scholarly journals A Visualization of Air Pollution Distribution over an Observed Area Surrounded by Mountains: A Computational Approach

2021 ◽  
Vol 18 (12) ◽  
Author(s):  
Udomsak RAKWONGWAN ◽  
Piyanut TANGMANUSSUKUM ◽  
Sanae RUJIVAN
Keyword(s):  
Air Pollution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Agricultural Land ◽  
Stokes Equations ◽  
Average Concentration ◽  
Windward Side ◽  
Partial Differential ◽  
Advection Diffusion Equation ◽  
Pollution Modeling

We study the propagation of pollutants emitted from a single generator such as a factory chimney located between 2 mountains as well as its effects on an observed area such as a village or agricultural land. The problem is formulated as a system of partial differential equations, composed of Navier-Stokes equations and an advection-diffusion equation, and is solved by the finite element method. We visualize the propagation of the pollutants for several variants of the problem depending on the heights of the mountains and investigate their negative effects on the observed area by computing an average concentration of the pollutants over the observed area. We found that the observed area between the two mountains experienced a long-term negative effect compared with those located on flat land. This is because the mountain on the side, where the wind is blowing, obstructs the wind resulting in air recirculation. In contrast, the other mountain bounces some pollutants back to the observed area, preventing them from leaving the domain. The higher the mountains, the longer the time the pollutants remain in the observed area. If the heights of the mountains encircling the observed area are not equal, the residual remains in the area longer if the taller mountain is on the windward side. HIGHLIGHTS Air Visualization of air pollution between two mountains Air pollution propagation modeling A system of partial differential equations for air pollution modeling with FEM GRAPHICAL ABSTRACT

scholarly journals Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Vipul K. Baranwal ◽  
Ram K. Pandey ◽  
Om P. Singh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Method ◽  
Nonlinear Partial Differential Equations ◽  
Variational Iteration Method ◽  
Partial Differential ◽  
Advection Diffusion Equation ◽  
Variational Asymptotic ◽  
Variational Asymptotic Method ◽  
Optimal Values

We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ0,γ1,γ2,… and auxiliary functions H0(x),H1(x),H2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

scholarly journals Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

2013 ◽  
Vol 469 (2158) ◽  
pp. 20130001 ◽  
Author(s):  
D. Venturi ◽  
X. Wan ◽  
R. Mikulevicius ◽  
B. L. Rozovskii ◽  
G. E. Karniadakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Stokes Equations ◽  
Wick Product ◽  
Navier Stokes ◽  
Quantum Field ◽  
Partial Differential ◽  
Stochastic Approximations ◽  
Navier Stokes Equations

Approximating nonlinearities in stochastic partial differential equations (SPDEs) via the Wick product has often been used in quantum field theory and stochastic analysis. The main benefit is simplification of the equations but at the expense of introducing modelling errors. In this paper, we study the accuracy and computational efficiency of Wick-type approximations to SPDEs and demonstrate that the Wick propagator, i.e. the system of equations for the coefficients of the polynomial chaos expansion of the solution, has a sparse lower triangular structure that is seemingly universal, i.e. independent of the type of noise. We also introduce new higher-order stochastic approximations via Wick–Malliavin series expansions for Gaussian and uniformly distributed noises, and demonstrate convergence as the number of expansion terms increases. Our results are for diffusion, Burgers and Navier–Stokes equations, but the same approach can be readily adopted for other nonlinear SPDEs and more general noises.

Symmetry-invariant conservation laws of partial differential equations

2017 ◽  
Vol 29 (1) ◽  
pp. 78-117 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
ABDUL H. KARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Stokes Equations ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations ◽  
General Method

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, theb-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

scholarly journals An improved data-free surrogate model for solving partial differential equations using deep neural networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xinhai Chen ◽  
Rongliang Chen ◽  
Qian Wan ◽  
Rui Xu ◽  
Jie Liu
Keyword(s):  
Neural Networks ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Surrogate Model ◽  
Deep Neural Networks ◽  
Stokes Equations ◽  
Universal Function ◽  
Neural Structure ◽  
Partial Differential ◽  
Weighted Residuals

AbstractPartial differential equations (PDEs) are ubiquitous in natural science and engineering problems. Traditional discrete methods for solving PDEs are usually time-consuming and labor-intensive due to the need for tedious mesh generation and numerical iterations. Recently, deep neural networks have shown new promise in cost-effective surrogate modeling because of their universal function approximation abilities. In this paper, we borrow the idea from physics-informed neural networks (PINNs) and propose an improved data-free surrogate model, DFS-Net. Specifically, we devise an attention-based neural structure containing a weighting mechanism to alleviate the problem of unstable or inaccurate predictions by PINNs. The proposed DFS-Net takes expanded spatial and temporal coordinates as the input and directly outputs the observables (quantities of interest). It approximates the PDE solution by minimizing the weighted residuals of the governing equations and data-fit terms, where no simulation or measured data are needed. The experimental results demonstrate that DFS-Net offers a good trade-off between accuracy and efficiency. It outperforms the widely used surrogate models in terms of prediction performance on different numerical benchmarks, including the Helmholtz, Klein–Gordon, and Navier–Stokes equations.

