Difference Method For Solution Research Articles

The plant growth, water and electricity consumption, and nutrients uptake are influenced by different light spectra and nutrition of lettuce

The aim of this study was to investigate the effect of different replacement methods of nutrient solution (complete replacement, electrical conductivity (EC)- based replacement, and replacing based on the plant needs) and different LED light spectra (monochromic white, red, blue, and a combination of red/blue) on the uptake of mineral nutrients, water and electricity consumption and biomass production of two varieties of lettuce (Lollo Rossa and Lollo Bionda; Lactuca sativa var. crispa) in the hydroponic systems. The results showed that replacement methods based on the plant needs and based on EC increased shoot fresh mass and yield index in the NFT system. Also, results showed that the combination of red/blue light increased shoot fresh mass and yield index in the NFT system and in the plant factory under treatment by replacement method based on plant needs. Increasing the concentrations of N, K, and Zn and loss of Fe in nutrient solution were observed in all three replacement methods of nutrient solution in the NFT system. Water consumption was decreased under plant nutrition based on plant needs and based on EC. In the plant factory, the application of LED light spectrum also decreased electricity consumption and cost against fluorescent lamps. In general, it is concluded that nutrient solution replacement based on the plant needs and based on EC and the use of different LED light spectra (especially the combination of red and blue light) can be used to reduce the consumption of water and nutrients in the hydroponic cultivation of lettuce.

Analytical solution of the fractional neutron point kinetic equations using the Mittag-Leffler function

A new analytical solution of the fractional point kinetic equations is developed in the present work, which considers one group of delayed neutrons as well as a constant reactivity. One of the main novelties and theoretical contributions of the developed solution is that it does not require approximating the Inverse Laplace transform by numerical means, as other solutions do, and instead it is obtained analytically using the Green and the Mittag-Leffler functions. A set of algorithms and MATLAB codes are developed with the purpose to implement and compute the developed solution in an adequate way, which represents the major computational contribution. Numerical experiments show that the developed solution is very accurate solving short-time dynamics phenomena, being able to reproduce, in a precise way, data that is reported in literature. The development of the proposed analytic solution represents an important contribution to validate numerical methods that are currently used to solve the neutron fractional point kinetic equations. Program SummaryProgram Title: Mittag-Leffler FNPKECPC Library link to program files:https://doi.org/10.17632/v4kfhzw7sf.1Developer's repository link:https://github.com/Cruz-Lopez-Carlos-Antonio/Mittag-Leffler-FNPKE/tree/mainLicensing provisions: Creative Commons by 4.0Programming language: MATLAB (2021a) / Python 3Nature of problem:Fractional Neutron Point Kinetic Equations describe the neutron population in a nuclear reactor, considering the changes on transport and thermohydraulic properties of the system. For such task, a mass balance approach based on fractional and integral derivatives is used, which improves and generalizes the integer case that uses the classical definitions of these mathematical operators. From a physical point of view, this new approach takes into account memory and non-local effects that are omitted in the standard neutron diffusion theory and which are relevant in scenarios where the current neutron density undergoes strongly variations on time. From a mathematical point of view, using fractional derivatives leads to a new and more general type of differential equations of arbitrary order that requires different methods of solution and approximation, and which must include the integer approach as particular case. In such context, the development of analytical solutions is a fundamental problem because it provides a framework to carry out verifications of numerical methods, as well as an insight to perform theoretical comparisons with the classical model. The developed Mittag-Leffler FNPK codes analytically solve the Fractional Neutron Point Kinetic Equations defined in terms of the Caputo's derivatives and subject to classical initial conditions.Solution method:The developed codes are written in MATLAB and Python programming languages, and solve the mentioned fractional differential equation system using a power series expansion as well as the Laplace transform. Such solution considers a single group of precursors of delayed neutrons and it is expressed in terms of the 2-parameter Mittag-Leffler function, being accurately computed considering a small number of terms of the infinite sum. Unlike other approaches, in the present case it is not necessary to use a numerical procedure to find the inverse Laplace transform, because it is analytically found using the Green function. Even when the developed solution is found for a constant reactivity case, the Mittag-Leffler FNPKE codes can extended it to more general scenarios where the reactivity is a function of time or where it depends on the neutron density itself. In these cases, it is necessary to apply a subinterval methodology where the time is divided in small steps, assuming a constant reactivity in each of them. The smaller of the intervals, the more accurate the approximation. With the purpose to overcome the numerical issues related to the computation of the 2-parameter Mittag-Leffler function, the developed codes integrate the most advanced algorithm reported in literature, which is useful also to estimate the high-order derivatives of such function.Additional comments including restrictions and unusual features:The developed codes do not face issues related to arithmetical precision digits, as other solutions do, because it is not expressed in terms of high order polynomials and instead it is written as a linear combination of the Green function. Additionally, several steps can be precomputed without regard the neutron parameters and initial conditions, which represents an important advantage in terms of the computational time.

