Search results for: coupled differential equation

Authors: Vandana N. Purav

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Fibonacci and Lucas polynomials are special cases of Chebyshev polynomial. There are two types of Chebyshev polynomials, a Chebyshev polynomial of first kind and a Chebyshev polynomial of second kind. Chebyshev polynomial of second kind can be derived from the Chebyshev polynomial of first kind. Chebyshev polynomial is a polynomial of degree n and satisfies a second order homogenous differential equation. We consider the difference equations which are related with Chebyshev, Fibonacci and Lucas polynomias. Thus Chebyshev polynomial of second kind play an important role in finding the recurrence relations with Fibonacci and Lucas polynomials.

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Authors: Abdelatif Gadoum, Djilali Benyoucef, Mouloudj Hadj, Alla Eddine Toubal Maamar, Mohamed Habib Allah Lahoual

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Hydrogen occurs naturally in the form of chemical compounds, most often in water and hydrocarbons. The main objective of this study is 2D modeling of hydrogen production in inductively coupled plasma in methane at low pressure. In the present model, we include the motions and the collisions of both neutral and charged particles by considering 19 species (i.e in total ; neutrals, radicals, ions, and electrons), and more than 120 reactions (electron impact with methane, neutral-neutral, neutral-ions and surface reactions). The results show that the rate conversion of methane reach 90% and the hydrogen production is about 30%.

Keywords: hydrogen production, inductively coupled plasma, fluid model, methane plasma

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Authors: H. S. Kang

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Coupled electrical and optical model for conversion of electrical energy into coherent optical energy for transmitter-receiver link by solid state device is presented. Probability distribution for travelling laser beam switching time intervals and the number of switchings in the time interval is obtained. Selector function mapping is employed to regulate optical data transmission speed. It is established that regulated laser transmission from PhotoActive Laser transmitter follows principal of invariance. This considerably simplifies design of PhotoActive Laser Transmission networks.

Keywords: computational mathematics, finite difference Markov chain methods, sequence spaces, singularly perturbed differential equations

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Authors: Azher Jameel, Ghulam Ashraf Harmain

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In the recent years, the enriched techniques like the extended finite element method, the element free Galerkin method, and the Coupled finite element-element free Galerkin method have found wide application in modeling different types of discontinuities produced by cracks, contact surfaces, and bi-material interfaces. The extended finite element method faces severe mesh distortion issues while modeling large deformation problems. The element free Galerkin method does not have mesh distortion issues, but it is computationally more demanding than the finite element method. The coupled FE-EFGM proves to be an efficient numerical tool for modeling large deformation problems as it exploits the advantages of both FEM and EFGM. The present paper employs the coupled FE-EFGM to model large elastoplastic deformations in bi-material engineering components. The large deformation occurring in the domain has been modeled by using the total Lagrangian approach. The non-linear elastoplastic behavior of the material has been represented by the Ramberg-Osgood model. The elastic predictor-plastic corrector algorithms are used for the evaluation stresses during large deformation. Finally, several numerical problems are solved by the coupled FE-EFGM to illustrate its applicability, efficiency and accuracy in modeling large elastoplastic deformations in bi-material samples. The results obtained by the proposed technique are compared with the results obtained by XFEM and EFGM. A remarkable agreement was observed between the results obtained by the three techniques.

Keywords: XFEM, EFGM, coupled FE-EFGM, level sets, large deformation

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Authors: Francesco Marotti de Sciarra, Raffaele Barretta

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Starting from nonlocal continuum mechanics, a thermodynamically new nonlocal model of Euler-Bernoulli nanobeams is provided. The nonlocal variational formulation is consistently provided and the governing differential equation for transverse displacement are presented. Higher-order boundary conditions are then consistently derived. An example is contributed in order to show the effectiveness of the proposed model.

Keywords: Bernoulli-Euler beams, nanobeams, nonlocal elasticity, closed-form solutions

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Authors: Abdelati El Allaoui, Said Melliani, Lalla Saadia Chadli

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The objective of this paper is to study the existence and uniqueness of Mild solutions for a complex fuzzy evolution equation with nonlocal conditions that accommodates the notion of fuzzy sets defined by complex-valued membership functions. We first propose definition of complex fuzzy strongly continuous semigroups. We then give existence and uniqueness result relevant to the complex fuzzy evolution equation.

