•Random non-autonomous logistic-type differential equations are studied.•Random Variable Transformation method and Karhunen–Love expansion are combined.•First probability density function of the ...solution stochastic process is determined.•Numerical simulations for the mean, variance and PDF of the solution are performed.•A wide range of PDFs for input data are considered in numerical experiments.
This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f1(p, t), of the solution stochastic process, P(t, ω). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen–Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f1N(p,t), of f1(p, t) are constructed. Afterwards, we rigorously prove that f1N(p,t)⟶f1(p,t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.
Atomization involves complex physical processes and gas–liquid interaction. Primary atomization on diesel spray is not well understood due to the difficulties to perform experimental measurements in ...the near nozzle field. Hence computational fluid dynamics (CFD) has been used as a key element to understand and improve diesel spray.
A recent new code for incompressible multiphase flow with adaptive octree mesh refinement has been used to perform simulations of atomization at low injection pressure conditions. The multiphase flow strategy to manage different flows is the volume of fluid (VOF) method. The adaptive mesh allows to locally refine the mesh at each time step where a better resolution is needed to capture important gradients instead of using a static mesh with a fixed and high number of cells which, in turn, would lead to an unaffordable computational cost. Even with this approach, the cell number is very high to achieve a Direct Numerical Simulation (DNS) at reasonable computational cost. To reduce the computational cost, an idea has been explored, the possibility of setting a maximum number of cells of the domain. Following this idea, the code has been tested with different configurations to understand their effects on numerical stability, the change in different spray parameters and the benefits achieved in terms of execution time. The outcomes have been validated against a theoretical model.
Commercial thyme and lavender essential oils were analysed by GC/MS. Sixty-six compounds accounting for 98.6-99.6% of total essential oil were identified. Thymol (52.14 ± 0.21%), followed by p-cymene ...(32.24 ± 0.16%), carvacrol (3.71 ± 0.01%) and γ-terpinene (3.34 ± 0.02%), were the main compounds in thyme essential oil, while large amounts of oxygenated monoterpenes linalool acetate (37.07 ± 0.24%) and linalool (30.16 ± 0.06%) were found in lavender one. In vitro antifungal activity of the essential oils was evaluated at 200 and 300 μg/mL against 10 phytopathogenic and post-harvest fungi, which significantly affect agriculture. Micelial growth inhibition was calculated for each tested fungus and dose. Thyme essential oil showed satisfactory results with 90-100% growth inhibition in almost all the assayed fungi at 300 μg/mL, while lavender essential oil showed no noteworthy inhibition data at either dose, and its growth was even enhanced. Thyme essential oil represents a natural alternative to control harvest and post-harvest fungi, and to extend the shelf-life of agriculture products.
This paper is devoted to study random linear control systems where the initial condition, the final target, and the elements of matrices defining the coefficients are random variables, while the ...control is a stochastic process. The so-called Random Variable Transformation technique is adapted to obtain closed-form expressions of the probability density functions of the solution and of the control. The theoretical findings are applied to study the dynamics of a damped oscillator subject to parametric noise.
Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random ...differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.
In this paper we perform a complete probabilistic study of a finite dimensional linear control system with uncertainty. The controllability condition with random initial data and final target is ...analysed. To conduct this investigation we determine the first probability density function of the control and the solution of the random control problem under different scenarios. To achieve this objective, Random Variable Transformation technique is extensively applied. Several examples illustrate the theoretical results.
The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation ...depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and
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catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.
In this paper, the behaviour of the internal nozzle flow and cavitation phenomenon are numerically studied for non-conventional Diesel convergent–divergent nozzles in order to assess their potential ...in terms of flow characteristics. The used nozzles differ each other in the convergence–divergence level of the orifices but all of them keep the same diameter at the middle of the nozzle orifice. The calculations have been performed using a code previously validated and able to simulate cavitation phenomenon using a homogeneous equilibrium model for the biphasic fluid and using a RANS method (RNG k-ε) as a turbulence modelling approach. For the simulations, one injection pressure and different discharge pressures were used in order to assess the characteristics of nozzles for different Reynolds conditions involving cavitating and non-cavitating conditions.
The comparison of the nozzles has been carried out in terms of flow characteristics such as mass flow, momentum flux, effective velocity and other important dimensionless parameters which help to describe the behaviour of the inner flow: discharge coefficient (Cd), area coefficient (Ca) and velocity coefficient (Cv). Additionally, the nozzles have been compared in terms of cavitation inception conditions and cavitation development.
The study has shown a high influence on the results of the level of convergence–divergence used in the nozzles. In these nozzles, the vapour originated from cavitation phenomenon came from the throttle of the orifice at the midpoint, and it extended along the whole wall of the divergent nozzle part towards the outlet of the orifice. The main results of the investigation have shown how the different geometries modify the cavitation conditions as well as the discharge coefficient and effective velocity. In particular, the nozzle with highest convergence–divergence level showed cavitation for all the tested conditions while for the nozzle with lowest convergence–divergence level, the cavitation phenomenon could be avoided for high discharge pressures. Additionally, the nozzle with highest convergence–divergence level showed the lowest discharge coefficient values but similar effective injection velocity than the nozzle with lowest level of convergence–divergence level despite of its higher orifice outlet area.
•Three non-conventional convergent-divergent diesel nozzles are compared.•The nozzle with higher convergence-divergence level shows the higher mass flow and momentum flux.•The nozzle with higher convergence-divergence level exhibits similar effective velocity than the nozzle with lower level.•The nozzle with higher convergence-divergence level is more prone to cavitate.•Better mixing process is expected for the nozzle with highest convergence-divergence level.
This paper is addressed to give a generalization of the classical Markov methodology allowing the treatment of the entries of the transition matrix and initial condition as random variables instead ...of deterministic values lying in the interval
. This permits the computation of the first probability density function (1-PDF) of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. From the 1-PDF relevant probabilistic information about the evolution of Markov models can be calculated including all one-dimensional statistical moments. We are also interested in determining the computation of distribution of some important quantities related to randomized Markov chains (steady state, hitting times, etc.). All theoretical results are established under general assumptions and they are illustrated by modelling the spread of a technology using real data.
•gPC and Random Variable Transformation methods are combined.•Nonlinear random differential equations (RDEs) are solved.•Nonlinear uncertainties are considered as inputs in nonlinear RDEs.•Mean and ...variance of the solution stochastic process are computed.•The method works for high oscillatory systems.
Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.
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