Question

Consider the problem: $$ \frac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \frac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0 $$ a. Solve by direct integration. b. Compare this result to the Green's function in part b of Problem \(31 .\) c. Verify that \(G\) is symmetric in its arguments.

Short answer

The solution to the given partial differential equation is \(G(x,x_{0}) = (x-x_{0})H(x - x_{0}) - x H(-x_{0}) + x_{0} H(-x_{0}) - (\pi - x_{0}) H(\pi - x_{0})\). However, the comparison with the Green's function in Problem 31b and the verification of symmetry depend on the specifics of the named problem and the solutions are qualitative in nature.

Step-by-step solution

- Integrate the Equation

The given equation is a partial differential equation. Firstly, let's integrate the equation with respect to \(x\). The resulting equation after performing the first integration is \(\frac{\partial G}{\partial x} = \int{\delta(x - x_{0}) dx} = H(x - x_{0}) + C\), where \(H\) is the Heaviside step function.

- Apply boundary condition and integrate again

Now we will apply the boundary condition \(\frac{\partial G}{\partial x}(0, x_{0}) = 0\) to determine the constant \(C\). This gives \(C = -H(-x_{0})\). Hence, \(\frac{\partial G}{\partial x} = H(x - x_{0}) - H(-x_{0})\). Now integrate again with respect to \(x\), yielding \(G(x,x_{0}) = (x-x_{0})H(x - x_{0}) - x H(-x_{0}) + D\).

- Apply second boundary condition

Apply the second boundary condition \(G(\pi,x_{0}) = 0\) to find the constant \(D\). This can be expanded as \(D= x_{0} H(-x_{0}) - (\pi - x_{0}) H(\pi - x_{0})\), yielding the solution for \(G(x,x_{0}) = (x-x_{0})H(x - x_{0}) - x H(-x_{0}) + x_{0} H(-x_{0}) - (\pi - x_{0}) H(\pi - x_{0})\).

- Compare with Green Function

Now, for part \(b\), we need to compare this solution with the Green's function found in problem \(31b\). The result can be a recognition of shape, form, operation or relation.

- Verify Symmetry

In the step \(c\), check whether \(G(x,x_{0})\) is symmetric or not. To check the symmetry of \(G(x,x_0)\), swap \(x\) and \(x_0\) and check if the expression remains the same or not. If it does, the function is symmetric.

Key concepts

Partial Differential Equations (PDEs)

Partial Differential Equations (PDEs) are mathematical equations that involve the partial derivatives of a multivariable function. They are fundamentally used to describe various physical phenomena such as heat conduction, wave propagation, and fluid dynamics. A classic example is the heat equation, which describes how temperature changes over time and space.
In the given problem, we are dealing with a second-order linear PDE. Understanding PDEs requires not only knowing how to solve equations involving derivatives with respect to multiple variables but also understanding the physical context they are modeling. In the context of Green's functions, PDEs represent the response of a system to a point source or a 'localized' disturbance, depicted by the delta function \( \delta \).

Direct Integration

Direct integration in the context of PDEs is a method where we integrate the equation with respect to one of its variables to reduce the order of the differential equation. For simple differential equations, this method can be straightforward, but it often requires careful consideration of integrating factors or methods to simplify the integration process.
For the given Green's function problem, direct integration is used to integrate with respect to \(x\) the delta function, which leads to the introduction of the Heaviside step function \(H\). Additionally, direct integration considers boundary conditions to find the constants of integration that arise. This results in a Green's function that satisfies both the original PDE and the specified boundary conditions.

Boundary Conditions

Boundary conditions are constraints necessary for the unique solution of PDEs. These conditions are typically specified on the edges of the domain where the solution to the PDE is being sought. There are several types of boundary conditions:

• Dirichlet boundary conditions - specify the value of the function at the boundary.
• Neumann boundary conditions - specify the value of the derivative of the function at the boundary.
• Robin boundary conditions - specify a linear combination of the function and its derivative at the boundary.

In our scenario, we are given Neumann boundary condition at \(x=0\) and a Dirichlet boundary condition at \(x=\pi\). Applying these conditions helps in determining the constants that arise during the integration process of the PDE, leading to a complete solution that fulfills both the PDE and the boundary constraints.

Heaviside Step Function

The Heaviside step function \(H(x)\) plays a crucial role in solving differential equations, especially when dealing with discontinuities or 'steps' in function values. Defined as \(H(x) = 0\) for \(x < 0\) and \(H(x) = 1\) for \(x \geq 0\), it effectively turns 'on' at a certain point, which makes it extremely useful in modeling situations that are activated at a set point.
In the context of Green's functions, the Heaviside step function allows us to account for the sudden 'turn-on' of a source term in a PDE represented by a delta function. By integrating the delta function, we get a step function that indicates the presence of the source. In this problem, it helps in representing the solution in a piecewise function form, making it easier to impose boundary conditions and interpret the physical meaning of the solution.