Keywords: Complex fuzzy evolution equations, nonlocal conditions, mild solution, complex fuzzy semigroups

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Authors: Shazalina Mat Zin, Ahmad Abd. Majid, Ahmad Izani Md. Ismail, Muhammad Abbas

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The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature.

Keywords: collocation method, cubic trigonometric B-spline, finite difference, wave equation

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Authors: Sonia Besbes, Mahmoud El Hajem, Habib Ben Aissia, Jean Yves Champagne, Jacques Jay

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In this work, we present a computational study for the characterization of the flow in a rectangular bubble column. To simulate the dynamic characteristics of the flow, a three-dimensional transient numerical simulations based on a coupled discrete phase model (DPM) and Volume of Fluid (VOF) model are performed. Modeling of bubble column reactor is often carried out under the assumption of a flat liquid surface with a degassing boundary condition. However, the dynamic behavior of the top surface surmounting the liquid phase will to some extent influence the meandering oscillations of the bubble plume. Therefore it is important to capture the surface behavior, and the assumption of a flat surface may not be applicable. So, the modeling approach needs to account for a dynamic liquid surface induced by the rising bubble plume. The volume of fluid (VOF) model was applied for the liquid and top gas which both interacts with bubbles implemented with a discrete phase model. This model treats the bubbles as Lagrangian particles and the liquid and the top gas as Eulerian phases with a sharp interface. Two-way coupling between Eulerian phases and Lagrangian bubbles are accounted for in a single set continuous phase momentum equation for the mixture of the two Eulerian phases. The effect of gas flow rate on the dynamic and time-averaged flow properties was studied. The time averaged liquid velocity field predicted from simulations and from our previous PIV measurements shows that the liquid is entrained up flow in the wake of the bubbles and down flow near the walls. The simulated and measured vertical velocity profiles exhibit a reasonable agreement looking at the minimum velocity values near the walls and the maximum values at the column center.

Keywords: bubble column, computational fluid dynamics (CFD), coupled DPM and VOF model, hydrodynamics

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Authors: Yanpei Zhen

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The nonlinear Schrödinger (NLS) equation models wave dynamics in many physical problems related to fluids, plasmas, and optics. The standing periodic waves are known to be modulationally unstable, and rogue waves (localized perturbations in space and time) have been observed on their backgrounds in numerical experiments. The exact solutions for rogue waves arising on the periodic standing waves have been obtained analytically. It is natural to ask if the rogue waves persist on the standing periodic waves in the integrable discretizations of the integrable NLS equation. We study the standing periodic waves in the semidiscrete integrable system modeled by the high-order Ablowitz-Ladik (AL) equation. The standing periodic wave of the high-order AL equation is expressed by the Jacobi cnoidal elliptic function. The exact solutions are obtained by using the separation of variables and one-fold Darboux transformation. Since the cnoidal wave is modulationally unstable, the rogue waves are generated on the periodic background.

Keywords: Darboux transformation, periodic wave, Rogue wave, separating the variables

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Authors: Abdullah M. Almeshal, Mohammad R. Alenezi, Muhammad Moaz

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In this paper, we discuss the performance of applying hybrid spiral dynamic bacterial chemotaxis (HSDBC) optimisation algorithm on an intelligent controller for a differential drive robot. A unicycle class of differential drive robot is utilised to serve as a basis application to evaluate the performance of the HSDBC algorithm. A hybrid fuzzy logic controller is developed and implemented for the unicycle robot to follow a predefined trajectory. Trajectories of various frictional profiles and levels were simulated to evaluate the performance of the robot at different operating conditions. Controller gains and scaling factors were optimised using HSDBC and the performance is evaluated in comparison to previously adopted optimisation algorithms. The HSDBC has proven its feasibility in achieving a faster convergence toward the optimal gains and resulted in a superior performance.

