The current paper explores the flow of dusty nanoliquid over a rotating and stretchable disk with non-uniform heat sink/source. Further, we have done a comparative study on Single wall carbon ...nanotubes (SWCNT)-water and multi wall carbon nanotubes (MWCNT)-water based dusty fluid flows. By means of apt similarity variables, the governing equations are converted to set of nonlinear ordinary differential equations and then they are numerically tackled using Runge-Kutta-Fehlberg's fourth fifth order (RKF45) method by adopting shooting technique. The influence of non-dimensional parameters on the heat transfer fields are incorporated and extensively discussed by means of appropriate graphs. Further, the reduced shear stresses at the disk in the tangential direction, in the radial direction and the heat transference rates of the fluid and particles are deliberated graphically. Results reveal that, the escalating values of space and temperature dependent heat source/sink parameters improves the heat transference of both liquids. The SWCNT-water based fluid shows improved shear stress in tangential and radial direction when compared to MWCNT-water based fluid for both the phases. The SWCNT-water based fluid shows enhanced heat transfer rate than MWCNT-water based fluid for both fluid and dust phases.
Boundary conditions for fractional diffusion Baeumer, Boris; Kovács, Mihály; Meerschaert, Mark M. ...
Journal of computational and applied mathematics,
07/2018, Volume:
336
Journal Article
Peer reviewed
Open access
This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are ...reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
In this paper, we present Chebyshev and ultraspherical polynomials method to obtain numerical solutions of Cauchy-type singular integro-differential equations. We prove that the singular ...integro-differential equations with smooth/non-smooth bivariate kernels and general boundary conditions can be transformed into a constrained least square problem and solved via generalized QR factorization. By using open-source Chebfun code, low-rank approximate solutions of singular integro-differential equations can be obtained efficiently in the spectral accuracy range. The method can be easily extended to solve those singular integro-differential equations with smooth bivariate kernels, or convolution kernels with absolute value symbols (non-smooth cases), or when the solutions have (inverse) square root singularities.
•Solutions for Cauchy type singular integro-differential equations are derived.•Bivariate kernels are considered.•Absolute convolution kernels are considered.•Chebyshev and ultrasphereical polynomial methods are used.•Numerical results with corresponding Chebfun code are given.
•The internal force analytical solution of lining structure under P-ware action was obtained by using the analytical method.•The influence of the hardness of surrounding rock and peak acceleration on ...the analytical solution were analyzed.•Through the engineering, the reasons for the large error between numerical solution and analytical solution was analyzed.•The most unfavorable position of lining structure under P-wave seismic load was proposed.
In an earthquake, the strong interaction between the surrounding rock and the lining structure causes the lining structure susceptible to extrusion or shear damage, and predicting the internal force distribution trend of the lining structure by the analytical method was advantageous for the preliminary design of the tunnel structure. In this work, the quasi-static method was used to approximate the displacement and deformation caused by the P-wave seismic load as far-field compressive stress. The analytical solution of the internal force for the lining structure was obtained by the analytical method, and the reliability of the analytical method was further verified. The dynamic response law of the lining structure was analyzed using numerical solutions in conjunction with engineering examples. The results show that the analytical method can be used to predict the trend of internal force distribution in the lining structure under seismic action, which has important theoretical guiding significance for the preliminary design of the tunnel structure. With the decrease in the hardness of the loess surrounding, the relative deviation of the theoretical, numerical, and literature results gradually decreases. With the increase of ground motion intensity, the interaction between the loess surrounding and the lining structure becomes more intense, and the internal force of the lining structure gradually increases. The numerical results were greater than the analytical results due to the consideration of the influence of initial stress. Under the action of the P-wave earthquake, the extreme values of the internal force with the lining structure mainly occurred at the location of the vault, arch waist, and inverted, and the most unfavorable positions of circular loess tunnel lining structures under P-wave seismic load was proposed.
•Develops a general framework for macroscopic pedestrian flow simulation, which is the first attempt to solve this problem under the physics-informed neural network (PINN) framework.•Proposes the ...reduced-order PINN to decompose the higher-order PDE and improve the model performance by reducing the derivative order of the auto-differentiation part.•Compares three PINN schemes, namely, vanilla PINN, the extended-variable PINN, and the reduced-order PINN.•Performs a sensitivity analysis on the key parameters of the PINN schemes while solving a particular macroscopic pedestrian flow model.
Given the importance of pedestrian flow simulation in reproducing pedestrian flows and optimizing pedestrian facilities, it is crucial to accurately simulate macroscopic pedestrian flows. Physics-informed neural network (PINN) is a recently proposed advanced scheme for solving partial differential equations (PDE) that can approximate the solution values by minimizing the initial values, boundary values, and residuals of the PDE. However, few studies have discussed how the performance of PINN changes in the face of macroscopic pedestrian flow equations with high-order derivatives and multiple variables. In this study, we propose to compare the performance of three schemes, i.e., the vanilla PINN, the extended-variable PINN (ev-PINN), and the reduced-order PINN (ro-PINN), by solving the macroscopic pedestrian flow equations coupled by the conservation equation and Eikonal equation in the steady-state and transient cases. The results show that the comparison of the PINN models under different hyperparameters indicates that the ro-PINN is the most stable in training, while the other two schemes have a certain degree of fluctuation in the face of different hyperparameters. Secondly, when the traditional numerical solution scheme is used as the reference solution, the ro-PINN solution works best in the steady-state and transient cases, and the solution results of different solution variables such as density and flow are closest to the reference solution, which can make the whole solution process easier and the solution results more accurate by reducing the derivative order in the automatic differentiation part and increasing the output dimension of the neural network. On the contrary, compared with vanilla PINN, ev-PINN not only did not improve the solution results, but even reduced the performance. Finally, we compare the running time of the three PINN schemes, and although increasing the output dimension and decreasing the order of PDE increase the computing time, it is not significant. Therefore, the ro-PINN can be used as an effective alternative model for simulating macroscopic pedestrian flow evolution.
