Question

Use eigenfunction expansion of the GF to solve the \(\mathrm{BVP} u^{\prime \prime}=x\), \(u(0)=0, u(1)-2 u^{\prime}(1)=0 .\)

Short answer

The solution to the boundary value problem is \(u(x) = \frac{x^3}{6}\).

Step-by-step solution

Obtain the Homogeneous Equation and Its Solution

The given equation is \(u^{\prime \prime} = x\). Consider the homogeneous equation \(u^{\prime \prime} = 0\) with the same boundary conditions. Its general solution is \(u(x)=Ax+B\). Applying \(u(0)=0\) we get \(B=0\), so our homogeneous solution is \(u(x)=Ax\).

Calculate Eigenfunctions

To find the eigenfunctions we solve the homogeneous equation under the boundary conditions. Applying \(u(1)=2u’(1)\) gives \(A=2A\), only possible if \(A=0\). So, there are no nontrivial solutions and no eigenvalues.

Compute Green's Function

The Green's function for this problem is given by \(G(x, x') = \frac{x(x'-1)}{2}\), for \(0 < x' < x < 1\), and \(G(x, x') = \frac{x'(x-1)}{2}\), for \(0 < x < x' < 1\). This satisfies our original second order linear differential equation, \(G''(x, x') = \delta(x - x')\), and the boundary conditions, \(G(0, x') = G(1, x') = 0\).

Solve for \(u(x)\) Using Eigenfunction Expansion

Given that there are no nontrivial solutions and no eigenvalues, the solution \(u(x)\) can be expressed using Green's function as \(u(x) = \int_0^1 G(x, x') u(x') dx'\). Substituting \(G(x, x')\) and \(u'(x')\), and performing the integration yields the final solution: \(u(x) = \frac{x^3}{6}\).

Key concepts

Boundary Value Problem

A boundary value problem (BVP) involves finding a function that satisfies a differential equation along with specific boundary conditions. These conditions are set constraints at the endpoints of the interval over which the solution is defined. For example, in the given exercise, the differential equation is \( u'' = x \) and the boundary conditions are \( u(0) = 0 \) and \( u(1) - 2u'(1) = 0 \). These conditions specify how the solution should behave at \( x=0 \) and \( x=1 \).
This form of problem is central in many physical and engineering applications, where the solution must adhere to certain criteria at boundaries, such as temperature or stress at the edges of a material.

• When dealing with a BVP, always confirm the differential equations and boundary conditions are correctly understood.
• Differences in boundary conditions (e.g., Dirichlet, Neumann) demand different approaches for solving the BVP.
• Solutions can often be verified through substituting back into the original differential equation and boundary conditions.

Green's Function

Green's function is a powerful tool to solve differential equations, especially linear ones. It builds a bridge between a differential operator and solution behavior over an interval. In our exercise, the Green's function, \( G(x, x') \), is used to turn the boundary value problem into an integral equation.
For the given problem with \( u'' = x \), the Green's function is explicitly given by:

• \( G(x, x') = \frac{x(x' - 1)}{2} \) for \( 0 < x' < x < 1 \)
• \( G(x, x') = \frac{x'(x - 1)}{2} \) for \( 0 < x < x' < 1 \)

This function must both satisfy the original differential equation \( G''(x, x') = \delta(x - x') \) and comply with the boundary conditions imposed on \( u \).
Having the Green's function allows us to express the solution as \( u(x) = \int_0^1 G(x, x') u(x') dx' \), making complicated differential equations more manageable through integration.

Homogeneous Equation

A homogeneous equation relates to different kinds of differential equations that equal zero. In our case, the homogeneous counterpart of \( u'' = x \) is \( u'' = 0 \). When solving this, we aim to figure out the fundamental solutions without any non-zero external forces.
The significance of solving a homogeneous equation lies in its role in forming the basis for the broader solution to the non-homogeneous equation. Here, the steps involve:

• Solving \( u'' = 0 \), which leads to general solutions like \( u(x) = Ax + B \).
• Applying boundary conditions to determine constants. The condition \( u(0) = 0 \) gives \( B = 0 \).
• Recognizing when \( u(x) = Ax \) helps us solve or simplify the solution to a more complex, non-homogeneous equation.

Homogeneous solutions are vital in understanding the complete nature of differential equations as they describe equilibrium or non-perturbed states.

Eigenvalues and Eigenfunctions

In boundary value problems, eigenvalues and eigenfunctions often arise when considering homogeneous linear differential equations. They are essential when there are nontrivial solutions that satisfy both the equation and the boundary conditions.
For the exercise we've tackled, solving the homogeneous equation with boundary conditions reveals that there are no nontrivial solutions; hence, there are no eigenvalues or eigenfunctions.

• Usually, finding eigenvalues involves determining solutions where certain parameters yield nontrivial solutions.
• Corresponding eigenfunctions are functions that describe the behavior of these solutions, shaped by specific boundary constraints.
• In more complex formulations, these concepts contribute to expanding functions into a series of components that simplify calculation.

Understanding how eigenvalues and eigenfunctions work equips us with greater tools to solve and intuit complex systems within mathematics and physics.