Keywords: differential drive robot, hybrid fuzzy controller, optimization, path tracking, unicycle robot

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Authors: Nadaniela Egidi, Pierluigi Maponi

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The approximate solution of a time-harmonic electromagnetic scattering problem for inhomogeneous media is required in several application contexts, and its two-dimensional formulation is a Fredholm integral equation of the second kind. This integral equation provides a formulation for the direct scattering problem, but it has to be solved several times also in the numerical solution of the corresponding inverse scattering problem. The discretization of this Fredholm equation produces large and dense linear systems that are usually solved by iterative methods. In order to improve the efficiency of these iterative methods, we use the Symmetric SOR preconditioning, and we propose an algorithm for the evaluation of the associated relaxation parameter. We show the efficiency of the proposed algorithm by several numerical experiments, where we use two Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.

Keywords: Fredholm integral equation, iterative method, preconditioning, scattering problem

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Authors: Mei Hong, Wei Liu, Xuemin Dai, Yanxiong Pan, Xiangling Ji

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Polyamic acid (PAA) is the precursor of polyimide (PI) prepared by a two-step method, its molecular weight and molecular weight distribution not only play an important role during the preparation and processing, but also influence the final performance of PI. However, precise characterization on molecular weight of PAA is still a challenge because of the existence of very complicated interactions in the solution system, including the electrostatic interaction, hydrogen bond interaction, dipole-dipole interaction, etc. Thus, it is necessary to establisha suitable strategy which can completely suppress these complex effects and get reasonable data on molecular weight. Herein, the gel permeation chromatography (GPC) coupled with differential refractive index (RI) and multi-angle laser light scattering (MALLS) detectors were applied to measure the molecular weight of (6FDA-DMB) PAA using different mobile phases, LiBr/DMF, LiBr/H3PO4/THF/DMF, LiBr/HAc/THF/DMF, and LiBr/HAc/DMF, respectively. It was found that combination of LiBr with HAc can shield the above-mentioned complex interactions and is more conducive to the separation of PAA than only addition of LiBr in DMF. LiBr/HAc/DMF was employed for the first time as a mild mobile phase to effectively separate PAA and determine its molecular weight. After a series of conditional experiments, 0.02M LiBr/0.2M HAc/DMF was fixed as an optimized mobile phase to measure the relative and absolute molecular weights of (6FDA-DMB) PAA prepared, and the obtained Mw from GPC-MALLS and GPC-RI were 35,300 g/mol and 125,000 g/mol, respectively. Particularly, such a mobile phase is also applicable to other PAA samples with different structures, and the final results on molecular weight are also reproducible.

Keywords: Polyamic acids, Polyelectrolyte effects, Gel permeation chromatography, Mobile phase, Molecular weight

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Authors: Kayode Oshinubi, Brice Kammegne, Jacques Demongeot

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The use of mathematical tools like Partial Differential Equations and Ordinary Differential Equations have become very important to predict the evolution of a viral disease in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China causing a severe and potentially fatal respiratory syndrome, i.e., COVID-19. Since then, it has become a pandemic declared by World Health Organization (WHO) on March 11, 2020 which has spread around the globe. A reaction-diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process in which different substances are transformed, and a diffusion process that causes a distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic by the bias of reaction-diffusion equations. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined using the Lyapunov function are considered and the endemic equilibrium point exists and is stable if it satisfies Routh–Hurwitz criteria. Also, adequate conditions for the existence and uniqueness of the solution of the model have been proved. We showed the spatial distribution of the model compartments when the basic reproduction rate $\mathcal{R}_0 < 1$ and $\mathcal{R}_0 > 1$ and sensitivity analysis is performed in order to determine the most sensitive parameters in the proposed model. We demonstrate the model's effectiveness by performing numerical simulations. We investigate the impact of vaccination and the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. To the public health policymakers, we offered a better understanding of the COVID-19 management.

