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Representation of Solutions and Numerical Calculation

The Sturm-Liouville differential equation (1) with boundary conditions may be solved in practice by a variety of numerical methods. In difficult cases, one may need to carry out the intermediate calculations to several hundred decimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places.

1. Shooting methods. These methods proceed by guessing a value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [a,b], comparing the value this solution takes at the other endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues.

2. Finite difference method.

3. The Spectral Parameter Power Series (SPPS) method makes use of a generalization of the following fact about second order ordinary differential equations: if y is a solution which does not vanish at any point of [a,b], then the function

is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ₀^* (often λ₀^*=0; it does not need to be an eigenvalue) and any solution y₀ of (1) with λ=λ₀^* which does not vanish at on [a,b]. (Discussion below of ways to find appropriate y₀ and λ₀^*.) Two sequences of functions

, on [a,b], referred to as iterated integrals, are defined recursively as follows. First when n=0, they are taken to be identically equal to 1 on [a,b]. To obtain the next functions they are multiplied alternately by 1/(py₀²) and wy₀² and integrated, specifically

for n odd, for n even, 5

for n odd, for n even, 6

when n>0. The resulting iterated integrals are now applied as coefficients in the following two power series in λ:

and

Then for any λ (real or complex), u₀ and u₁ are linearly independent solutions of the corresponding equation (1). (The functions p(x) and w(x) take part in this construction through their influence on the choice of y₀.)

Next one chooses coefficients c₀, c₁ so that the combination y=c₀u₀(b) + c₁u₁(b) satisfies the first boundary condition (2). This is simple to do since (a)=0 and (a)=0 for n>0. The values of (b) and (b) provide the values of u₀(b) and u₁(b) and the derivatives u₀'(b) and u₁'(b), so the second boundary condition (3) becomes an equation in a power series in λ. For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the sought-after eigenvalues.

When λ= λ₀, this reduces to the original construction described above for a solution linearly independent to a given one. The representations ('5'),('6') also have theoretical applications in Sturm-Liouville theory.

Construction of a nonvanishing solution

The SPPS method can be itself used to find a starting solution y₀. Consider the equation (p'y)'=μqy; i.e., q, w, and λ are replaced in (1) by 0, -q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ₀=0. While there is no guarantee that u₀ or u₁ will not vanish, the complex function y₀=u₀+iu₁ will never vanish because two linearly independent solutions of a regular S-L equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y₀ of (1) for the value λ₀=0. In practice if (1) has real coefficients, the solutions based on y₀ will have very small imaginary parts which must be discarded.

See also

• Variation of Parameters
• Spectral theory of ordinary differential equations