Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

Solubility and Diffusivity of Water in Lunar Basalt, and Chemical Zonation in Olivine-Hosted Melt Inclusions

I report the solubility and diffusivity of water in lunar basalt and an iron-free basaltic analogue at 1 atm and 1350 °C. Such parameters are critical for understanding the degassing histories of lunar pyroclastic glasses. Solubility experiments have been conducted over a range of fO₂ conditions from three log units below to five log units above the iron-wüstite buffer (IW) and over a range of pH₂/pH₂O from 0.03 to 24. Quenched experimental glasses were analyzed by Fourier transform infrared spectroscopy (FTIR) and secondary ionization mass spectrometry (SIMS) and were found to contain up to ~420 ppm water. Results demonstrate that, under the conditions of our experiments: (1) hydroxyl is the only H-bearing species detected by FTIR; (2) the solubility of water is proportional to the square root of pH₂O in the furnace atmosphere and is independent of fO₂ and pH₂/pH₂O; (3) the solubility of water is very similar in both melt compositions; (4) the concentration of H₂ in our iron-free experiments is <3 ppm, even at oxygen fugacities as low as IW-2.3 and pH₂/pH₂O as high as 24; and (5) SIMS analyses of water in iron-rich glasses equilibrated under variable fO₂ conditions can be strongly influenced by matrix effects, even when the concentrations of water in the glasses are low. Our results can be used to constrain the entrapment pressure of the lunar melt inclusions of Hauri et al. (2011).

Diffusion experiments were conducted over a range of fO₂ conditions from IW-2.2 to IW+6.7 and over a range of pH₂/pH₂O from nominally zero to ~10. The water concentrations measured in our quenched experimental glasses by SIMS and FTIR vary from a few ppm to ~430 ppm. Water concentration gradients are well described by models in which the diffusivity of water (D^*_water) is assumed to be constant. The relationship between D^*_water and water concentration is well described by a modified speciation model (Ni et al. 2012) in which both molecular water and hydroxyl are allowed to diffuse. The success of this modified speciation model for describing our results suggests that we have resolved the diffusivity of hydroxyl in basaltic melt for the first time. Best-fit values of D^*_water for our experiments on lunar basalt vary within a factor of ~2 over a range of pH₂/pH₂O from 0.007 to 9.7, a range of fO₂ from IW-2.2 to IW+4.9, and a water concentration range from ~80 ppm to ~280 ppm. The relative insensitivity of our best-fit values of D^*_water to variations in pH₂ suggests that H₂ diffusion was not significant during degassing of the lunar glasses of Saal et al. (2008). D^*_water during dehydration and hydration in H₂/CO₂ gas mixtures are approximately the same, which supports an equilibrium boundary condition for these experiments. However, dehydration experiments into CO₂ and CO/CO₂ gas mixtures leave some scope for the importance of kinetics during dehydration into H-free environments. The value of D^*_water chosen by Saal et al. (2008) for modeling the diffusive degassing of the lunar volcanic glasses is within a factor of three of our measured value in our lunar basaltic melt at 1350 °C.

In Chapter 4 of this thesis, I document significant zonation in major, minor, trace, and volatile elements in naturally glassy olivine-hosted melt inclusions from the Siqueiros Fracture Zone and the Galapagos Islands. Components with a higher concentration in the host olivine than in the melt (MgO, FeO, Cr₂O₃, and MnO) are depleted at the edges of the zoned melt inclusions relative to their centers, whereas except for CaO, H₂O, and F, components with a lower concentration in the host olivine than in the melt (Al₂O₃, SiO₂, Na₂O, K₂O, TiO₂, S, and Cl) are enriched near the melt inclusion edges. This zonation is due to formation of an olivine-depleted boundary layer in the adjacent melt in response to cooling and crystallization of olivine on the walls of the melt inclusions concurrent with diffusive propagation of the boundary layer toward the inclusion center.