Manipulation of light spectrum is an effective tool to regulate biochemical traits and gene expression in lettuce under different replacement methods of nutrient solution

The use of light-emitting diode (LED) technology represents a promising approach to improve plant growth and metabolic activities. The aim of this study was to investigate the effect of different light spectra: red (656 nm), blue (450 nm), red/blue (3:1), and white (peak at 449 nm) on biochemical properties, photosynthesis and gene expression in two lettuce cultivars (Lollo Rossa and Lollo Bionda) grown under different methods of nutrient solution replacement in hydroponics. Complete replacement and EC-based replacement of nutrient solution increased content of proline and soluble sugars and activity of antioxidant enzymes (CAT, GPX and SOD) under the red/blue LED and red LED light treatments in both cultivars. In addition, the red/blue and the monochromatic red light increased the soluble protein content and the antioxidant activity in the Lollo Rosa cultivar under the replacement method according to the needs of the plant. An increase in flavonoid content in the EC-based method in the Lollo Rosa variety treated with a combination of red and blue light was also observed. The red/blue light had the greatest induction effect on anthocyanin content, expression of the UFGT, CHS, and Rubisco small subunit genes, and the net photosynthetic rate. Data presented here will directly contribute to the development of nutrient solution and LED spectrum management strategies to significantly improve plant growth and metabolism, while avoiding water and nutrient waste, and environmental pollution.

Sand production by hydraulic erosion during multicycle steam stimulation: An analytical study

The production of sand particles associated with the reservoir hydrocarbons becomes one of the most common problems a well of multi cycle steam stimulation may experience during reservoir lifetime. Therefore, it is essential to the investigation of sand particles migration and sand production mechanisms coupled with fluid flow for optimization design of injection and profile control parameters. This paper presents a model based on hydraulic erosion with heat transfer that can analysis release of sand fines under high temperature. The model describes the continuous hydraulic erosion of rock core by the mass conservation equation and particle transport equation, moreover, the sand production and heat transfer are connected through it is coupled with steam injection model. Then, a coupled nonlinear governing equation is derived by using porosity and liquefied sand concentration to evaluate pore enlargement behavior of flow channel and predict sand production behavior under different steam breakthrough channel. Finally, in terms of numerical processing for model, an effectively difference solution method is adopted based on PDE module in COMSOL Multiphysics. The simulation results show that the sand production model can effectively couple heat and sand production, and the relationship between sand production and channeling flow can be studied through seepage process. The degree of steam channeling affects sand production, that is, sand production increases with the increase of the number of steam breakout channels. In addition, it reveals why the sand production can be effectively slowed down by adjusting reasonable injection parameters for multicycle steam stimulation.

Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model

We study the evolution of a single-particle wave function during the collision of a one dimensional potential well by another well, which can be regarded as a simple model for the problem of the scattering of a one-neutron halo nucleus by another nucleus. This constitutes an effective three-body problem, whose solution in three dimensions can be extremely complicated, particularly when breakup and rearrangement channels are to be considered. Our one-dimensional model provides the essential three-body nature of this problem, and allows for a much simpler application and assessment of different methods of solution. To simplify further the problem, we assume that the potential well representing the projectile moves according to a predetermined classical trajectory, although the internal motion of the "valence" particle is treated fully quantum-mechanically. This corresponds to a semiclassical approach of the scattering problem. Different approaches are investigated to understand the dynamics involving one-body halo-like systems: the "exact" time-dependent solution of the Schr\"odinger equation is compared to a numerical continuum-discretized coupled-channels (CC) calculation presenting various model cases including different reaction channels. This framework allows us to discuss the reaction mechanism and the role of continuum, whose inclusion in the CC calculation results to be crucial to reproduce the "exact" solution, even when the initial and final states are well bound. We also link each dynamical situation with analogous problem solved in a three dimensional (3D) CC framework, discussing the main challenges experienced in the usual 3D models.

