Optimal Feedback Solutions for a Class of Distributed Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 337-342 ◽  
Author(s):  
H. C. Khatri ◽  
R. E. Goodson
Keyword(s):  
Heat Transfer ◽  
Integral Equation ◽  
Control System ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lagrange Equations ◽  
Partial Differential ◽  
Approximate Methods ◽  
Exact And Approximate Methods ◽  
Good Agreement

In the design of controllers for heat transfer systems, one must often describe the plant dynamics by partial differential equations. The problem of optimizing a controller for a system described by partial differential equations is considered here using exact and approximate methods. Results equivalent to the Euler-Lagrange equations are derived for the minimization of an index of performance with integral equation constraints. These integral equation constraints represent the solution of the partial differential equations and the associated boundary conditions. The optimization of the control system using a product expansion as an approximation to the transcendental transfer function of the system is also considered. The results using the two methods are in good agreement. Two examples are given illustrating the application of both the exact and approximate methods. The approximate method requires less computation.

scholarly journals Nonlinear Resonance Interaction between Conjugate Circumferential Flexural Modes in Single-Walled Carbon Nanotubes

2019 ◽  
Vol 2019 ◽  
pp. 1-33 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano
Keyword(s):  
Carbon Nanotubes ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Single Walled Carbon Nanotubes ◽  
Resonance Interaction ◽  
Phase Portraits ◽  
Lagrange Equations ◽  
Partial Differential ◽  
Single Walled Carbon ◽  
Walled Carbon Nanotubes

This paper presents an investigation on the dynamical properties of single-walled carbon nanotubes (SWCNTs), and nonlinear modal interaction and energy exchange are analysed in detail. Resonance interactions between two conjugate circumferential flexural modes (CFMs) are investigated. The nanotubes are analysed through a continuous shell model, and a thin shell theory is used to model the dynamics of the system; free-free boundary conditions are considered. The Rayleigh–Ritz method is applied to approximate linear eigenfunctions of the partial differential equations that govern the shell dynamics. An energy approach, based on Lagrange equations and series expansion of the displacements, is considered to reduce the initial partial differential equations to a set of nonlinear ordinary differential equations of motion. The model is validated in linear field (natural frequencies) by means of comparisons with literature. A convergence analysis is carried out in order to obtain the smallest modal expansion able to simulate the nonlinear regimes. The time evolution of the nonlinear energy distribution over the SWCNT surface is studied. The nonlinear dynamics of the system is analysed by means of phase portraits. The resonance interaction and energy transfer between the conjugate CFMs are investigated. A travelling wave moving along the circumferential direction of the SWCNT is observed.

Element-Free Galerkin (EFG) Method for Time Fractional Partial Differential Equations

2011 ◽  
Vol 101-102 ◽  
pp. 343-347
Author(s):  
Yong Qing Liu ◽  
Rong Jun Cheng ◽  
Hong Xia Ge
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Time Derivative ◽  
Discrete Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Element Free Galerkin ◽  
Element Free ◽  
Caputo Fractional Order Derivative ◽  
Good Agreement

In this paper, the first order time derivative of time fractional partial differential equations are replaced by the Caputo fractional order derivative. We derive the numerical solution of this equation using the Element-free Galerkin (EFG) method. In order to obtain the discrete equation, a various method is used and the essential boundary conditions are enforced by the penalty method. Numerical examples are presented and the results are in good agreement with exact solutions.

scholarly journals Compressible, Turbulent, Viscous Flow Computations for Blade Aerodynamic and Heat Transfer

1996 ◽  
Author(s):  
A. A. Boretti
Keyword(s):  
Heat Transfer ◽  
Experimental Data ◽  
Reynolds Number ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Viscous Flow ◽  
Three Dimensional ◽  
Time Dependent ◽  
Navier Stokes ◽  
Partial Differential

The paper presents a computer code for steady and unsteady, three-dimensional, compressible, turbulent, viscous flow simulations. The mathematical model is based on the Favre-averaged Navier-Stokes conservation equations, closed by a statistical model of turbulence. Turbulence effects are represented by using a low Reynolds number K-ω model. The numerical model makes use of a finite difference approximation in generalized coordinates for space discretization. The solution of time-dependent, three-dimensional, non-homogeneous, partial differential equations is obtained by solving, in a prescribed, symmetric pattern, three time-dependent, one-dimensional, homogeneous partial differential equations, representing convection and diffusion along each generalized coordinate direction, and one ordinary differential equation, representing generation and destruction. An explicit, multi-step, dissipative, Runge-Kutta scheme is finally adopted for time discretization. The code is applied to simulate the flow through a linear cascade of turbine rotor blades, where detailed experimental data are available. Blade aerodynamic and heat transfer are computed, at variable Reynolds and Mach numbers and turbulence levels, and compared with experimental data. While the aerodynamic prediction is relatively unaffected by the properties of both mathematical and numerical models, the heat transfer prediction proves to be extremely sensitive to models details. Low Reynolds number K-ω turbulence models theoretically reproduce laminar, turbulent and transitional boundary layers. However, their practical use in a Navier-Stokes code does not allow to entirely capture the effects of turbulence intensity and Mach and Reynolds numbers on blade heat transfer.

scholarly journals Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs

2020 ◽  
Author(s):  
Matteo Petrera ◽  
Mats Vermeeren
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integrable Hierarchies ◽  
Dimensional Space ◽  
Original System ◽  
Lagrange Equations ◽  
Partial Differential ◽  
Dimensional Space Time ◽  
Variational Symmetries ◽  
Nonlinear Math

Abstract We investigate the relation between pluri-Lagrangian hierarchies of 2-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings in Petrera and Suris (Nonlinear Math. Phys. 24(suppl. 1):121–145, 2017) for ordinary differential equations. We consider hierarchies of 2-dimensional Lagrangian PDEs (many of which have a natural $$(1\,{+}\,1)$$ ( 1 + 1 ) -dimensional space-time interpretation) and show that if the flow of each PDE is a variational symmetry of all others, then there exists a pluri-Lagrangian 2-form for the hierarchy. The corresponding multi-time Euler–Lagrange equations coincide with the original system supplied with commuting evolutionary flows induced by the variational symmetries.

Solving One-Dimensional Partial Differential Equations

2021 ◽  
pp. 191-215
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Material Properties ◽  
Final Part ◽  
Temperature Dependent ◽  
Partial Differential ◽  
One Dimensional

This chapter describes the pdepe command, which is used to solve spatially one-dimensional partial differential equations (PDEs). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the pdepe solver uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics are presented in the final part of the chapter. They illustrate how to solve: heat transfer PDE with temperature dependent material properties, startup velocities of the fluid flow in a pipe, Burger's PDE, and coupled FitzHugh-Nagumo PDE.

Method of Separation of Variables for the Solution of Certain Nonlinear Partial Differential Equations

1971 ◽  
Vol 93 (2) ◽  
pp. 162-164
Author(s):  
V. A. Bapat ◽  
P. Srinivasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Separation Of Variables ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Partial Differential Equation ◽  
Exact Methods ◽  
Partial Differential ◽  
Approximate Methods

A method for the solution of a certain class of nonlinear partial differential equations by the method of separation of variables is presented. The method enables the nonlinear partial differential equation to be reduced to ordinary nonlinear differential equations, which can be solved by exact methods (or by approximate methods if an exact solution is not possible).

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