Solution Of Such Models Research Articles

Correlation equation for evaluating energy consumption and process performance of brackish water desalination by electrodialysis

Electrodialysis (ED) is an electro-driven desalination technology that relies on the selective transport of ions through ion exchange membranes. Though several approaches have been developed to model and evaluate the performance of ED, mechanistic ion-transport models, which rigorously solve the fundamental Nernst-Planck equation, remain some of the most reliable and utilized. However, complexity of the involved transport phenomena prevents analytical solutions of such models, and numerical solutions can be prohibitively intensive. Here, we use an equivalent circuit analogue to derive a simple correlation equation that predicts the energetic performance of ED for brackish water desalination. Specifically, our correlation equation predicts the specific energy consumption of ED for a given productivity, set of desalination parameters (i.e., feed salinity, salt removal, water recovery), and system properties. The correlation equation demonstrates robustness in predicting the specific energy consumption across a wide range of operational parameters, showing excellent agreement with a Nernst-Planck ion-transport model and literature-reported experimental data. Furthermore, we use the developed correlation equation to show the dependence of the specific energy consumption on the productivity, highlighting the tradeoff between the thermodynamic energy efficiency and desalination rate of the ED process. Overall, our developed correlation equation provides a convenient alternative to computationally intensive mechanistic models for performance analysis of the ED process.

Numerical study for unsteady Casson fluid flow with heat flux using a spectral collocation method

There is a crucial necessity for enhancing the efficiency of the solution of problem describing the non-Newtonian fluid flow over a stretching sheet to improve proficient the mechanism of heat transfer in numerous practical applications. For this aim, an efficient spectral collocation method is implemented in this paper to present the numerical solution for the flow and heat transfer of a non-Newtonian Casson model together with the presence of viscous dissipation and variable heat flux, with reference herein to the slip velocity and the heat generation or absorption condition. The absence of the magnetic field phenomenon in some studies is an essential restriction in the processes of development for the energy-efficient heat transfer mechanism that is desired in numerous industrial applications. So, the magnetic field is implemented here for this Casson model. Industrially, this type of fluid can describe the flow of blood in an industrial artery, which can be polished by a material governing the blood flow. The spectral collocation method based on Chebyshev polynomials of the third kind is employed to solve the resulting system of ODEs which describes the problem. This scientific study encompasses many parameters such as unsteadiness parameter, slip velocity parameter, Casson parameter, local Eckert number, heat generation parameter, and the Prandtl number. From this study, it has been consummated that the Casson model is superior to others that are much used as the industrial fluid. Finally, the given results show that the spectral collocation method is an easy and efficient tool to investigate the solution for such models.

Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

Highly accurate technique for studying some chaotic models described by ABC-fractional differential equations of variable-order

The propose of this paper is to introduce and investigate a highly accurate technique for solving the fractional Logistic and Ricatti differential equations of variable-order. We consider these models with the most common nonsingular Atangana–Baleanu–Caputo (ABC) fractional derivative which depends on the Mittag–Leffler kernel. The proposed numerical technique is based upon the fundamental theorem of the fractional calculus as well as the Lagrange polynomial interpolation. We satisfy the efficiency and the accuracy of the given procedure; and study the effect of the variation of the fractional-order [Formula: see text] on the behavior of the solutions due to the presence of ABC-operator by evaluating the solution with different values of [Formula: see text]. The results show that the given procedure is an easy and efficient tool to investigate the solution for such models. We compare the numerical solutions with the exact solution, thereby showing excellent agreement which we have found by applying the ABC-derivatives. We observe the chaotic solutions with some fractional-variable-order functions.

A Guide on Solving Non-convex Consumption-Saving Models

Consumption-saving models with adjustment costs or discrete choices are typically hard to solve numerically due to the presence of non-convexities. This paper provides a number of tools to speed up the solution of such models. Firstly, I use that many consumption models have a nesting structure implying that the continuation value can be efficiently pre-computed and the consumption choice solved separately before the remaining choices. Secondly, I use that an endogenous grid method extended with an upper envelope step can be used to solve efficiently for the consumption choice. Thirdly, I use that the required pre-computations can be optimized by a novel loop reordering when interpolating the next-period value function. As an illustrative example, I solve a model with non-durable consumption and durable consumption subject to adjustment costs. Combining the provided tools, the model is solved almost 50 times faster than with standard value function iteration for a given level of accuracy. Software is provided in both Python and C++.

