A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations
A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm-Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter the difference between the exact solution and the approximate one (the truncated NSBF) depends on (the truncation parameter) and the coefficients of the equation and does not depend on . A similar result is valid when belongs to a strip . This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of . Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm-Liouville equation is developed and illustrated on a test problem.
In the recent work  a new representation for solutions of the one-dimensional Schrödinger equation
was obtained in terms of so-called (see, e.g., ,  and ) Neumann series of Bessel functions (NSBF). The representation possesses an attractive feature for applications: for all the difference between the exact solution and the approximate one (the truncated NSBF) depends on (the truncation parameter) and and does not depend on . A similar result is valid when belongs to a strip . This feature makes the NSBF representation especially useful for applications requiring computation of solutions of (1.1) for large intervals of . For example, as was shown in , the NSBF representation allows one to compute hundreds or if necessary even thousands of eigendata with a nondeteriorating for large and remarkable accuracy. In  the NSBF representation was extended onto perturbed Bessel equations.
In the present work we derive an NSBF representation for solutions of the Sturm-Liouville equation
The coefficients are assumed to admit the application of the Liouville transformation (see, e.g., , ). The main result consists in an NSBF representation for solutions of (1.2) and for their derivatives, preserving the same attractive feature described above. Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for (1.2) is developed and illustrated on a test problem.
Besides this Introduction the paper contains the following sections. Section 2 presents some well known facts concerning the Liouville transformation together with recent results from  showing how the system of formal powers associated with (1.2) is transformed by the Liouville transformation. In Section 3 we recall some relevant results from  which are used in Section 4 and Section 5 for obtaining the main results of this work, the NSBF representation for solutions of (1.2) as well as for their derivatives. In Section 6 convenient for computation formulas for the coefficients of the NSBF representations are derived. In Section 7 a computational algorithm based on the NSBF representation is formulated and discussed. