Geometric Analysis of a Nonlinear Boundary Value Problem from Physical Oceanography

In this paper, Lyapunov functions are generated to determine the regions of asymptotic stability of power systems under transient disturbances. With suitable assumptions, the swing equations of the synchronous machines connected to a power system are second- or third-order nonlinear autonomous differential equations. The Lyapunov functions V, employed to determine the domains of asymptotic stability of these nonlinear differential equations, are simple quadratic forms, whose coefficients are chosen so that the Routh-Hurwitz criteria are satisfied for the corresponding linear differential equations. For a synchronous machine swinging against an infinite bus, three typical Lyapunov functions are generated, taking transient saliency and positive-sequence damping into account. The application of Aizerman's method to the same problem leads to a fourth Lyapunov function. The domains of stability given by these Lyapunov functions are compared with the actual stability region obtained by numerical integration. Consideration of one time constant of the prime-mover governor leads to a third-order differential equation. A Lyapunov function is generated, and the stability surface obtained is verified by computing the swing curves numerically for various initial conditions. Finally, it is explained how some of these techniques could be extended for generating Lyapunov functions for multimachine systems.

A systematic way of generating sets of local boundary conditions on the gauge\nfields in a path integral is presented. These boundary conditions are suitable\nfor one--loop effective action calculations on manifolds with boundary and for\nquantum cosmology. For linearised gravity, the general proceedure described\nhere leads to new sets of boundary conditions.\n

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

For the first time, an accurate analytical solution, based on coupled three-dimensional (3D) piezoelasticity equations, is presented for free vibration analysis of the angle-ply elastic and piezoelectric flat laminated panels under arbitrary boundary conditions. The present analytical solution is applicable to composite, sandwich and hybrid panels having arbitrary angle-ply lay-up, material properties, and boundary conditions. The modified Hamiltons principle approach has been applied to derive the weak form of governing equations where stresses, displacements, electric potential, and electric displacement field variables are considered as primary variables. Thereafter, multi-term multi-field extended Kantorovich approach (MMEKM) is employed to transform the governing equation into two sets of algebraic-ordinary differential equations (ODEs), one along in-plane (x) and other along the thickness (z) direction, respectively. These ODEs are solved in closed-form manner, which ensures the same order of accuracy for all the variables (stresses, displacements, and electric variables) by satisfying the boundary and continuity equations in exact manners. A robust algorithm is developed for extracting the natural frequencies and mode shapes. The numerical results are reported for various configurations such as elastic panels, sandwich panels and piezoelectric panels under different sets of boundary conditions. The effect of ply-angle and thickness to span ratio (s) on the dynamic behavior of the panels are also investigated. The presented 3D analytical solution will be helpful in the assessment of various 1D theories and numerical methods.

Remarks on Barbashin–Ezeilo Problem on third-order nonlinear differential equations

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

The steady boundary layer problems with buoyancy effect over a moving and permeable plate are studied in the present paper. By using the boundary layer approximations with similarity variables, we obtain coupled nonlinear velocity and temperature equations. Then, we derive boundary shape functions for velocity and temperature, which automatically satisfy the specified boundary conditions, including a convective boundary condition. Generally, numerical method is difficult to keep all the specific boundary conditions exactly and accurately. Because this will lead to the decline of the accuracy of numerical solution, we are motivated to develop a novel iterative algorithm which is based on the boundary shape functions. It transforms the nonlinear boundary layer problems into the initial value problems for new variables. The unknown terminal values of these variables can be determined by iteration, while the initial values of them can be arbitrarily given. The high-performance of the iterative boundary shape functions method (BSFM), which is convergent very fast, can be confirmed by numerical examples. The parametric studies in terms of the Grashof number, Prandtl number and Biot number with different suction/injection and velocity ratio values, as well as the flow instability area in terms of the Biot number and velocity ratio are carried out by the BSFM.

Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di- stability) in problem discretizations. However, computational cate- gories are not precise, and qualitative descriptors such as of moderate size cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive New- ton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate bound- ary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.

It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false–periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third–order differential equation y'''(x) = λy(x) are described. The general form of degenerate boundary conditions for the fourth–order differentiation operator D4 is found. 12 classes of boundary value eigenvalue problems are described for the operator D4, the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): are there similar problems for odd–order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): can a spectral boundary–value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary–value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker’s tenth problem is resolved.