Quasi-Invariants of Chemical Reactions in the Ideal Displacement Reactor

Mathematical models of nonequilibrium processes in distributed systems are described by partial differential equations with given boundary conditions and, as a rule, cannot be solved analytically. The absence of exact solutions of such models does not allow one to find exact invariants; i.e., functions are strictly constant on these solutions. This article has developed a method for determining the approximate spatiotemporal kinetic invariants (quasi-invariants) of chemical reactions that do not take into account diffusion in an ideal displacement reactor. Spacetime quasi-invariants are explicit algebraic expressions that relate the nonequilibrium values of the concentrations of the reagents and/or temperatures measured in two or more experiments with various specially selected initial and input conditions (multiexperiments). The selection of suitable initial conditions makes it possible to find such functions of solutions that weakly depend on the time and length of the reactor, i.e., remain almost constant throughout the reaction in the entire reactor volume. It is shown that the number of quasi-invariants is determined by the number of independent model variables (reagent concentrations and temperature). The application of the method is illustrated by the example of a one-stage reaction occurring in a reactor of ideal displacement. The quasi-invariant 3D surfaces found for this reaction are compared with 3D plots of concentration and temperature changes during the reaction. It is shown that quasi-invariants vary in a smaller range (along the ordinate axis) than the corresponding concentrations and temperatures in different experiments, i.e., are spacetime approximate invariants. Quasi-invariants of this kind can be observed on the dependences of the model variables (concentrations and temperatures) on the time and length of the reactor in the form of various flattened three-dimensional structures (“lying” waves). The method expands the tools for analyzing the unsteady kinetics of chemical reactions in distributed systems of ideal displacement.

Numerical treatment for studying the blood ethanol concentration systems with different forms of fractional derivatives

The purpose of this paper is to implement an approximate method for obtaining the solution of a physical model called the blood ethanol concentration system. This model can be expressed by a system of fractional differential equations (FDEs). Here, we will consider two forms of the fractional derivative namely, Caputo (with singular kernel) and Atangana–Baleanu–Caputo (ABC) (with nonsingular kernel). In this work, we use the spectral collocation method based on Chebyshev approximations of the third-kind. This procedure converts the given model to a system of algebraic equations. The implementation of the proposed method to solve fractional models in ABC-sense is the first time. We satisfy the efficiency and the accuracy of the given procedure by evaluating the relative errors. The results show that the implemented technique is an easy and efficient tool to simulate the solution of such models.

Dynamics of fractional-order delay differential model for tumor-immune system

Abstract Time-delays and fractional-order play a vital role in modeling biological systems with memory. In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. Non-negativity of the solution of such model has been investigated. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two differenttumor time-delays τ1 and τ2. The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ 1 * , or τ 2 * . Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model.

Cointegrated continuous-time linear state-space and MCARMA models

In this paper, we define and characterize cointegrated solutions of continuous-time linear state-space models driven by Lévy processes. Cointegrated solutions of such models are shown to be representable as a sum of a Lévy process and a stationary solution of a linear state-space model, analogous to the Granger representation for cointegrated VAR models. Moreover, we prove that the class of cointegrated multivariate Lévy-driven autoregressive moving-average (MCARMA) processes, the continuous-time analogues of the classical vector ARMA processes, is equivalent to the class of cointegrated solutions of continuous-time linear state-space models. Necessary conditions for MCARMA processes to be cointegrated are given as well extending the results of Comte [Discrete and continuous time cointegration, J. Econometrics 88 (1999), pp. 207–226] for MCAR processes. The conditions depend only on the autoregressive polynomial if we have a minimal model. Finally, we investigate cointegrated continuous-time linear state-space models observed on a discrete time-grid and calculate their linear innovations. Based on the representation of the linear innovations, we derive an error correction form. The error correction form uses an infinite linear filter in contrast to the finite linear filter for VAR models. Some sufficient identifiable criteria are also given.

