Search results for: non-local in time equations

Authors: Geeta Partap, Nitika Chugh

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The present paper deals with the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler-Bernoulli beam resonators, due to Exponential time varying load. Both the axial ends of the beam are assumed to be at simply supported conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection.The analytical results have been numerically analyzed with the help of MATLAB software for magnesium like material. The graphical representations and interpretations have been discussed for Deflection of beam under Simply Supported boundary condition and for distinct considered values of time and space as well. The obtained results are easy to implement for engineering analysis and designs of resonators (sensors), modulators, actuators.

Keywords: microstretch, deflection, exponential load, Laplace transforms, residue theorem, simply supported

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Authors: Bolatbek Rysbaiuly, Aliya S. Azhibekova

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Problems concerning the interpretation of the well testing results belong to the class of inverse problems of subsurface hydromechanics. The distinctive feature of such problems is that additional information is depending on the capabilities of oilfield experiments. Another factor that should not be overlooked is the existence of errors in the test data. To determine reservoir properties, some inverse problems for parabolic equations were investigated. An approach to solving the inverse problems based on the method of regularization is proposed.

Keywords: iterative approach, inverse problem, parabolic equation, reservoir properties

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Authors: Phuoc Si Nguyen

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Frequency transformation with Pascal matrix equations is a method for transforming an electronic filter (analogue or digital) into another filter. The technique is based on frequency transformation in the s-domain, bilinear z-transform with pre-warping frequency, inverse bilinear transformation and a very useful application of the Pascal’s triangle that simplifies computing and enables calculation by hand when transforming from one filter to another. This paper will introduce two methods to transform a filter into a digital filter: frequency transformation from the s-domain into the z-domain; and frequency transformation in the z-domain. Further, two Pascal matrix equations are derived: an analogue to digital filter Pascal matrix equation and a digital to digital filter Pascal matrix equation. These are used to design a desired digital filter from a given filter.

Keywords: frequency transformation, bilinear z-transformation, pre-warping frequency, digital filters, analog filters, pascal’s triangle

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Authors: Charline David, Alexandre Blondin Massé, Arnaud Zinflou

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In 2016, Clements, Hurn, and Li proposed a multiple equation time series approach for the short-term load forecasting, reporting an average mean absolute percentage error (MAPE) of 1.36% on an 11-years dataset for the Queensland region in Australia. We present an adaptation of their model to the electrical power load consumption for the whole Quebec province in Canada. More precisely, we take into account two additional meteorological variables — cloudiness and wind speed — on top of temperature, as well as the use of multiple meteorological measurements taken at different locations on the territory. We also consider other minor improvements. Our final model shows an average MAPE score of 1:79% over an 8-years dataset.

Keywords: short-term load forecasting, special days, time series, multiple equations, parallelization, clustering

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Authors: Abhilash Sahu, Amit Priyadarshi

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In this paper, we will discuss the existence of weak solutions of the Kirchhoff type boundary value problem on the Sierpinski gasket. Where S denotes the Sierpinski gasket in R² and S₀ is the intrinsic boundary of the Sierpinski gasket. M: R → R is a positive function and h: S × R → R is a suitable function which is a part of our main equation. ∆p denotes the p-Laplacian, where p > 1. First of all, we will define a weak solution for our problem and then we will show the existence of at least two solutions for the above problem under suitable conditions. There is no well-known concept of a generalized derivative of a function on a fractal domain. Recently, the notion of differential operators such as the Laplacian and the p-Laplacian on fractal domains has been defined. We recall the result first then we will address the above problem. In view of literature, Laplacian and p-Laplacian equations are studied extensively on regular domains (open connected domains) in contrast to fractal domains. In fractal domains, people have studied Laplacian equations more than p-Laplacian probably because in that case, the corresponding function space is reflexive and many minimax theorems which work for regular domains is applicable there which is not the case for the p-Laplacian. This motivates us to study equations involving p-Laplacian on the Sierpinski gasket. Problems on fractal domains lead to nonlinear models such as reaction-diffusion equations on fractals, problems on elastic fractal media and fluid flow through fractal regions etc. We have studied the above p-Laplacian equations on the Sierpinski gasket using fibering map technique on the Nehari manifold. Many authors have studied the Laplacian and p-Laplacian equations on regular domains using this Nehari manifold technique. In general Euler functional associated with such a problem is Frechet or Gateaux differentiable. So, a critical point becomes a solution to the problem. Also, the function space they consider is reflexive and hence we can extract a weakly convergent subsequence from a bounded sequence. But in our case neither the Euler functional is differentiable nor the function space is known to be reflexive. Overcoming these issues we are still able to prove the existence of at least two solutions of the given equation.

Keywords: Euler functional, p-Laplacian, p-energy, Sierpinski gasket, weak solution

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Authors: Abdul Hakim Chikho

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Most international codes allow the use of an equivalent lateral load method for designing practical buildings to withstand earthquake actions. This method requires calculating an approximation to the fundamental mode period of vibrations of these buildings. Several empirical equations have been suggested to calculate approximations to the fundamental periods of different types of structures. Most of these equations are knowing to provide an only crude approximation to the required fundamental periods and repeating the calculation utilizing a more accurate formula is usually required. In this paper, a new formula to calculate a satisfactory approximation of the fundamental period of a practical building is proposed. This formula takes into account the mass and the stiffness of the building therefore, it is more logical than the conventional empirical equations. In order to verify the accuracy of the proposed formula, several examples have been solved. In these examples, calculating the fundamental mode periods of several farmed buildings utilizing the proposed formula and the conventional empirical equations has been accomplished. Comparing the obtained results with those obtained from a dynamic computer has shown that the proposed formula provides a more accurate estimation of the fundamental periods of practical buildings. Since the proposed method is still simple to use and requires only a minimum computing effort, it is believed to be ideally suited for design purposes.

Keywords: earthquake, fundamental mode period, design, building

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Authors: Mohammad Sahadet Hossain, Shamsil Arifeen, Mehrab Hossian Likhon

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In this paper, we analyze a linear time invariant (LTI) descriptor system of large dimension. Since these systems are difficult to simulate, compute and store, we attempt to reduce this large system using Low Rank Cholesky Factorized Alternating Directions Implicit (LRCF-ADI) iteration followed by Square Root Balanced Truncation. LRCF-ADI solves the dual Lyapunov equations of the large system and gives low-rank Cholesky factors of the gramians as the solution. Using these cholesky factors, we compute the Hankel singular values via singular value decomposition. Later, implementing square root balanced truncation, the reduced system is obtained. The bode plots of original and lower order systems are used to show that the magnitude and phase responses are same for both the systems.

Keywords: low-rank cholesky factor alternating directions implicit iteration, LTI Descriptor system, Lyapunov equations, Square-root balanced truncation

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Authors: Utkan Caliskan

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Two types of numerical codes based on finite volume method are developed in order to solve compressible Euler equations to simulate the flow through forward facing step channel. Both algorithms have AUSM+- up (Advection Upstream Splitting Method) scheme for flux splitting and two-stage Runge-Kutta scheme for time stepping. In this study, the flux calculations differentiate between the algorithm based on OpenFOAM mesh format which is called 'face-based' algorithm and the basic algorithm which is called 'element-based' algorithm. The face-based algorithm avoids redundant flux computations and also is more flexible with hybrid grids. Moreover, some of OpenFOAM’s preprocessing utilities can be used on the mesh. Parallelization of the face based algorithm for which atomic operations are needed due to the shared memory model, is also presented. For several mesh sizes, 2.13x speed up is obtained with face-based approach over the element-based approach.

