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  • Articles  (8)
  • numerical methods  (5)
  • Chebyshev
  • mathematics
  • 2020-2023
  • 2010-2014
  • 1990-1994  (2)
  • 1970-1974  (6)
  • 1965-1969
  • Mathematics  (8)
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  • Articles  (8)
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  • 1
    Electronic Resource
    Electronic Resource
    Error estimates of Hermite interpolation (1973)
    Springer
    BIT 13 (1973), S. 338-343 
    Details
    ISSN: 1572-9125
    Keywords: Hermite interpolation ; Chebyshev ; order of continuity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a procedure for obtaining error estimates for Hermite interpolation at the Chebyshev nodes {cos ((2j+1)π/2n)} j =0n−1 −1≦x≦1, for functionsf(x) of various orders of continuity. The procedure is applicable in many cases when the usual Lagrangian error bound is not, and is a better bound, in general, when both are applicable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01951944
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  • 2
    Electronic Resource
    Electronic Resource
    Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis (1993)
    Springer
    Journal of nonlinear science 3 (1993), S. 477-539 
    Details
    ISSN: 1432-1467
    Keywords: nearly integrable systems ; spectral transform ; attractors ; traveling waves ; stability ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary In this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02429875
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  • 3
    Electronic Resource
    Electronic Resource
    A piecewise exponential model for seismic well-logging data (1974)
    Springer
    Mathematical geology 6 (1974), S. 33-45 
    Details
    ISSN: 1573-8868
    Keywords: inversion of data ; mathematics ; numerical analysis ; regression analysis ; geophysics ; petroleum ; well logging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Conventional methods of analyzing sonic log data do not always yield accurate information on each velocity segment of a well. It is shown here that the velocity-depth parameters and the sections of approximately constant velocity may be more precisely defined by using an exponential spline to model the data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02111925
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  • 4
    Electronic Resource
    Electronic Resource
    Equations for stress and displacement fields around pressurized elliptical holes in elastic solids (1973)
    Springer
    Mathematical geology 5 (1973), S. 11-25 
    Details
    ISSN: 1573-8868
    Keywords: mathematics ; structural geology
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A mechanical model based upon elasticity theory for the deformation around a pressurized elliptical hole in an homogeneous, isotropic solid has found application in many areas of engineering, rock mechanics, and structural geology. The explicit equations for stress and displacement around such a hole are given. An apl computer program for calculating these stresses and displacements also is presented. These equations and this program should ease future usage of this model by engineers and geologists.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02114084
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  • 5
    Electronic Resource
    Electronic Resource
    A second-order iterative technique for differential-integral systems (1974)
    Springer
    Journal of optimization theory and applications 13 (1974), S. 179-185 
    Details
    ISSN: 1573-2878
    Keywords: Optimal control theory ; functional differential equations ; numerical methods ; hereditary processes ; contraction mapping principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This note considers the optimal control of a system represented by nonlinear differential-integral equations with a general cost function. A second-order iterative method of solution based on the fixed-point contraction mapping principle is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935538
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  • 6
    Electronic Resource
    Electronic Resource
    Parameter variation for the solution of two-point boundary-value problems and applications in the calculus of variations (1974)
    Springer
    Journal of optimization theory and applications 13 (1974), S. 164-178 
    Details
    ISSN: 1573-2878
    Keywords: Two-point boundary-value problems ; calculus of variations ; numerical methods ; differential equations ; computing methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the parameter variation method, a scalar parameterk, kε[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method. The method is applied to the solution of problems with various types of boundary-value specifications and is further extended to take account of situations arising in the solution of problems from variational calculus (e.g., total elapsed time not specified, optimum control not a simple function of the variables).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935537
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  • 7
    Electronic Resource
    Electronic Resource
    Decomposition technique for minimum time trajectories (1974)
    Springer
    Journal of optimization theory and applications 14 (1974), S. 233-250 
    Details
    ISSN: 1573-2878
    Keywords: Time-optimal control ; decomposition methods ; two-point boundary-value problems ; trajectories ; numerical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A decomposition technique is presented for minimum-time trajectories which are characterized by intermediate constraints and discontinuities. The optimization of such multiple are trajectories is usually a formidable task. One optimization method, trajectory decomposition, breaks the original trajectory at points of discontinuity into separate arcs and then optimizes each are subject to prescribed boundary conditions. This constitutes a first level of control. Each first-level solution is evaluated by a second-level controller, which iteratively specifies new are boundary conditions in order to achieve an optimum solution. Unfortunately, this two-level method cannot be applied directly to minimum-time trajectories. The two-level trajectory decomposition method is extended here to a three-level technique for treating the minimum-time trajectory. The first level again optimizes each are for specified intervention parameters. The new second level, the time interface controller, exploits certain homogeneity properties to satisfy time transversality conditions at all boundaries and to couple the first-level solution arcs in time. The third level, the state interface controller, satisfies state transversality conditions at the arc junctions iteratively while driving the trajectory to its optimum. The new three-level procedure represents a feasible decomposition because each solution trajectory in the iterative sequence is physically realizable. The technique also offers a decentralization of control effort and reduction of initial-value sensitivities. An example problem is formulated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932942
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  • 8
    Electronic Resource
    Electronic Resource
    Control of an aircraft landing in windshear (1994)
    Springer
    Journal of optimization theory and applications 83 (1994), S. 237-267 
    Details
    ISSN: 1573-2878
    Keywords: Flight mechanics ; landing ; feedback control ; windshear problems ; differential games ; longitudinal motion ; lateral motion ; numerical methods ; linear differential games ; switch surfaces
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of the feedback control of an aircraft landing in the presence of windshear is considered. The landing process is investigated up to the time when the runway threshold is reached. It is assumed that the bounds on the wind velocity deviations from some nominal values are known, while information about the windshear location and wind velocity distribution in the windshear zone is absent. The methods of differential game theory are employed for the control synthesis. The complete system of aircraft dynamic equations is linearized with respect to the nominal motion. The resulting linear system is decomposed into subsystems describing the vertical (longitudinal) motion and lateral motion. For each subsystem, an, auxiliary antagonistic differential game with fixed terminal time and convex payoff function depending on two components of the state vector is formulated. For the longitudinal motion, these components are the vertical deviation of the aircraft from the glide path and its time derivative; for the lateral motion, these components are the lateral deviation and its time derivative. The first player (pilot) chooses the control variables so as to minimize the payoff function; the interest of the second player (nature) in choosing the wind disturbance is just opposite. The linear differential games are solved on a digital computer with the help of corresponding numerical methods. In particular, the optimal (minimax) strategy is obtained for the first player. The optimal control is specified by means of switch surfaces having a simple structure. The minimax control designed via the auxiliary differential game problems is employed in connection with the complete nonlinear system of dynamical equations. The aircraft flight through the wind downburst zone is simulated, and three different downburst models are used. The aircraft trajectories obtained via the minimax control are essentially better than those obtained by traditional autopilot methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02190056
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