Problem 4 - Advanced Engineering Mathematics

Question

Solve the given boundary-value problem by an appropriate integral transform. Make assumptions about boundedness where necessary. $$ \begin{aligned} &\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=e^{-|x|},-\infty < x < \infty, t>0 \\ &u(x, 0)=u_{0},-\infty< x < \infty \end{aligned} $$

Step-by-Step Solution

Answer

Apply Fourier transform to the PDE, solve in Fourier space, and then inverse transform to find \(u(x,t)\).

Step 1: Identify the Problem Type

This is a linear partial differential equation known as the heat equation, with a source term \(e^{-|x|}\). The solution is sought for all \(x\) in the real line for \(t > 0\).

Step 2: Apply Fourier Transform

To solve this PDE, apply the Fourier transform with respect to \(x\). The Fourier transform of \(u(x,t)\) is denoted by \(\hat{u}(k,t)\). The Fourier transform of \(e^{-|x|}\) is \(\frac{2}{1+k^2}\). Therefore, applying the Fourier transform to both sides gives:\[ \frac{\partial \hat{u}}{\partial t} + k^2 \hat{u} = \frac{2}{1+k^2} \]

Step 3: Initial Condition in Fourier Space

Transform the initial condition \(u(x,0) = u_0\). Its Fourier transform is \(\hat{u}(k,0) = \hat{u}_0(k)\). This will be used to solve the ordinary differential equation obtained after the Fourier transform.

Step 4: Solve the Transformed ODE

Solve the linear ODE: \[ \frac{\partial \hat{u}}{\partial t} + k^2 \hat{u} = \frac{2}{1+k^2} \] Using the integrating factor \(e^{k^2 t}\), multiply through the equation and integrate with respect to \(t\):\[ \hat{u}(k,t) = e^{-k^2 t} \hat{u}_0(k) + \frac{2}{1+k^2} \frac{1 - e^{-k^2 t}}{k^2} \]

Step 5: Inverse Fourier Transform

To obtain \(u(x,t)\), apply the inverse Fourier transform to \(\hat{u}(k,t)\). This involves computing integrals of the form:\[ u(x,t) = \int_{-\infty}^{\infty} \left( e^{-k^2 t} \hat{u}_0(k) + \frac{2}{1+k^2} \frac{1 - e^{-k^2 t}}{k^2} \right) e^{ikx} \, dk \]

Step 6: Evaluate Solutions

Evaluating the inverse Fourier transform yields the solution \(u(x,t)\). For specific initial conditions, \(\hat{u}_0(k)\) must be explicitly computed, but the general form considers decaying terms involving the exponential of \(-k^2 t\) and shifts applied via the source term.

Key Concepts

Heat Equation

The heat equation is a type of partial differential equation (PDE) that describes how heat diffuses through a given region over time. In its simplest form, it involves the function \( u(x, t) \), which represents the temperature at a point \( x \) and time \( t \). The standard form of the heat equation is \( \frac{\partial u}{\partial t} - \alpha \frac{\partial^2 u}{\partial x^2} = 0 \), where \( \alpha \) is the thermal diffusivity constant.

• It models how heat, or other substances, spreads over time.
• Applies to any dimension, but solutions often require boundary-value constraints for practical problems.
• The presence of a source term \( e^{-|x|} \) in the given problem modifies the equation to include an external heat input.

The key challenge is finding a solution \( u(x, t) \) that satisfies both the heat equation and any given initial or boundary conditions.

Boundary-Value Problems

Boundary-value problems (BVPs) require a solution to a differential equation that satisfies certain specified conditions at the boundaries of the domain. In the context of the heat equation, these conditions generally involve the temperature at the edges of the region under consideration.

• For the heat equation, initial conditions specify the temperature distribution at \( t = 0 \).
• Boundary conditions might be defined at spatial boundaries, which in this case extend to infinity on the real line.
• Solving BVPs accurately connects the abstract mathematical solution to real-world physical interpretation.

In this problem, the boundary condition initially provides temperature distribution \( u(x,0) = u_0 \). Solutions require a method like Fourier transforms to handle infinite domain boundaries efficiently.

Integral Transforms

Integral transforms, like the Fourier transform, are mathematical techniques that convert functions into a different domain, often to simplify the analysis of differential equations. They are particularly useful in solving linear PDEs such as the heat equation.

• The Fourier transform converts a function of time and space into a frequency domain representation.
• This transformation can simplify the PDE by converting it from a function of space and time into a function of frequency and time alone.
• The inclusion of a source term, like \( e^{-|x|} \), may also be efficiently handled through its transform.

Applying the Fourier transform to the heat equation gives a new equation \( \frac{\partial \hat{u}}{\partial t} + k^2 \hat{u} = \frac{2}{1+k^2} \), making it easier to solve as an ordinary differential equation for \( \hat{u}(k,t) \).

Inverse Fourier Transform

The inverse Fourier transform is a mathematical operation that allows you to convert data from the frequency domain back into the original spatial domain. This step is crucial in obtaining the actual solution to the PDE after applying the Fourier transform.

• It takes a transformed function \( \hat{u}(k,t) \) back into \( u(x,t) \).
• Requires evaluating integrals that involve exponentials and can be computationally intensive.
• Restores the physical interpretation of the solution, mapping frequencies back to spatial variables.

For our specific problem, this involves computing an integral that reconstructs \( u(x,t) \) from \( \hat{u}(k,t) \), providing a complete picture of heat distribution with respect to space and time.