Second-order Equations Research Articles

Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space

Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.

Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type

In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvz′v)′ + qvFxgv = 0, where v≥v0 and the corresponding function z = x + px∘h. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.

Neimark–Sacker bifurcation of two second-order rational difference equations

We investigate the Neimark–Sacker bifurcation of the equilibrium of two special cases of the difference Equation x n + 1 = β x n x n − 1 + γ x n − 1 2 + δ x n B x n x n − 1 + C x n − 1 2 + D x n where the parameters β, γ, δ, B, C, D are non-negative numbers which satisfy B + C + D>0 and the initial conditions x − 1 and x 0 are arbitrary nonnegative numbers such that B x n x n − 1 + C x n − 1 2 + D x n > 0 for all n ≥ 0 . More precisely, we consider special cases where either γ = D = 0 or β = D = 0 . As we will show both equations exhibit Neimark–Sacker bifurcation, where one of equations ( γ = D = 0 ) probably exhibits Chenciner bifurcation, with two invariant curves, while another Equation ( β = D = 0 ) exhibits simple Neimark–Sacker bifurcation with one invariant curve. We will also obtain some global asymptotic stability result for each equation in the subset of the parametric region of local stability.

Optimization of poplar wood shavings bio-pretreated with Coriolus versicolor to produce binderless fiberboard using response surface methodology

Free formaldehyde is released due to the addition of aldehyde-based adhesives during the production of fiberboard. This is harmful to human health and pollutes the environment, and for that reason binderless fiberboard has become a research hotspot. There have been reports about pretreatments with white-rot fungi or lignocellulase to produce binderless fiberboard, but there have been no such reports about optimizing the bio-pretreatment conditions. In this study, poplar wood shavings were used for fiberboard production, and the bio-pretreatment conditions with Coriolus versicolor were studied using response surface methodology. After single-factor optimization, the central levels of bran, molasses, and magnesium sulfate were obtained. Further optimization was carried out using Box-Behnken design to study the influence of the factors. A second-order polynomial equation was obtained, and the low p-value (0.001) implied that the model was highly significant. The optimized bio-pretreatment conditions for modulus of rupture (MOR) of the fiberboard were obtained by ridge analysis as 3.021 g of bran, 8.907 g of molasses, and 0.27 g of magnesium sulfate. Under the optimized conditions, MOR of fiberboard reached 27.21±0.64 MPa, which was 2.2 times that of the control fiberboard. Bio-pretreatment with C. versicolor should be a good choice to produce a high-strength binderless fiberboard.

Adsorption of heavy metal ions by carbon-based adsorbent using magnetic nitrogen-doped carbon and graphene oxide

ABSTRACT The adsorption of weighty metal particles from contaminated water sources has garnered significant attention due to its critical role in environmental remediation and ensuring safe drinking water. Heavy metal ions can be removed from water using conventional adsorbents such as activated zeolites; however, these materials have low absorption and slow kinetics. To solve these issues, carbon-based adsorbents that exhibit easy synthesis, high porosity, design ability, and stability have been proposed. In this review, a carbon-based adsorbent, named M-NC, and graphene oxide were created for the particular evacuation of weighty metal particles. To increase the potential for Heavy Metals (HM) immobilization, sulfide-modified biochar was created via a process called synchronous carbon layer epitome. A hypothetical physicochemical and thermodynamic examination of the adsorption of weighty metals Zn2+, Cd2+, Ni2+, Ag2+, Pb2+ and Cu2+ on carbon-based adsorbents was done with factual material science fundaments. The biochar with large surface areas is utilized to eliminate weighty metal particles, quite possibly the most significant heavy metal pollutants, from aqueous solutions. The limit of the adsorbent for eliminating weighty metal ions was concentrated on utilizing Langmuir adsorption isotherm under ultrasound-helped conditions. The Mesoporous Silica Nanoparticles (MNCs) can be applied to the Langmuir model and pseudo-second-order kinetics. It is possible to use the Langmuir and second-order kinetic equations to accurately explain the adsorption method. Thermodynamic limitations were also envisioned because sorption is exothermic when it happens spontaneously. A homogeneous measurable physical science adsorption typically was utilized to describe and analyze the experimental heavy metal removal isotherms at 30°C and pH5 utilizing adsorbents produced by pyrolysis of biomasses (broccoli stalks). The experimental results were investigated in terms of Langmuir and pseudo-2nd-order kinetic equations, Freundlich and isotherm models. The outcome of pH, initial heavy metal ion concentration, contact time, and adsorbent dosage regarding the adsorption capacity and removal efficiency of Pb2+ and Cd2+ on the hydrogel was examined. This study contributes to the advancement of information in the ground of environmentally friendly heavy metal removal techniques, specifically focusing on the usage of biomass-based adsorbents. These findings have the potential to address the need for effective solutions in water purification and environmental cleanup efforts.