Keywords: cell centered finite volume method, compressible Euler equations, OpenFOAM mesh format, OpenMP

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Authors: Ali Kezmane, Said Boukais, Mohand Hamizi

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This paper presents an analytical study on the behavior of reinforced concrete walls with rectangular cross section. Several experiments on such walls have been selected to be studied. Database from various experiments were collected and nominal shear wall strengths have been calculated using formulas, such as those of the ACI (American), NZS (New Zealand), Mexican (NTCC), and Wood and Barda equations. Subsequently, nominal shear wall strengths from the formulas were compared with the ultimate shear wall strengths from the database. These formulas vary substantially in functional form and do not account for all variables that affect the response of walls. There is substantial scatter in the predicted values of ultimate shear strength. Two new semi empirical equations are developed using data from tests of 57 walls for transitions walls and 27 for slender walls with the objective of improving the prediction of peak strength of walls with the most possible accurate.

Keywords: shear strength, reinforced concrete walls, rectangular walls, shear walls, models

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Authors: Hanan Rizk

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A heat exchanger is a device used to mix liquids having different temperatures. In this case, the temperature control becomes a critical objective. This research work presents the temperature control of the double-pipe heat exchanger (multi-input multi-output (MIMO) system), which is modeled as first-order coupled hyperbolic partial differential equations (PDEs), using conventional and advanced control techniques and develops appropriate robust control strategy to meet stability requirements and performance objectives. We designed a PID controller and H-infinity controller for a heat exchanger (HE) system. Frequency characteristics of sensitivity functions and open-loop and closed-loop time responses are simulated using MATLAB software, and the stability of the system is analyzed using Kalman's test. The simulation results have demonstrated that the H-infinity controller is more efficient than PID in terms of robustness and performance.

Keywords: heat exchanger, multi-input multi-output system, MATLAB simulation, partial differential equations, PID controller, robust control

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Authors: Yilmaz Simsek

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In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values.

Keywords: generating functions, Bernstein basis functions, Bernstein polynomials, Bezier curves, differential equations

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Authors: Boguslaw Schreyer

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In this paper, a general dynamical model is derived using the Lagrange formalism. The two masses: sprang and unsprang are included in a six-degree of freedom model for a sprung mass. The unsprung mass is included and shown only in a simplified model, although its equations have also been derived by an author. The simplified equations, more suitable for the computer model of robot’s dynamics are also shown.

Keywords: dynamics, mobile, robot, wheeled mobile robots

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Authors: Keltoum Bouherine, Olivier Leroy

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The present work is dedicated to study a three–dimensional (3D) self-consistent fluid simulation of microwave discharges of argon plasma in PECVD reactor. The model solves the Maxwell’s equations, continuity equations for charged species and the electron energy balance equation, coupled with Poisson’s equation, and Navier-Stokes equations by finite element method, using COMSOL Multiphysics software. In this study, the simulations yield the profiles of plasma components as well as the charge densities and electron temperature, the electric field, the gas velocity, and gas temperature. The results show that the microwave plasma reactor is outside of local thermodynamic equilibrium.The present work is dedicated to study a three–dimensional (3D) self-consistent fluid simulation of microwave discharges of argon plasma in PECVD reactor. The model solves the Maxwell’s equations, continuity equations for charged species and the electron energy balance equation, coupled with Poisson’s equation, and Navier-Stokes equations by finite element method, using COMSOL Multiphysics software. In this study, the simulations yield the profiles of plasma components as well as the charge densities and electron temperature, the electric field, the gas velocity, and gas temperature. The results show that the microwave plasma reactor is outside of local thermodynamic equilibrium.

Keywords: electron density, electric field, microwave plasma reactor, gas velocity, non-equilibrium plasma

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Authors: Ather Saeed, Arif Khan, Jeffrey Gosper

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Sensor devices are prone to errors and sudden node failures, which are difficult to detect in a timely manner when deployed in real-time, hazardous, large-scale harsh environments and in medical emergencies. Therefore, the loss of data can be life-threatening when the sensed phenomenon is not disseminated due to sudden node failure, battery depletion or temporary malfunctioning. We introduce a set of partial differential equations for localizing faults, similar to Green’s and Maxwell’s equations used in Electrostatics and Electromagnetism. We introduce a node organization and clustering scheme for self-stabilizing sensor networks. Green’s theorem is applied to regions where the curve is closed and continuously differentiable to ensure network connectivity. Experimental results show that the proposed GTFD (Green’s Theorem fault-detection and Self-stabilization) protocol not only detects faulty nodes but also accurately generates network stability graphs where urgent intervention is required for dynamically self-stabilizing the network.

