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Spatial Models Using Partial Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
The wave equation describes the propagation of oscillations and is represented by a linear second-order partial differential equation. Consider a homogeneous string of length L is tied at both ends. We assume that the string offers no resistance due to bending; that is, it is thin and flexible; the tension in the string is much greater than the gravitational force, and hence, it can be neglected; the motion of the string takes place in the vertical plane only (fig. 4.6).
An introduction to partial differential equations
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Partial differential equations occur in many areas of engineering and technology; electrostatics, heat conduction, magnetism, wave motion, hydrodynamics and aerodynamics all use models that involve partial differential equations. Such equations are difficult to solve, but techniques have been developed for the simpler types. In fact, for all but the simplest cases, there are a number of numerical methods of solutions of partial differential equations available.
Classification of PDEs
Published in Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin, Finite Analytic Method in Flows and Heat Transfer, 2020
Ching Jen Chen, Richard Bernatz, Kent D. Carlson, Wanlai Lin
A partial differential equation (PDE) is an equation involving one or more partial derivatives of an unknown function of several independent variables. Suppose the dependent variable u is a function of the spatial variables x, y and z, and time t so that u = u(x,y,z,t). The general form of a PDE for u is shown in Equation (3.1). F(x,y,z,t;u,ux,uy,uz,ut,uxx,uyy,…)=0
Thermal Performance Analysis of a Commercial Space Exposed to Solar Radiations in the Composite Climatic Conditions
Published in Heat Transfer Engineering, 2023
Shubham Kumar Verma, Vibhushit Gupta, Sahil Thappa, Sanjeev Anand, Navin Gupta, Yatheshth Anand
In the present study, a field study along with a set of numerical assessments was conducted to assess the dwelling conditions of the commercial space. An appropriate isolated space whose roof is directly exposed to solar radiation is selected for visualizing the real behavior of airflow, temperature, and humidity. Physical measurements of various parameters in and around the space have been performed on 21st June 2021 at 1300 h. This is followed by the development of a similar model in the Design modeler, a module of the Ansys workbench. Further, a set of numerical methods has been utilized to solve and approximate the partial differential equations using the computational fluid dynamics (CFD) package. All the CFD simulations in this examination had been carried out by utilizing Ansys Fluent. It works on the Navier stokes equation [21] which is a fundamental partial differential equation that governs the fluid motion and can be seen as Newton’s second law of motion for fluid. The basic Navier Stokes equation is given as: where
Optimal control of coupled parabolic–hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach
Published in International Journal of Systems Science, 2018
I. Aksikas, A. Alizadeh Moghadam, J. F. Forbes
Partial differential equations (PDEs) can describe the dynamics of many phenomena such as chemical processes, fluid dynamics, quantum mechanics, electrodynamics and much more. These dynamical systems are called distributed parameter systems since their dynamics are distributed over space (or another parameter) beside the variation in time. Control problems of such systems are usually expressed as abstract differential equations on infinite-dimensional state space in order to keep the distributed nature of the system and then avoid discretisation-based control techniques. The theory of infinite-dimensional linear system represents a natural extension of the finite-dimensional case and is now a well-established area of research (Bensoussan, Da Prato, Delfour, & Mitter, 2007; Curtain & Zwart, 1995). Control of infinite-dimensional systems has increasingly received more attention during the last decades and all the standard control techniques such as model predictive control (Dubljevic, El-Farra, Mhaskar, & Christofides, 2006), sliding mode control (Orlov & Utkin, 1987) and linear-quadratic (LQ)-control methods (Aksikas, 2005; Bensoussan et al., 2007; Da Prato & Ichikawa, 1990; Pandolfi, 1992), have been developed and/or extended for distributed parameter systems. Moreover, LQ control has been developed for many kinds of PDEs, including hyperbolic PDEs (Aksikas & Forbes, 2010), parabolic PDEs (Ng, Aksikas, & Dubljevic, 2013) and also PDEs coupled with ordinary differential equations (ODEs) (Alizadeh Moghadam, Aksikas, Dubljevic, & Forbes, 2010, 2013).
Sliding window method for vehicles moving on a long track
Published in Vehicle System Dynamics, 2018
Shuqi Song, Weihua Zhang, Peng Han, Dong Zou
The rail in the sliding window is also modelled with a fourth-order partial differential equation because it is assumed to be a simply supported beam. We can use the separation of variables method to solve the partial differential equations. So, we need to establish the coordinates of the discrete points, which will be used in the mode-superposition method. The origin of the calculation coordinate is set at the left-hand edge of the window and x1 is the coordinate of the first sleeper from the left. By this analogy, the first sleeper coordinate of the nth window is xi and the last one is xj. When the time reaches T+ and the window moves a distance of S1 metres, the original m sleepers are replaced by m new sleepers moving into the window. Thus, the first and last sleeper coordinates at time T+ in the window are xi+m and xj+m, respectively. Because the sleepers and ballast are discrete, the window edge should not be truncated as is the rail. This process can be expressed with the following equations: