SYSTEM CHARACTERISTICS OF DISTRIBUTED PARAMETER SYSTEMS

2020 ◽  
Vol 70 (3) ◽  
pp. 34-44
Author(s):  
Kamen Perev
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Distributed Parameter Systems ◽  
Point Of View ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Special Cases ◽  
The Green Function ◽  
Set Up

The paper considers the problem of distributed parameter systems modeling. The basic model types are presented, depending on the partial differential equation, which determines the physical processes dynamics. The similarities and the differences with the models described in terms of ordinary differential equations are discussed. A special attention is paid to the problem of heat flow in a rod. The problem set up is demonstrated and the methods of its solution are discussed. The main characteristics from a system point of view are presented, namely the Green function and the transfer function. Different special cases for these characteristics are discussed, depending on the specific partial differential equation, as well as the initial conditions and the boundary conditions.

Exact and approximate controllability for distributed parameter systems

Acta Numerica ◽  
1994 ◽  
Vol 3 ◽  
pp. 269-378 ◽  
Author(s):  
R. Glowinski ◽  
J.L. Lions
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Approximate Controllability ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Control Functions

We consider a system whose state is given by the solution y to a Partial Differential Equation (PDE) of evolution, and which contains control functions, denoted by v.

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

2016 ◽  
Vol 53 (3) ◽  
pp. 938-945 ◽  
Author(s):  
K. Bruce Erickson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Null Solution ◽  
Nonlinear Parabolic ◽  
Branching Diffusion ◽  
Creation Rate

AbstractThe explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

scholarly journals XX.—A Demonstration of Lagrange's Rule for the Solution of a Linear Partial Differential Equation, with some Historical Remarks on Defective Demonstrations hitherto Current

1892 ◽  
Vol 36 (2) ◽  
pp. 551-562 ◽  
Author(s):  
G. Chrystal
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
English Text ◽  
Exact Nature ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Special Cases ◽  
Linear Differential ◽  
Historical Remarks

It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

scholarly journals Finite Analytic Method for One-Dimensional Nonlinear Consolidation under Time-Dependent Loading

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Dawei Cheng ◽  
Wenke Wang ◽  
Xi Chen ◽  
Zaiyong Zhang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Dependent ◽  
Analytic Method ◽  
Partial Differential ◽  
One Dimensional ◽  
Finite Analytic Method ◽  
Nonlinear Consolidation

For one-dimensional (1D) nonlinear consolidation, the governing partial differential equation is nonlinear. This paper develops the finite analytic method (FAM) to simulate 1D nonlinear consolidation under different time-dependent loading and initial conditions. To achieve this, the assumption of constant initial effective stress is not considered and the governing partial differential equation is transformed into the diffusion equation. Then, the finite analytic implicit scheme is established. The convergence and stability of finite analytic numerical scheme are proven by a rigorous mathematical analysis. In addition, the paper obtains three corrected semianalytical solutions undergoing suddenly imposed constant loading, single ramp loading, and trapezoidal cyclic loading, respectively. Comparisons of the results of FAM with the three semianalytical solutions and the result of FDM, respectively, show that the FAM can obtain stable and accurate numerical solutions and ensure the convergence of spatial discretization for 1D nonlinear consolidation.

scholarly journals Exploring Capabilities within ForTrilinos by Solving the 3D Burgers Equation

2012 ◽  
Vol 20 (3) ◽  
pp. 275-292 ◽  
Author(s):  
Karla Morris ◽  
Damian W.I. Rouson ◽  
M. Nicole Lemaster ◽  
Salvatore Filippone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Message Passing ◽  
Message Passing Interface ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Abstract Data Type ◽  
Partial Differential ◽  
Sparse Linear Solvers ◽  
Set Up

We present the first three-dimensional, partial differential equation solver to be built atop the recently released, open-source ForTrilinos package (http://trilinos.sandia.gov/packages/fortrilinos). ForTrilinos currently provides portable, object-oriented Fortran 2003 interfaces to the C++ packages Epetra, AztecOO and Pliris in the Trilinos library and framework [ACM Trans. Math. Softw.31(3) (2005), 397–423]. Epetra provides distributed matrix and vector storage and basic linear algebra calculations. Pliris provides direct solvers for dense linear systems. AztecOO provides iterative sparse linear solvers. We demonstrate how to build a parallel application that encapsulates the Message Passing Interface (MPI) without requiring the user to make direct calls to MPI except for startup and shutdown. The presented example demonstrates the level of effort required to set up a high-order, finite-difference solution on a Cartesian grid. The example employs an abstract data type (ADT) calculus [Sci. Program.16(4) (2008), 329–339] that empowers programmers to write serial code that lower-level abstractions resolve into distributed-memory, parallel implementations. The ADT calculus uses compilable Fortran constructs that resemble the mathematical formulation of the partial differential equation of interest.

scholarly journals A generalized Hankel transform and its use for solving certain partial differential equations

1999 ◽  
Vol 41 (1) ◽  
pp. 105-117 ◽  
Author(s):  
I. Ali ◽  
S. Kalla
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Integral Transform ◽  
Diffusion Theory ◽  
Ground Surface ◽  
Ground Level ◽  
Hankel Transform ◽  
Partial Differential ◽  
Special Cases

AbstractWe introduce a generalized form of the Hankel transform, and study some of its properties. A partial differential equation associated with the problem of transport of a heavy pollutant (dust) from the ground level sources within the framework of the diffusion theory is treated by this integral transform. The pollutant concentration is expressed in terms of a given flux of dust from the ground surface to the atmosphere. Some special cases are derived.

PID Control System for a Distributed Parameter Process

2014 ◽  
Vol 555 ◽  
pp. 222-231 ◽  
Author(s):  
Mihaela Ligia Ungureşan ◽  
Vlad Mureşan
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Partial Differential Equation ◽  
Control System ◽  
Control Systems ◽  
Pid Control ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Pid Algorithm

This paper presents the numerical simulation of a control system, with PID algorithm, for a process modeled through a partial differential equation of second order (PDE II.2), with respect to time (t) and to a spatial variable (p). Because these types of control systems are less usual, this paper develops a case study, with a program run on the computer. The details of using the PID control are pointed out, for an example of a system which contains a process with PDE II.2 structure.

Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups

2019 ◽  
Vol 69 (1) ◽  
pp. 111-124 ◽  
Author(s):  
Xuping Zhang ◽  
Pengyu Chen ◽  
Ahmed Abdelmonem ◽  
Yongxiang Li
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Partial Differential Equation ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Growth Conditions ◽  
Partial Differential ◽  
Nonlocal Initial Conditions ◽  
Noncompactness Measure

Abstract The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

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