Keywords: Green’s Theorem, self-stabilization, fault-localization, RSSI, WSN, clustering

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Authors: Jorge Gonzalez Camus, Valentin Keyantuo, Mahamadi Warma

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In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula.

Keywords: discrete fractional Laplacian, explicit representation of solutions, fractional heat and wave equations, fundamental

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Authors: Eugene Stepanov, Arcady Ponosov

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To model gene regulatory networks one uses ordinary differential equations with switching nonlinearities, where the initial value problem is known to be well-posed if the trajectories cross the discontinuities transversally. Otherwise, the initial value problem is usually ill-posed, which lead to theoretical and numerical complications. In the presentation, it is proposed to apply the theory of hybrid dynamical systems, rather than switched ones, to regularize the problem. 'Hybridization' of the switched system means that one attaches a dynamic discrete component ('automaton'), which follows the trajectories of the original system and governs its dynamics at the points of ill-posedness of the initial value problem making it well-posed. The construction of the automaton is based on the classification of the attractors of the specially designed adjoint dynamical system. Several examples are provided in the presentation, which support the suggested analysis. The method can also be of interest in other applied fields, where differential equations contain switchings, e.g. in neural field models.

Keywords: hybrid dynamical systems, ill-posed problems, singular perturbation analysis, switching nonlinearities

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Authors: Mujde Turkkan, Nurkan Yagiz

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In this study an active controller is presented for vibration suppression of a full-bus model. The bus is modelled having seven degrees of freedom. Using the achieved model via Lagrange Equations the system equations of motion are derived. The suspensions of the bus model include air springs with two auxiliary chambers are used. Fuzzy logic controller is used to improve the ride comfort. The numerical results, verifies that the presented fuzzy logic controller improves the ride comfort.

Keywords: ride comfort, air spring, bus, fuzzy logic controller

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Authors: Amal Machtalay

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In this survey, we aim to review the rapidly growing literature on numerical methods to solve different forms of mean field problems, namely mean field games (MFG), mean field controls (MFC), potential MFGs, and master equations, as well as their corresponding recent applications. Here, we distinguish two families of numerical methods: iterative methods based on mesh generation and those called mesh-free, normally related to neural networking and learning frameworks.

Keywords: mean-field games, numerical schemes, partial differential equations, complex systems, machine learning

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Authors: Artion Kashuri, Rozana Liko

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In this paper, the authors establish an integral identity using delta differentiable functions. By applying this identity, some new results via a general class of convex functions with respect to two nonnegative functions on a time scale are given. Also, for suitable choices of nonnegative functions, some special cases are deduced. Finally, in order to illustrate the efficiency of our main results, some applications to special means are obtained as well. We hope that current work using our idea and technique will attract the attention of researchers working in mathematical analysis, mathematical inequalities, numerical analysis, special functions, fractional calculus, quantum mechanics, quantum calculus, physics, probability and statistics, differential and difference equations, optimization theory, and other related fields in pure and applied sciences.

Keywords: convex functions, Hermite-Hadamard inequality, special means, time scale

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Authors: Mohammadreza Akbari, Pooya Soleimani Besheli, Reza khalili, Sara Akbari, Davood Domiri Ganji

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In this symposium, our aims are accuracy, capabilities and power at solving of the complicate non-linear differential at the reaction chemical in the catalyst reactor (heterogeneous reaction). Our purpose is to enhance the ability of solving the mentioned nonlinear differential equations at chemical engineering and similar issues with a simple and innovative approach which entitled ‘’Akbari-Ganji's Method’’ or ‘’AGM’’. In this paper we solve many examples of nonlinear differential equations of chemical reactions and its investigate. The chemical reactor with the energy changing (non-isotherm) in two reactors of mixed and plug are separately studied and the nonlinear differential equations obtained from the reaction behavior in these systems are solved by a new method. Practically, the reactions with the energy changing (heat or cold) have an important effect on designing and function of the reactors. This means that possibility of reaching the optimal conditions of operation for the maximum conversion depending on nonlinear nature of the reaction velocity toward temperature, results in the complexity of the operation in the reactor. In this case, the differential equation set which governs the reactors can be obtained simultaneous solution of mass equilibrium and energy and temperature changing at concentration.