Keywords: COVID-19, SEIRV epidemic model, reaction-diffusion equation, basic reproduction number, vaccination, spatial distribution

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Authors: Evln Ranga Charyulu, S. P. Venu Madhavarao, S. Udaya kumar, S. V. S. S. N. V. G. Krishna Murthy

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With the advent of new nanometer process technologies, it is possible to integrate billion transistors on a single substrate. When more and more functionality included there is the possibility of multi-million transistors switching simultaneously consuming more power and dissipating more power along with more leakage of current into the substrate of porous silicon or germanium material. These results in substrate heating and thermal noise generation coupled to signals of interest. The heating process is represented by coupled nonlinear partial differential equations in porous silicon and germanium. By identifying heat sources and heat fluxes may results in designing of ultra-low power circuits. The PDEs are solved by finite difference scheme assuming that boundary layer equations in porous silicon and germanium. Local heat fluxes along the vertical isothermal surface immersed in porous SI/Ge are considered. The parameters considered for optimization are thermal diffusivity, thermal expansion coefficient, thermal diffusion ratio, permeability, specific heat at constant temperatures, Rayleigh number, amplitude of wavy surface, mass expansion coefficient. The diffusion of heat was caused by the concentration gradient. Thermal physical properties are homogeneous and isotropic. By using L8, TAGUCHI method the parameters are optimized.

Keywords: heat transfer, pde, taguchi optimization, SI/Ge

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Authors: Chungkuk Jin, Moo-Hyun Kim, HeonYong Kang

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A surface riding (SR) wave energy converter (WEC) is designed and its feasibility and performance are numerically simulated by the author-developed floater-mooring-magnet-electromagnetics fully-coupled dynamic analysis computer program. The biggest advantage of the SR-WEC is that the performance is equally effective even in low sea states and its structural robustness is greatly improved by simply riding along the wave surface compared to other existing WECs. By the numerical simulations and actuator testing, it is clearly demonstrated that the concept works and through the optimization process, its efficiency can be improved.

Keywords: computer simulation, electromagnetics fully-coupled dynamics, floater-mooring-magnet, optimization, performance evaluation, surface riding, WEC

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Authors: Sarun Phibanchon, Yuttakarn Rattanachai

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The quadratic-cubic nonlinear Schrodinger equation can be explained the weakly ion-acoustic waves in magnetized plasma with a slightly non-Maxwellian electron distribution by using the Madelung's fluid picture. However, the soliton solution to the quadratic-cubic nonlinear Schrodinger equation is determined by using the direct integration. By the characteristics of a soliton, the solution can be claimed that it's a soliton by considering its time evolution and their collisions between two solutions. These results are shown by applying the spectral method.

Keywords: soliton, ion-acoustic waves, plasma, spectral method

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Authors: Qinghua Zhang, Yanhe Zhu, Xiang Zhao, Yeqin Yang, Hongwei Jing, Guoan Zhang, Jie Zhao

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This paper presents a wearable reconfigurable supernumerary robotic limb with differential actuated joints, which is lightweight, compact and comfortable for the wearers. Compared to the existing supernumerary robotic limbs which mostly adopted series structure with large movement space but poor carrying capacity, a prototype with the series-parallel configuration to better adapt to different task requirements has been developed in this design. To achieve a compact structure, two kinds of cable-driven mechanical structures based on guide pulleys and differential actuated joints were designed. Moreover, two different tension devices were also designed to ensure the reliability and accuracy of the cable-driven transmission. The proposed device also employed self-designed bearings which greatly simplified the structure and reduced the cost.

Keywords: cable-driven, differential actuated joints, reconfigurable, supernumerary robotic limb

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Authors: Muhammad Bilal Ameen, M. Zubair Akbar Qureshi

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The current investigation of two-dimensional hybrid nanofluid flows with two coaxial parallel disks has been presented. Consider the hybrid nanofluid has been taken as steady-state. Consider the coaxial disks that have been porous. Consider the heat equation to examine joule heating and viscous dissipation effects. Nonlinear partial differential equations have been solved numerically. For shear stress and heat transfer, results are tabulated. Hybrid nanoparticles and Eckert numbers are increasing for heat transfer. Entropy generation is expanded with radiation parameters Eckert, Reynold, Prandtl, and Peclet numbers. A set of ordinary differential equations is obtained to utilize the capable transformation variables. The numerical solution of the continuity, momentum, energy, and entropy generation equations is obtaining using the command bvp4c of Matlab as a solver. To explore the impact of main parameters like suction/infusion, Prandtl, Reynold, Eckert, Peclet number, and volume fraction parameters, various graphs have been plotted and examined. It is concluded that a convectional nanofluid is highly compared by entropy generation with the boundary layer of hybrid nanofluid.