Concentration profiles of some components in the melt inclusions exhibit multicomponent diffusion effects such as uphill diffusion (CaO, FeO) or slowing of the diffusion of typically rapidly diffusing components (Na₂O, K₂O) by coupling to slow diffusing components such as SiO₂ and Al₂O₃. Concentrations of H2O and F decrease towards the edges of some of the Siqueiros melt inclusions, suggesting either that these components have been lost from the inclusions into the host olivine late in their cooling histories and/or that these components are exhibiting multicomponent diffusion effects.

A model has been developed of the time-dependent evolution of MgO concentration profiles in melt inclusions due to simultaneous depletion of MgO at the inclusion walls due to olivine growth and diffusion of MgO in the melt inclusions in response to this depletion. Observed concentration profiles were fit to this model to constrain their thermal histories. Cooling rates determined by a single-stage linear cooling model are 150–13,000 °C hr^-1 from the liquidus down to ~1000 °C, consistent with previously determined cooling rates for basaltic glasses; compositional trends with melt inclusion size observed in the Siqueiros melt inclusions are described well by this simple single-stage linear cooling model. Despite the overall success of the modeling of MgO concentration profiles using a single-stage cooling history, MgO concentration profiles in some melt inclusions are better fit by a two-stage cooling history with a slower-cooling first stage followed by a faster-cooling second stage; the inferred total duration of cooling from the liquidus down to ~1000 °C is 40 s to just over one hour.

Based on our observations and models, compositions of zoned melt inclusions (even if measured at the centers of the inclusions) will typically have been diffusively fractionated relative to the initially trapped melt; for such inclusions, the initial composition cannot be simply reconstructed based on olivine-addition calculations, so caution should be exercised in application of such reconstructions to correct for post-entrapment crystallization of olivine on inclusion walls. Off-center analyses of a melt inclusion can also give results significantly fractionated relative to simple olivine crystallization.

All melt inclusions from the Siqueiros and Galapagos sample suites exhibit zoning profiles, and this feature may be nearly universal in glassy, olivine-hosted inclusions. If so, zoning profiles in melt inclusions could be widely useful to constrain late-stage syneruptive processes and as natural diffusion experiments.

Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.

On Nonnegative Solutions of Fractional q-Linear Time-Varying Dynamic Systems with Delayed Dynamics

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based on q-calculus and Caputo fractional derivatives on any order.

Simulação do processo de transferência de calor não linear condução-radiação por meio de um esquema linear em diferenças finitas.

Neste trabalho o processo não linear de transmissão de calor condução-radiação é abordado num contexto bidimensional plano e simulado com o uso de um esquema linear em diferenças finitas. O problema original é tratado como o limite de uma sequencia de problemas lineares, do tipo condução-convecção. Este limite, cuja existência é comprovada, é facilmente obtido a partir de procedimentos básicos, accessíveis a qualquer estudante de engenharia, permitindo assim o emprego de hipóteses mais realistas, já que não se tem o limitante matemático para a abordagem numérica de uma equação diferencial parcial elíptica. Neste trabalho foi resolvido o problema de condução de calor em regime permanente em uma placa com condições de contorno convectivas e radioativas utilizando-se o software MatLab, vale ressaltar, que a mesma metodologia é aplicável para geometrias mais complexas.

The post-processed Galerkin method applied to non-linear shell vibrations

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary differential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the post-processing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-difference technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

Numerical and theoretical analysis of the crosstalk in linear optical amplifiers

The dynamic characteristics, including the crosstalk and relaxation oscillation, of linear optical amplifiers (LOAs) are investigated by small-signal analysis under an averaging carrier density approximation and compared with the results of numerical simulation. The good agreement between the numerical simulation and the small-signal analysis indicated the averaging carrier density is an appropriate approximation for analyzing LOAs. Theoretical analyzes also show that the dynamic properties of the vertical laser fields dominate the dynamic performance of LOAs. Based on the small-signal analysis, a concise equation for the crosstalk under high bit rate was derived, which can be applied to measure the differential gain of LOAs.