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

This paper studies the semiclassical approximation of simple supergravity in Riemannian four-manifolds with a boundary, within the framework of $\ensuremath{\zeta}$-function regularization. The massless nature of gravitinos, jointly with the presence of a boundary and a local description in terms of potentials for spin 3/2, force the background to be totally flat. First, nonlocal boundary conditions of the spectral type are imposed on spin-3/2 potentials, jointly with boundary conditions on metric perturbations which are completely invariant under infinitesimal diffeomorphisms. The axial gauge-averaging functional is used, which is then sufficient to ensure self-adjointness. One, thus, finds that the contributions of ghost and gauge modes vanish separately. Hence the contributions to the one-loop wave function of the universe reduce to those $\ensuremath{\zeta}(0)$ values resulting from physical modes only. Another set of mixed boundary conditions, motivated instead by local supersymmetry and first proposed by Luckock, Moss, and Poletti, is also analyzed. In this case the contributions of gauge and ghost modes do not cancel each other. Both sets of boundary conditions lead to a nonvanishing $\ensuremath{\zeta}(0)$ value, and spectral boundary conditions are also studied when two concentric three-sphere boundaries occur. These results seem to point out that simple supergravity is not even one-loop finite in the presence of boundaries.

Gaussian process (GP) regression is a nonparametric Bayesian approach that has been used successfully in various astronomical domains, especially in time-domain astronomy. The most common applications are the smoothing of data for interpolation and the detection of periodicities. The ability to create unbiased data-driven models without a predefined physical model can be a major advantage over conventional regression methods. Prior knowledge can be included by setting boundary conditions or constraining hyperparameter values, while unknown hyperparameters are optimized during the conditioning of the model. We have adapted and transformed previous approaches of GP regression and introduce three new applications for this regression method, especially in the context of stellar occultations: the modeling of occultation light curves, the correction of public JPL ephemerides of minor planets based on publicly available image data of the Zwicky Transient Facility, and the detection of natural satellites. We used data from observations of stellar occultations to validate the models and achieved promising results in all cases, and thus we confirmed the flexibility of GP regression models. Considering various existing use cases in addition to our novel applications, GP regression can be used to model diverse data sets addressing a wide range of problems. The accuracy of the model depends on the input data and on the set boundary conditions. Generally, high-quality data allow the usage of loose boundary conditions, while low-quality data require more restrictive boundary conditions to avoid overfitting.

Chapter 5 - Life Is Really Nonlinear

The CFD simulation accuracy mostly depends on the appropriate setting of boundary conditions and numerical simulation parameters. This study shows the influence of two types of boundary condition settings on the CFD simulation results of Double-Skin Facade (DSF) for a specific problem. These two boundary settings are the constant temperature on the DSF surfaces called Boundary A, and Boundary B is defined via solar radiation using the Discrete Ordinate radiation Model (DOM). The paper verified both the numerical simulations using the experimental data. Comparing the numerical results of two types of boundaries with experimental data shows that both cases underestimated the values lower than 5.2 K and 0.1 m/s for the temperature and velocity respectively at the regarded measured points. Boundary A gives more accurate temperature prediction results, while Boundary B shows velocity magnitude closer to the measurements in the middle height of the cavity; the average temperature and velocity differences between the two boundary types are 0.6 K and 0.003 m/s respectively which are negligible. Finally, the selection of boundary conditions depends on study purposes, however, when the DSF is equipped with blinds and if there is not enough data in hand but the exact value of solar irradiation, using the Boundary B approach is suggested; it can provide reasonable results associated with multi-type of thermal boundary conditions at the same time. Furthermore, if the goal is to investigate the flow pattern in the DSF, Boundary B is argued to perform better than the constant temperature boundary condition.

Robust and flexible numerical methodologies for the imposition of boundary conditions are required to formulate well-posed problems. A boundary condition should be Robust and flexible numerical methodologies for the imposition of boundary conditions are required to formulate well-posed problems. A boundary condition should be nonreflecting, to avoid spurious perturbations that can provocate unsteadiness or instabilities. The reflectiveness of various boundary conditions is analyzed in the context of the Godunov methods. A nonlinear, isentropic wave propagation model is used to investigate the reflection mechanism on the flowfield borders, and a parameter τ is defined to give a measure of the boundary reflectiveness. A new set of boundary conditions, in which τ =0, that is, totally nonreflecting, is then proposed. The approach has been integrated in an aerodynamic design procedure using a distributed boundary control.