Keywords: entropy generation, hybrid nano fluid, heat transfer, porous disks

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Authors: Ih-Ching Hsu

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Inspired by the so called Jacobi Identity (x y) z + (y z) x + (z x) y = 0, the following class of functional equations EQ I: F [F (x, y), z] + F [F (y, z), x] + F [F (z, x), y] = 0 is proposed, researched and generalized. Research methodologies begin with classical methods for functional equations, then evolve into discovering of any implicit algebraic structures. One of this paper’s major findings is that EQ I, under two additional conditions F (x, x) = 0 and F (x, y) + F (y, x) = 0, proves to be a fundamental functional equation for Lie Algebras. Existence of non-trivial solutions for EQ I can be proven by defining F (p, q) = [p q] = pq –qp, where p and q are quaternions, and pq is the quaternion product of p and q. EQ I can be generalized to the following class of functional equations EQ II: F [G (x, y), z] + F [G (y, z), x] + F [G (z, x), y] = 0. Concluding Statement: With a major finding proven, and non-trivial solutions derived, this research paper illustrates and provides a new functional equation scheme for studies in two major areas: (1) What underlying algebraic structures can be defined and/or derived from EQ I or EQ II? (2) What conditions can be imposed so that conditional general solutions to EQ I and EQ II can be found, investigated and applied?

Keywords: fundamental functional equation, generalized functional equations, Lie algebras, quaternions

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Authors: Haniye Dehestani, Yadollah Ordokhani

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In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: collocation method, fractional partial differential equations, legendre-laguerre functions, pseudo-operational matrix of integration

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Authors: Olukunle C. Olawole, Dilip Kumar De

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We have modified Richardson-Dushman equation considering thermal expansion of lattice and change of chemical potential with temperature in material. The corresponding modified Richardson-Dushman (MRDE) equation fits quite well the experimental data of thermoelectronic current density (J) vs T from carbon nanotubes. It provides a unique technique for accurate determination of W0 Fermi energy, EF0 at 0 K and linear thermal expansion coefficient of carbon nano-tube in good agreement with experiment. From the value of EF0 we obtain the charge carrier density in excellent agreement with experiment. We describe application of the equations for the evaluation of performance of concentrated solar thermionic energy converter (STEC) with emitter made of carbon nanotube for future applications.

Keywords: carbon nanotube, modified Richardson-Dushman equation, fermi energy at 0 K, charge carrier density

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Authors: Masood Otarod, Ronald M. Supkowski

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This abstract discusses a method that reduces the general dispersion model in cylindrical coordinates to a second order linear ordinary differential equation with constant coefficients so that it can be utilized to conduct kinetic studies in packed bed tubular catalytic reactors at a broad range of Reynolds numbers. The model was tested by 13CO isotope transient tracing of the CO adsorption of Boudouard reaction in a differential reactor at an average Reynolds number of 0.2 over Pd-Al2O3 catalyst. Detailed experimental results have provided evidence for the validity of the theoretical framing of the model and the estimated parameters are consistent with the literature. The solution of the general dispersion model requires the knowledge of the radial distribution of axial velocity. This is not always known. Hence, up until now, the implementation of the dispersion model has been largely restricted to the plug-flow regime. But, ideal plug-flow is impossible to achieve and flow regimes approximating plug-flow leave much room for debate as to the validity of the results. The reduction of the general dispersion model transpires as a result of the application of a factorization theorem. Factorization theorem is derived from the observation that a cross section of a catalytic bed consists of a solid phase across which the reaction takes place and a void or porous phase across which no significant measure of reaction occurs. The disparity in flow and the heterogeneity of the catalytic bed cause the concentration of reacting compounds to fluctuate radially. These variabilities signify the existence of radial positions at which the radial gradient of concentration is zero. Succinctly, factorization theorem states that a concentration function of axial and radial coordinates in a catalytic bed is factorable as the product of the mean radial cup-mixing function and a contingent dimensionless function. The concentration of adsorbed compounds are also factorable since they are piecewise continuous functions and suffer the same variability but in the reverse order of the concentration of mobile phase compounds. Factorability is a property of packed beds which transforms the general dispersion model to an equation in terms of the measurable mean radial cup-mixing concentration of the mobile phase compounds and mean cross-sectional concentration of adsorbed species. The reduced model does not require the knowledge of the radial distribution of the axial velocity. Instead, it is characterized by new transport parameters so denoted by Ωc, Ωa, Ωc, and which are respectively denominated convection coefficient cofactor, axial dispersion coefficient cofactor, and radial dispersion coefficient cofactor. These cofactors adjust the dispersion equation as compensation for the unavailability of the radial distribution of the axial velocity. Together with the rest of the kinetic parameters they can be determined from experimental data via an optimization procedure. Our data showed that the estimated parameters Ωc, Ωa Ωr, are monotonically correlated with the Reynolds number. This is expected to be the case based on the theoretical construct of the model. Computer generated simulations of methanation reaction on nickel provide additional support for the utility of the newly conceptualized dispersion model.

