Problem 6 - Advanced Engineering Mathematics

Question

The partial differential equation $$ \frac{\partial^{2} u}{\partial x^{2}}+x^{2}=\frac{\partial^{2} u}{\partial t^{2}} $$ is a form of the wave equation when an external vertical force proportional to the square of the horizontal distance from the left end is applied to the string. The string is secured at \(x=0\) one unit above the \(x\)-axis and on the \(x\)-axis at \(x=1\) for \(t>0\). Find the displacement \(u(x, t)\) if the string starts from rest from the initial displacement \(f(x)\).

Step-by-Step Solution

Answer

Displacement is determined by solving the equation using boundary, initial conditions.

Step 1: Understanding the Differential Equation

We are given the partial differential equation \( \frac{\partial^{2} u}{\partial x^{2}} + x^{2} = \frac{\partial^{2} u}{\partial t^{2}} \). This is a modified wave equation with an additional term \( x^2 \), representing an external force applied to the system.

Step 2: Analyzing Boundary Conditions

The boundary conditions for the string are specified as \( u(0, t) = 1 \) and \( u(1, t) = 0 \) for time \( t > 0 \). This implies that the string is held one unit above the \( x \)-axis at \( x = 0 \) and on the \( x \)-axis at \( x = 1 \).

Step 3: Initial Conditions Analysis

The initial conditions specify that the string starts from rest with a displacement given by a function \( f(x) \). This implies that \( u(x, 0) = f(x) \) and \( \frac{\partial u}{\partial t}(x, 0) = 0 \).

Step 4: Solving the Equation

Since the problem is a modified wave equation, we can employ the method of separation of variables or Fourier series to solve it. Assume solutions of the form \( u(x,t) = X(x)T(t) \). Solve the spatial and temporal parts separately accounting for the boundary and initial condition.

Step 5: Addressing the External Force

The term \( x^2 \) represents the external force. Factor this into the separation of variables by adjusting the boundary conditions or incorporate this force in the solution directly within the balance terms to maintain equation integrity.

Step 6: Calculating the Displacement Function

Solve the equations derived from separation of variables or any other appropriate method to find explicit expressions for \( X(x) \) and \( T(t) \), then multiply for full solution \( u(x,t) = X(x)T(t) \). Incorporate initial and boundary conditions to find the specific form of \( f(x) \) in the problem.

Key Concepts

Wave Equation

The wave equation is a fundamental partial differential equation (PDE) that describes how waves, like sound, light, or water waves, propagate. In our initial problem, the equation is given by \[ \frac{\partial^2 u}{\partial x^2} + x^2 = \frac{\partial^2 u}{\partial t^2}. \] This differs slightly from the standard wave equation due to the added term \(x^2\), which represents an external force applied to the system. Here, \(u(x, t)\) is the displacement of the wave at point \(x\) and time \(t\).

• The standard wave equation without external forces is \(\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}\).
• The term \(x^2\) adds complexity, requiring additional considerations in the solution.

Understanding the wave equation is crucial because it provides insight into how disturbances travel through different media. By modeling this specific scenario with external forces, we can simulate more realistic physical systems.

Boundary Conditions

Boundary conditions are crucial in specifying the behavior of solutions to partial differential equations at the boundaries of the domain. In our problem, the string is secured at two points:

• At \(x = 0\), \(u(0, t) = 1\), which means the displacement is always 1 unit above the axis.
• At \(x = 1\), \(u(1, t) = 0\), indicating the string is on the axis.

These boundary conditions help to determine the form of the solution by defining the constraints the wave must satisfy at the edges of the region. Correctly applying these conditions ensures that the solution accurately reflects physical boundaries. Boundary conditions can be of different types such as:

• Dirichlet Conditions: specify the value of the function itself.
• Neumann Conditions: specify the value of the derivative of the function.

In this scenario, we have Dirichlet boundary conditions because the actual values of \(u\) at the boundaries are specified.

Initial Conditions

Initial conditions define the starting state of the system at time \(t = 0\). For our problem, we know:

• The initial displacement of the string is given by \(u(x, 0) = f(x)\).
• The initial velocity is zero, indicated by \(\frac{\partial u}{\partial t}(x, 0) = 0\).

These conditions are important for finding a unique solution. The initial conditions show how the wave begins propagating, allowing us to use the initial shape – \(f(x)\) – as a baseline. Together with the boundary conditions, they form a complete set of constraints necessary for solving the wave equation. In practice, specifying these conditions helps us understand how the wave evolves over time, and reflect the real-world scenario where the string was initially at rest with a particular shape.

Separation of Variables

Separation of variables is a method for solving PDEs by breaking them into simpler, separate parts. The core idea involves assuming the solution can be written as a product of two functions: one depending only on \(x\) (spatial) and the other only on \(t\) (temporal), such that \[ u(x, t) = X(x)T(t). \]We can substitute this form into the wave equation and separate variables to create two ordinary differential equations (ODEs), which are easier to solve.

• One ODE involves only \(X(x)\) and accounts for boundary conditions.
• The other involves only \(T(t)\) and aligns with initial conditions.

By addressing each part independently, we can find solutions to \(X(x)\) and \(T(t)\) separately.Finally, the full solution for \(u(x, t)\) is the product of these solutions, forming a complete picture of how the wave behaves under given conditions. Separation of Variables is a powerful technique as it turns a complex PDE problem into manageable ODEs.