Keywords: new method (AGM), nonlinear differential equation, tubular and mixed reactors, catalyst bed

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Authors: Ainul Haque, Ameeye Kumar Nayak

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Flow enhancement and species transport in a slit hydrophobic microchannel is studied for non-Newtonian fluids with the externally imposed electric field and pressure gradient. The incompressible Poisson-Nernst-Plank equations and the Navier-Stokes equations are approximated by lubrication theory to quantify the flow structure due to hydrophobic and hydrophilic surfaces. The analytical quantification of velocity and pressure of electroosmotic flow (EOF) is made with the numerical results due to the staggered grid based finite volume method for flow governing equations. The resistance force due to fluid friction and shear force along the surface are decreased by the hydrophobicity, enables the faster movement of fluid particles. The resulting flow enhancement factor Ef is increased with the low viscous fluid and provides maximum species transport. Also, the analytical comparison of EOF with pressure driven EOF justifies the flow enhancement due to hydrophobicity and shear impact on flow variation.

Keywords: electroosmotic flow, hydrophobic surface, power-law fluid, shear effect

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Authors: Alex Fedoseyev

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This study explores Generalized Hydrodynamic Equations (GHE) for the simulation of turbulent flows. The GHE was derived from the Generalized Boltzmann Equation (GBE) by Alexeev (1994). GBE was obtained by first principles from the chain of Bogolubov kinetic equations and considered particles of finite dimensions, Alexeev (1994). The GHE has new terms, temporal and spatial fluctuations compared to the Navier-Stokes equations (NSE). These new terms have a timescale multiplier τ, and the GHE becomes the NSE when τ is zero. The nondimensional τ is a product of the Reynolds number and the squared length scale ratio, τ=Re*(l/L)², where l is the apparent Kolmogorov length scale, and L is a hydrodynamic length scale. The turbulence phenomenon is not well understood and is not described by NSE. An additional one or two equations are required for the turbulence model, which may have to be tuned for specific problems. We show that, in the case of the GHE, no additional turbulence model is needed, and the turbulent velocity profile is obtained from the GHE. The 2D turbulent channel and circular pipe flows were investigated using a numerical solution of the GHE for several cases. The solutions are compared with the experimental data in the circular pipes and 2D channels by Nicuradse (1932, Prandtl Lab), Hussain and Reynolds (1975), Wei and Willmarth (1989), Van Doorne (2007), theory by Wosnik, Castillo and George (2000), and the relevant experiments on Superpipe setup at Princeton, data by Zagarola (1996) and Zagarola and Smits (1998), the Reynolds number is from Re=7200 to Re=960000. The numerical solution data compared well with the experimental data, as well as with the approximate analytical solution for turbulent flow in channel Fedoseyev (2023). The obtained results confirm that the Alexeev generalized hydrodynamic theory (GHE) is in good agreement with the experiments for turbulent flows. The proposed approach is limited to 2D and 3D axisymmetric channel geometries. Further work will extend this approach by including channels with square and rectangular cross-sections.

Keywords: comparison with experimental data. generalized hydrodynamic equations, numerical solution, turbulent boundary layer, turbulent flow in channel

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Authors: Mohamed Suleiman, Zarina Bibi Ibrahim, Nor Ain Azeany, Khairil Iskandar Othman

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In this paper, a class of variable order fully implicit multistep Block Backward Differentiation Formulas (VOBBDF) using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs) is developed. The code will combine three multistep block methods of order four, five and six. The order selection is based on approximation of the local errors with specific tolerance. These methods are constructed to produce two approximate solutions simultaneously at each iteration in order to further increase the efficiency. The proposed VOBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with single order Block Backward Differentiation Formula (BBDF). Numerical results shows the advantage of using VOBBDF for solving ODEs.