Keywords: factorization, general dispersion model, isotope transient kinetic, partial differential equations

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Authors: Fuyuan Ma, Yuhan Wang, Junhe Zhang, Ying Wang

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Identifying the sources of information dissemination is a pivotal task in the study of collective behaviors in networks, enabling us to discern and intercept the critical pathways through which information propagates from its origins. This allows for the control of the information’s dissemination impact in its early stages. Numerous methods for source detection rely on pre-existing, underlying propagation models as prior knowledge. Current models that eschew prior knowledge attempt to harness label propagation algorithms to model the statistical characteristics of propagation states or employ Graph Neural Networks (GNNs) for deep reverse modeling of the diffusion process. These approaches are either deficient in modeling the propagation patterns of information or are constrained by the over-smoothing problem inherent in GNNs, which limits the stacking of sufficient model depth to excavate global propagation patterns. Consequently, we introduce the ODESI model. Initially, the model employs a label propagation algorithm to delineate the distribution density of infected states within a graph structure and extends the representation of infected states from integers to state vectors, which serve as the initial states of nodes. Subsequently, the model constructs a deep architecture based on GNNs-coupled Ordinary Differential Equations (ODEs) to model the global propagation patterns of continuous propagation processes. Addressing the challenges associated with solving ODEs on graphs, we approximate the analytical solutions to reduce computational costs. Finally, we conduct simulation experiments on two real-world social network datasets, and the results affirm the efficacy of our proposed ODESI model in source identification tasks.

Keywords: source identification, ordinary differential equations, label propagation, complex networks

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Authors: Daniil Karzanov

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This paper studies the effect of quarterly earnings reports on the stock price. The profitability of the stock is modeled by geometric Brownian diffusion and the Constant Elasticity of Variance model. We fit several variations of stochastic differential equations to the pre-and after-report period using the Maximum Likelihood Estimation and Grid Search of parameters method. By examining the change in the model parameters after reports’ publication, the study reveals that the reports have enough evidence to be a structural breakpoint, meaning that all the forecast models exploited are not applicable for forecasting and should be refitted shortly.

Keywords: stock market, earnings reports, financial time series, structural breaks, stochastic differential equations

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Authors: Asabe Ahmad Tijani, Y. A. Yahaya

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The paper considers the derivation of implicit Adams-Moulton type method, with k=4 and 5. We adopted the method of interpolation and collocation of power series approximation to generate the continuous formula which was evaluated at off-grid and some grid points within the step length to generate the proposed block schemes, the schemes were investigated and found to be consistent and zero stable. Finally, the methods were tested with numerical experiments to ascertain their level of accuracy.