Applicability of vapor pressure models on the prediction of evaporation and motion of sulfuric and hydrochloric droplets in free-falling process

In acid pickling and electroplating processes, a large number of acid droplets, such as sulfuric droplets and hydrochloric droplets, are emitted from the open tank surface. These two types of droplets are typical indoor pollutants in industrial buildings. They are harmful to personnel, devices, and structures. To effectively remove droplets and evaluate their damage to indoor environments, it is important to accurately predict the evaporation and motion of droplets. Partial vapor pressure at the droplet surface significantly affects droplet evaporation and motion. Hence, in this study, the applicability of different droplet surface partial vapor pressure models on the evaporation and motion prediction of two droplets was numerically investigated. Two widely used models, namely osmotic coefficient method and Van't Hoff factor method, were studied for predicting droplet behavior in a free-falling process. The results indicated that the Van't Hoff factor method exhibited a large deviation in predicting the evaporation and motion of the two droplets. The osmotic coefficient method could accurately predict the evaporation and motion of hydrochloric droplets. However, the prediction of sulfuric droplets demonstrated a sharp change (data mutation), thereby resulting in serious deviation. Furthermore, the results indicated that the conditions that led to data mutation corresponded to a relative humidity less than 62% and a solute mass fraction (ω) greater than 40%. Additionally, different ionization methods of sulfuric solution affected the accuracy of the prediction results. The results can provide guidance for the accurate prediction of the evaporation and motion of sulfuric and hydrochloric droplets.

The Dirichlet problem for rectangle and new identities for elliptic integrals and functions

In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables’ separation. By means of this procedure normal derivative of Green’s function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle’s boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.

Comparative analysis for partial slip flow of ferrofluid [formula omitted] nanoparticles in a semi-porous channel

The present study develops the equations that govern a steady flow of ferrofluid in a semi-porous channel (SPC) under the effects of Lorentz force. Three different base fluids namely water, kerosene and blood and magnetite as ferroparticles are used in the analysis. These equations are modeled by using the Cartesian coordinate system with the help of Berman’s similarity transformation. The partial slip condition is also considered at the lower plate of the channel. Three different methods of solution, namely the method of homotopy analysis (HAM) (analytical technique), the method of differential transformation (DTM) (semi-numerical technique) and the method of Runge-Kutta (numerical technique), are used to achieve the solution of non-dimensional, non-linear ordinary differential equations. For HAM solution, the auxiliary parameter ħ delivers an effective way to regulate the convergence range of solution series whereas in DTM an approximation to the solution is obtained without any auxiliary parameter. The impact of Hartman and Reynolds number on the flow velocity are shown graphically and discussed. Finally, the comparison between the solution methods is given and found in very good agreement.

Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium

In this paper, vibration and buckling of two-phase nanobeams embedded in size dependent elastic medium and under thermal load are analyzed. Due to paradoxes of common differential nonlocal elasticity, such as neglecting the size effect of uniform loads, failure to satisfy the constitutive boundary conditions, associated to transformation of integral nonlocal equation to differential one, and incompatibility between the results of differential nonlocal with those of integral nonlocal, the size dependent effects of nanobeam, elastic medium and thermal load are taken into account simultaneously by using two-phase local/nonlocal Eringen's elasticity, for the first time. Governing equations and corresponding boundary conditions are derived using Hamilton's principle. To obtain natural vibration frequencies as well as critical buckling temperature, three different methods of solution are presented, i.e. exact solution, Generalized Differential Quadrature Method (GDQM) and Finite Element Method (FEM) which is based on the integral form of two-phase elasticity. Several comparison studies are conducted to examine the validity of the present formulation and results. The effects of applying two-phase elasticity on elastic medium and thermal load in vibration and buckling of nanobeams with different boundary conditions are investigated in details. Differences appeared in present results, especially in the cases with higher temperature and nonlocality as well as stiffer elastic medium, reveal that the size dependency of elastic medium and uniform thermal load, which is neglected by differential nonlocal, must be considered by employing two-phase elasticity.

Equations of higher orders

A complicated system of two equations with two variables is considered. The theory of equations of higher orders required for solution is given. Many different methods of solution are indicated. 
 