Keywords: block backward differentiation formulas, uniform step size, ordinary differential equations

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Authors: Nilay K. Doshi

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A theoretical probe describing the excited energy states of the electron density surrounding a nanoparticle (NP) is presented. An electromagnetic (EM) wave interacts with a NP much smaller than the incident wavelength. The plasmon that oscillates locally around the NP comprises of excited conduction electrons. The system is based on the Jellium model of a cluster of metal atoms. Hohenberg-Kohn (HK) equations and the variational Kohn-Sham (SK) scheme have been used to obtain the NP electron density in the ground state. Furthermore, a time-dependent density functional (TDDFT) theory is used to treat the excited states in a density functional theory (DFT) framework. The non-interacting fermionic kinetic energy is shown to be a functional of the electron density. The time dependent potential is written as the sum of the nucleic potential and the incoming EM field. This view of the quantum oscillation of the electron density is a part of the localized surface plasmon resonance.

Keywords: electron density, energy, electromagnetic, DFT, TDDFT, plasmon, resonance

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Authors: Youb Said, Fourar Ali

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To describe the influence of bottom roughness on the free surface flows by numerical modeling, a two-dimensional model was developed. The equations of continuity and momentum (Naviers Stokes equations) are solved by the finite volume method. We considered a turbulent flow in an open channel with a bottom roughness. For our simulations, the K-ε model was used. After setting the initial and boundary conditions and solve the equations set, we were able to achieve the following results: vortex forming in the hollow causing substantial energy dissipation in the obstacle areas that form the bottom roughness. The comparison of our results with experimental ones shows a good agreement in terms of the results in the rough area. However, in other areas, differences were more or less important. These differences are in areas far from the bottom, especially the free surface area just after the bottom. These disagreements are probably due to experimental constants used by the k-ε model.

Keywords: modeling, free surface flow, turbulence, bottom roughness, finite volume, K-ε model, energy dissipation

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Authors: A. Abbasi Moshaii, M. Soltan Rezaee, M. Mohammadi Moghaddam

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Control of some mechanisms is hard because of their complex dynamic equations. If part of the complexity is resulting from uncertainties, an efficient way for solving that is robust control. By this way, the control procedure could be simple and fast and finally, a simple controller can be designed. One kind of these mechanisms is 3-RRR which is a parallel mechanism and has three revolute joints. This paper aims to robust control a 3-RRR planner mechanism and it presents that this could be used for other mechanisms. So, a significant problem in mechanisms control could be solved. The relevant diagrams are drawn and they show the correctness of control process.

Keywords: 3-RRR, dynamic equations, mechanisms control, structural uncertainty

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Authors: Youssef Abdelli, Rachid Nasri

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The large amplitude free vibration analysis of three-layered symmetric sandwich beams is carried out using two different approaches. The governing nonlinear partial differential equations of motion in free natural vibration are derived using Hamilton's principle. The formulation leads to two nonlinear partial differential equations that are coupled both in axial and binding deformations. In the first approach, the method of multiple scales is applied directly to the governing equation that is a nonlinear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained by two approaches; they are compared with the solutions obtained numerically by the finite difference method.

Keywords: finite difference method, large amplitude vibration, multiple scales, nonlinear vibration

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Authors: Youngji Lee, Yonghyeon Jeon, Sunyoung Bu, Philsu Kim

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The aim of this paper is to construct an algorithm of time step control for the error correction method most recently developed by one of the authors for solving stiff initial value problems. It is achieved with the generalized Chebyshev polynomial and the corresponding error correction method. The main idea of the proposed scheme is in the usage of the duplicated node points in the generalized Chebyshev polynomials of two different degrees by adding necessary sample points instead of re-sampling all points. At each integration step, the proposed method is comprised of two equations for the solution and the error, respectively. The constructed algorithm controls both the error and the time step size simultaneously and possesses a good performance in the computational cost compared to the original method. Two stiff problems are numerically solved to assess the effectiveness of the proposed scheme.

