Loading...
Please wait, while we are loading the content...
Similar Documents
One-Dimensional Hyperbolic Equations
| Content Provider | Scilit |
|---|---|
| Author | Henner, Victor Belozerova, Tatyana Nepomnyashchy, Alexander |
| Copyright Year | 2019 |
| Description | In Chapter 3 (Example 3.1) we introduced the wave equation as an example of hyperbolic PDE and obtained its general solution. In the present chapter, we study that equation in more detail. Let us start with the following physical example. Consider the problem of small transverse oscillations of a thin, stretched string. The transverse oscillations mean that the movement of each point of the string is perpendicular to the x axis with no displacements or velocities along this axis. Let u(x, t) represent displacements of the points of the string from the equilibrium at location x and time t (u plays the role of the y coordinate, see Figure 5.1). Small oscillations mean that the displacement amplitude u(x, t) is small relative to the string length, and what is important is our assumption that the partial derivative u$ _{ x }$ (x, t) is small for all values of x and t (i.e. the slope is small everywhere during the string's motion), and its second power can be neglected: (u$ _{ x $}$)^{2}$ ≪ 1 (u$ _{ x }$ has no dimension). With these assumptions which are justified in many applications, the equations that we will derive will be linear partial differential equations. Book Name: Partial differential equations |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2018-0-86657-4&isbn=9780429440908&doi=10.1201/9780429440908-5&format=pdf |
| Ending Page | 99 |
| Page Count | 57 |
| Starting Page | 43 |
| DOI | 10.1201/9780429440908-5 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2019-11-20 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Partial differential equations Mathematical Physics Differential Oscillations |
| Content Type | Text |
| Resource Type | Chapter |
