Search results for: ODEs

Authors: Preeti Aghalayam, Jay Shah

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Underground coal gasification (UCG) is the in-situ gasification of unmineable coals to produce syngas. In UCG, gasifying agents are injected into the coal seam, and a reactive cavity is formed due to coal consumption. The cavity formed is typically hemispherical, and this report consists of the MATLAB model of the UCG cavity to predict the composition of the output gases. There are seven radial and two time-variant ODEs. A MATLAB solver (ode15s) is used to solve the radial ODEs from the above equations. Two for-loops are implemented in the model, i.e., one for time variations and another for radial variation. In the time loop, the radial odes are solved using the MATLAB solver. The radial loop is nested inside the time loop, and the density odes are numerically solved using the Euler method. The model is validated by comparing it with the literature results of laboratory-scale experiments. The model predicts the radial and time variation of the product gases inside the cavity.

Keywords: gasification agent, MATLAB model, syngas, underground coal gasification (UCG)

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Authors: S. A. M. Yatim, Z. B. Ibrahim, K. I. Othman, M. Suleiman

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The method of solving second order stiff ordinary differential equation (ODEs) that is based on backward differentiation formula (BDF) is considered in this paper. We derived the method by increasing the order of the existing method using an improved strategy in choosing the step size. Numerical results are presented to compare the efficiency of the proposed method to the MATLAB’s suite of ODEs solvers namely ode15s and ode23s. The method was found to be efficient to solve second order ordinary differential equation.

Keywords: backward differentiation formulae, block backward differentiation formulae, stiff ordinary differential equation, variable step size

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Authors: Armin Ardekani, Mohammad Akbari

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We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields.

Keywords: computational mathematics, differential equations, engineering, series

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Authors: H. N. Agiza, M. A. Sohaly, M. A. Elfouly

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Parkinson's disease (PD) is a heterogeneous disorder with common age of onset, symptoms, and progression levels. In this paper we will solve analytically the PD model as a non-linear delay differential equation using the steps method. The step method transforms a system of delay differential equations (DDEs) into systems of ordinary differential equations (ODEs). On some numerical examples, the analytical solution will be difficult. So we will approximate the analytical solution using Picard method and Taylor method to ODEs.

Keywords: Parkinson's disease, step method, delay differential equation, two delays

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Authors: Khairil I. Othman, Nur N. Kamal, Zarina B. Ibrahim

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In this paper, we present modified Newton method as a new strategy for improving the efficiency of Two Point Block Backward Differentiation Formulas (BBDF) when solving stiff systems of ordinary differential equations (ODEs). These methods are constructed to produce two approximate solutions simultaneously at each iteration The detailed implementation of the predictor corrector BBDF with PE(CE)2 with modified Newton are discussed. The proposed modification of BBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with the existing Block Backward Differentiation Formula. Numerical results show the advantage of using the new strategy for solving stiff ODEs in improving the accuracy of the solution.

Keywords: newton method, two point, block, accuracy

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Authors: A. M. Sagir

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The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software.

Keywords: block method, first order ordinary differential equations, linear multistep, self-starting

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Authors: Shohel Ahmed, Md. Abdul Alim, Sumaiya Rahman

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Optimal control can be helpful to test and compare different vaccination strategies of a certain disease. The mathematical model of HIV we consider here is a set of ordinary differential equations (ODEs) describing the interactions of CD4+T cells of the immune system with the human immunodeficiency virus (HIV). As an early treatment setting, we investigate an optimal chemotherapy strategy where control represents the percentage of effect the chemotherapy has on the system. The aim is to obtain a new optimal chemotherapeutic strategy where an isoperimetric constraint on the chemotherapy supply plays a crucial role. We outline the steps in formulating an optimal control problem, derive optimality conditions and demonstrate numerical results of an optimal control for the model. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on cell-virus interactions.

Keywords: chemotherapy of HIV, optimal control involving ODEs, optimality conditions, Pontryagin’s maximum principle

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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: Davood Yazdani Cherati, Ali Pak, Mehrdad Jafarzadeh

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This research contains the formulation of a two-dimensional analytical solution to transient heat, and moisture flow in a semi-infinite unsaturated soil environment under the influence of daily temperature changes. For this purpose, coupled energy conservation and mass fluid continuity equations governing hydrothermal behavior of unsaturated soil media are presented in terms of temperature and volumetric moisture content. In consideration of the soil environment as an infinite half-space and by linearization of the governing equations, Laplace–Fourier transformation is conducted to convert differential equations with partial derivatives (PDEs) to ordinary differential equations (ODEs). The obtained ODEs are solved, and the inverse transformations are calculated to determine the solution to the system of equations. Results indicate that heat variation induces moisture transport in both horizontal and vertical directions.

