ON SINGULAR PERTURBATIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

SIMPLE SUMMARY: The use of flaxseed in complete pet foods as a source of omega-3 fatty acids, and in particular alpha-linolenic acid, has been studied extensively in dogs. However, to date, no scientific published literature exists on the impacts of feeding whole ground flaxseed as an ingredient in cat food on nutrient digestibility, fatty acid bioavailability, and overall health in cats. In addition, dietary flaxseed inclusion has been challenged as providing little nutritional value for cats by not changing serum fatty acid levels and lowering nutrient digestibility. Therefore, the objective of this study was to directly evaluate the effects of adding ground flaxseed to dry cat food on fatty acid bioavailability, stool consistency, nutrient digestibility, and overall health in adult cats. These results clearly exhibit that a flaxseed-supplemented diet resulted in no changes in stool quality or nutrient digestibility, and elicited an increase in serum alpha-linolenic acid concentrations in healthy cats. Overall, these results clearly demonstrate that ground flaxseed can be used to modulate serum fatty acid concentrations and serve as a bioavailable source of alpha-linolenic acid in commercial cat food. ABSTRACT: We evaluated effects of dietary ground flaxseed on fecal and serum alpha-linolenic acid (ALA) concentrations, nutrient digestibility, and stool quality in female and male adult cats (n = 20 (8 males, 12 females); 3.95 ± 1.49 years of age (mean ± SD); 3.88 ± 0.82 kg BW). We hypothesized that adding ground flaxseed would increase serum ALA compared with feeding no flax, without changing nutrient digestibility. Cats were fed as-is 2.6% added-flaxseed (flax, n = 10) or no-flax (control, n = 10) diets (2.66 vs. 0.78% ALA of total fatty acids; crude protein 35%, fat 20%, fiber 3% as-fed) twice daily to maintain body weight for 28 days. Fecal collections were conducted on days 23–27 for total-tract nutrient digestibility, stool quality (scale 1–5; 1 = watery diarrhea, 5 = hard, dry, crumbly) and long-chain fatty acid (LCFA) analyses. Blood was collected on days 0, 14, and 28 for serum LCFA and chemistry analysis. Digestibility and fecal data were analyzed by ANOVA (SAS v9.4, Cary, NC, USA) and a repeated measures ANOVA for serum ALA. Flax-fed cats, compared with control-fed, had greater (p < 0.05) serum ALA after 14 days (4.00 vs. 0.71 µg/mL) and 28 days (7.83 and 3.67 µg/mL). No differences were observed in stool quality, and dry matter, protein, fat, and ALA digestibility. However, metabolizable energy was greater in the flax vs. control diet (4.18 vs. 3.91 kcal/g; p < 0.05). Overall, these data demonstrate that ground flaxseed added to cat diets increases serum ALA within 14 days, with no detriments to nutrient digestibility. We conclude that flaxseed can be used as a bioavailable source of ALA in cat diets.

The objective of this investigation is to examine the buoyancy-driven gravitational spreading currents, especially as applied to ocean disposal of wastewater and the accidental release of hazardous fluids, such as liquefied natural gas. A series of asymptotic solutions are used to describe the displacement of a gravitationally driven spreading front during an inertial phase of motion and the subsequent viscous phase. Solutions are derived by a force scale analysis and a self-similar technique for flows in stagnant, homogeneous, or linearly density-stratified environments. The self-similar solutions for inertial-buoyancy currents are found using an analogy to the well-known shallow-water wave propagation equations and also to those applicable to a blast wave in gasdynamics. For the viscous-buoyancy currents the analogy is to the viscous long wave approximation to a nonlinear diffusive wave, or thermal wave propagation. Other similarity solutions describing the initial stage of motion of the flow formed by the collapse of a finite volume fluid are developed by analogy to the expansion of a gas cloud into a vacuum. For the case of a continuous discharge there is initially a starting jet flow followed by the buoyancy-driven spreading flow. The jet mixing zone in such flows is described using Prandtl's mixing length theory. Dimensional analysis is used to derive the relevant scaling factors describing these flows.

In this paper the effect of axial load on dynamic behavior of a simple frame, subjected to harmonic, seismic and earthquake excitation is investigated. The equations of motion are considered for two types of small and large deformations. The method of multiple scales is applied to solve the differential equations of motion with harmonic loading and for small and large deformations. Then, the steady state response near one-to-one resonance condition is studied. The results show that the dynamic behavior of the frame under axial load is completely different in resonance and non-resonance cases. The equations of motion with earthquake loading is also considered and the effect of axial load in the frame behavior under the time history and the response spectrum of the model is studied. Although white noise as a stochastic loading is applied to the model and, the results are approximated using the method of stochastic differential equations so, the mean value and covariance are calculated and the effect of axial force on them is investigated.

