Problem 14 - Advanced Engineering Mathematics

Question

The concentration \(c(x, t)\) of a substance that both diffuses in a medium and is convected by the currents in the medium satisfies the partial differential equation $$ k \frac{\partial^{2} c}{\partial x^{2}}-h \frac{\partial c}{\partial x}=\frac{\partial c}{\partial t}, 00 $$ where \(k\) and \(h\) are constants. Solve the PDE subject to $$ \begin{aligned} &c(0, t)=0, \quad c(1, t)=0, t>0 \\ &c(x, 0)=c_{0}, 0

Step-by-Step Solution

Answer

The solution is a series: \( c(x,t) = \sum_{n=1, \text{odd}}^{\infty} \frac{4c_0}{n\pi} \sin(n\pi x) e^{-\left( \frac{n\pi}{1} \right)^2 t} \).

Step 1: Formulate the Problem

Consider the given partial differential equation (PDE): \( k \frac{\partial^{2} c}{\partial x^{2}} - h \frac{\partial c}{\partial x} = \frac{\partial c}{\partial t} \). This is a linear convection-diffusion equation that needs to be solved with the boundary conditions \( c(0, t) = 0 \) and \( c(1, t) = 0 \), and initial condition \( c(x, 0) = c_{0} \).

Step 2: Assume Solution Form

Assume a solution of the form \( c(x, t) = X(x)T(t) \). This is called separation of variables, allowing us to transform the PDE into an ordinary differential equation (ODE).

Step 3: Separate Variables

Substitute \( c(x, t) = X(x)T(t) \) into the PDE to get \( k X''(x)T(t) - h X'(x)T(t) = X(x)T'(t) \). Dividing through by \( X(x)T(t) \) gives: \( \frac{k X''(x)}{X(x)} - \frac{h X'(x)}{X(x)} = \frac{T'(t)}{T(t)} \). Set each side equal to a constant \( - \lambda \) to separate the variables.

Step 4: Solve the Spatial ODE

Solving \( k X''(x) - h X'(x) + \lambda X(x) = 0 \) gives the spatial part of the solution. Assume \( X(x) = e^{rx} \), substitute this into the spatial ODE to find that \( r = \frac{h \pm \sqrt{h^2 + 4k\lambda}}{2k} \).

Step 5: Solve the Temporal ODE

Solve the temporal ODE \( \frac{dT}{dt} = -\lambda T \), giving \( T(t) = T_0 e^{-\lambda t} \).

Step 6: Apply Boundary Conditions

Apply the boundary conditions to find that the only valid values for \( \lambda \) are \( \lambda_n = \left( \frac{n\pi}{1}\right)^2, n=1,2,3,... \) leading to solutions \( X(x) = \sin(n\pi x) \).

Step 7: Construct General Solution

The general solution is \( c(x,t) = \sum_{n=1}^{\infty} A_n \sin(n\pi x) e^{-\lambda_n t} \).

Step 8: Apply Initial Condition

Apply the initial condition \( c(x,0) = c_0 \) which gives \( c_0 = \sum_{n=1}^{\infty} A_n \sin(n\pi x) \). Determine coefficients \( A_n \) using Fourier sine series by integrating: \( A_n = \int_0^1 c_0 \sin(n\pi x) \, dx \). This simplifies to \( A_n = \frac{2c_0}{n\pi} \left(1 - \cos(n\pi)\right) \), as a result, for n odd \( A_n = \frac{4c_0}{n\pi} \) and for n even \( A_n = 0 \).

Step 9: Final Solution

Substitute the coefficients \( A_n \) back into the general solution to get the full solution: \( c(x,t) = \sum_{n=1, \text{odd}}^{\infty} \frac{4c_0}{n\pi} \sin(n\pi x) e^{-\left( \frac{n\pi}{1} \right)^2 t} \).

Key Concepts

Separation of Variables

To solve the convection-diffusion equation, we employ a clever technique called separation of variables. This method assumes that a complex problem can be broken down into simpler, independent parts. In this case, we imagine our solution as the product of two functions: one depending only on position, and the other only on time. By doing so, we substitute the form \( c(x, t) = X(x)T(t) \) into the given partial differential equation (PDE).
This substitution transforms our PDE into two ordinary differential equations (ODEs), one involving space \(x\) and the other involving time \(t\). The benefit of this is to reduce the challenge of solving a PDE into solving simpler ODEs, which are often easier to handle. Once we have these ODEs, each part can be addressed independently, and then combined to form the complete solution.

Boundary Conditions

Boundary conditions provide essential rules that the solution to a differential equation must follow. In our convection-diffusion problem, these rules are \( c(0, t) = 0 \) and \( c(1, t) = 0 \) for any time \( t > 0 \). They essentially tell us what must happen at the spatial boundaries of our domain at any given time.
Applying these conditions, especially in the separation of variables approach, is important because it helps us determine particular forms of solutions that satisfy these boundary rules. Here, they lead us to specific solutions with functions that vanish at the boundaries, such as sine functions in this specific problem.
These sine functions are particularly handy because they naturally equal zero at the boundaries being considered, which mathematically agrees with the stipulations set by our boundary conditions.

Fourier Sine Series

In the context of our convection-diffusion equation, the Fourier sine series comes into play when we try to construct the general solution that satisfies the boundary conditions. This series is a collection of sine functions that are summed up to approximate our solution as accurately as needed.
Each function in the series has its own coefficient \( A_n \), which is determined by fitting the initial condition to the series. For our equation, because our boundary condition solutions are sine terms, a Fourier sine series naturally fits. We apply it by expressing the initial condition function \( c(x, 0) = c_0 \) as a sum of sine functions.
This approach not only fulfills the boundary conditions but also allows handling of varying initial conditions. By solving integrals of our initial conditions against these sine functions, we find the coefficients \( A_n \) for each term in the series.

Partial Differential Equation

A partial differential equation (PDE) is a mathematical equation where the unknown function depends on multiple independent variables. Our example involves the concentration \(c(x, t)\), which changes both over space \(x\) and time \(t\). PDEs are essential tools in describing phenomena in physics and engineering that vary continuously in space and time.
The convection-diffusion equation we are working with combines diffusion, the process of substance spreading out in space, with convection, the transport due to fluid motion. This PDE tells us how the concentration of a substance disperses and moves.
To solve such equations, methods like separation of variables are employed, transforming a complex, multi-variable problem into more manageable problems that deal with individually changing aspects.