Lattice Boltzmann Method without Invoking the M << 1 Assumption

When a Maxwellian distribution is assumed for the distribution function in the BGK-type modelled BE, it will give rise to the Euler equations if it is the first-order approximation in the Chapman-Enskog method. Then the second-order equations will yield the N-S equations. Most LBM developed to date are formulated based on the second-order equations. Consequently, the assumption of a flow Mach number M << 1 is inherent in this formulation. This approach creates an unnecessary restriction on the LBM that should be avoided if possible. An alternative approach is to formulate a new LBM by considering an equilibrium distribution function where the first-order approximations give rise to the N-S equations. Adopting this approach, a new LBM has been formulated. This new LBM gives reliable results when applied to simulate aeroacoustics, incompressible flows, and compressible flows with and without shocks. Good agreement with measurements and numerical data derived from DAS/DNA calculations is obtained.

Revisiting shear stress tensor evolution: Nonresistive magnetohydrodynamics with momentum-dependent relaxation time

This study aims to develop second-order relativistic viscous magnetohydrodynamics (MHD) derived from kinetic theory within an extended relaxation time approximation (momentum/energy dependent) for the collision kernel. The investigation involves a detailed examination of shear stress tensor evolution equations and associated transport coefficients. The Boltzmann equation is solved using a Chapman-Enskog-like gradient expansion for a charge-conserved conformal system, incorporating a momentum-dependent relaxation time. The derived relativistic nonresistive, viscous second-order MHD equations for the shear stress tensor reveal significant modifications in the coupling with dissipative charge current and magnetic field due to the momentum dependence of the relaxation time. By utilizing a power law parametrization to quantify the momentum dependence of the relaxation time, the anisotropic magnetic field-dependent shear coefficients in the Navier-Stokes limit have been investigated. The resulting viscous coefficients are seen to be sensitive to the momentum dependence of the relaxation time. Published by the American Physical Society 2024

Bounded and homoclinic-like solutions of second-order singular difference equations

Bounded and homoclinic-like solutions of second-order singular difference equations

Linear and nonlinear transport equations with coordinate-wise increasing velocity fields

We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which unique minimal and maximal solutions exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit.

Evaluation of Decay Kinetics of Black Elderberry Antioxidants from Fruits and Flowers.

The health-promoting properties of black elderberry are related to its high content of polyphenols (natural antioxidants), which eliminate free radicals and prevent the formation of oxidative stress responsible for many diseases. The aim of this work was to determine, the anti-radical effect of Sambucus nigra infusions based on the reaction with 2,2-diphenyl-1-picrylhydrazyl (DPPH) and galvinoxyl (Glv) radicals and to determine the function describing the disappearance curves of these radicals. The antioxidant properties of infusions obtained from the flowers and fruits of this plant were tested using the modified Brand-Williams method using DPPH and Glv radicals. Higher antioxidant activity towards both the DPPH and Glv radicals was found in flowers compared to fruits. In addition, it was found that the process of quenching radicals in the reaction with Sambucus nigra infusions proceeds in accordance with the assumptions of second-order reaction kinetics. The infusion obtained from flowers quenched radicals faster than fruit infusions. The applied second-order kinetics equation may enable estimation of antioxidants levels in natural sources of radicals.

Second-order numerical method for a neutral Volterra integro-differential equation

This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.

Compressibility and pressure-induced structural evolution of kokchetavite, hexagonal polymorph of KAlSi3O8, by single-crystal X-ray diffraction

Abstract Compressibility and pressure-induced structural evolution of kokchetavite, the hexagonal polymorph of KAlSi3O8, has been studied up to 11.8 GPa using synchrotron single-crystal X-ray diffraction. Two phase transitions were observed at pressures of ~0.3 and 10.4 GPa. Kokchetavite-I (as-synthesized, P6/mcc) transforms into kokchetavite-II with the P6c2 space group. Kokchetavite-II → kokchetavite-III phase transition at ~10.4 GPa is accompanied by a change of symmetry to probably orthorhombic. After pressure release, kokchetavite reverts to the initial single-crystal state with P6/mcc space group. A second-order Birch-Murnaghan equation of state was calculated for phase kokchetavite-II with coefficients V0 = 1486(3) Å3, K0 = 59(2) GPa.

