Mathematics
Particular Solutions to Differential Equations
Particular solutions to differential equations are specific solutions that satisfy the given initial conditions or boundary conditions. They are obtained by integrating the general solution and then applying the given conditions to determine the specific values of the arbitrary constants. These particular solutions provide the exact solution to the differential equation for the given scenario.
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6 Key excerpts on "Particular Solutions to Differential Equations"
- Yuefan Deng(Author)
- 2014(Publication Date)
- World Scientific(Publisher)
A particular solution (P.S.) is a subset of the general solutions whose arbitrary constant(s) are determined by initial or boundary conditions or both. A singular solution is a solution that is singular. In a less convoluting term, a singular solution is one for which the DE (or the initial value problem or the Cauchy problem) fails to have a unique solution at some point on the solution. The set on which the solution is singular can be a single point or the entire real line. Chapter 1 First-Order Differential Equations 6 Examples Try to guess the solutions of the following DEs: (1) + = 0 (2) + = (3) + = 0 (4) + = sin ( where, in general, ( ≠ (5) + = sin As briefly mentioned before, there are several methods to find the solution of DEs. The following two are common: 1. To obtain analytical (closed form) solutions. Only a small percentage of linear DEs and a few special nonlinear DEs are simple enough to allow findings of analytical solutions. In such studies, basic concepts and theorems concerning the properties of the DEs or their solutions are introduces. 2. To obtain numerical solutions. Most DEs in science, engineering, and finance are too complicated to allow findings of analytical solutions and numerical methods are the only approach to finding approximate numerical solutions. Unfortunately, most of these DEs are the most important for practical purposes. Indeed, every rose has its thorn. To solve DEs numerically, one has to acquire a different set of skills: numerical analysis and computer programming. Problems Problem 1.1.1 Verify by substitution that each given function is a solution of the given DE.
- 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.
- Yogesh Jaluria(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
445 10 Numerical Solution of Partial Differential Equations 10.1 INTRODUCTION In the preceding chapter, the numerical solution of ODEs, which involve a single independent variable, was discussed. However, for a wide variety of problems in science and engineering, the dependent variables are functions of two or more independent variables, such as time and the spatial coordinate distances. Consequently, the differential equations that govern such problems involve partial derivatives and are known as partial differential equations (PDEs). These equations arise in almost all areas of engineering, for instance, in fluid mechanics, elasticity, heat transfer, energy systems, environmental flows, hydraulics, neutron diffusion in nuclear reac-tors, and structural analysis. The numerical solution of PDEs is generally more involved than that of ODEs because of the presence of several independent vari-ables, each with its own initial and boundary conditions. Therefore, effort is often made, whenever possible, by the use of simplifying approximations and transforma-tions, to reduce the governing PDE to an ODE. However, this simplification is pos-sible in only a limited number of cases. Because of the complicated nature of PDEs, analytical solutions are rarely obtained, and numerical methods are necessary for most problems of practical interest. 10.1.1 C LASSIFICATION Many of the classifications outlined in Chapter 9 for ODEs also apply for PDEs. Therefore, the equations may be linear or nonlinear, homogeneous or inhomo-geneous, of first or higher order, and may involve a single equation or a system of equations. The initial and boundary conditions are specified in terms of the vari-ous independent variables, making it possible for the problem to be an initial-value problem in relation to one independent variable and a boundary value problem in relation to another variable.