Geophysical granular flows, such as snow avalanches and debris flows, represent a great hazard to man and infrastructures. These natural phenomena are characterized by the rapid flow of a granular solid phase, embedded in an ambient fluid. Since such events occur with little warning, in addition to investigating the triggering conditions, it is of crucial interest to predict their propagation and run-out distances. The main purpose of the present research is to better understand the flow dynamics of dry granular flows, in the presence of no-slip bottom boundary conditions, and predict their propagation by using a depth-averaged approach. In this case, a stratification of different flow regimes, such as quasi-static and dense collisional, occurs and makes the utilization of the classical single layer depth-averaged models insufficient, because of important uncertainties about the velocity and shear stress distributions along the flow depth. In the first part of the present work, an experimental-numerical study on dam-break flows of dry granular material is reported. It has been found that a modification to the formula, proposed in the Savage-Hutter model for calculating the earth-pressure coefficient, leads to an improved agreement with experimental data, in presence of no-slip bottom conditions. In the second part, an experimental study on steady state velocity profiles of dry granular flows is reported. The granular material, used in this experimental research, was Ottawa sand (ASTM C-778 20/30). The velocity profiles at the side walls and free surfaces have been obtained, through granular PIV techniques. The measurements are in accordance with other experimental works on different granular materials and suggest the occurrence of a rheological stratification along the flow depth, in case of no-slip bottom condition. In order to better describe such a complex flow dynamics, a two-layer depth-averaged model has been proposed. The dynamics of the two layers, ideally corresponding to dense-collisional and quasi-static regimes, have been considered independently. As well, mass exchanges between the layers have calculated through a physically based closure equation. The well-known mathematical issue of the hyperbolicity loss in two-layer models has been carefully addressed. In order to overcome it, a local modification of source terms of the original mathematical model has been proposed. Such a treatment consists of introducing an extra resistance at the interface, if necessary, to avoid the hyperbolicity loss. This approach has been found to be robust and yields reasonable results, as long as the asymptotic steady state solution and boundary conditions fulfil the hyperbolicity requirements of the original model. Through comparisons with experimental data, the two-layer approach turns out to be very promising to describe the complex flow dynamics in case of no-slip bottom boundary conditions, although a more detailed description of the basal shear stress seems to be required. In the third part of the present dissertation, the same two-layer approach has been used in order to rewrite the mathematical model in a curvilinear coordinates system, attached to the interface between the two layers. The model equations have been obtained, by using the same approach proposed in Tai and Kuo (2008). In the model derivation, the physically negligible quantities are identified in a more rational way than the previous model, by means of a scaling approximation.

In this paper, we study the existence of coupled solutions of anti-periodic boundary value problems for impulsive differential equations with ϕ-Laplacian operator. Based on a pair of coupled lower and upper solutions and appropriate Nagumo condition, we prove the existence of coupled solutions for anti-periodic impulsive differential equations boundary value problems with ϕ-Laplacian operator.

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

The low-thrust guidance problem is defined as the minimum terminal variance (MTV) control of a space vehicle subjected to random perturbations of its trajectory. To accomplish this control task, only bounded thrust level and thrust angle deviations are allowed, and these must be calculated based solely on the information gained from noisy, partial observations of the state. In order to establish the validity of various approximations, the problem is first investigated under the idealized conditions of perfect state information and negligible dynamic errors. To check each approximate model, an algorithm is developed to facilitate the computation of the open loop trajectories for the nonlinear bang-bang system. Using the results of this phase in conjunction with the Ornstein-Uhlenbeck process as a model for the random inputs to the system, the MTV guidance problem is reformulated as a stochastic, bang-bang, optimal control problem. Since a complete analytic solution seems to be unattainable, asymptotic solutions are developed by numerical methods. However, it is shown analytically that a Kalman filter in cascade with an appropriate nonlinear MTV controller is an optimal configuration. The resulting system is simulated using the Monte Carlo technique and is compared to other guidance schemes of current interest.

An existence theorem for a class of nonlinear singular boundary value problems

This paper is concerned with the numerical solution of the singularly perturbed differential-difference equations with small shifts called delay and advanced parameters. A fourth order finite difference method with a fitting factor is proposed for the solution of the singularly perturbed differential-difference equations with mixed shifts. The delay and advanced shifts are managed by Taylor series and an asymptotically equivalent singularly perturbed two-point boundary value problem is obtained. A fitting factor is introduced in the fourth order finite difference scheme for the problem which takes care of the small values of the perturbation parameter. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the discrete system of the difference scheme. Convergence of the proposed method is analyzed. Maximum absolute errors in comparison with the several numerical experiments aretabulated to illustrate the proposed method.