Keywords: analytical solution, heat conduction, hydrothermal analysis, laplace–fourier transformation, two-dimensional

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Authors: Mostafa Shahriari, Sergio Rojas, David Pardo, Angel Rodriguez- Rozas, Shaaban A. Bakr, Victor M. Calo, Ignacio Muga

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Logging-While Drilling (LWD) is a technique to record down-hole logging measurements while drilling the well. Nowadays, LWD devices (e.g., nuclear, sonic, resistivity) are mostly used commercially for geo-steering applications. Modern borehole resistivity tools are able to measure all components of the magnetic field by incorporating tilted coils. The depth of investigation of LWD tools is limited compared to the thickness of the geological layers. Thus, it is a common practice to approximate the Earth’s subsurface with a sequence of 1D models. For a 1D model, we can reduce the dimensionality of the problem using a Hankel transform. We can solve the resulting system of ordinary differential equations (ODEs) either (a) analytically, which results in a so-called semi-analytic method after performing a numerical inverse Hankel transform, or (b) numerically. Semi-analytic methods are used by the industry due to their high performance. However, they have major limitations, namely: -The analytical solution of the aforementioned system of ODEs exists only for piecewise constant resistivity distributions. For arbitrary resistivity distributions, the solution of the system of ODEs is unknown by today’s knowledge. -In geo-steering, we need to solve inverse problems with respect to the inversion variables (e.g., the constant resistivity value of each layer and bed boundary positions) using a gradient-based inversion method. Thus, we need to compute the corresponding derivatives. However, the analytical derivatives of cross-bedded formation and the analytical derivatives with respect to the bed boundary positions have not been published to the best of our knowledge. The main contribution of this work is to overcome the aforementioned limitations of semi-analytic methods by solving each 1D model (associated with each Hankel mode) using an efficient multi-scale finite element method. The main idea is to divide our computations into two parts: (a) offline computations, which are independent of the tool positions and we precompute only once and use them for all logging positions, and (b) online computations, which depend upon the logging position. With the above method, (a) we can consider arbitrary resistivity distributions along the 1D model, and (b) we can easily and rapidly compute the derivatives with respect to any inversion variable at a negligible additional cost by using an adjoint state formulation. Although the proposed method is slower than semi-analytic methods, its computational efficiency is still high. In the presentation, we shall derive the mathematical variational formulation, describe the proposed multi-scale finite element method, and verify the accuracy and efficiency of our method by performing a wide range of numerical experiments and comparing the numerical solutions to semi-analytic ones when the latest are available.

Keywords: logging-While-Drilling, resistivity measurements, multi-scale finite elements, Hankel transform

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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: Khairil I. Othman, Mahfuzah Mahayaddin, Zarina Bibi Ibrahim

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We develop and demonstrate a computationally efficient numerical technique to solve first order stiff differential equations. This technique is based on block method whereby three approximate points are calculated. The Cholistani of varied step sizes are presented in divided difference form. Stability regions of the formulae are briefly discussed in this paper. Numerical results show that this block method perform very well compared to existing methods.

Keywords: block method, divided difference, stiff, computational

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Authors: Khairil Iskandar Othman, Nadhirah Kamal, Zarina Bibi Ibrahim

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Block methods for solving stiff systems of ordinary differential equations (ODEs) are based on backward differential formulas (BDF) with PE(CE)2 and Newton method. In this paper, we introduce Modified Newton as a new strategy to get more efficient result. The derivation of BBDF using modified block Newton method is presented. This new block method with predictor-corrector gives more accurate result when compared to the existing BBDF.

Keywords: modified Newton, stiff, BBDF, Jacobian matrix

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Authors: A. I. Ma’ali, R. B. Adeniyi, A. Y. Badeggi, U. Mohammed

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An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples.

Keywords: approximant, error estimate, tau method, overdetermination

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Authors: A. M. Sagir

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Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

Keywords: block method, first order ordinary differential equations, hybrid, self-starting

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Authors: M. S. H. Chowdhury, Ishak Hashim

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In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multi-stage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs).