eBook - PDFPartial Differential Equations
A First Course
- Rustum Choksi(Author)
- 2022(Publication Date)
- American Mathematical Society(Publisher)
a par-ticular solution . An ODE has an infinite number of solutions and this infinite class of solutions (called the general solution) is parametrized (labeled) via arbitrary con-stants . For example when we say to find the general solution to ? ′ (?) = ?(?), we are asking to describe all possible solutions . We do this by saying that the general so-lution is ?(?) = ?? ? , for any constant ? . Given an initial condition, i.e., the value of the solution at a single point, we determine which constant we want for our particular solution. A second-order ODE would require two constants to parametrize (label) all solutions. PDEs will also have an infinite number of solutions but they will now be param-etrized (labeled) via arbitrary functions . This type of parametrization or labeling can easily become the source of great confusion, especially if one becomes guided by notation as opposed to concept. It is important to digest this now as we illustrate with a few examples where we “find” the general solution by describing, or more precisely characterizing, all solutions. Example 1.5.1. Find the general solution on the full domain ℝ 2 for ?(?, ?) solving ? ? = 0. The only restriction that the PDE places on the solution is that the derivative with respect to one of the variables ? is 0 . Thus, the solution cannot vary with ? . On the 8 1. Basic Definitions other hand, the PDE says nothing about how ? varies with ? . Hence: • If ? is any function of one variable, then ?(?, ?) = ?(?) solves the PDE. • On the other hand, any solution of the PDE must have the form ?(?, ?) = ?(?) , for some function ? of one variable. These two statements together mean that ?(?, ?) = ?(?) for any function ? of one variable is the general solution to the PDE ? ? = 0 . If we want to solve the same PDE but in three independent variables (i.e., solve for ?(?, ?, ?) ), then the general solution would be ?(?, ?, ?) = ?(?, ?), for any function ? of two variables.
eBook - PDFComputational Mathematics in Engineering and Applied Science
ODEs, DAEs, and PDEs
- W.E. Schiesser(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
chapter one The General Problems in Ordinary, Differential Algebraic and Partial Differential Equations Ordin~~ry differential equations (ODEs), differential algebraic equations (DAEs), and partial differential equations (PDEs) are mathematical forms that have broad application in sci-ence and engineering. The formulation of mathematical models expressed as sets of ODEs/DAEs/PDEs is frequently the starting point for quantatitive studies of the be-havior and performance of scientific and engineering systems. Mathematical models, however, have limited utility unless solutions to the models can be produced and studied with reasonable effort. Before the general purpose scientific computer became available, models were typically manipulated analytically, often with major assump-tions made during the analysis so that the mathematics would be tractable. Practically, this meant that models consisting of at most a few ODEs/DAEs/PDEs could be solved, and this process often required great ingenuity, particularly in the case of nonlinear equations. With the availability of the scientific computer, this general requirement for mod-els to be highly simplified was completely circumvented; now large sets of nonlinear ODEs/DAEs/PDEs, in principle, can be integrated numerically. In practice, the cod-ing (programming) of a numerical algorithm to solve a particular ODE/DAE/PDE problem system can appear daunting to the scientist or engineer who has limited knowledge and experience in numerical mathematics, and who wishes primarily to arrive at a useful solution without a major investment of time and effort to learn the details of numerical analysis and programming. Additionally, even if the analyst is willing to make this investment of time and effort, the direction this effort should take is often far from clear.
- Joe D. Hoffman, Joe D. Hoffman, Steven Frankel(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
II Ordinary Differential Equations 11.1. Introduction 11.2. General Features of Ordinary Differential Equations II.3. Classification of Ordinary Differential Equations 11.4. Classification of Physical Problems 11.5. Initial-Value Ordinary Differential Equations 11.6. Boundary-Value Ordinary Differential Equations 11.7. Summary 11.1 INTRODUCTION Differential equations arise in all fields of engineering and science. Most real physical processes are governed by differential equations. In general, most real physical processes involve more than one independent variable, and the corresponding differential equations are partial differential equations (PDEs). In many cases, however, simplifying assump-tions are made which reduce the PDEs to ordinary differential equations (ODEs). Part II is devoted to the solution of ordinary differential equations. Some general features of ordinary differential equations are discussed in Part II. The two classes of ODEs (i.e., initial-value ODEs and boundary-value ODEs) are introduced. The two types of physical problems (i.e., propagation problems and equilibrium problems) are discussed. The objectives of Part II are (a) to present the general features of ordinary differential equations; (b) to discuss the relationship between the type of physical problem being solved, the class of the corresponding governing ODE, and the type of numerical method required; and (c) to present examples to illustrate these concepts. 11.2 GENERAL FEATURES OF ORDINARY DIFFERENTIAL EQUATIONS An ordinary differential equation (ODE) is an equation stating a relationship between a function of a single independent variable and the total derivatives of this function with respect to the independent variable. The variable y is used as a generic dependent variable throughout Part II. The dependent variable depends on the physical problem being modeled. In most problems in engineering and science, the independent variable is either time t or space x.
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