Lamb’s problem: a brief history

In this review note, a historical scientific investigation is presented for Lamb’s problem in the mathematical theory of elasticity. This problem first appeared in 1904 in the pioneering paper of Professor Sir Horace Lamb (Lamb, H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc Lon 1904; 203: 1–42). Of special interest here are the analytical studies of the three-dimensional version of Lamb’s problem, which consists of a semi-infinite, homogeneous, isotropic elastic solid that is set in motion by the exertion of a dynamical point force applied suddenly on the surface of the domain. The objective of this paper is to offer a comprehensive introduction to Lamb’s problem for the reader, along with discussing its mathematical complexities. An account is given of the history of this ever-significant problem from its earlier stages to the more recent investigations via outlining and discussing different rigorous approaches and methods of solution that have been hitherto suggested. The limitations of different methods, if they exist, are also discussed. Eventually, various solution methods are compared considering their nature, advantages, and restrictions.

Semi analytical method for solving lymphatic filariasis epidemic model

In this paper, we present a deterministic model on the transmission dynamics of Lymphatic Filariasis. Non-Standard Finite Difference Method (NSFDM) is employed to attempt the solution of the model. The validity of the NSFDM in solving the model is established by using the computer in-built classical fourth-order Runge-Kutta method. The comparism between Non-Standard Finite Difference Method solution and Runge-Kutta (RK4) were performed which were found to be efficient, accurate and rapidly convergence. Keywords: Lymphatic Filariasis, Non-Standard Finite Difference, Runge-Kutta; Mathematical model

Construction of solvable non-central potential using vector superpotential: a new approach

We introduce here vector superpotential which is useful to find general potential form using supersymmetric quantum mechanics (SUSY QM) approach. Using the vector superpotential, we reconstruct different solved central potentials and non-central shape variant or shape invariant potentials. To construct and reconstruct the central potentials and non-central potentials, appropriate choice of radial part of the superpotential as well as angular and azimuthal parts of the superpotential is required. Our main aim is to construct angular and azimuthal parts of potential directly from the appropriate choice of vector superpotential. We have reconstructed few potentials which are solved by different researchers using different methods of solution and also constructed new non-central potentials.

Spectral graph wavelet optimized finite difference method for solution of Burger's equation with different boundary conditions

ABSTRACTIn this paper, spectral graph wavelet optimized finite difference method (SPGWOFD) has been proposed for solving Burger's equation with distinct boundary conditions. Central finite difference approach is utilized for the approximations of the differential operators and the grid on which the numerical solution is obtained is chosen with the help of spectral graph wavelet. Four test problems (with Dirichlet, Periodic, Robin and Neumann's boundary conditions) are considered and the convergence of the technique is checked. For assessing the efficiency of the developed technique, the computational time taken by the developed technique is compared to that of the finite difference method. It has been observed that developed technique is extremely efficient.

On Lithuanian mathematical olympiads

Some problems of Lithuanian school mathematical olympiad 2018 and Vilnius University mathematicalolympiad 2018 are considered. Different methods of solution are compared. Some usefulcommon advices for solving mathematical problems are given.

Theoretical modelling for calculation of the energy densities of adsorption sites using inverse gas chromatography

The inverse gas chromatography is used to determine the energy densities of the adsorption sites of the stationary solid phase. The use of this technique is old and dates back to the 1940s. The many possibilities offered by this method are described in several works. This work is an attempt to explore some adsorption local isotherm models in order to determine the energy density of the adsorption sites. It involves the use of integral equations of the first kind which are known to be numerically instable. These integral equations were solved by two different methods of solution. One is based on the use of Taylor series expansions and the other uses the Stieltjes transform. Some interesting theoretical and numerical results are presented.

Optimization of building envelope design for nZEBs in Mediterranean climate: Performance analysis of residential case study

Design criteria for a residential nZEB in Mediterranean climate are discussed. The integrated design procedure focuses on the problem of a large number of available building variants concerning the building envelope. The aim is to search the ones that minimize winter and summer energy demand without compromising thermal comfort.The adopted methodological approach combines the use of dynamic energy simulation tool (EnergyPlus), based on a one-dimensional conduction finite difference solution method, and a constrained multi-objective optimization algorithm. For four cities (Madrid, Nice, Naples, Athens), several passive strategies are compared: thermal properties of the building envelope, adoption of phase change materials with different melting temperatures, cool roof solutions, several window/wall ratio values, some external and internal shading systems. The results allow to evidence that it is difficult to understand the best trade-off between summer and winter performance, by assuring high standard of thermal comfort when the aim is to reach NZEB objectives in Mediterranean climate. However, some guidelines are indicated, starting from the discussed results.

Polarization of two-level atom in a weak polychromatic field

Numerical solutions have been obtained for 2-level atom driven by a polychromatic field. Two different methods of analytical solution were proposed for any case of position of the field's components.