This topic addresses the influence of binary chemical reaction and activation energy in hydromagnetic flow of third grade nanofluid associated with convective conditions. Flow is developed through ...nonlinearly stretched surface. Nanoparticles concentration and temperature profiles are considered in the presence of Brownian dispersion and thermophoresis effects. Third grade liquid is electrically conducted via uniform applied magnetic field. Assumption of boundary layer has been used in the problem development. Governing differential systems have been computed in frame of NDsolve. The graphical illustrations explore influences of various sundry variables. Further surface drag force, heat and mass transfer rate are sketched and analyzed. Temperature and concentration distributions are declared increasing functions of Hartman number while reverse trend is seen for velocity distribution. Furthermore an enhancement is observed in temperature and concentration distributions for the higher values of thermal and concentration Biot numbers respectively.
•Magnetohydrodynamic flow of third grade nanofluid is modeled.•Binary chemical reaction and Arrhenius activation energy aspects are utilized.•Heat and mass transfer attributes are analyzed through Brownian motion and thermophoresis effects.•Convective heat and mass conditions are also implemented at the surface.•Numerical solutions are developed by shooting technique.
We introduce a new generalized Caputo-type fractional derivative which generalizes Caputo fractional derivative. Some characteristics were derived to display the new generalized derivative features. ...Then, we present an adaptive predictor corrector method for the numerical solution of generalized Caputo-type initial value problems. The proposed algorithm can be considered as a fractional extension of the classical Adams-Bashforth-Moulton method. Dynamic behaviors of some fractional derivative models are numerically discussed. We believe that the presented generalized Caputo-type fractional derivative and the proposed algorithm are expected to be further used to formulate and simulate many generalized Caputo type fractional models.
•The stagnation point flow of hybrid nano fluid over a stretching cylinder is the key highlight.•Effects of inclined magnetic field are useful for the boundary layer control in hybrid nano ...fluid.•Numerical solutions are computed for the complicated problem.•The extended versions of Xue model and Yamada Ota model for hybrid nanofluid is taken into account.
Focus of the present analysis is on the stagnation point flow of hybrid nanofluid with inclined magnetic field over a moving cylinder. The extended version of two models (e.g. Xue model and Yamada-Ota model for hybrid nanofluids) are considered in this study). A mathematical model of hybrid nanofluid flow is developed under certain flow assumptions. Boundary layer approximations are also utilized to model a system of partial differential equations. The systems of partial differential equations are further converted to dimensionless systems of ordinary differential equations by means of suitable similarity transformations. A numerical solution is obtained by applying bv4c technique. Effects of variation in physical parameters involved are depicted through graphs. Skin friction coefficient and Nusselt number are highlighted through tables. Our main objective is to investigate the heat transfer rate on the surface of the nonlinear stretching cylinder. The results of Xue model and Yamada-Ota model for the hybrid nanofluid due to nonlinear stretching cylinder are computed for comparison. In both cases, velocity and temperature profiles are best compared to the decay results.
•Numerical computation of SDEs in high dimension is approached via Gaussian analysis.•We make use of the Kolmogorov equation associated to the SDE to compute probabilities.•A novel strategy that ...allow to reuse samples from a Gaussian process is applied to study nonlinear problems.
Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to numerically compute the expected values and probabilities associated to their solutions, by solving the corresponding Kolmogorov equations, with a partial use of Monte Carlo strategy - precisely, using Monte Carlo only for the linear part of the SDE. The basic idea was presented in 16, but here we strongly improve the numerical results by means of a shift of the auxiliary Gaussian process. For relatively simple nonlinearities, we have good results in dimension of the order of 100.
•Maxwell fluid in the presence of nanoparticle is presented first time.•Governing nonlinear partial differential equations are transformed into system ordinary differential equations.•Obtaining ...coupled ordinary differential equations are investigated numerically.•Comparative study with the previous literature is presented.
In the present article, two dimensional boundary-layer flows and the heat transfer of a Maxwell fluid past a stretching sheet are studied numerically. The effects of magnetohydrodynamics (MHD) and elasticity on the flow are considered. Moreover, the effects of nanoparticles are also investigated. Similarity transformations are presented to convert the governing nonlinear partial differential equation into coupled ordinary differential equations. The reduced boundary layer equations of the Maxwell nanofluid model are solved numerically. The effects of the emerging parameters, namely, the magnetic parameter M, the elastic parameter K, the Prandtl parameter Pr, the Brownian motion Nb, the thermophoresis parameter Nt and the Lewis number Le on the temperature and the concentration profile are discussed. Interesting results are shown graphically. The skin friction coefficient, the dimensionless heat transfer rate and the concentration rate are also plotted against the flow control parameters.
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