Problem 22 - Applied Partial Differential Equations

Question

Solve $$ \frac{d G}{d x}+G=\delta\left(x-x_{0}\right) \quad \text { with } \quad G\left(0, x_{0}\right)=0 . $$ Show that \(G\left(x, x_{0}\right)\) is not symmetric even though \(\delta\left(x-x_{0}\right)\) is.

Step-by-Step Solution

Answer

The solution of the differential equation is \(G(x,x_0) = e^{x_0 - x} \Theta(x-x0)\), and is not symmetric even though the Dirac delta function is.

Step 1: Solve the differential equation

To solve this differential equation, we realize that the homogeneous solution of \(\frac{d G}{d x}+G=0\) is \(G(x, x_0) = A e^{-x}\). Using the given initial condition \(G(0, x_0) = 0\), we obtain \(A=0\). Therefore, the particular solution is given by \(G(x,x_0) = \int_0^x e^{y-x} \delta(y-x_0) dy\).

Step 2: Evaluate the integral

To evaluate this integral, we use the property of the Dirac delta function \(\int f(y) \delta(y-a) dy = f(a)\). Applying this to our integral, we get \(G(x,x_0) = e^{x_0 - x} \Theta(x-x_0)\), where \(\Theta(x-x0)\) is the Heaviside step function which is 0 for \(x < x0\) and 1 for \(x \geq x0\).

Step 3: Show asymmetry of \(G\left(x, x_{0}\right)\)

Notice that \(\delta\left(x-x_{0}\right)=\delta\left(x_{0}-x\right)\) because the Dirac delta function is symmetric. However, \(G(x,x_0)\) is not symmetric because it also includes the factor \(e^{x_0 - x}\) which is not symmetric. Also, the Heaviside function \(\Theta(x-x0)\) makes sure that \(G(x,x0)\) is non-zero only for \(x \geq x0\), further breaking the symmetry.

Key Concepts

Dirac Delta Function

The Dirac delta function, denoted as \( \delta(x) \), is a mathematical concept that is not a function in the traditional sense, but rather a distribution with a very specific property. It is defined to be zero for all values of \(x\) except at \(x = 0\), where it is conceptually infinite. The key feature of the Dirac delta function is that it acts as an identity under the integral for continuous functions: \[ \int_{-\infty}^{\infty} f(t) \delta(t-a) dt = f(a) \.\]

This property makes it incredibly useful in solving differential equations where the Dirac delta appears as a forcing function. When encountering a Dirac delta in such equations, one can think of it as a pulse or a spike that delivers an instant impact at a specific point. In the context of a homework exercise, understanding the integral property of the Dirac delta is crucial for steps that involve solving for particular solutions.

Heaviside Step Function

The Heaviside step function, \( \Theta(x) \), is another pivotal concept when dealing with differential equations, and it serves as a mathematical representation of a 'step'. For a given value \( x_0 \), the Heaviside function is defined as zero for \( x < x_0 \) and one for \( x \geq x_0 \). In essence, it 'turns on' at \( x = x_0 \), creating a discontinuity that is characteristic of step functions.

This function is particularly useful in modeling scenarios where a sudden change occurs, such as the switching on of a current in an electric circuit. In solving differential equations, the Heaviside function helps express the piecewise nature of certain solutions, especially in conjunction with the Dirac delta function, illustrating how systems respond to instantaneous impulses.

Partial Differential Equations

Partial differential equations (PDEs) are used to describe a variety of phenomena in engineering, physics, and other sciences. A PDE involves unknown multivariable functions and their partial derivatives. Solving a PDE means finding a function that satisfies the equation, often subject to certain boundary or initial conditions.

These equations can often be much more complex than ordinary differential equations because they deal with functions of several variables, as opposed to just one. For example, in the context of an educational problem, if students are dealing with a PDE, they must be comfortable with concepts such as boundary value problems and be capable of using techniques like separation of variables or Fourier series to find a solution.

Asymmetry in Solutions

Asymmetry in solutions of differential equations can arise in various contexts. It means that the solution does not exhibit the same behavior or structure on both sides of a central point or axis. This can be due to the nature of the differential equation itself or the boundary/initial conditions applied.

In the given exercise, the asymmetry is introduced because of the Heaviside step function within the solution. While the Dirac delta function is symmetric, the presence of the step function and the exponential modifies the solution in a way that it is no longer mirror-image identical on either side of the point \( x = x_0 \). Understanding asymmetry is important for students as it can have significant implications in physical systems where non-symmetric behaviors lead to different outcomes depending on the variable's domain.