Keywords: stiff system of ODEs, Runge-Kutta Type Method, Adomian decomposition method, Multistage ADM

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Authors: Martin K. Steiger, Lukas Heisler, Hans-Georg Brachtendorf

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Deep neural networks and their variants form the backbone of many AI applications. Based on the so-called residual networks, a continuous formulation of such models as ordinary differential equations (ODEs) has proven advantageous since different techniques may be applied that significantly increase the learning speed and enable controlled trade-offs with the resulting error at the same time. For the evaluation of such models, high-performance numerical differential equation solvers are used, which also provide the gradients required for training. However, whether classical gradient-based methods are even applicable or which one yields the best results has not been discussed yet. This paper aims to redeem this situation by providing empirical results for different applications.

Keywords: deep neural networks, gradient-based learning, image processing, ordinary differential equation networks

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Authors: A. A. James, A. O. Adesanya, M. R. Odekunle, D. G. Yakubu

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This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compete favorably with the existing methods.

Keywords: interpolation, approximate solution, collocation, differential system, half step, converges, block method, efficiency

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Authors: Khawar Ali

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We present the numerical study of unsteady hydromagnetic (MHD) flow and heat transfer characteristics of a viscous incompressible electrically conducting water-based hybrid metallic nanofluid (containing Cu-Au/ H₂O nanoparticles) between two orthogonally moving porous coaxial disks with suction. Different from the classical shooting methodology, we employ a combination of a direct and an iterative method (SOR with optimal relaxation parameter) for solving the sparse systems of linear algebraic equations arising from the FD discretization of the linearized self similar nonlinear ODEs. Effects of the governing parameters on the flow and heat transfer are discussed and presented through tables and graphs. The findings of the present investigation may be beneficial for the electronic industry in maintaining the electronic components under effectiveand safe operational conditions.

Keywords: heat transfer enhancement, hybrid metallic nanofluid, viscous dissipation and joule heating effect , Two dimensional flow

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Authors: Gleda Kutrolli, Maksi Kutrolli, Etjon Meco

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SARS-CoV-2 virus is currently one of the most infectious pathogens for humans. It started in China at the end of 2019 and now it is spread in all over the world. The origin and diffusion of the SARS-CoV-2 epidemic, is analysed based on the discussion of viral phylogeny theory. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the spread of the virus and simulate its activity. In this paper, the prediction of coronavirus outbreak is done by using SIR model without vital dynamics, applying different numerical technique solving ordinary differential equations (ODEs). We find out that ABM and MRT methods perform better than other techniques and that the activity of the virus will decrease in April but it never cease (for some time the activity will remain low) and the next cycle will start in the middle July 2020 for Norway and Denmark, and October 2020 for Sweden, and September for Finland.

Keywords: forecasting, ordinary differential equations, SARS-COV-2 epidemic, SIR model

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Authors: Guesh Simretab Gebremedhin, Saumya Rajan Jena

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In this work, a septic B-spline scheme has been used to simplify the process of solving an approximate solution of the generalized Rosenau-regularized long-wave equation (GR-RLWE) with initial boundary conditions. The resulting system of first-order ODEs has dealt with Butcher’s fifth order Runge-Kutta (BFRK) approach without using finite difference techniques for discretizing the time-dependent variables at each time level. Here, no transformation or any kind of linearization technique is employed to tackle the nonlinearity of the equation. Two test problems have been selected for numerical justifications and comparisons with other researchers on the basis of efficiency, accuracy, and results of the two invariants Mᵢ (mass) and Eᵢ (energy) of some motion that has been used to test the conservative properties of the proposed scheme.

Keywords: septic B-spline scheme, Butcher's fifth order Runge-Kutta approach, error norms, generalized Rosenau-RLW equation

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Authors: Jana Abu Ahmada, Zaineb Mohamed, Ilyasse Aksikas

