Question

The equation $$ u_{t t}+c u_{t}+k u=a^{2} u_{x x}+F(x, t) $$ where \(a^{2}>0, c \geq 0,\) and \(k \geq 0\) are constants, is known as the telegraph equation. It arises in the study of an elastic string under tension (see Appendix \(\mathrm{B}\) of Chapter 10 ). Equation (i) also occurs in other applications. Assuming that \(F(x, t)=0,\) let \(u(x, t)=X(x) T(t),\) separate the variables in Eq. (i), and derive ordinary differential equations for \(X\) and \(T\)

Short answer

Question: Given the telegraph equation involving a second-order partial differential equation (PDE), separate variables to obtain ordinary differential equations (ODEs) for the functions X(x) and T(t). Answer: The ordinary differential equations for X(x) and T(t) are as follows: For X(x): \(a^2 X''(x) - \lambda X(x) = 0\) For T(t): \(T''(t) + c T'(t) + (\lambda - k) T(t) = 0\)

Step-by-step solution

Substitute the given solution into the equation

Substitute \(u(x, t) = X(x)T(t)\) into the telegraph equation: $$ X(x)T''(t)+cX(x)T'(t)+kX(x)T(t)=a^{2}X''(x)T(t) $$

Divide by \(X(x)T(t)\)

Divide both sides of the equation by \(X(x)T(t)\) to separate the variables: $$ \frac{T''(t)}{T(t)} + \frac{c T'(t)}{T(t)} + k = a^2 \frac{X''(x)}{X(x)} $$

Set each side to a constant

Since the left-hand side is a function of time \(t\), and the right-hand side is a function of position \(x\), both sides must be equal to some constant value, say \(\lambda\): $$ \frac{T''(t)}{T(t)} + \frac{c T'(t)}{T(t)} + k = \lambda = a^2 \frac{X''(x)}{X(x)} $$

Derive the ODEs for X and T

Rearrange the equations obtained in Step 3 to get the ODEs for X(x) and T(t): For X(x): $$ a^2 X''(x) - \lambda X(x) = 0 $$ For T(t): $$ T''(t) + c T'(t) + (\lambda - k) T(t) = 0 $$ These are the desired ordinary differential equations for X(x) and T(t).

Key concepts

Separation of Variables

Separation of variables is a mathematical technique used to solve partial differential equations (PDEs), like the telegraph equation. It involves expressing a function as a product of functions, each depending on a single independent variable. In this case, we assumed that the solution to the telegraph equation could be written as a product of two functions: one depending on space \(x\) and the other on time \(t\), i.e., \(u(x,t) = X(x)T(t)\). This assumption simplifies the PDE into ordinary differential equations (ODEs) for \(X(x)\) and \(T(t)\). The process begins by substituting the assumed form \(u(x,t) = X(x)T(t)\) into the given PDE. By dividing through by the product \(X(x)T(t)\), we separate the variables such that each side of the equation is only a function of one variable. This allows us to equate each side to a constant \(\lambda\), leading us to individual ODEs. This is crucial because solving these simpler ODEs individually is typically more manageable than solving the original PDE. The ultimate goal of separation of variables is to reduce the complexity of solving differential equations by breaking them down into smaller, more manageable equations.

Ordinary Differential Equations

Once we separate variables in a partial differential equation (PDE) like the telegraph equation, we are left with ordinary differential equations (ODEs) to solve. These ODEs are easier to handle as each involves only one independent variable.For the exercise provided, the separation process leads to two ODEs: one for the spatial component \(X(x)\) and another for the temporal component \(T(t)\). Specifically, the equations derived are:

• \( a^{2}X''(x) - \lambda X(x) = 0 \) for the spatial part, and
• \( T''(t) + cT'(t) + (\lambda - k) T(t) = 0 \) for the temporal part.

These equations are classified as second-order linear differential equations. Each of them can be solved using various techniques depending on the boundary or initial conditions provided. Solving these ODEs yields solutions for \(X(x)\) and \(T(t)\), which can then be combined to form the full solution \(u(x,t)\) for the original PDE. Understanding how to handle and solve these ODEs is key to tackling problems involving PDEs like the telegraph equation.

Elastic String Under Tension

The telegraph equation models various physical phenomena, one of which is the behavior of an elastic string under tension. This model helps us understand the displacement of a string fixed at both ends and subjected to external forces. An elastic string, in this context, can be thought of as a flexible and stretchable object that can vibrate when disturbed. The tension in the string plays a significant role in its behavior, influencing the speed at which waves travel along its length. When a string is under tension, energy can propagate through it, and the telegraph equation offers a way to describe this wave-like motion mathematically. The constants \(a^2\), \(c\), and \(k\) in the telegraph equation help to parameterize these influences. Here, \(a^2\) is related to the wave speed, \(c\) is associated with damping (representing potential friction or resistance), and \(k\) may represent stiffness or another restoring force. Through the insights provided by the equation, one can study how these factors impact the vibrations and waves traveling through the string. This theoretical framework aids in predicting real-world behaviors and properties of elastic materials under tension.