Flow in a two dimensional channel with deforming and peristaltically moving walls

In this paper, effects of deforming walls on peristaltic flow in a two dimensional channel have been investigated. The two dimensional form of the governing equations is simplified by using appropriate transformations and well established approximations, which are used extensively for solution of such models. The transformations are designed so that the complex problem is reduced into an ordinary differential equation (ODE). New and simple non-linear ODE is formed in view of adopted procedures and techniques. Its solutions are exactly matched with the solutions of classical problems. Solutions of the final problem are provided for small values of the surface expansion (contraction) ratio and Reynolds number with the help of the perturbation technique and non-linear shooting method. The velocity field, pressure and shear stress are evaluated analytically and numerically. Meanwhile, effects of all parameters are observed on the velocity field, pressure rise per wavelength and shear stress profiles with the help of tables and different figures. Excellent agreement between solutions is found. Current results are apparently matched with the classical problems of peristaltic flow in deforming and non-deforming walls.

The novel approach for electric power system simulation tools validation

The reliability and efficiency of electric power system (EPS) operation depends on the adequacy of the information about processes in the EPS. The complexity of obtaining this information lies in the need to use large-scale EPS mathematical model for calculations. Within the framework of numerical integration, a satisfactory solution of such model is hardly to be ensured; consequently, the simplifications and limitations are applied. In this regard, every obtained solution should be validated. The most reliable way of validation is to compare simulation results with field data. However, it is not always possible to receive the necessary amount of field data. The paper proposes an alternative approach for validation: the adequate reference model (RM) usage instead of field data. The prototype of hybrid real-time power system simulator having the necessary properties and capabilities is used as the RM in this paper. The adequacy of proposed approach is illustrated by the fragments of the experimental studies.

Exact results on the large-scale stochastic transport of inertial particles including the Basset history term

The Maxey-Riley equation and its simplified versions represent the most widespread tool to investigate dynamics and dispersion of inertial small particles in turbulent flows. The numerical solution of such models is often very challenging, and some of their terms, such as the molecular diffusivity or the Basset history force, are often neglected to reduce the complexity upon suitable approximations. Here, we propose exact results with regard to the rate of transport on large time scales in random shear flows. These can be expediently used as a benchmark to develop and assess algorithms when solving this class of stochastic integrodifferential problems on large time scales.

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [-1,1]. For the evaluation of the integral term efficiently a Legendre–Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed.

Structure and thermodynamics of charged nonrotating black holes in higher dimensions

We analyze the structural and thermodynamic properties of $D$-dimensional ($D \geq 4$), asymptotically flat or Anti-de-Sitter, electrically charged black hole solutions, resulting from the minimal coupling of general nonlinear electrodynamics to General Relativity. This analysis deals with static spherically symmetric (elementary) configurations with spherical horizons. Our methods are based on the study of the behaviour (in vacuum and on the boundary of their domain of definition) of the Lagrangian density functions characterizing the nonlinear electrodynamic models in flat spacetime. These functions are constrained by some admissibility conditions endorsing the physical consistency of the corresponding theories, which are classified in several families, some of them supporting elementary solutions in flat space which are non topological solitons. This classification induces a similar one for the elementary black hole solutions of the associated gravitating nonlinear electrodynamics, whose geometrical structures are thoroughly explored. A consistent thermodynamic analysis can be developed for the subclass of families whose associated black hole solutions behave asymptotically as the Schwarzschild metric (in absence of a cosmological term). In these cases we obtain the behaviour of the main thermodynamic functions, as well as important finite relations among them. In particular, we find the general equation determining the set of extreme black holes for every model, and a general Smarr formula, valid for the set of elementary black hole solutions of such models. We also consider the one-parameter group of scale transformations, which are symmetries of the field equations of any nonlinear electrodynamics in flat spacetime.

Simple modeling techniques for base-stock inventory systems with state dependent demand rates

In this paper new modeling techniques of (S-1,S) inventory systems (continuous review base-stock inventory systems) with state dependent demand rates are proposed. Examples of single-location (S-1,S) inventory systems where the demand, experienced by the system, varies due to the state of the system are, e.g., inventory models with partial backorders, inventory models with lost sales, inventory models with perishable items, inventory models with emergency replenishments etc. Models of such inventory systems are in general hard to solve due to the fact that the Markov property is often lost, and the prevalent tool used in the literature for providing exact solutions of such models is the theory of partial differential equations. Instead of using partial differential equations with rather complicated analysis of boundary conditions, we suggest considerably simpler techniques which are based on elementary theory of queueing and renewal processes. First, we show that it is possible to use Markov theory in order to prove certain statistical properties of the limiting distribution of the ages of the items in the system. Secondly, we develop a corresponding procedure based on renewal theory, which forms a basis for more complicated models assuming non-Poisson customer demand processes.