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The linear quadratic control system of hyperbolic first order partial differential equations (PDEs) are presented. The aim of this research is to control chemical reactions. This is achieved by converting the PDEs system to ordinary differential equations (ODEs) using the method of characteristics to reduce the system to control it by using the integral reinforcement learning. The designed controller is applied to a catalytic cracking reactor. Background—Transport-Reaction systems cover a large chemical and bio-chemical processes. They are best described by nonlinear PDEs derived from mass and energy balances. As a main application to be considered in this work is the catalytic cracking reactor. Indeed, the cracking reactor is widely used to convert high-boiling, high-molecular weight hydrocarbon fractions of petroleum crude oils into more valuable gasoline, olefinic gases, and others. On the other hand, control of PDEs systems is an important and rich area of research. One of the main control techniques is feedback control. This type of control utilizes information coming from the system to correct its trajectories and drive it to a desired state. Moreover, feedback control rejects disturbances and reduces the variation effects on the plant parameters. Linear-quadratic control is a feedback control since the developed optimal input is expressed as feedback on the system state to exponentially stabilize and drive a linear plant to the steady-state while minimizing a cost criterion. The integral reinforcement learning policy iteration technique is a strong method that solves the linear quadratic regulator problem for continuous-time systems online in real time, using only partial information about the system dynamics (i.e. the drift dynamics A of the system need not be known), and without requiring measurements of the state derivative. This is, in effect, a direct (i.e. no system identification procedure is employed) adaptive control scheme for partially unknown linear systems that converges to the optimal control solution. Contribution—The goal of this research is to Develop a characteristics-based optimal controller for a class of hyperbolic PDEs and apply the developed controller to a catalytic cracking reactor model. In the first part, developing an algorithm to control a class of hyperbolic PDEs system will be investigated. The method of characteristics will be employed to convert the PDEs system into a system of ODEs. Then, the control problem will be solved along the characteristic curves. The reinforcement technique is implemented to find the state-feedback matrix. In the other half, applying the developed algorithm to the important application of a catalytic cracking reactor. The main objective is to use the inlet fraction of gas oil as a manipulated variable to drive the process state towards desired trajectories. The outcome of this challenging research would yield the potential to provide a significant technological innovation for the gas industries since the catalytic cracking reactor is one of the most important conversion processes in petroleum refineries.

Keywords: PDEs, reinforcement iteration, method of characteristics, riccati equation, cracking reactor

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Authors: Elmongi Elbellili, Ben Lauwens, Daan Huybrechs

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The increasing complexity of modern engineering systems presents a challenge to the digital simulation of these systems which usually can be represented by differential equations. The Linearly Implicit Quantized State System (LIQSS) offers an alternative approach to traditional numerical integration techniques for solving Ordinary Differential Equations (ODEs). This method proved effective for handling discontinuous and large stiff systems. However, the inherent discrete nature of LIQSS may introduce oscillations that result in unnecessary computational steps. The current oscillation detection mechanism relies on a condition that checks the significance of the derivatives, but it could be further improved. This paper describes a different cycle detection mechanism and presents the outcomes using LIQSS order one in simulating the Advection Diffusion problem. The efficiency of this new cycle detection mechanism is verified by comparing the performance of the current solver against the new version as well as a reference solution using a Runge-Kutta method of order14.

Keywords: numerical integration, quantized state systems, ordinary differential equations, stiffness, cycle detection, simulation

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Authors: Muhammad Faraz, Talat Rafique, Jang Min Park

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In this article, rotational flow above a permeable surface with a variable free stream angular velocity is considered. Main interest is to solve the associated heat/mass transport equations under different situations. Firstly, heat transport phenomena occurring in generalized vortex flow are analyzed under two altered heating processes, namely, the (i) prescribed surface temperature and (ii) prescribed heat flux. The vortex motion imposed at infinity is assumed to follow a power-law form 〖(r/r_0)〗^((2n-1)) where r denotes the radial coordinate, r_0 the disk radius, and n is a power-law parameter. Assuming a similar solution, the governing Navier-Stokes equations transform into a set of coupled ODEs which are treated numerically for the aforementioned thermal conditions. Secondly, mass transport phenomena accompanied by activation energy are incorporated into the generalized vortex flow situation. After finding self-similar equations, a numerical solution is furnished by using MATLAB's built-in function bvp4c.

Keywords: bödewadt flow, vortex flow, rotating flows, prescribed heat flux, permeable surface, activation energy

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Authors: Randy Price, Harbir Antil, Rainald Löhner, Fumiya Togashi

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Chemically reacting flows are common in engineering applications such as hypersonic flow, combustion, explosions, manufacturing process, and environmental assessments. The number of reactions in combustion simulations can exceed 100, making a large number of flow and combustion problems beyond the capabilities of current supercomputers. Motivated by this, deep neural networks (DNNs) will be introduced with the goal of eventually replacing the existing chemistry software packages with DNNs. The DNNs used in this paper are motivated by the Residual Neural Network (ResNet) architecture. In the continuum limit, ResNets become an optimization problem constrained by an ODE. Such a feature allows the use of ODE control techniques to enhance the DNNs. In this work, DNNs are constructed, which update the species un at the nᵗʰ timestep to uⁿ⁺¹ at the n+1ᵗʰ timestep. Parallel DNNs are trained for each species, taking in uⁿ as input and outputting one component of uⁿ⁺¹. These DNNs are applied to multiple species and reactions common in chemically reacting flows such as H₂-O₂ reactions. Experimental results show that the DNNs are able to accurately replicate the dynamics in various situations and in the presence of errors.

