5.5 Bessel's Equation

5.5 Bessel's Equation

  • There is a problem in place of Eqs.

  • The tableau is added vertically.
    • The differential equation is equal to zero on the left-hand sides.

  • In order for the equality to hold, each term in this power series must be zero.

  • We use this series to evaluate the function and get its properties.
  • Variation of parameters can be used to find a second independent solution of Bessel's equation.
  • There are two kinds of Bessel functions.
  • You can also look at the CD.
  • There are books of tables with further information.

  • The origin of the Bessel function is not known.

  • The number of solutions is infinite.

  • Both the heat and wave equations have a lot in common, especially the equilibrium solution and the eigenvalue problem.
    • To reinforce that observation, we will solve a heat problem and a wave problem with similar conditions.
  • Problems that are two-dimensional in one coordinate system may become onedimensional in another system.

  • a.

  • a.
  • We can recognize it.

  • The CD has these shown.
  • Returning to the beginning.
  • a.
  • It is true that the eigenfunctions of Eqs can be used in this problem, even though it is not a regular exercise in the series.

  • Explanation for Eq.
    • is given by the following theorem.

  • An animation of a Bessel series is shown on the CD.
  • We can proceed with the problem now.

  • Put together the numerator and denominator.

  • There is an animation on the CD.

  • The first three terms of the series should be written.
  • The heat problem has Eqs.

  • Exercise 5 can be used to verify Eq.
  • We will try to solve the problem of describing the displacement of a circular membrane that is fixed at its edge.

  • a.

  • a.
  • We are aware of that.
    • It is necessary to satisfy the boundary condition.
  • The rest of our problem is no longer a problem.
    • Returning to the beginning.

  • a.

  • The function given in Eq.
    • is determined by the coefficients determined by these formulas.
  • We went back to the more general case after seeing the simplest case of the vibrations.

  • a.

  • a.

  • There are a few standing waves on the CD.

  • It's according to a principle.
  • All functions from one series are related to all functions in the other series and the rest of the functions in their own series.

  • Functions from two different series are involved in this relation.

  • The functions within the second series are shown together.
    • The proof of orthogonality is the same for the functions of the last series.

  • The second initial condition is used to calculate's.
  • The computation of the solution to the original problem will be very painful in practice.
    • We can sketch out some of the fundamental modes of vibration of the membrane.
    • The nodal curves represent points with zero displacement.
  • The functions in the series should be verified.
  • The sign says that the Adjacent regions bulge up or down.
  • List the five lowest frequencies.

  • Refer to the derivation of Eqs.

  • Take a look at the nodal curves of the eigenfunctions.
  • There are circles.