Technology & Engineering
Partial Differential Equations
Partial Differential Equations (PDEs) are mathematical equations that involve multiple independent variables and their partial derivatives. They are used to model physical phenomena in engineering and technology, such as heat conduction, fluid dynamics, and electromagnetic fields. Solving PDEs is essential for understanding and predicting the behavior of complex systems in various engineering applications.
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9 Key excerpts on "Partial Differential Equations"
eBook - PDF- D. Vaughan Griffiths, I.M. Smith(Authors)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 8 Introduction to Partial Differential Equations 8.1 Introduction A partial differential equation (PDE) contains derivatives involving two or more independent variables. This is in contrast to ordinary differential equa-tions (ODE) as described in Chapter 7, which involve only one independent variable. Many phenomena in engineering and science are described by PDEs. For example, a dependent variable, such as a pressure or a temperature, may vary as a function of time ( t ) and space ( x, y, z ). Two of the best known numerical methods for solving PDEs are the Finite Difference and Finite Element methods, both of which will be covered in this chapter. Nothing more than an introductory treatment is attempted here, since many more advanced texts are devoted to this topic, including one by the authors themselves on the Finite Element Method (Smith and Griffiths 2004). This and other references on the subject are included in the Bibliography at the end of the text. The aim of this chapter is to familiarize the student with some important classes of PDE, and to give insight into the types of physical phenomena they describe. Techniques for solving these problems will then be described through simple examples. 8.2 Definitions and types of PDE Consider the following second order PDE in two independent variables a ∂ 2 u ∂x 2 + b ∂ 2 u ∂x∂y + c ∂ 2 u ∂y 2 + d ∂u ∂x + e ∂u ∂y + fu + g = 0 (8.1) Note the presence of “mixed” derivatives such as that associated with the coefficient b . 393 394 Numerical Methods for Engineers If a, b, c, . . . , g are functions of x and y only, the equation is “linear”, but if these coefficients contain u or its derivatives, the equation is “nonlinear”. The degree of a PDE is the power to which the highest derivative is raised, thus equation (8.1) is first degree. Only first degree equations will be consid-ered in this chapter.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
10 Partial Differential Equations In this chapter and the next, the solution of differential equations of types typically encountered in the physical sciences and engineering is extended to situations involving more than one independent variable. A partial differential equation (PDE) is an equation relating an unknown function (the dependent variable) of two or more variables to its partial derivatives with respect to those variables. The most commonly occurring independent variables are those describing position and time, and so we will couch our discussion and examples in notation appropriate to them. As in the rest of this book, we will focus our attention on the equations that arise most often in physical situations. We will restrict our discussion, therefore, to linear PDEs, i.e. those of first degree in the dependent variable. Furthermore, we will discuss primarily second-order equations. The solution of first-order PDEs will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third-and higher-order equations. We shall also see that many ideas developed for ODEs can be carried over directly into the study of PDEs. Initially, in the current chapter, we will concentrate on general solutions of PDEs in terms of arbitrary functions of particular combinations of the independent variables, and on the solutions that may be derived from them in the presence of boundary conditions. We also discuss the existence and uniqueness of the solutions to PDEs under given boundary conditions. In the following chapter the methods most commonly used in practice for obtaining solutions to PDEs subject to given boundary conditions will be considered. These methods include the separation of variables, integral transforms and Green’s functions.
