽  
Initial Value Problems ◽  
Linear Operators ◽  
Point Boundary ◽  
Initial Value ◽  
Quantities Of Interest

SynopsisDual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

Invariant Imbedding Applied to Eigenvalue Problems in Mechanics

1965 ◽  
Vol 32 (1) ◽  
pp. 47-50 ◽  
Author(s):  
E. M. Shoemaker
Keyword(s):  
Wave Propagation ◽  
Eigenvalue Problem ◽  
Initial Value Problem ◽  
Present Method ◽  
Eigenvalue Problems ◽  
Transport Theory ◽  
Formal Method ◽  
Point Boundary ◽  
Invariant Imbedding ◽  
Initial Value

A formal method is presented for obtaining eigenvalues of ordinary differential equations associated with problems of buckling and vibration. The method utilizes the idea of invariant imbedding which has previously been applied to two point boundary value problems in transport theory and wave propagation. The present method reduces the eigenvalue problem to an initial value problem for a matrix of Riccati equations. The numerical solution of such formulations has proved to be generally more efficient than known methods.

The initial value problem for a nonlinear semi-infinite string

1978 ◽  
Vol 82 (1-2) ◽  
pp. 19-26 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Galerkin Procedure ◽  
Initial Boundary Value ◽  
Infinite String

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Existence and uniqueness for two-point boundary value problems

1981 ◽  
Vol 88 (3-4) ◽  
pp. 219-234 ◽  
Author(s):  
John V. Baxley ◽  
Sarah E. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Point Boundary ◽  
Initial Value

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

scholarly journals Initial and Boundary Value Problems for Fractional Differential Equations Involving Atangana-Baleanu Derivative

2018 ◽  
Vol 23 (2) ◽  
pp. 137 ◽  
Author(s):  
Al-Musalhi Fatma S. ◽  
Al-Salti Nasser S. ◽  
Karimov Erkinjon
Keyword(s):  
Boundary Value Problem ◽  
Initial Value Problem ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Integral Form ◽  
Orthogonal Basis ◽  
Initial Value ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
The Given

In the present work, an initial value problem involving the Atangana-Baleanu derivative is considered. An explicit solution of the given problem in integral form is obtained by using the Laplace transform. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of a series expansion using an orthogonal basis obtained by separation of variables. Some examples are also given to illustrate the obtained results.

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