Question

Solve the initial-boundary value problem. Perform numerical experiments for specific values of \(L\) and \(a\). $$ \begin{array}{l} \text { C } u_{t}=a^{2} u_{x x}, \quad 00, \\ u_{x}(0, t)=0, \quad u_{x}(L, t)=0, \quad t>0 \\ u(x, 0)=\left\\{\begin{array}{ll} 1, & 0 \leq x \leq \frac{L}{2} \\ 0, & \frac{L}{2}

Short answer

Answer: The temperature distribution, u(x,t), changes over time according to the solution found using separation of variables technique and eigenfunction expansion. To analyze the evolution of the temperature, we can plot the approximate solution u(x,t) for different time instants while considering the truncation of the infinite series to n_max = 100.

Step-by-step solution

Rewrite the heat equation in the standard form

Divide the given heat equation by \(C\) to get the standard form: $$ u_{t} = a^{2} u_{xx}, \quad 00 $$

Perform separation of variables technique

Assume a solution \(u(x,t) = X(x)T(t)\). Substituting it into the heat equation, we get: $$ X(x)T'(t) = a^{2} X''(x)T(t) $$ Divide by \(X(x)T(t)\) to separate the variables: $$ \frac{T'(t)}{a^{2} T(t)} = \frac{X''(x)}{X(x)} = \lambda $$ Now, we have two ordinary differential equations (ODEs) to solve: $$ T'(t) - a^{2} \lambda T(t) = 0 $$ $$ X''(x) - \lambda X(x) = 0 $$

Calculate the solution \(u(x,t)\) using the eigenfunction expansion

We need to solve these ODEs with the given boundary and initial conditions. a) Boundary Conditions: $$ u_x(0,t) = X'(0)T(t) = 0 \Rightarrow X'(0) = 0 $$ $$u_x(L,t) = X'(L)T(t) = 0 \Rightarrow X'(L) = 0 $$ b) Initial Condition: $$ u(x,0) = X(x)T(0) = \begin{cases} 1, & 0 \leq x \leq \frac{L}{2} \\ 0, & \frac{L}{2}<x<L \end{cases} $$ Now we must solve the separated ODEs with these conditions. Implementing these conditions, we can now find the series solution. Let \(u_n(x,t)\) be the eigenfunction in the expansion such that: $$ u_n(x,t) = X_n(x)T_n(t) $$ where $$ T_n(t) = e^{-a^2 \lambda_n t} $$ and \(X_n(x)\) is the solution of the ODE satisfying the Neumann boundary conditions. The total solution \(u(x,t)\) can now be found as: $$ u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) $$

Perform numerical experiments

For specific values of \(L\) and \(a\), we can now find the approximation to the solution \(u(x,t)\) using a finite number of terms of the eigenfunction expansion series. Let's choose specific values for the constants: $$ L = 10, \quad a = 2 $$ To approximate the solution numerically, we can truncate the infinite series and for example take \(n_{\text{max}} = 100\). Then the approximate solution is: $$ u(x,t) \approx \sum_{n=1}^{100} u_n(x,t) $$ Now, we can plot the temperature distribution \(u(x,t)\) over the domain \(0<x<L\) and for different time instants to investigate the evolution of the temperature in detail.

Key concepts

Separation of Variables

When solving partial differential equations (PDEs) like the heat equation, the technique known as separation of variables is a powerful method for finding solutions that satisfy both the differential equation and its imposed boundary conditions. This method breaks down complex PDEs into simpler ordinary differential equations (ODEs) by assuming that the solution can be written as a product of functions, each depending on a single variable.

In the context of our problem, the solution for the heat equation in the form of a PDE, \( u_t = a^2 u_{xx} \), is assumed to be a product of two functions: \( X(x) \) depending only on the spatial variable \( x \) and \( T(t) \) depending only on time \( t \): \( u(x, t) = X(x)T(t) \). By substituting this form into the heat equation and isolating the variables associated with space on one side of the equation and time on the other side, it simplifies the problem to solving for the functions \( X(x) \) and \( T(t) \) separately.

The beauty of this technique is that it reduces the dimensions of the problem and converts it into a set of ODEs, which are usually more straightforward to solve. The constant \( \lambda \) that appears in the process is called the separation constant and leads to different solutions depending on its value, which is determined by the boundary conditions of the problem.

Eigenfunction Expansion

The term eigenfunction expansion refers to expressing the solution of a PDE as a sum of eigenfunctions. An eigenfunction is a non-zero function that, when applied to an operator, yields a scalar multiple of itself, called an eigenvalue. In our initial-boundary value problem, we're dealing with Neumann boundary conditions, which involve the derivative of the function at the boundaries.

Each eigenfunction \( X_n(x) \) satisfies the spatial ODE and the boundary conditions, producing a series of functions that correspond to discrete eigenvalues \( \lambda_n \). The time-dependent part \( T_n(t) \) involves these eigenvalues and models the temporal evolution of the system via an exponential decay.

By considering a superposition of these eigenfunctions multiplied by their respective time functions, the solution to the initial-boundary problem is constructed as an infinite series: \( u(x,t) = \sum_{n=1}^{\infty} u_n(x,t) \). However, for practical purposes and numerical simulations, this series is often truncated to include a finite number of terms, giving an approximate yet accurate representation of the actual temperature distribution over time.

Heat Equation

The heat equation is a fundamental PDE that describes the distribution of heat (or variations in temperature) in a given region over time. Mathematically, it's formulated as \( u_t = a^2 u_{xx} \) where \( u_t \) is the time derivative representing the rate of change of the temperature, \( u_{xx} \) is the second space derivative representing the curvature of the temperature profile, and \( a^2 \) is the thermal diffusivity of the material, dictating how quickly heat spreads.

The heat equation lies at the heart of Fourier's law of heat conduction and is essential in fields such as physics, engineering, and environmental science. In our case, solving this heat equation with specific initial and boundary conditions provides insight into how temperature evolves in a one-dimensional rod with insulated ends over time. The initial condition represents a step temperature profile, introducing a challenge: as time progresses, heat diffuses, and the temperature becomes more uniform.

Understanding the solution to the heat equation helps predict how long it would take for the rod to reach a near-uniform temperature and guides the design of systems requiring precise thermal control. The numerical experiments approximating the solution show the practical application of the theoretical principles and align with real-life observations.