eBook - PDF- Kuzman Adzievski, Abul Hasan Siddiqi(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
CHAPTER 4 Partial Differential Equations Many problems in physics and engineering are described by partial differen-tial equations (PDEs) with appropriate boundary and initial value conditions. Partial Differential Equations have a major role in electromagnetic theory, fluid dynamics, traffic flow and many other disciplines. Therefore there is a need to study the theory of Partial Differential Equations and the rest of the book is devoted entirely to this important topic. A major part of this chapter is devoted to the study of a particular class of first order partial differential equation, namely, linear Partial Differential Equations. We will start with basic concepts and terminology and finally we will give some generalities of linear Partial Differential Equations of the second order and their classification. In this chapter, as well as in the remaining chapters, all functions will be real-valued functions of one or more real variables unless we indicate otherwise. 4.1 Basic Concepts and Terminology. A domain Ω in the plane R 2 or in the space R 3 is a nonempty, open and connected set in R 2 or in R 3 . A set Ω is called open if for every point P ∈ Ω, there is an open disc/ball (without the circle/sphere) with center at P which is a subset of Ω. A set Ω is called connected if any two points in Ω can be joined by a polygonal line which lies entirely in Ω. If u = u ( x, y, . . . ), ( x, y, . . . ) ∈ Ω is a function of two or more variables, then the partial derivatives of u of the first order will be denoted by u x , u y , instead of ∂u ∂x , ∂u ∂y . Similarly, for the partial derivatives of the second order we will use the nota-tion u xx , u yx , u yy , instead of ∂ 2 u ∂x 2 , ∂ 2 u ∂y∂x , ∂ 2 u ∂y 2 . 204 4.1 BASIC CONCEPTS AND TERMINOLOGY 205 We say that a function u = u ( x, y ), ( x, y ) ∈ Ω ⊆ R 2 belongs to the class C k (Ω), k = 1 , 2 , . . . , if all partial derivatives of u up to the order of k are continuous functions in Ω.
eBook - PDFPartial Differential Equations
A First Course
- Rustum Choksi(Author)
- 2022(Publication Date)
- American Mathematical Society(Publisher)
We will most often use ? to denote the unknown function, i.e., the dependent variable. So, for example, in purely spatial variables we would be dealing with ?(?, ?), ?(?, ?, ?) , or more generally ?( x ) . When time is also relevant, we will deal with ?(?, ?), ?(?, ?, ?), ?(?, ?, ?, ?) , or more generally ?( x , ?) . 1.2. • What Are Partial Differential Equations and Why Are They Ubiquitous? “ The application of calculus to modern science is largely an exercise in the formulation, solution, and interpretation of Partial Differential Equations . ... Even at the cutting edge of modern physics, partial dif-ferential equations still provide the mathematical infrastructure. ” -Steven Strogatz a a Strogatz is an applied mathematician well known for his outstanding undergraduate book on the qualitative theory of ODEs: Nonlinear Dynamics and Chaos , Westview Press. This quote is taken from his popular science book Infinite Powers: How Calculus Reveals the Secrets of the Universe , Houghton Mifflin Harcourt Publishing Company, 2019. We begin with a precise definition of a partial differential equation. Definition of a Partial Differential Equation (PDE) Definition 1.2.1. A partial differential equation (PDE) is an equation which relates an unknown function ? and its partial derivatives together with inde-pendent variables. In general, it can be written as ?( independent variables , ?, partial derivatives of ?) = 0, for some function ? which captures the structure of the PDE. For example, in two independent variables a PDE involving only first-order partial derivatives is described by ?(?, ?, ?, ? ? , ? ? ) = 0, (1.1) where ? ∶ ℝ 5 → ℝ . Laplace’s equation is a particular PDE which in two independent variables is associated with ?(? ?? , ? ?? ) = ? ?? + ? ?? = 0. Note that often there is no explicit appearance of the independent variables, and the function ? need not depend on all possible derivatives.
- Lennart Edsberg(Author)
- 2015(Publication Date)
- Wiley(Publisher)
5 Partial Differential Equations This chapter is not intended to be an extensive treatment of mathematical properties of Partial Differential Equations (PDEs) but rather a survey of important definitions, concepts, and results. For a more careful description, the reader may consult some of the mathematical textbooks on PDEs referenced in the end of this chapter. A general formulation of a system of PDEs expressed in vector form is u t = f ( t, x, y, z, u, u x , u y , u z , 2 u x 2 , 2 u y 2 , 2 u z 2 , … ) (5.1) The variables t, x, y, z, the independent variables, are defined in some region, bounded or unbounded and the variable u, the dependent variable, is a solution of (5.1) if it satisfies the PDEs for all t, x, y, z in the region. Among the independent variables, t is used to denote time and x, y, z denote the space variables. If time is among the independent variables, the PDE problem is called a time-dependent or an evolution problem, if not the problem is called an equilibrium or a steady-state problem. The solution of a PDE is called a field, a function that depends on time and space or just space. The field is scalar valued if the PDE (5.1) is scalar and vector valued if (5.1) is a system of PDEs. A problem formulated in three space dimensions x, y, z is called a 3D problem (three dimension). The mathematical and numerical treatment of a PDE is often easier when formulated in 2Ds x, y or 1D x. As was illustrated in Chapter 1, a PDE has infinitely many solutions. To obtain a unique solution, we need to have initial and boundary conditions. It depends on the character of the PDE what conditions should be given in order to have a well-posed problem, i.e., a problem the solution of which is stable with respect to perturbations in boundary and initial data. Introduction to Computation and Modeling for Differential Equations, Second Edition. Lennart Edsberg. © 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.
