Initial value/boundary value problem for composite fractional relaxation equation

The “classical” chaos of deterministic systems is characteristic for the motion of dynamical systems. Recently, some attempts were made to find static analogies of chaos [Thompson & Virgin, 1988; Naschie & Athel, 1989; Naschie, 1989]. However, this was considered for structures in specific artificial conditions (for example, infinitely long bars with sinusoidal geometric imperfections) transferring de facto the boundary value problem (which always describes static deformation of structures) into an initial value problem characteristic for problems of motion. In this article, chaotic (unpredictable) behavior is described for a usual (not special) nonlinear structure in statics, which is governed, naturally, by a boundary value problem in a finite interval of the argument. The behavior of this structure (geometrically nonlinear plate), which is an example of the class of static chaotic structures, is investigated by a new geometrical approach called the “deformation map.” The presented results are one of the first steps in the chapter of chaos in statics, and therefore the link between “classical” and static chaos needs further investigations.

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SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE

International Journal of Computational Methods ◽

10.1142/s0219876213500412 ◽

2013 ◽

Vol 10 (06) ◽

pp. 1350041 ◽

Cited By ~ 2

Author(s):

J. CHRISTY ROJA ◽

A. TAMILSELVAN

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Outer Region ◽

Coupled System ◽

Shishkin Mesh ◽

Singularly Perturbed ◽

Initial Value ◽

Weakly Coupled ◽

Weakly Coupled System ◽

The Given

A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.

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The initial value problem for a nonlinear semi-infinite string

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500011008 ◽

1978 ◽

Vol 82 (1-2) ◽

pp. 19-26 ◽

Cited By ~ 21

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.

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AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

International Journal of Modern Physics A ◽

10.1142/s0217751x13400150 ◽

2013 ◽

Vol 28 (22n23) ◽

pp. 1340015 ◽

Cited By ~ 23

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.

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Existence and uniqueness for two-point boundary value problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500020072 ◽

1981 ◽

Vol 88 (3-4) ◽

pp. 219-234 ◽

Cited By ~ 7

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.

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