The slow translation of a sphere in a rotating viscous fluid

1982 ◽  
Vol 117 ◽  
pp. 251-267 ◽  
Author(s):  
S. C. R. Dennis ◽  
D. B. Ingham ◽  
S. N. Singh
Keyword(s):  
Reynolds Number ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Angular Velocity ◽  
Finite Difference ◽  
Differential Equations ◽  
Viscous Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Partial Differential

The motion of a sphere along the axis of rotation of an incompressible viscous fluid that is rotating as a solid mass is investigated by means of numerical methods for small values of the Reynolds and Taylor numbers. The Navier–Stokes equations governing the steady axisymmetric flow can be written as three coupled, nonlinear, elliptic partial differential equations for the stream function, vorticity and rotational velocity component. Two numerical methods are employed to solve these equations. The first is the method of series truncation in which the dependent variables are expressed as series of orthogonal Gegenbauer functions and the equations of motion are then reduced to three coupled sets of ordinary differential equations, which are integrated numerically subject to their boundary conditions. In the second method, specialized finite–difference techniques of solution are applied to the two-dimensional partial differential equations. These techniques employ finite-difference equations with coefficients that depend upon the exponential function; a particularly suitable form of approximation for use in calculating numerical solutions is obtained by expanding the exponential coefficients in powers of their exponents.Calculated results obtained by the two methods are in good agreement with each other. The calculations have been carried out according to theoretical assumptions that simulate the experiments of Maxworthy (1965) in which the sphere experiences no resultant torque exerted by the surrounding fluid and is free to rotate with constant angular velocity. Numerical estimates of this angular velocity and of the drag exerted by the fluid on the sphere are found to agree well with the experimental results for Reynolds and Taylor numbers in the range from zero to unity. The results for small values of the Reynolds number are also consistent with theoretical work of Childress (1963, 1964) which is valid as the Reynolds number tends to zero.

scholarly journals Transfer learning for deep neural network-based partial differential equations solving

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Xinhai Chen ◽  
Chunye Gong ◽  
Qian Wan ◽  
Liang Deng ◽  
Yunbo Wan ◽  
...  
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Transfer Learning ◽  
Stokes Equations ◽  
Surrogate Models ◽  
Training Data ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

AbstractDeep neural networks (DNNs) have recently shown great potential in solving partial differential equations (PDEs). The success of neural network-based surrogate models is attributed to their ability to learn a rich set of solution-related features. However, learning DNNs usually involves tedious training iterations to converge and requires a very large number of training data, which hinders the application of these models to complex physical contexts. To address this problem, we propose to apply the transfer learning approach to DNN-based PDE solving tasks. In our work, we create pairs of transfer experiments on Helmholtz and Navier-Stokes equations by constructing subtasks with different source terms and Reynolds numbers. We also conduct a series of experiments to investigate the degree of generality of the features between different equations. Our results demonstrate that despite differences in underlying PDE systems, the transfer methodology can lead to a significant improvement in the accuracy of the predicted solutions and achieve a maximum performance boost of 97.3% on widely used surrogate models.

Exact Solutions of the Unsteady Navier-Stokes Equations

1989 ◽  
Vol 42 (11S) ◽  
pp. S269-S282 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

The unsteady Navier-Stokes equations are a set of nonlinear partial differential equations with very few exact solutions. This paper attempts to classify and review the existing unsteady exact solutions. There are three main categories: parallel, concentric and related solutions, Beltrami and related solutions, and similarity solutions. Physically significant examples are emphasized.

scholarly journals New Fractional Complex Transform for Conformable Fractional Partial Differential Equations

2016 ◽  
Vol 12 (2) ◽  
pp. 41-47 ◽  
Author(s):  
Y. Çenesiz ◽  
A. Kurt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Telegraph Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Complex Transform ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Fractional Telegraph Equation ◽  
Advanced Calculus

Abstract Conformable fractional complex transform is introduced in this paper for converting fractional partial differential equations to ordinary differential equations. Hence analytical methods in advanced calculus can be used to solve these equations. Conformable fractional complex transform is implemented to fractional partial differential equations such as space fractional advection diffusion equation and space fractional telegraph equation to obtain the exact solutions of these equations.

Sign in / Sign up

Export Citation Format

Share Document