eBook - PDFPartial Differential Equations
An Introduction
- Walter A. Strauss(Author)
- 2012(Publication Date)
- Wiley(Publisher)
1 WHERE PDEs COME FROM After thinking about the meaning of a partial differential equation, we will flex our mathematical muscles by solving a few of them. Then we will see how naturally they arise in the physical sciences. The physics will motivate the formulation of boundary conditions and initial conditions. 1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? The key defining property of a partial differential equation (PDE) is that there is more than one independent variable x , y , . . . . There is a dependent variable that is an unknown function of these variables u ( x , y , . . . ). We will often denote its derivatives by subscripts; thus ∂ u /∂ x = u x , and so on. A PDE is an identity that relates the independent variables, the dependent variable u , and the partial derivatives of u . It can be written as F ( x , y , u ( x , y ) , u x ( x , y ) , u y ( x , y )) = F ( x , y , u , u x , u y ) = 0 . (1) This is the most general PDE in two independent variables of first order. The order of an equation is the highest derivative that appears. The most general second -order PDE in two independent variables is F ( x , y , u , u x , u y , u xx , u xy , u yy ) = 0 . (2) A solution of a PDE is a function u ( x , y , . . . ) that satisfies the equation identically, at least in some region of the x , y , . . . variables. When solving an ordinary differential equation (ODE), one sometimes reverses the roles of the independent and the dependent variables—for in-stance, for the separable ODE du dx = u 3 . For PDEs, the distinction between the independent variables and the dependent variable (the unknown) is always maintained. 1 2 CHAPTER 1 WHERE PDEs COME FROM Some examples of PDEs (all of which occur in physical theory) are: 1. u x + u y = 0 (transport) 2. u x + yu y = 0 (transport) 3. u x + uu y = 0 (shock wave) 4. u xx + u yy = 0 (Laplace’s equation) 5. u tt − u xx + u 3 = 0 (wave with interaction) 6. u t + uu x + u xxx = 0 (dispersive wave) 7.
eBook - PDF- William Bober, Chi-Tay Tsai, Oren Masory(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
187 10 Chapter Partial Differential Equations 10.1 The Classification of Partial Differential Equations The mathematical modeling of many types of engineering-type problems involves Partial Differential Equations (PDEs). PDEs of the general form as given by Equation (10.1) fall into one of three categories. These categories are listed below: A u x B u x y C u y f x y u u x u y ∂ ∂ + ∂ ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ 2 2 2 2 2 , , , , (10.1) where A , B, and C are constants. If B AC 2 4 0 -< , the equation is said to be elliptic . If B AC 2 4 0 -= , the equation is said to be parabolic. If B AC 2 4 0 -> , the equation is said to be hyperbolic. The steady-state heat conduction problem in two dimensions is an example of an elliptic PDE. Laplace’s PDE falls into this category. The parabolic PDE is also called the diffusion equation. The unsteady heat conduction problem is an example of a parabolic PDE. The hyperbolic PDE is also called the wave equation. Sound waves and vibration problems, such as the vibrating string, fall into this category. How a PDE is treated numerically depends into which category it falls. However, there are cases in all three categories where a closed-form solution can be obtained by a method called separation of variables . This solution method is discussed in the next section. 188 ◾ Numerical and Analytical Methods with MATLAB 10.2 Solution by Separation of Variables 10.2.1 The Vibrating String The first problem to be considered is the vibrating string, such as a violin or a viola string (see Figure 10.1). We will assume that 1. The string is elastic. The string motion is vertical. The gravitational forces are negligible compared to the tension in the string. 2. The displacement, Y ( x , t ), from the horizontal is small and the angle that the string makes with the horizontal is small. Then ∂ ∂ Y t is the vertical velocity of the string and ∂ ∂ 2 2 Y t is the acceleration of the string at position x .