Keywords: stiff initial value problem, error correction method, generalized Chebyshev polynomial, node points

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Authors: R. Haoui

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The purpose of this work is to simulate the flow at the exit of Vulcan 1 engine of European launcher Ariane 5. The geometry of the propellant nozzle is already determined using the characteristics method. The pressure in the outlet section of the nozzle is less than atmospheric pressure on the ground, causing the existence of oblique and normal shock waves at the exit. During the rise of the launcher, the atmospheric pressure decreases and the shock wave disappears. The code allows the capture of shock wave at exit of nozzle. The numerical technique uses the Flux Vector Splitting method of Van Leer to ensure convergence and avoid the calculation instabilities. The Courant, Friedrichs and Lewy coefficient (CFL) and mesh size level are selected to ensure the numerical convergence. The nonlinear partial derivative equations system which governs this flow is solved by an explicit unsteady numerical scheme by the finite volume method. The accuracy of the solution depends on the size of the mesh and also the step of time used in the discretized equations. We have chosen in this study the mesh that gives us a stationary solution with good accuracy.

Keywords: finite volume, lunchers, nozzles, shock wave

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Authors: Alexander Norbach

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The script in this paper describes the use of advanced simulation environment using electronic systems (Microcontroller, Operational Amplifiers, and FPGA). The simulation may be used for all dynamic systems with the diffusion and the ionisation behaviour also. By additionally required observer structure, the system works with parallel real-time simulation based on diffusion model and the state-space representation for other dynamics. The proposed deposited model may be used for electrodynamic effects, including ionising effects and eddy current distribution also. With the script and proposed method, it is possible to calculate the spatial distribution of the electromagnetic fields in real-time. For further purpose, the spatial temperature distribution may be used also. With upon system, the uncertainties, unknown initial states and disturbances may be determined. This provides the estimation of the more precise system states for the required system, and additionally, the estimation of the ionising disturbances that occur due to radiation effects. The results have shown that a system can be also developed and adopted specifically for space systems with the real-time calculation of the radiation effects only. Electronic systems can take damage caused by impacts with charged particle flux in space or radiation environment. In order to be able to react to these processes, it must be calculated within a shorter time that ionising radiation and dose is present. All available sensors shall be used to observe the spatial distributions. By measured value of size and known location of the sensors, the entire distribution can be calculated retroactively or more accurately. With the formation, the type of ionisation and the direct effect to the systems and thus possible prevent processes can be activated up to the shutdown. The results show possibilities to perform more qualitative and faster simulations independent of kind of systems space-systems and radiation environment also. The paper gives additionally an overview of the diffusion effects and their mechanisms. For the modelling and derivation of equations, the extended current equation is used. The size K represents the proposed charge density drifting vector. The extended diffusion equation was derived and shows the quantising character and has similar law like the Klein-Gordon equation. These kinds of PDE's (Partial Differential Equations) are analytically solvable by giving initial distribution conditions (Cauchy problem) and boundary conditions (Dirichlet boundary condition). For a simpler structure, a transfer function for B- and E- fields was analytically calculated. With known discretised responses g₁(k·Ts) and g₂(k·Ts), the electric current or voltage may be calculated using a convolution; g₁ is the direct function and g₂ is a recursive function. The analytical results are good enough for calculation of fields with diffusion effects. Within the scope of this work, a proposed model of the consideration of the electromagnetic diffusion effects of arbitrary current 'waveforms' has been developed. The advantage of the proposed calculation of diffusion is the real-time capability, which is not really possible with the FEM programs available today. It makes sense in the further course of research to use these methods and to investigate them thoroughly.

Keywords: advanced observer, electrodynamics, systems, diffusion, partial differential equations, solver

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