In the present work, a wedge structure made of absorbing panels has been analyzed by making use of the matrizant analysis with the help of the Boundary-Condition-Transfer (BCT) algorithm. The rectangular panel wedge, as it is called in this manuscript, is simple in geometry. The theoretical model, based on the plane wave acoustical coupling between multiple interacting ducts of variable cross sectional area, is applied to predict the pressure reflection coefficient of the present wedge configuration. Bulk reaction and hence wave propagation in the wedge material has been assumed in the proposed model. An asymptotic solution using the Peano-Baker series of matrix calculus is derived for different variable area ducts and then solved with the aid of the boundary condition transfer algorithm. A brief parametric study is also included as an aid to designers.

Bifurcation analysis on a turning system with large and state-dependent time delay

The existence and uniqueness of the solution to the boundary value problem for a third-order mixed-type equation with variable coefficients in the lower terms is proven, with conditions for the gluing of the function itself and its first- and second-order derivatives on the line y=0, where the type of equation changes when a second-order mixed parabolic-hyperbolic operator is applied to a first-order linear differential operator with constant coefficients. By reducing the order of the equation, the problem was reduced to the Tricomi problem for a second-order mixed parabolic-hyperbolic equation with continuous conditions for the function itself and its first-order derivative with respect to y on the line of change of the equation type. By the method of elimination of the system of equations obtained from the parabolic and hyperbolic parts of the domains, the solvability of the problem was reduced to the solvability of the Fredholm integral equation of the second kind. A sufficient condition for the solvability of the integral equation was obtained through estimates of the kernel of the equation. The solution of the problem was split into two problems in the regions under consideration: in the parabolic part of the region, the first boundary value problem for the heat conduction equation was solved using the Green's function method, and in the hyperbolic part of the region, bounded by the characteristics of the equation and the line y=0, the solution of the problem using the Riemann function construction method was determined as the solution of the Cauchy problem. By applying a curvilinear integral, the solution to the problem in the areas under consideration was found. The necessity of the requirement of continuity of the function itself and its first two derivatives with respect to y on the line of change of the equation type was justified. Sufficient conditions for the unique classical solvability of the boundary value problem were established. The obtained conditions for the solvability of the boundary value problem provided a theoretical basis for the development of numerical methods for solving applied problems in aerohydrodynamics, geophysics, and engineering thermodynamics

We set forth solvability conditions and construction of the generalized Green’s operator for Noetherian boundary value problem for the matrix differential-algebraic equations and solvability conditions and the constructive scheme for constructing solutions of nonlinearNoetherian boundary value problem for matrix differential-algebraic equation. We show that the principal results in the theory of weakly nonlinear periodic oscillations remain valid for nonlinear Noetherian boundary value problem for matrix differential-algebraic equation. The study is illustrated by the periodic problem for a Duffing-type matrix differential equation.

In this dissertation, we investigate some boundary value problems for complex partial differential equations in fan-shaped domains. First of all, we establish the Schwarz-Poisson representation in fan-shaped domains with angle π/n (n ∈ N) by the reflection method, and study the corresponding Schwarz and Dirichlet problems respectively. Further, the Schwarz-Poisson formula is extended to the general fan-shaped domains with angle π/α (α ≥ 1/2) by proper conformal mappings, and then the Schwarz and Dirichlet problems for the Cauchy-Riemann equation are solved. Next, we also establish a bridge between the unit disc and the fan-shaped domain with α = 1/2, and the Schwarz-Poisson formula for the unit disc is derived from the Schwarz-Poisson formula for α = 1/2. Then, we firstly obtain a harmonic Green function and a harmonic Neumann function in the fan-shaped domain with angle π/α (α ≥ 1/2), and then investigate the Dirichlet and Neumann problems for the Poisson equation. In particular, the outward normal derivative at the three corner points is properly defined. Next, a biharmonic Green function, a biharmonic Neumann function, a triharmonic Green function, a triharmonic Neumann function and a tetra-harmonic Green function are constructed for the fan-shaped domain with angle π/n (n ∈ N) in explicit form respectively. Moreover, we give the process of constructing a tetra-harmonic Neumann function and the expression of the tetra-harmonic Neumann function with integral representation. Accordingly, the Dirichlet and Neumann problems are discussed. Finally, we establish the iterated expressions and the solvability conditions of polyharmonic Dirichlet and Neumann problems for the higher order Poisson equation in the fan-shaped domain with angle π/n (n ∈ N) respectively. In the meantime, the boundary behavior of polyharmonic Green and polyharmonic Neumann functions by convolution are discussed in detail. Besides, in the Appendix, the tetra- harmonic Green function and the triharmonic Neumann function for the unit disc are constructed in explicit form.