eBook - PDF- Franco Tomarelli(Author)
- 2019(Publication Date)
- Società Editrice Esculapio(Publisher)
By finding solutions we mean, either obtaining a simple, explicit solution, or computing a series expansion converging to the solutions, or, failing all these approaches, showing at least the existence and some qualitative properties of solutions. 2.1.1 Examples of PDEs No general theory is available for finding solutions of PDEs, due to the huge variety of physical, geometric and probabilistic phenomena which can be modeled by PDEs and due to the complexity of the natural related prob-lems. Therefore, in the present chapter we will focus only on some important cases which are important for the applications: basic problems for first or-der transport equation and constant coe ffi cients second order linear PDEs which are relevant in Mathematical Physics, namely heat equation, Laplace equation and wave equation. In subsequent chapters we will develop suitable technical tools for the analysis of these PDEs and related problems also in spatially unbounded domains and in the distributional framework. The standard notations are listed below: • x = ( x 1 , x 2 , · · · , x n ) 2 U denote the space variable and U is an open subset of R n , • the real variable t ≥ 0 (if present) denotes time, in such case U = ⌦ ⇥ I , x 2 ⌦ ⇢ R n , t 2 I ⇢ R , • Du = D x u = ( u x 1 , u x 2 , . . . , u x n ) denotes the gradient with respect to the space variables only, • u x j is often used as a short notation in place of @ u @ x j , • u t is often used as a short notation in place of derivative @ u @ t , the partial with respect to time t , • Δ u = Δ x u = P n j =1 u x j x j = div x D x u is the Laplacean operator . In order to get familiarity with terminology, in the sequel of present Sub-section 2.1.1 we introduce a wide list of PDEs which play a crucial role in Engineering and Mathematical Physics. Starting form Section 2.2, we intro-duce some toools for solving meaningful associated problems.
eBook - PDF- Peter Deuflhard, Martin Weiser(Authors)
- 2012(Publication Date)
- De Gruyter(Publisher)
Chapter 2 Partial Differential Equations in Science and Technology In the preceding chapter we discussed some elementary examples of PDEs. In the present chapter we introduce problem classes from science and engineering where these elementary examples actually arise, although in slightly modified shapes. For this purpose we selected problem classes from the application areas electrodynamics (Section 2.1), fluid dynamics (Section 2.2) and elastomechanics (Section 2.3). Typ-ical for these areas is the existence of deeply ramifying hierarchies of mathematical models where different elementary PDEs dominate on different levels. Moreover, a certain knowledge of the fundamental model connections is also necessary for numer-ical mathematicians, in order to be able to maintain an interdisciplinary dialog. 2.1 Electrodynamics As its name says, electro dynamics describes dynamical, i.e., time-dependent electric and magnetic phenomena. It includes (in contradiction to its name) as a subset elec-tro statics for static or stationary phenomena. 2.1.1 Maxwell Equations Fundamental concepts of electrodynamics are the electric charge as well as electric and magnetic fields . From their physical properties Partial Differential Equations can be derived, as we will summarize here in the necessary brevity. Conservation of Charge. The electric charge is a conserved quantity – in the frame-work of this theory, charge can neither be generated nor extinguished. It is described as a continuum by some charge density .x/ > 0; x 2 . From the conservation property one obtains a special differential equation, which we will now briefly derive. Let denote an arbitrary test volume. The charge contained in this volume is Q D Z d x: A change of charge in is only possible via some charge flow u through the surface @ , where u denotes a local velocity: Q t D @ @t Z d x C Z @ n T u d s D Z . t C div .u// d x D 0:
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