Effects of Ionizing Radiation on Three-Dimensional Human Vessel Models: Differential Effects According to Radiation Quality and Cellular Development

Little is known about the effects of space radiation on the human body. There are a number of potential chronic and acute effects, and one major target for noncarcinogenic effects is the human vasculature. Cellular stress, inflammatory response, and other radiation effects on endothelial cells may affect vascular function. This study was aimed at understanding the effects of space ionizing radiation on the formation and maintenance of capillary-like blood vessels. We used a 3D human vessel model created with human endothelial cells in a gel matrix to assess the effects of low-LET protons and high-LET iron ions. Iron ions were more damaging and caused significant reduction in the length of intact vessels in both developing and mature vessels at a dose of 80 cGy. Protons had no effect on mature vessels up to a dose of 3.2 Gy but did inhibit vessel formation at 80 cGy. Comparison with gamma radiation showed that photons had even less effect, although, as with protons, developing vessels were more sensitive. Apoptosis assays showed that inhibition of vessel development or deterioration of mature vessels was not due to cell death by apoptosis even in the case of iron ions. These are the first data to show the effects of radiation with varying linear energy transfer on a human vessel model. (C) 2011 In Radiation Research Society

Preparation and properties of a reactive type nonionic surfactant grafted linear low density polyethylene

A reactive type nonionic surfactant, monostearic acid monomaleic acid glycerol diester (MMGD) was synthesized in our laboratory. Grafting-copolymerization of linear low density polyethylene ( LLDPE) with MMGD was carried out by using beta ray irradiation in air in a twin-screw extruder. Evidence of the grafting of MMGD as well as its extent was determined by Fourier-transformed infrared (FT-IR) spectroscopy. The effects of monomer concentration, reaction temperature and screw run speed on degree of grafting were studied systematically. The thermal behavior of LLDPE-g-MMGD was investigated by using differential scanning calorimety ( DSC). Compared with neat LLDPE, the crystallization temperature ( Tc) of LLDPE-g-MMGD increased about 3 degrees C, and the melting enthalpy (Delta H-m) decreased with increase of MMGD content. It showed that the grafted MMGD monomer onto LLDPE acted as a nucleating agent. The tensile properties and light transmission of blown films were determined. Comparing with neat LLDPE film, no obvious changes could be found for the tensile strength, elongation at break and right angle tearing strength of LLDPE-g-MMGD film. The wettability is expressed by the water contact angle. With an increasing percentage of MMGD, the contact angles of water on film surface of LLDPE- g-MMGD decrease monotonically.

Preparation and properties of a novel nonionic surfactant grafted linear low density polyethylene

A novel nonionic surfactant, glycerol monostearic acid monomaleic acid diester (GMMD) was synthesized in our laboratory. Grafting-copolymerization of linear low density polyethylene (LLDPE) with GMMD was carried out by using P-ray irradiation in a twin-screw extruder. Evidence of the grafting of GMMD, as well as its extent, was determined by FT-IR. The effects of monomer concentration, reaction temperature and screw run speed on degree of grafting were studied systematically. The thermal behavior of LLDPE-g-GMMD was investigated by using differential scanning calorimety (DSC). Compared with neat LLDPE, the crystallization temperature (T,) of LLDPE-g-GMMD increased about 3 degrees C, and the melting enthalpy (Delta H-m) decreased with increase of GMMD content. It showed that the arafted GMMD monomer onto LLDPE acted as a nucleating agent. The tensile properties and light transmission of blown films were determined. Comparing with neat LLDPE film, no obvious changes could be found for the tensile strength, elongation at break and right angle tearing strength of LLDPE-g-GMMD film. Accelerated dripping property of film samples was investigated. The dripping duration of LLDPE-g-GMMD film and commercial anti-fog dripping film at 60 degrees C were 52 days and 17 days, respectively.

