Question

Use eigenfunction expansions to find the solution of the given boundary value problem. $$ \begin{array}{l}{u_{t}=u_{x x}+e^{-t}, \quad u_{x}(0, t)=0, \quad u_{x}(1, t)+u(1, t)=0, \quad u(x, 0)=1-x} \\ {\text { see Section } 11.2, \text { Problems } 10 \text { and } 12 .}\end{array} $$

Short answer

Answer: The main goal when solving a boundary value problem using eigenfunction expansions is to find the solution for u(x,t) that satisfies the given conditions in the problem by finding the eigenfunctions and eigenvalues, constructing the eigenfunction expansion of the initial condition, and solving the resulting coefficients.

Step-by-step solution

Eigenfunctions and Eigenvalues

To obtain the eigenfunctions and eigenvalues, we look at the given boundary conditions. Let's consider the homogeneous boundary condition in x. $$ \frac{d\phi}{dx}(0) = 0 $$ The boundary condition is homogenous, so our eigenvalue problem corresponding to this boundary condition is: $$ \phi''(x) + \lambda\phi(x) = 0 $$ Solving this eigenvalue problem with the given boundary condition will lead us to the eigenfunctions and eigenvalues for our problem.

Constructing Eigenfunction Expansion

Now, let's express the initial condition u(x,0) in terms of the eigenfunctions we found in step 1. u(x,0) is given as: $$ u(x, 0) = 1 - x $$ We can express u(x,0) as a Fourier sine series: $$ u(x, 0) = \sum_{n=1}^{\infty}a_{n}\phi_{n}(x) $$ To find the coefficients \(a_n\), we must solve the following equation: $$ a_n = \int_0^1 (1-x)\phi_n(x) dx $$

Solving for Coefficients

Now that we have our eigenfunction expansion and the coefficients, it is time to solve for them. Once we have the coefficients, we can construct the solution to the PDE. The solution has the form: $$ u(x, t) = \sum_{n=1}^{\infty} a_n\phi_n(x)e^{-\lambda_n t} $$ Here, the coefficients \(a_n\) were found in step 2 and the \(\phi_n(x)\) and \(\lambda_n\) were found in step 1. Now, simply substitute the coefficients, eigenfunctions, and eigenvalues into the equation, and this will give the solution to the given boundary value problem.

Key concepts

Eigenfunction Expansions

Imagine that you want to solve a complex problem for a function, but the solution seems elusive and tangled. In boundary value problems, one effective approach is to break the big problem into smaller, more manageable parts—this is where eigenfunction expansions come in. Think of eigenfunctions as "building blocks" that help us construct the solution for complex equations. They correspond to certain "critical" states of the system described by the boundary conditions.

These eigenfunctions, when combined with the concept of eigenvalues, can expand any function into a series that suits the specific boundary and initial conditions of a problem. This method simplifies the process of solving partial differential equations (PDEs), allowing us to express the solution as a sum of these eigenfunction series. Hence, working with eigenfunction expansions often transforms complicated problems into simpler algebraic tasks.

Fourier Series

Sometimes, complex problems require refined tools to tackle them well. In the field of boundary value problems, Fourier series are incredibly useful. They allow us to express complex functions as an infinite sum of sines and cosines, forming what we call the Fourier series representation.

By doing so, the problem is transformed from a complex function into individual sine and cosine terms that are much simpler to work with. In our specific problem, the initial condition is expressed as a Fourier series. This means we decompose the initial condition into a series of periodic functions (sines, cosines) that respect the problem's boundary conditions.

• Fourier series consists of periodic functions.
• They make the math involved more manageable.
• They allow us to find solutions that align with symmetry and periodicity of the problem.

By expressing functions in such a series, you ease the pathway to finding the coefficients necessary for solving these types of PDEs. This approach is intuitive once you understand that complex motions or changes often mirror simple repetitive patterns.

Eigenvalues and Eigenvectors

You're in a room with multiple doors, each leading to a potential solution. Here, eigenvalues and eigenvectors act as clues that guide you through the correct door. In mathematical problems like boundary value challenges, an eigenvalue is a number that indicates how the eigenvector is scaled during transformations.

Eigenvectors point you in "directions" that remain unchanged by a given transformation, save for being multiplied by an eigenvalue. In our specific problem, recognizing the eigenvalues and eigenvectors helps us gauge the behavior of the system outlined by the PDE. They indicate the scales and directions that preserve the characteristics of the problem across transformations.

• Eigenvalues tell us about scale transformations.
• Eigenvectors show invariant directions in the transformation.
• Using both helps to solve PDEs by reducing them to simpler, solvable parts.

Thus, understanding eigenvalues and eigenvectors enhances our capability to dissect and resolve boundary value problems efficiently.