Silk fibroin membranes for controlled release of drugs: incorporation method, chemical and morphological characteristics

Event Abstract Back to Event Silk fibroin membranes for controlled release of drugs: incorporation method, chemical and morphological characteristics Bruno Tomoda1, Marisa Beppu1, Mariana Agostini De Moraes1, 2 and Giovana M. Genevro1 1 University of Campinas, School of Chemical Engineering, Brazil 2 Federal University of São Paulo, Department of Exact and Earth Science, Brazil Introduction: Silk fibroin (SF) is a natural polymer present in the cocoons of silkworms, which has been studied in the biomaterials area[1],[2]. SF membranes have good water vapor and oxygen permeability, blood compatibility and improve collagen formation and fibroblast proliferation[3], which makes them a promising material for application as wound dressings. The aim of this work is to study the incorporation of a model drug in dense membranes of silk fibroin and evaluate the drug release kinetics, aiming application as high performance wound dressing. Materials and Methods: Silk fibroin solution was prepared from cocoons of Bombyx mori silkworm (provided by Bratac, Bastos/ SP/Brazil) according to previously published procedure[4],[5]. SF membranes were prepared by casting SF aqueous solution. Diclofenac sodium (DS), an anti-inflammatory and analgesic drug used worldwide, was chosen as drug model. Two different methods for DS solution (10 mg/mL) incorporation were studied: 1) Incorporation of DS solution directly in SF solution, prior to membrane preparation; 2) Incorporation of DS on SF membrane by adsorption. The membranes were characterized by FTIR and SEM-EDS. Drug release was performed in phosphate buffer saline (PBS) and in water. Samples were collected at fixed time intervals and analyzed by UV/vis spectroscopy, at 276 nm. Results and Discussion: FTIR results (Figure 1), show that pure SF membranes undergo conformation transition from random coil/α-helix to stable β-sheets when ethanol post-treatment is performed[5]. Interestingly, when DS solution is incorporated in SF solution, prior to membrane preparation, the same conformational transition is observed. For the incorporation method by adsorption, ethanol post-treatment of SF membranes was necessary, to avoid membrane degradation prior to the DS incorporation. Thus, ethanol post-treatment is a step of production that can be eliminated when DS solution incorporation method is performed. SEM micrographs (Figure 2) revealed that SF membranes prepared by DS incorporation in SF solution present globular structures of regular size and shape and uniformly distributed within the membranes. The process of globule formation was spontaneous and allowed the encapsulation of DS inside the globules, as confirmed by EDS mapping. On the contrary, when the adsorption method was performed, DS was found just on the surface of the membrane. Drug release results (Figure 3) demonstrate that DS has a faster initial release in water than in PBS. Moreover, DS was completely released in water, whereas in PBS just 70% of DS was released. This difference is related to DS solubility, which is higher in water than in PBS. The release equilibrium was reached after 100 minutes for both media. Conclusion: The method of DS solution incorporation directly in SF solution, prior to membrane preparation, was the most promising one. By this method SF underwent conformation transition to a more stable form, avoiding additional post-treatments. Furthermore, SF self-assembled into globular structures containing the DS encapsulated, which is usually desired for controlled release applications. SF membranes showed a fast initial release and equilibrium was reached after approximately 100 minutes of release. These results revealed an excellent drug incorporation method and confirm the potential application of SF membranes in wound dressings; however, further studies should be carried out in order to obtain a slow release kinetics. CNPq; FAPESP

Phase separation in pre-ceramic polymers blends and its influence on the microstructure of ZrC/SiC ceramic derivatives

Ultra-high temperature ceramics (UHTCs) composed of ZrC and SiC were fabricated via pyrolysis of hybrid polymeric precursors comprising of polycarbosilane (PCS) and polyzirconobutanediol (PZC). Phase separation in these polymers blends during different removal methods of toluene solution by freezing drying, vacuum drying and rotary evaporation was studied. Elements and phase distribution as well as molecular aggregation state in the hybrid polymeric precursors and the pyrolyzed composite ceramics were characterized with AFM and SEM. It was found that macromolecules of pre-ceramic polymers aggregated spontaneously in the hybrid precursor solution and formed molecular clusters within dozens of nanometers, and the resultant phase separation during removal of toluene solution exhibits strongly impact on the microstructure of the derived UHTCs after being pyrolyzed at temperature above 1000°C, which was controlled by solvent removing time.