A modified cell-to-cell simulation model to predict oil-gas minimum miscibility pressure

Calculating the minimum miscibility pressure (MMP) between crude oil and carbon dioxide (CO2) is critical for optimizing injection parameters, designing schemes, and predicting production capacity in CO2 injection projects for enhancing oil recovery. However, an accurate approach for obtaining this parameter is not yet established. In order to tackle this issue, a novel approach is suggested, based on the original cell-to-cell model, to determine the MMP and the 97% oil recovery rate as the standard. Using the volume-transformed Peng-Robinson equation of state enhances the precision of fluid volume estimation, as it mainly relies on predicting fluid volume within each cell. Furthermore, to ensure a precise estimation of the ultimate oil recovery rate, it is imperative to employ a total cell count of 500 in all simulations to avoid the problem of numerical dispersion. Finally, a second-order polynomial equation more accurately predicts the infinite-cell oil recovery factor. The accuracy of the modified model is verified by comparing MMP values from five oil and gas systems in the literature. The computational results of the modified multiple-mixing-cell (MMC) approach exhibit a higher level of concordance with the MMPs in the literature. The average relative error is less than 3.96%. The improved MMC algorithm can quickly determine the miscibility mechanism and visually represent the dynamic miscibility process involving multiple oil-gas contacts in a slim tube. This study provides a theoretical and practical basis for addressing the critical scientific issues of CO2-safe storage technology.

Existence and non-existence results for a singular second-order elliptic equation

Using the variational methods, this paper is concerned with the existence and non-existence results to a class of elliptic equation with singular term on compact Riemannian manifold. In the context of complete, non-compact and non parabolic manifold, we obtain the non-existence result of solution for this type of elliptic problems.

Computational decomposition and composition technique for approximate solution of nonstationary problems

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit–implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition–composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Influence of the Acceptor Fluid on the Bupivacaine Release from the Prospective Intra-Articular Methylcellulose Hydrogel.

Injections are one way of delivering drugs directly to the joint capsule. Employing this possibility, local anesthetic, such as bupivacaine (Bu), in the form of the suspension can be administered. The aim of this work was to propose a methylcellulose-based hydrogel-incorporated bupivacaine for intra-articular injections and to study the release kinetics of the drug from the hydrogel to different acceptor media, reflecting the synovial fluid of a healthy joint and the synovial fluid of an inflamed joint. The drug release studies were performed employing the flow apparatus. The drug was released to four different acceptor fluids: phosphate buffer pH = 7.4 (PBS7.4), phosphate buffer pH = 6.8 (PBS6.8), phosphate buffer pH = 7.4 with the high-molecular-weight sodium hyaluronate (PBS7.4H), and phosphate buffer pH = 6.8 with the low-molecular-weight sodium hyaluronate (PBS6.8L). The investigation was carried out at the temperature of 37 °C. The absorbance of the Bu released was measured at the wavelength of 262 nm every 2 min for 24 h. The release profiles of Bu to the acceptor media PBS7.4, PBS6.8, PBS7.4H, and PBS6.8L were described best by the first-order kinetics and the second-order equation. According to these models, the release rate constants were the highest when Bu was released to the fluid PBS7.4 and were k1 = (7.20 ± 0.01) × 10-5 min-1 and k2 = (3.00 ± 0.04) × 10-6 mg-1 × min-1, respectively. The relative viscosity of the acceptor medium, its pH, and the addition of high-molecular-weight or low-molecular-weight sodium hyaluronate (HAH or HAL) to the acceptor fluid influenced the drug dissolution. The release of Bu into the medium reflecting healthy synovial fluid takes a different pattern from its release into the fluid of an inflamed joint.

Factors, Mechanisms, and Kinetics of Spontaneous Emulsification for Heavy Oil-in-Water Emulsions.

In challenging reservoirs where thermal recovery falls short, cold or chemical oil recovery methods are crucial. Spontaneous emulsification (SE), triggered by gentle disturbance, significantly enhances oil recovery. In elucidating SE mechanisms and kinetics, SE processes via direct contact between oil and aqueous phases without stirring were conducted. The effects of temperature, emulsifier concentration, pH, NaCl concentration, and the oil-to-water ratio on SE were investigated through droplet size analysis and turbidity measurements. Furthermore, the emulsification mechanism and derived emulsification kinetics based on turbidity data were obtained. The results underscore the feasibility of SE for oil-water systems, reducing viscous and capillary resistances without agitation. The emulsified oil mass increased with the temperature, pH, and aqueous-to-oil phase volume ratio while decreasing with the NaCl concentration. In this study, for GD-2 crude oil, the optimal emulsified oil amount occurred at a betaine surfactant (BetS-2) emulsifier concentration of 0.45%. Microscopic photo analysis indicated narrow particle size distributions and small droplets, which remained stable over time under various experimental conditions. A combined SE mechanism involving ultralow interfacial tension, interfacial turbulence due to Marangoni effects, and "diffusion and stranding" due to in situ emulsifier hydrophilicity, was speculated. Additionally, an analogous second-order kinetic equation for SE was proposed, indicating exceptional correlation with calculated and experimentally measured values. This study offers theoretical insight for enhancing oil recovery in chemical and cold production of heavy oil in oilfields.

