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It is shown that a general class of nonlinear integral equations may be transformed into a Cauchy system. That this leads to an effective numerical scheme is demonstrated by solving the Ambarzumian integral equation. The new method does not involve successive approximations or series expansions.
The computation of inviscid supersonic flow over a two-dimensional airfoil is considered. There are two main nonlinear approaches which lead to approximate solutions. Small-amplitude theory gives valid solutions provided the airfoil thickness is not too great and the Mach number is not too high. The second type of approximation, shock expansion theory, employs the fact that even for flows with strong shocks the effect of the down-running characteristics remains small. This leads to an analytic solution at the airfoil. It is pointed out that the approximate solution is accurate enough for many cases of interest. The numerical method furnishes a rapid correction to the solution in those cases where it is not. The characteristic-streamline coordinate system is useful both for the computation of the approximate solution and the corrections, and is also convenient for displaying and interpreting the results.
In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.
In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.
The overall objective of the program is to provide the engineering tools to reduce the analysis, design, and development effort, and thus the cost, in achieving the required performances for switching regulators and dc-dc converter systems. The program was both tutorial and application oriented. Various analytical methods were described in detail and supplemented with examples, and those with standardization appeals were reduced into computer-based subprograms. Major program efforts included those concerning small and large signal control-dependent performance analysis and simulation, control circuit design, power circuit design and optimization, system configuration study, and system performance simulation. Techniques including discrete time domain, conventional frequency domain, Lagrange multiplier, nonlinear programming, and control design synthesis were employed in these efforts. To enhance interactive conversation between the modeling and analysis subprograms and the user, a working prototype of the Data Management Program was also developed to facilitate expansion as future subprogram capabilities increase.
As the nuclear fleet in the United States ages and subsequent license renewal applications grow, the prediction of concrete durability at extended operation becomes more important. To address this issue, a Fast-Fourier Transform (FFT) method is utilized to simulate aging-related degradation of concrete within the Microstructure Oriented Scientific Analysis of Irradiated Concrete (MOSAIC) software. MOSAIC utilizes compositional phase maps to simulate damage from radiation-induced volumetric expansion (RIVE), applied force, creep, and thermal expansion. This compositional detail allows each mineral in the microstructure to be assigned specific material properties, allowing the simulation to be as accurate and representative as possible. The principal goal of MOSAIC is to simulate the effects of nonlinear aging mechanisms occurring in nuclear concrete on the macroscopic mechanical properties, using only the aggregate microstructure compositional information as a starting point. Here, several realistic example simulations are shown to demonstrate the utility and uniqueness of the MOSAIC software.
The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by gravitational and aerodynamic forces. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. A rigorous analytic solution is used to integrate this equation with a high degree of accuracy, using Poincare's method of small parameters and Lagrange's expansion to explicitly express the major axis as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. Numerical solutions were generated to display the accuracy of the analytic theory.
We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINNs posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI with mean field Gaussian approximation for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Loève (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.
An aerodynamic shape optimization method has previously been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs. This method has also included a flowfield extrapolation (or flow prediction) method based on the Taylor series expansion of an existing CFD solution. The present paper reports on the extension of this method to the thin-layer Navier-Stokes equations in order to account for the viscous effects. Also, to test the method under highly nonlinear conditions, it has been applied to the transonic flows. Initially, the success of the flow prediction method is tested. Then, the overall method is demonstrated by optimizing the shapes of two supercritical transonic airfoils at zero angle of attack. The first one is shape optimized to achieve a minimum drag while obtaining a lift above a specified value. Whereas, the second one is shape optimized for a maximum lift while attaining a drag below a specified value. The results of these two cases indicate that the present method can produce successfully optimized aerodynamic shapes.
The motion of charged particles in a stochastic magnetic field with nonzero mean is considered via a generalized quasi-linear expansion of Liouville's equation. The general result is an equation relating cosmic ray scintillations to magnetic fluctuations and to cosmic ray gradients. The resonant interaction between particles and the random magnetic field is considered in detail, and the effect of nonlinear terms in the equations is considered. The nonlinear terms are important in damping out initial conditions and in determining conditions near cyclotron resonances. The application of the theory to the propagation of cosmic rays during quiet times in interplanetary space is considered. It is concluded that cosmic ray scintillations in interplanetary space may provide useful information about interplanetary particles and fields and also about nonlinear plasma interactions.
