Application of singular eigenfunction expansions to the propagation of periodic disturbances in a radiating grey gas
Periodic disturbances propagation in radiating gray gas, using singular eigenfunction expansions
Cheng, P.↗
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Smoothness of solutions of Volterra integral equations with weakly singular kernels
Differentiability of nonlinear Volterra integral equations of second kind with convolutional weakly singular kernels
Feldstein, A.↗
Necessary conditions for joining optimal singular and nonsingular subarcs
Continuity and smoothness properties of piecewise optimal control at junction between singular and nonsingular subarcs, developing necessary conditions
Mcdanell, J. P.↗
A numerical method for solving Fredholm integral equations of the first kind using singular values
First kind Fredholm integral equation approximation by numerical quadrature formulas plus collocation, using singular value decomposition for solution
Hanson, R. J.↗
Multiple scattering theory of radiative transfer in the light of singular modes
Multiple scattering theory of radiative transfer boundary value problems, showing Neumann intensity expansion coefficients relation to singular normal modes
Kanal, M.↗
Singular perturbation methods and the warm plasma model
The application of techniques of the singular perturbation theory to the analysis of warm plasma is discussed. Typically, the cold plasma model can be applied over wide ranges of parameters and only over narrow ranges forming so-called boundary layers is the warm plasma model used. Simplified equations can be used and the solutions matched on both sides of the layer's boundary. Simple examples to illustrate the solution are presented. The analysis confirms that some results are highly sensitive to the values of: (1) wire radius or gap size for an antenna, (2) temperature of the medium, and (3) incident angle of a plane wave.
Lee, S. W.↗
Evaluation of the use of a singularity element in finite element analysis of center-cracked plates
Two different methods are applied to the analyses of finite width linear elastic plates with central cracks. Both methods give displacements as a primary part of the solution. One method makes use of Fourier transforms. The second method employs a coarse mesh of triangular second-order finite elements in conjunction with a single singularity element subjected to appropriate additional constraints. The displacements obtained by these two methods are in very good agreement. The results suggest considerable potential for the use of a cracked element for related crack problems, particularly in connection with the extension to nonlinear material behavior.
Mendelson, A.↗
Singular perturbations in the state regulator problem
Most of the results of singular perturbation theory have been concerned with initial value problems whereas optimal control problems are of two-point boundary value type. The portions of this theory applicable to the open loop state regulator problem are reviewed. For obtaining approximate solutions to the state regulator problem the method of matched asymptotic expansions is employed. This method has been developed in connection with certain fluid mechanics problems and is applicable to nonlinear as well as linear problems. It has been found in the past to be advantageous not to formulate this method generally but to apply it to each individual problem and this approach is adopted here. A general recipe for the method is given and its application is illustrated by using the method to obtain an approximate solution to a simple, specific state regulator problem.
Ardema, M. D.↗
Singular perturbation of absolute stability.
The influence of a small parameter at the higher derivatives in the differential equations describing nonlinear systems of the Lur'e-Postnikov class on absolute stability in the parameter space is investigated. The conditions leading to singular perturbations of absolute stability are examined.
Siljak, D. D.↗
Eigenvalues of singular differential operators by finite difference methods. II.
Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.
Baxley, J. V.↗
Singular perturbation of absolute stability.
It was previously shown (author, 1969) that the regions of absolute stability in the parameter space can be determined when the parameters appear on the right-hand side of the system equations, i.e., the regular case. Here, the effect on absolute stability of a small parameter attached to higher derivatives in the equations (the singular case) is studied. The Lur'e-Postnikov class of nonlinear systems is considered.
Siljak, D. D.↗
Aerodynamic influence coefficient method using singularity splines.
A new numerical formulation with computed results, is presented. This formulation combines the adaptability to complex shapes offered by paneling schemes with the smoothness and accuracy of the loading function methods. The formulation employs a continuous distribution of singularity strength over a set of panels on a paneled wing. The basic distributions are independent, and each satisfies all of the continuity conditions required of the final solution. These distributions are overlapped both spanwise and chordwise (termed 'spline'). Boundary conditions are satisfied in a least square error sense over the surface using a finite summing technique to approximate the integral.
Mercer, J. E.↗
Solution of coupled and singular perturbation methods using duality theory.
Dual variational techniques developed by Chan and Leininger (1972) are summarized, and duality theory in the form of the Complementary Variational Principle is employed to provide a suboptimal measure for the singular and epsilon-coupled perturbation methods proposed by Kokotovic and Cruz. The suboptimal measure is independent of any a priori knowledge of the optimal solution, thereby providing an absolute estimate of the performance loss rather than an estimate relative to the unknown optimal solution.
Chan, W. L.↗
Function space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs
Two existing function space algorithms, Davidon and projected gradient, are modified so that they may handle directly control variable inequality constraints. A third quasi-Newton type algorithm developed by C. G. Broyden is extended to optimal control problems. The Broyden algorithm is further modified so that it also may handle directly control variable inequality constraints. These methods along with a pure gradient and two conjugate gradient algorithms are simulated on three relatively simple yet representative bounded control problems, two of which have singular subarcs. Overall the Broyden algorithm was found to be superior. The most notable result of the study was the clear superiority of the Broyden and Davidon algorithms in producing a sharp interior control subarc.
Edge, E. R.↗
Conical flow near singular rays
The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.
Zahalak, G. I.↗
Solution of linear systems by a singular perturbation technique
An approximate solution is obtained for a singularly perturbed system of initial valued, time invariant, linear differential equations with multiple boundary layers. Conditions are stated under which the approximate solution converges uniformly to the exact solution as the perturbation parameter tends to zero. The solution is obtained by the method of matched asymptotic expansions. Use of the results for obtaining approximate solutions of general linear systems is discussed. An example is considered to illustrate the method and it is shown that the formulas derived give a readily computed uniform approximation.
Ardema, M. D.↗
Analysis of generalized forces in the singular perturbation equations of motion of flexible satellites
The formulation and existence of a generalized force in the singularly perturbed formulation of flexible satellites is described. The concept of this force sharply reduces the number of degrees of freedom and the equations of motion of satellites with a large number of flexible elements. The force is analyzed to demonstrate its existence and convergence criteria. The complete solution has been obtained in three time zones - the inner boundary layer, the outer boundary layer, and the large time extending beyond these boundary layers. A stability criterion is proposed for this generalized force.
Huang, T. C.↗
Generalized Theodorsen solution for singular integral equations of the airfoil class
A class of singular integral equations is considered which arise in various two-dimensional mixed boundary-value problems with simple harmonic time variation. A problem typical of this class is that of determining the lifting pressure distribution on an oscillating airfoil in an unbounded incompressible potential flow. It is shown that Theodorsen's (1935) solution to this problem, with some modification, is valid for a general class of unsteady kernel functions. The technique employed is to consider an equivalent steady problem and then show that the unsteady resolvent and unsteady homogeneous solution can be written directly in terms of the steady solutions and a single frequency-dependent function which reduces to the Theodorsen function for the steady kernel.
Williams, M. H.↗