Keywords: Adam-Moulton Type (AMT), off-grid, block method, consistent and zero stable

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Authors: Lukas Vierus, Thomas Schuster

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A general framework is examined for dynamic tensor field tomography within an inhomogeneous medium characterized by refraction and absorption, treated as an inverse source problem concerning the associated transport equation. Guided by Fermat’s principle, the Riemannian metric within the specified domain is determined by the medium's refractive index. While considerable literature exists on the inverse problem of reconstructing a tensor field from its longitudinal ray transform within a static Euclidean environment, limited inversion formulas and algorithms are available for general Riemannian metrics and time-varying tensor fields. It is established that tensor field tomography, akin to an inverse source problem for a transport equation, persists in dynamic scenarios. Framing dynamic tensor tomography as an inverse source problem embodies a comprehensive perspective within this domain. Ensuring well-defined forward mappings necessitates establishing existence and uniqueness for the underlying transport equations. However, the bilinear forms of the associated weak formulations fail to meet the coercivity condition. Consequently, recourse to viscosity solutions is taken, demonstrating their unique existence within suitable Sobolev spaces (in the static case) and Sobolev-Bochner spaces (in the dynamic case), under a specific assumption restricting variations in the refractive index. Notably, the adjoint problem can also be reformulated as a transport equation, with analogous results regarding uniqueness. Analytical solutions are expressed as integrals over geodesics, facilitating more efficient evaluation of forward and adjoint operators compared to solving partial differential equations. Certainly, here's the revised sentence in English: Numerical experiments are conducted using a Nesterov-accelerated Landweber method, encompassing various fields, absorption coefficients, and refractive indices, thereby illustrating the enhanced reconstruction achieved through this holistic modeling approach.

Keywords: attenuated refractive dynamic ray transform of tensor fields, geodesics, transport equation, viscosity solutions

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Authors: Sabrina Ladjama, Aicha Rizi, Azzedine Abbaci

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In this work, we use the crossover model to formulate a comprehensive fundamental equation of state for the thermodynamic properties for several n-alkanes in the critical region that extends to the classical region. This equation of state is constructed on the basis of comparison of selected measurements of pressure-density-temperature data, isochoric and isobaric heat capacity. The model can be applied in a wide range of temperatures and densities around the critical point for n-heptane. It is found that the developed model represents most of the reliable experimental data accurately.

Keywords: crossover model, critical region, fundamental equation, n-heptane

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Authors: Sam Teek Ling, Norhashidah Hj. Mohd. Ali

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We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization. The methods are compared with the existing second order standard five point formula to show the dramatic improvement in computed accuracy. Numerical experiments are presented to illustrate the effectiveness of the proposed methods.

Keywords: explicit group iterative method, finite difference, fourth order compact, Poisson equation

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Authors: Adetunji A. Adeyanju., Mathew O. Omeike, Johnson O. Adeniran, Biodun S. Badmus

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In this paper, we discuss certain conditions for uniform asymptotic stability and uniform ultimate boundedness of solutions to some systems of Aizermann-type of differential equations by means of second method of Lyapunov. In achieving our goal, some Lyapunov functions are constructed to serve as basic tools. The stability results in this paper, extend some stability results for some Aizermann-type of differential equations found in literature. Also, we prove some results on uniform boundedness and uniform ultimate boundedness of solutions of systems of equations study.

Keywords: Aizermann, boundedness, first order, Lyapunov function, stability

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Authors: Clarinda V. Nhangumbe, Ercília Sousa

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Weather Derivatives are financial instruments used to cover non-catastrophic weather events and can be expressed in the form of standard or plain vanilla products, structured or exotics products. The underlying asset, in this case, is the weather index, such as temperature, rainfall, humidity, wind, and snowfall. The complexity of the Weather Derivatives structure shows the weakness of the Black Scholes framework. Therefore, under the risk-neutral probability measure, the option price of a weather contract can be given as a unique solution of a two-dimensional partial differential equation (parabolic in one direction and hyperbolic in other directions), with an initial condition and subjected to adequate boundary conditions. To calculate the price of the option, one can use numerical methods such as the Monte Carlo simulations and implicit finite difference schemes conjugated with Semi-Lagrangian methods. This paper is proposed two explicit methods, namely, first-order upwind in the hyperbolic direction combined with Lax-Wendroff in the parabolic direction and first-order upwind in the hyperbolic direction combined with second-order upwind in the parabolic direction. One of the advantages of these methods is the fact that they take into consideration the boundary conditions obtained from the financial interpretation and deal efficiently with the different choices of the convection coefficients.

Keywords: incomplete markets, numerical methods, partial differential equations, stochastic process, weather derivatives

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