Problem 7 - Applied Partial Differential Equations

Question

Consider $$ \begin{aligned} \frac{d^{2} u}{d x^{2}}+4 u &=\cos x \\ u(0) &=u(\pi)=0 \end{aligned} $$ (a) Determine all solutions using the hint that a particular solution of the differential equation is in the form, \(u_{p}=A \cos x\). (h) Determine all solutions using the eigenfunction expansion method. (c) Apply the Fredholm alternative. Is it consistent with parts (a) and (b)?

Step-by-Step Solution

Answer

The second order differential equation can be solved by: (a) finding a particular solution using given hint, (b) using the Eigenfunction expansion method, and (c) applying the Fredholm alternative theorem to check if solutions from parts (a) and (b) are correct.

Step 1: Part a: Finding particular solution

Substitute \( u_{p} = A \cos(x) \) into the differential equation. After differentiation, we obtain: \( \frac{d^2}{dx^2} (A \cos x) + 4 A \cos x = \cos x \). Solve for A to find the value of the constant A which gives the particular solution.

Step 2: Part b: Eigenfunction Expansion Method

The eigenfunctions of this problem are \( \sin(nx) \) where n is an integer. The use of this orthonormal set of functions will be expanded on the forcing function \( \cos(x) \). This will result in a fourier series which will give us the \( u(x) \) in the form of a series. The coefficients of the series can be determined using integration and the orthonormal property of the eigenfunctions.

Step 3: Part c: Fredholm alternative principle

Based on the Fredholm alternative theorem, a homogeneous equation has only the trivial solution if and only if the right hand side of the original equation is orthogonal to the solutions of the homogeneous equation. This can be checked by taking the inner product of the right hand side (RHS) with the homogeneous solutions. If they are indeed orthogonal, this will confirm whether the solutions from parts (a) and (b) are correct.

Key Concepts

Eigenfunction Expansion Method

The Eigenfunction Expansion Method is an elegant and powerful technique utilized to solve boundary value problems, particularly within the realm of partial differential equations. Imagine you have a complex landscape and you want to describe every hill and valley with a combination of simple, well-known shapes. That's essentially what eigenfunction expansion is all about, but with functions and differential equations.

For the given exercise, the eigenfunctions are the sine functions, \( \text{sin}(nx) \), that form a basis set—much like building blocks—for our solution. At its core, this method involves expressing the unknown function, in this case \( u(x) \), as a sum of eigenfunctions multiplied by unknown coefficients. This transform the problem into finding the coefficients that make up our solution, which suit the given boundary conditions.

Here's the magic: because the eigenfunctions have a property called orthogonality—like perpendicular lines, they 'ignore' each other in a specific mathematical sense—we can isolate each coefficient by integrating. This process simplifies what could be a monstrously complex problem into a series of manageable chunks. By applying this method to our differential equation, we build a series that approximates your solution—one piece at a time.

Fredholm Alternative Theorem

The Fredholm Alternative Theorem is not so much an alternative choice, but rather an either-or situation regarding solutions to certain types of equations. Think of it as a fork in the road for mathematicians. When confronted with a nonhomogeneous differential equation, the theorem posits two scenarios:

Firstly, if the associated homogeneous equation has only the trivial solution (essentially nothing, or zero), then the original nonhomogeneous equation must have a unique solution. This is like saying if you have a box that can only be empty, then anything you find inside must be the one item that actually fits.

Secondly, if there is a nontrivial solution to the homogeneous equation, then for a solution to exist for the nonhomogeneous equation, the right hand side (what you're given to find your unknown function) must be orthogonal (independent, or, again, 'ignoring' each other) to the solutions of the transposed homogeneous equation. Therefore, applying this theorem to our exercise helps determine if the solutions we obtained from previous parts are consistent and viable by checking for orthogonality of the given functions.

Boundary Value Problems

Digging into Boundary Value Problems (BVPs) is like setting the rules for a game before you start playing. They are essential in ensuring that the solutions to differential equations make practical sense within a given context. A BVP defines what's happening on the edges of your domain, like the rules that define how a ball behaves when it hits the boundaries of a pinball machine.

In our textbook problem, the boundaries are set by the conditions \( u(0) = u(\pi) = 0 \). Any solution we come up with for our differential equation has to respect these rules and return 0 when x is either 0 or \( \pi \). These problems crop up everywhere in the physical sciences because they often model stable states or steady-state conditions, like the constant temperature distribution in a rod or the vibration modes of a drumhead.

Our exercise gives us an opportunity to reconcile the abstract math with concrete physical constraints. By including these boundary conditions, we're implicitly determining the 'shape' of the solution and ensuring that it's not just mathematically accurate, but also physically meaningful.