Keywords: chemical reacting flows, computational fluid dynamics, ODEs, residual neural networks, ResNets

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Authors: M. Osman Gani, M. Ferdows, Toshiyuki Ogawa

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In this work, we proposed a modified FHN-type reaction-diffusion system for a bistable excitable system by adding a scaled function obtained from a given function. We study the existence and the stability of the periodic traveling waves (or wavetrains) for the FitzHugh-Nagumo (FHN) system and the modified one and compare the results. The stability results of the periodic traveling waves (PTWs) indicate that most of the solutions in the fast family of the PTWs are stable for the FitzHugh-Nagumo equations. The instability occurs only in the waves having smaller periods. However, the smaller period waves are always unstable. The fast family with sufficiently large periods is always stable in FHN model. We find that the oscillation of pulse widths is absent in the standard FHN model. That motivates us to study the PTWs in the proposed FHN-type reaction-diffusion system for the bistable excitable media. A good agreement is found between the solutions of the traveling wave ODEs and the corresponding whole PDE simulation.

Keywords: bistable system, Eckhaus bifurcation, excitable media, FitzHugh-Nagumo model, periodic traveling waves

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Authors: Z. B. Ibrahim, N. Ismail, K. I. Othman

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Fuzzy Differential Equations (FDEs) play an important role in modelling many real life phenomena. The FDEs are used to model the behaviour of the problems that are subjected to uncertainty, vague or imprecise information that constantly arise in mathematical models in various branches of science and engineering. These uncertainties have to be taken into account in order to obtain a more realistic model and many of these models are often difficult and sometimes impossible to obtain the analytic solutions. Thus, many authors have attempted to extend or modified the existing numerical methods developed for solving Ordinary Differential Equations (ODEs) into fuzzy version in order to suit for solving the FDEs. Therefore, in this paper, we proposed the development of a fuzzy version of three-point block method based on Block Backward Differentiation Formulas (FBBDF) for the numerical solution of first order FDEs. The three-point block FBBDF method are implemented in uniform step size produces three new approximations simultaneously at each integration step using the same back values. Newton iteration of the FBBDF is formulated and the implementation is based on the predictor and corrector formulas in the PECE mode. For greater efficiency of the block method, the coefficients of the FBBDF are stored at the start of the program. The proposed FBBDF is validated through numerical results on some standard problems found in the literature and comparisons are made with the existing fuzzy version of the Modified Simpson and Euler methods in terms of the accuracy of the approximated solutions. The numerical results show that the FBBDF method performs better in terms of accuracy when compared to the Euler method when solving the FDEs.

Keywords: block, backward differentiation formulas, first order, fuzzy differential equations

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Authors: Ricardo Godoy, Herbert Venthur, Hector Jimenez, Andres Quiroz, Ana Mutis

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In agriculture, grape production has great economic importance at global level, considering that in 2013 it reached 7.4 million hectares (ha) covered by plantations of this fruit worldwide. Chile is the number one exporter in the world with 800,000 tons. However, these values have been threatened by the attack of the grapevine moth, Lobesia botrana (Denis & Schiffermuller) (Lepidoptera: Tortricidae), since its detection in 2008. Nowadays, the use of semiochemicals, in particular the major component of the sex pheromone, (E,Z)-7.9-dodecadienil acetate, are part of mating disruption methods to control L. botrana. How insect pests can recognize these molecules, is being part of huge efforts to deorphanize their olfactory mechanism at molecular level. Thus, an interesting group of proteins has been identified in the antennae of insects, where odorant-binding proteins (OBPs) are known by transporting molecules to odorant receptors (ORs) and a co-receptor (ORCO) causing a behavioral change in the insect. Other proteins such as chemosensory proteins (CSPs), ionotropic receptors (IRs), odorant degrading enzymes (ODEs) and sensory neuron membrane proteins (SNMPs) seem to be involved, but few studies have been performed so far. The above has led to an increasing interest in insect communication at a molecular level, which has contributed to both a better understanding of the olfaction process and the design of new pest management strategies. To date, it has been reported that the ORs can detect one or a small group of odorants in a specific way. Therefore, the objective of this study is the identification of genes that encode these ORs using the antennal transcriptome of L. botrana. Total RNA was extracted for females and males of L. botrana, and the antennal transcriptome sequenced by Next Generation Sequencing service using an Illumina HiSeq2500 platform with 50 million reads per sample. Unigenes were assembled using Trinity v2.4.0 package and transcript abundance was obtained using edgeR. Genes were identified using BLASTN and BLASTX locally installed in a Unix system and based on our own Tortricidae database. Those Unigenes related to ORs were characterized using ORFfinder and protein Blastp server. Finally, a phylogenetic analysis was performed with the candidate amino acid sequences for LbotORs including amino acid sequences of other moths ORs, such as Bombyx mori, Cydia pomonella, among others. Our findings suggest 61 genes encoding ORs and one gene encoding an ORCO in both sexes, where the greatest difference was found in the OR6 because of the transcript abundance according to the value of FPKM in females and males was 1.48 versus 324.00. In addition, according to phylogenetic analysis OR6 is closely related to OR1 in Cydia pomonella and OR6, OR7 in Epiphyas postvittana, which have been described as pheromonal receptors (PRs). These results represent the first evidence of ORs present in the antennae of L. botrana and a suitable starting point for further functional studies with selected ORs, such as OR6, which is potentially related to pheromonal recognition.

Keywords: antennal transcriptome, lobesia botrana, odorant receptors (ORs), phylogenetic analysis

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Authors: Anne Schattel, Mitja Echim, Christof Büskens

Abstract:

Trajectory planning for deep space missions has become a recent topic of great interest. Flying to space objects like asteroids provides two main challenges. One is to find rare earth elements, the other to gain scientific knowledge of the origin of the world. Due to the enormous spatial distances such explorer missions have to be performed unmanned and autonomously. The mathematical field of optimization and optimal control can be used to realize autonomous missions while protecting recourses and making them safer. The resulting algorithms may be applied to other, earth-bound applications like e.g. deep sea navigation and autonomous driving as well. The project KaNaRiA ('Kognitionsbasierte, autonome Navigation am Beispiel des Ressourcenabbaus im All') investigates the possibilities of cognitive autonomous navigation on the example of an asteroid mining mission, including the cruise phase and approach as well as the asteroid rendezvous, landing and surface exploration. To verify and test all methods an interactive, real-time capable simulation using virtual reality is developed under KaNaRiA. This paper focuses on the specific challenge of the guidance during the cruise phase of the spacecraft, i.e. trajectory optimization and optimal control, including first solutions and results. In principle there exist two ways to solve optimal control problems (OCPs), the so called indirect and direct methods. The indirect methods are being studied since several decades and their usage needs advanced skills regarding optimal control theory. The main idea of direct approaches, also known as transcription techniques, is to transform the infinite-dimensional OCP into a finite-dimensional non-linear optimization problem (NLP) via discretization of states and controls. These direct methods are applied in this paper. The resulting high dimensional NLP with constraints can be solved efficiently by special NLP methods, e.g. sequential quadratic programming (SQP) or interior point methods (IP). The movement of the spacecraft due to gravitational influences of the sun and other planets, as well as the thrust commands, is described through ordinary differential equations (ODEs). The competitive mission aims like short flight times and low energy consumption are considered by using a multi-criteria objective function. The resulting non-linear high-dimensional optimization problems are solved by using the software package WORHP ('We Optimize Really Huge Problems'), a software routine combining SQP at an outer level and IP to solve underlying quadratic subproblems. An application-adapted model of impulsive thrusting, as well as a model of an electrically powered spacecraft propulsion system, is introduced. Different priorities and possibilities of a space mission regarding energy cost and flight time duration are investigated by choosing different weighting factors for the multi-criteria objective function. Varying mission trajectories are analyzed and compared, both aiming at different destination asteroids and using different propulsion systems. For the transcription, the robust method of full discretization is used. The results strengthen the need for trajectory optimization as a foundation for autonomous decision making during deep space missions. Simultaneously they show the enormous increase in possibilities for flight maneuvers by being able to consider different and opposite mission objectives.

Keywords: deep space navigation, guidance, multi-objective, non-linear optimization, optimal control, trajectory planning.

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