Spectral moments of the real Ginibre ensemble

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments M2pr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{2p}^\ extrm{r}$$\\end{document}. The latter are expressed in terms of the 3F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_3 F_2$$\\end{document} hypergeometric functions, with a simplification to the 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2 F_1$$\\end{document} hypergeometric function possible for p=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=0$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence {M2pr}p=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{ M_{2p}^\ extrm{r} \\}_{p=0}^\\infty $$\\end{document}. Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.

On close-to-pseudoconvex Dirichlet series

For a Dirichlet series of form $F(s)=\exp\{s\lambda_1\}+\sum\nolimits_{k=2}^{+\infty}f_k\exp\{s\lambda_k\}$ absolutely convergent in the half-plane $\Pi_0=\{s\colon \mathop{\rm Re}s<0\}$ new sufficient conditionsfor the close-to-pseudoconvexity are found and the obtained result is applied to studying of solutions linear differential equations of second order with exponential coefficients. In particular, are proved the following statements: 1) Let $\lambda_k=\lambda_{k-1}+\lambda_1$ and $f_k>0$ for all $k\ge 2$. If $1\le\lambda_2f_2/\lambda_1\le 2$ and $\lambda_kf_k-\lambda_{k+1}f_{k+1}\searrow q\ge 0$ as $k\to+\infty$ then function of form {\bf(1)} is close-to-pseudoconvex in $\Pi_0$ (Theorem 3). This theorem complements Alexander's criterion obtained for power series.2) If either $-h^2\le\gamma\le0$ or $\gamma=h^2$ then differential equation $(1-e^{hs})^2w''-h(1-e^{2hs})w'+\gamma e^{2hs}=0$ $(h>0, \gamma\in{\mathbb R})$ has a solution $w=F$ of form {\bf(1)} with the exponents $\lambda_k=kh$ and the the abscissa of absolute convergence $\sigma_a=0$ that is close-to-pseudoconvex in $\Pi_0$ (Theorem 4).

Experimental investigation of performance, emissions, combustion characteristics, cyclic variations, and exergy efficiency of CI engine fueled with Karanja-camphor oil blends and diesel-camphor oil blends.

This article compares the influence of blending the low-viscous oxygenated camphor oil with hydrocarbon diesel fuel and high-viscous oxygenated Karanja oil. The experiment is conducted in a four-stroke one-cylinder naturally aspirated Kirloskar compression ignition (CI) engine coupled with an eddy current dynamometer. The three types of fuel blends are prepared by blending the camphor oil with Karanja oil on the volume ratio of 30:70 (C30K70), 50:50 (C50K50), and 70:30 (C70K30), and the other three types of fuels are prepared by blending the camphor oil with diesel on the volume ratio of 30:70 (C30D70), 50:50 (C50D50), and 70:30 (C70D30). The experimental efficiency results show higher thermal efficiency of 31.86% and 30.84% for C70D30 and C70K30 at rated operating conditions. The brake-specific energy consumptions of C70D30 and C70K30 were found to be 11.29 and 11.67MJ/kWh, respectively, at rated operating conditions. The lowest CO, CO2, HC, and smoke emissions are found for C70D30 at rated operating conditions, which are 96.58%, 6.15%, 34.20%, and 7.59% lower than diesel. However, the NO emissions were found to be 27.62% higher for C70D30 than diesel at full load. The rate of pressure rise, net heat release rate, and cyclic variations were found to increase with increase in proportion of the camphor oil. The P-v diagram also confirms the lower heat addition period for the C70D30 and C70K30 with an increase in brake thermal efficiency. The actual compression ratio and the actual cutoff ratio are found to have a reasonable correlation with the thermal efficiency of the engine. The exergy efficiency of C30K70 is found higher which is 2.11% higher than diesel fuel at rated power. Second-order polynomial equations were obtained for the engine characteristics using the curve fitting method, and the characteristic equations confirmed the confidence level of 95%.