Thermal properties and morphology of noncrosslinking linear low-density polyethylene-grafted acrylic acid

Noncrosslinking linear low-density polyethylene-grafted acrylic acid (LLDPE-g-AA) was prepared by melt-reactive extrusion in our laboratory. The thermal behavior of LLDPE-g-AA was investigated by using differential scanning calorimetry (DSC). Compared with neat linear low-density polyethylene (LLDPE), melting temperature (T-m) of LLDPE-g-AA increased a little, the crystallization temperature (T-c) increased about 4degreesC, and the melting enthalpy (DeltaH(m) ) decreased with an increase in acrylic acid content. Isothermal crystallization kinetics of LLDPE and LLDPE-g-AA samples were carried out by using DSC. The overall crystallization rate of LLDPE was smaller than that of grafted samples. It showed that the grafted acrylic acid monomer onto LLDPE acted as a nucleating agent. Crystal morphologies of LLDPE-g-AA and LLDPE were examined by using SEM. Spherulite sizes of LLDPE-g-AA samples were lower than that of LLDPE.

Blends of linear low-density polyethylene and a diblock copolymer of hydrogenated polybutadiene and methyl methacrylate

Blends of linear low-density polyethylene (LLDPE) and a diblock copolymer of hydrogenated polybutadiene and methyl methacrylate [P(HB-b-MMA)] were studied by transimission electron microscope (TEM), differential scanning calorimetry (DSC), and wide angle X-ray diffraction (WAXD). At 10 wt% block copolymer content, block copolymer chains exist as spherical micelles and cylindrical micelles in LLDPE matrix. At 50 wt% block copolymer content, block copolymer chains mainly form cylindrical micelles. The core and corona of micelles consist of PMMA and PHB blocks, respectively. DSC results show that the total enthalpy of crystallization of the blends varies linearly with LLDPE weight percent, indicating no interactions in the crystalline phase. In the blends, no distortion of the unit cell is observed in WAXD tests.

Crystallization behavior of blends of high-density polyethylene with novel linear low-density polyethylene

Blends of high-density polyethylene (HDPE) with novel linear low-density polyethylene (LLDPE) samples in the whole range of compositions were investigated by means of differential scanning calorimetry (DSC), small-angle X-ray scattering (SAXS) and wide-angle X-ray diffraction (WAXD). The LLDPEs are ethylene/octene-1 copolymers prepared with a single-site catalyst, with a narrower distribution of branches compared to Ziegler-Natta type polymers. It was found that cocrystallization or separate crystallization in the blends profoundly depends on the content of branches in the LLDPE, while the critical branch content of the novel LLDPE for separate crystallization is much lower than that of commercial LLDPE (prepared with Ziegler-Natta catalysts). This implies that the miscibility of linear and branched polyethylene is also affected by the distribution of branches. The marked expansion of the unit cell in cocrystals, which are formed by HDPE with the novel LLDPE, indicates that the branches are included in the crystal lattice during the cocrystallization process. The result is very helpful to understand the phenomenon that the unit cell dimensions of commercial branched polyethylene are larger than those of linear polyethylene.

Miscibility of linear low density polyethylene (LLDPE)-graft-poly(methyl methacrylate) with poly(vinylidene fluoride) (PVF2) and its application as compatibilizer LLDPE/PVF2 blends

Differential scanning calorimetry (DSC), Fourier-transform infrared spectroscopy (FTIR), scanning electron microscopy (SEM), and X-ray photoelectron spectroscopy (XPS) were used to study the miscibility of blends of a graft copolymer of poly(methyl methacrylate) on linear low density polyethylene (LLDPE-g-PMMA, G-3) with poly(vinylidene fluoride)(b) (PVF2) and the compatibilization of blends of LLDPE/PVF2. The specific interaction between PMMA side chains and PVF2 in G-3/PVF2 binary blends is weaker than that between the homopolymers PMMA and PVF2. There are two states of PVF2 in the melt of a G-3/PVF2 (60/40, w/w) blend, one as pure PVF2 and the other interacting with PMMA side chains. The miscibility between PMMA side chains and PVF2 affects the crystallization of PVF2. LLDPE-g-PMMA was demonstrated to be a good compatibilizer in LLDPE/PVF2 blends, improving the interfacial adhesion and dispersion in the latter. Diffusion of PMMA side chains into PVF2 in the interfacial region reduces the crystallization rate and lowers the melting point (T-m) and the crystallization temperature (T-c) of PVF2 in the blends.