Lyapunov inequalities for nabla Caputo boundary value problems

ABSTRACT

We will establish uniqueness of solutions to boundary value problems involving the nabla Caputo fractional difference under two-point boundary conditions and give an explicit expression for the Green's functions for these problems. Using the Green's functions for specific cases of these boundary value problems, we will then develop Lyapunov inequalities for certain nabla Caputo BVPs.     

Lipase-catalyzed green synthesis of starch–maleate monoesters and its characterization

The present study reports the green synthesis of starch–maleate (SM) at ambient temperature in solvent-free system using Rhizopus arrhizus lipase as a biocatalyst and maleic acid (MA) as an esterification agent. The synthetic scheme was found to be efficient, economical, and ecofriendly. The newly synthesized SM samples were characterized using Fourier transform infrared (FTIR) and proton nuclear magnetic resonance (¹H NMR) spectroscopic techniques. The degree of substitution (DS) was found in the range of 0.53–0.62. Moreover, DS was found to be temperature and time-dependent. X-ray diffraction (XRD) exhibited that maleation did not change the crystalline nature of native starch. Scanning electron microscopy (SEM) revealed that size of SM granules was in the range of 4–18 µm. The activation energy (E_a) of SM formation was calculated to be 42.94 kcal mol^?1 which clearly indicated the effective and rapid interaction of functional groups. Hence, the solvent-free solid-state synthetic methodology proved to be excellent for the synthesis of novel biomaterials with appreciable high DS for drug delivery and sorption of heavy metal ions from water.     

Forward and Inverse Viscoacoustic Modelling in a Tunnel Environment

In this work, the goal is to model forward acoustic waves in a tunnel environment with attenuation and to do full waveform inversion. In reality, there is no material without attenuation. Some materials, such as rocks, have so low attenuation that, in a small domain, the waves are almost not damped at all. At the same time, there are materials with high attenuation. In an environment with such materials, the attenuation has to be taken into account in order to model the waves properly. In this study, attenuation effect is integrated into acoustic equation by using Kolsky-Futterman model ( [1], [2]) which only replaces velocity field with a complex-valued field in frequency domain. Apart from attenuation, another objective is to consider an inhomogeneous density field. Mainly, acoustic equation with a constant density field is referred to in many studies. In many cases, it may suffice to model waves appropriately. However, in reality, the density field of ground can be highly inhomogeneous. The objective is to investigate the effect of the inhomogeneity in waves, and to search for density field ρ and attenuation parameter Q as well as pressure wave velocity c using full waveform inversion. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frequency Domain Waveform Inversion in a Tunnel Environment

The aim of the waveform inversion is to find an optimal model for the distribution of the elastic stiffness such that its response is consistent with the seismic data. In this study, a high-order finite element method is being used to solve the forward problem in the frequency domain and a gradient-based approach is followed to invert the model where an adjoint method eases the computational struggle by reducing the number of the required forward simulations significantly. Perfectly Matched Layers (PML) are implemented in order to absorb waves on the fictitious boundaries of the considered geometry. The model is inverted only over a limited number of frequencies. This reduces the number of the required forward simulations and the nonlinearity of the inverse problem. Every model is unique and has to be investigated before being inverted; appropriate boundary conditions, placement of the source and receiver locations, and an initial model close to the real model are crucial points to be considered separately and carefully. Applications to the reconnaissance problem in a tunnel environment are shown. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Innovative Green Chemistry Approach to Synthesis of Sn2+-Metal Complex and Design of Polymer Composites with Small Optical Band Gaps

In this work, the green method was used to synthesize Sn²⁺-metal complex by polyphenols (PPHs) of black tea (BT). The formation of Sn²⁺-PPHs metal complex was confirmed through UV-Vis and FTIR methods. The FTIR method shows that BT contains NH and OH functional groups, conjugated double bonds, and PPHs which are important to create the Sn²⁺-metal complexes. The synthesized Sn²⁺-PPHs metal complex was used successfully to decrease the optical energy band gap of PVA polymer. XRD method showed that the amorphous phase increased with increasing the metal complexes. The FTIR and XRD analysis show the complex formation between Sn²⁺-PPHs metal complex and PVA polymer. The enhancement in the optical properties of PVA was evidenced via UV-visible spectroscopy method. When Sn²⁺-PPHs metal complex was loaded to PVA, the refractive index and dielectric constant were improved. In addition, the absorption edge was also decreased to lower photon. The optical energy band gap decreases from 6.4 to 1.8 eV for PVAloaded with 30% (v/v) Sn²⁺-PPHs metal complex. The variations of dielectric constant versus wavelength of photon are examined to measure localized charge density (N/m*) and high frequency dielectric constant. By increasing Sn²⁺-PPHs metal complex, the N/m* are improved from 3.65 × 10⁵⁵ to 13.38 × 10⁵⁵ m⁻³ Kg⁻¹. The oscillator dispersion energy (E_d) and average oscillator energy (E_o) are measured. The electronic transition natures in composite films are determined based on the Tauc’s method, whereas close examinations of the dielectric loss parameter are also held to measure the energy band gap.     

Identification of the Velocity Field of 2D and 3D Tunnel Models with Frequency Domain Full Waveform Inversion

Full Waveform Inversion (FWI see [1]) has capability to identify the velocity field of a domain with a good precision. Its power comes from the fact that this approach does not only try to fit travel times of waves, but it tries to fit the whole seismogram. This work is about the application of acoustic full waveform inversion to 2D and 3D tunnel models. The necessary boundary conditions are applied to the models and the acoustic equation is solved by higher-order Finite Elements Method. The Conjugate Gradient (CG) method is utilized to minimize the misfit function. The results were verified with synthetic models. The synthetic tunnel models contain few bodies with different shapes and locations. Starting from a homogeneous velocity field, the synthetic model is sought over iterations. To avoid the ill-posedness, the locations and numbers of the source and receiver points have to be successfully chosen. Apart from this, the frequency set has also to be carefully constructed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recent Advances in the Applications of Nanotechnology and Nanomaterials in the Petroleum Industry: A Descriptive Review

Nanotechnology applications are rapidly expanding in various fields because of its unique qualities, such as a large surface area. Also, the synthetic changes can be utilized to alter nanomaterial to fit into specialized necessities. From the last decade there is a tremendous increase in the utilization of nanotechnology and nanomaterials in the petroleum industry. The current review's main objective is to summarize numerous nanoparticle applications in the field of petroleum, bio-fuel formation, and clean-up treatments of oil spill-related issues with their existing challenges that may help improve further research.