Solutions Of One-dimensional Differential Equations Research Articles

A New Wavelet-based Galerkin Method of Weighted Residual Function for The Numerical Solution of One-dimensional Differential Equations

A New Wavelet-based Galerkin Method of Weighted Residual Function for The Numerical Solution of One-dimensional Differential Equations

(Invited) Computational Electrochemical Engineering

In our introductory graduate course in electrochemistry and electrochemical engineering, the overall format is traditional: We start from fundamental thermodynamics, electrode kinetics, and ionic mass transport, and follow with detailed discussion of applications. The course is intended to aid graduate students in quantitative analysis of their research involving electrochemical technologies.However, an essential focus of the course is on detailed electrochemical kinetics and transport, and because these topics tend to be mathematically intensive, a significant number of computational components are included in the course. These components start with simple plotting of electrochemical data, and are gradually extended to include interpolation, curve fitting, solution of systems of nonlinear equations, and solution of one-dimensional differential equations. All of these skills find application in electrochemistry and electrochemical engineering.Computational components use the python programming language and the Jupyter programming environment, which provides a rich interface for computational content, including explanatory text, mathematics, and interactive graphics via the web browser. Using JupyterHub, students can access materials via the web alone and do not have to install or maintain any software. Frequent in-class active learning sessions provide opportunities for supervised practice, enabling students to build fluency in both electrochemistry and computation.Of equal importance, the jupyter interface provides a rich medium for interactive demonstration of electrochemical concepts in synchronous and asynchronous lecture settings. Concepts such as Marcus kinetics, potential variations across an electrochemical cell, and primary and secondary current distributions can be demonstrated in this way.Students complete the course with a capstone project consisting of a computer-based simulation of an electrochemical system. This project assignment is open-ended in the sense that students are asked to choose a topic of study that interests them. The project can involve design, analysis, or simulation of any electrochemical system. However, the project must be focused, and must involve quantitative modeling. Results are presented via a 15-minute in-class presentation and 5-page paper. Frequent topics include models of batteries, biosensors, electrolysis processes and analysis of experimental kinetic data.

Numerical modeling of surge wave in downstreams of the waterworks

Basic requirements for water measuring and water accounting devices developed to fulfil static control of water distribution have not changed virtually in recent decades. It is due primarily to the use of water accounting and measuring instruments developed and used in conditions of steady uniform water flow when unique dependence on depth and water discharge on the rise and decline of the waterline levels is observed. Under operation of watercourses in the unsteady water flow regimen, the existing means of water measurement are not applicable. Simultaneous direct measurement of hydraulic parameters in controlled cross sections of waterways is complicated even under steady water flow regimen. Nowadays there is the lack of data to predict extreme values of discharges and water depth in downstreams of the waterworks due to the absence of tested and feasible methods of water depth and discharge measurement in calculated cross sections of waterways. With the application of the complete integral theory, a hydraulic calculation of the changes in the flow rate and the depth of flow in fixed sections of the downstream of the waterworks was performed. It is based on the analytical method of linearized system solution of one-dimensional differential equations in partial derivatives of Saint-Venant hyperbolic type.

Numerical solving of hydrodynamic problems using machine learning methods

The use of the neural network approach in the process of finding a solution of a hydrodynamic problem by numerical methods is proposed. The two areas of possible application of the technology of neural networks are considered – the choice of initial approximation to the solution and the search for the next approximation. To select the initial approximation, it is proposed to solve the combined task of classification and regression based on the existing base of distributions samples and on the existing base of patterns of space transformations. An architecture of the combined neural network which solves this problem is proposed. For finding the transformation of the problem space, the use of radial-basis neural network is proposed. The mathematical apparatus for tuning the network with arbitrary number of neurons of the output layer is proposed. It is proposed to take the number of hidden layer neurons less than the number of educational transformation samples and find an approximate solution. The task of finding the optimal weight values in this case can be considered as a task of minimization of the target function, which describes the network outputt error. It is proposed to construct a neural network for finding the next approximation of a solution of a hydrodynamic problem based on a generalization of the principle previously proposed for solving solutions of one-dimensional differential equations. Finding the next approximation in the case of solving a task on a multiprocessor system is presented as a game with multiple players, each of which must find a compromise between local and global search purposes. It is proposed to replace one common neural network with a set of neural networks that interact with each other. The proposed approaches can reduce the amount of computation needed to find a solution.

On acoustic boundary-contact problems for a vertically stratified medium bounded from above by a plate with concentrated inhomogeneities

An approach is proposed to the description of diffraction fields of acoustic waves in a vertically stratified medium covered from above by an elastic plate. There is a finite number of point of parallel linear inhomogeneities (cracks, concentrated masses and moments of inertia, etc.) on the plate. Analogous investigations were performed earlier for a homogeneous medium by using integral transformation techniques. The procedure for the numerical solution of one-dimensional differential equations should also be relied upon for the media under consideration.