A procedure utilizing an expansion of proper orthogonal functions (or modes) to predict a fully developed flow in channel is derived. To examine numerical and conceptual difficulties, preliminary computations are performed with assigned mean velocity, and turbulence is expressed with only the first mode. The nonlinear interactions of the components of the first mode are treated specifically, with the influence of higher modes neglected; this treatment required adjustment of the skewness and effective Reynolds number to assure energy equilibrium of the first mode. Computational results show that the first mode possesses the structural character similar to that of the entire flow.
A procedure utilizing an expansion of proper orthogonal functions (or modes) to predict a fully developed flow in channel is derived. To examine numerical and conceptual difficulties, preliminary computations are performed with assigned mean velocity, and turbulence is expressed with only the first mode. The nonlinear interactions of the components of the first mode are treated specifically, with the influence of higher modes neglected; this treatment required adjustment of the skewness and effective Reynolds number to assure energy equilibrium of the first mode. Computational results show that the first mode possesses the structural character similar to that of the entire flow.
The moderate-amplitude axisymmetric oscillations of an inviscid liquid shell surrounding an incompressible gas bubble are calculated by a multiple-time-scale expansion for initial deformations composed of two-lobed perturbations of the shell and a displacement of the bubble from the center of mass of the liquid. Two types of small-amplitude motion are identified and lead to very different nonlinear dynamic interactions, as described by the results valid up to second order in the amplitude of the initial deformation. In the 'bubble mode', the oscillations of the captive bubble and the liquid shell are exactly in phase and the bubble vibrates about its initial eccentric location. The bubble moves toward the center of the drop when the shell is perturbed into a 'sloshing mode' of oscillation where both interfaces move out of phase. These results explain the centering of liquid shells observed in several experiments.
The paradigm of the inflationary universe provides a possible explanation for several observed cosmological properties. In order for such solutions to be successful, the universe must convert the energy stored in the inflaton potential into standard model particles through a process known as reheating. In this paper, we reconsider the reheating process for the case where the inflaton potential respects an approximate (but spontaneously broken) conformal symmetry during the reheating epoch. After reviewing the Effective Field Theory of Reheating, we present solutions for the nonlinear oscillations of the inflaton field, derive the corresponding Hill’s equation for the coupled reheating field, and determine the stability diagram for parametric resonance. For this class of models —the simplest realization being a scalar field with a quartic term—the expansion of the universe drives the coupled field toward a more unstable part of parameter space, in contrast to the standard case. We also generalize this class of models to include quadratic breaking terms in the potential during the reheating epoch and address the process of stability in that universality class of models.
A general method of generating optimal trajectories between an initial and a final position of an n degree of freedom manipulator arm with nonlinear equations of motion is proposed. The method is based on the assumption that the time history of each of the coordinates can be expanded in a series of simple time functions. By searching over the coefficients of the terms in the expansion, trajectories which minimize the value of a given cost function can be obtained. The method has been applied to a planar three degree of freedom arm.
It is shown that Witten's (1986) open-bosonic-string field-theory action and a closed-string analog can be written as a purely cubic interaction term. The conventional form of the action arises by expansion around particular solutions of the classical equations of motion. The explicit background dependence of the conventional action via the Becchi-Rouet-Stora-Tyutin operator is eliminated in the cubic formulation. A closed-form expression is found for the full nonlinear gauge-transformation law.
Here we propose an analytical Landau-Ginzburg (LG) theory of the charge density waves coupled with lattice and electronic long-range order parameters. Examples of long-range order include the electronic wave function of superconducting Cooper pairs, structural distortions, electric polarization, and magnetization. We formulate the LG free energy density as a power expansion with respect to the charge density and other long-range order parameters as well as their spatial gradients and biquadratic coupling terms. We introduced a biquadratic coupling between the charge density gradient and long-range order parameters as well as nonlinear higher gradients of the long-range order parameters. The biquadratic gradient coupling is critical to the appearance of different spatially modulated phases in charge-ordered ferroics and high-temperature superconductors. We derived the thermodynamic conditions for the stability of the spatially modulated phases, which are the intertwined spatial waves of charge density and lattice/electronic long-range order. The analytical expressions for the energies of different phases, corresponding order parameters, charge density waves amplitudes, and modulation periods obtained in this paper can be employed to guide the comprehensive physical explanation, deconvolution, and Bayesian analysis of experimental data on quantum materials ranging from charge-ordered ferroics to high-temperature superconductors.
The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.