Quenching Phenomenon of a Time-Fractional Kawarada Equation

In this paper, we introduce a class of time-fractional diffusion model with singular source term. The derivative employed in this model is defined in the Caputo sense to fit the conventional initial condition. With assistance of corresponding linear fractional differential equation, we verify that the solution of such model may not be globally well-defined, and the dynamics of this model depends on the order of fractional derivative and the volume of spatial domain. In simulation, a finite difference scheme is implemented and interesting numerical solutions of model are illustrated graphically. Meanwhile, the positivity, monotonicity, and stability of the proposed scheme are proved. Numerical analysis and simulation coincide the theoretical studies of this new model.

Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion π over space–time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, π is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are “twisted” by a flat symplectic vector bundle defined over the total space of π. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of π is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of π that global solutions of such models can be interpreted as classical “locally-geometric” U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are “locally non-geometric”.

几何稳定过程的性质

本文主要研究几何稳定过程的性质。首先，我们得到了由𝛼-𝑠𝑡𝑎𝑏𝑙𝑒过程驱动的几何稳定过程并给出几何稳定过程的解。其次，证明了在随机噪声较大时，几何稳定过程的解几乎处处以指数速率趋近于零。 The main purpose of this paper is to investigate the properties of geometric stable process. First, a model driven by 𝛼-𝑠𝑡𝑎𝑏𝑙𝑒 process is present. We obtain the solution of such model. Then, we prove that if the noise is sufficiently large, the solution of the geometric stable process will tend to zero at an exponential rate with probability one.

Digital Wave Simulation of Quasi-Static Partial Element Equivalent Circuit Method

The partial element equivalent circuit (PEEC) method is a well-established technique for obtaining a circuit equivalent for an electromagnetic problem. The time domain solution of such models is usually performed using nodal voltages and branch currents, or sometimes charge and currents. The present paper describes a possible alternative approach, which can be obtained expressing and solving the problem in the waves domain. The digital wave theory is used to find an equivalent representation of the PEEC circuit in the wave domain. Through a pertinent continuous to discrete time transformation, the constitutive relations for partial inductances, capacitances, and resistances are translated in an explicit form. The combination of such equations with Kirchhoff laws allows to achieve a semiexplicit resolution scheme. Three different physical configurations are analyzed and their extracted digital wave PEEC models are simulated at growing sizes using the general-purpose digital wave simulator. The results are compared to those obtained by using standard SPICE simulators in both linear and nonlinear cases. When the size of the model is manageable by SPICE, an excellent accuracy and a speed-up factor of up to three orders of magnitude are observed with much lower memory requirements. A comparative analysis of results including the effect of parameters like the simulation time step choice is also presented.

Modeling of drying process of wood raw material for its subsequent use at fish smoking

Models of many technological processes (drying, Smoking and drying fish) are described by equations. Such models and applies the model of drying process of raw materials for example wood drying. The article considered a model of drying of raw materials based on the heat equation with variable coefficients. Physical properties of wood during the drying process change, and therefore the equation coefficients are changed according to some given law. The model becomes essentially nonlinear. As a rule, to obtain the solution for such models explicitly impossible. In this case, it becomes the actual task of developing numerical methods for solving model equations and writing of appropriate programs for computer simulation of technological processes. The existence of such programs allows researchers to solve the problem of selecting the optimal parameters and the optimum conditions of the studied technological processes to determine the time required to achieve the desired values of the objective functions. Based on the calculation formulas developed simulation program of the thermal field in MatLab that can be used to simulate the drying process under different conditions. The article provides examples of model calculations for different values of parameters of raw materials. Processes associated with heating or drying of the feedstock, are observed when Smoking or drying fish. The suggested algorithm can be, after some modifications, used for their modeling.