Boundary Value Problems

1. Initial Value Problems.
2. Boundary Value Problems The solution of the differential equation is searched within certain constraints, called the boundary conditions. For example,
   $$\frac{d^2y}{dx^2}=f(x,y,y')$$
   • If the solution of the equation in the interval of $x:[0,L]$ is required,
   • the values at the y(0) and y(L) boundaries should be given.
3. Eigenvalue (characteristic-value) Problems

• Consider a 2nd-order differential equation in the interval [a,b]:

  $$y''=f(x,y,y')$$

  Two necessary conditions for the solution of this equation are given at two extremes:

  
  

  $$y(a) = A$$

  $$y(b) = B$$

  
  
  
  
  
• This problem is more difficult to solve than the initial value problem that we previously discussed.
  • In initial value problem, $y(0)$ and $y'(0)$ are both given at $x=0$.
  • It was possible to start with these two initial values and progress the solution through the interval.
  • In the boundary value problem, the number of initial conditions is insufficient.
• We cannot directly obtain the solution with methods such as Euler or Runge-Kutta.

• Eigenvalue problems are even more difficult.
• See the following 2nd degree differential equation:

  $$y''=f(x,y,y':\lambda)$$

• Again, let the boundary conditions of this equation to be given at both ends.
• If these conditions can only be satisfied for certain $\lambda$ values, we call it the eigenvalue problem.
  • e.g.: Vibrations of a wire with both ends fixed give stable solution only for certain wavelengths.
  • e.g.: Solutions of the Schrödinger equation that are zero at infinity exist only for certain energy eigenvalues.
• In terms of numerical solution, boundary value and eigenvalue problems are solved by the same method.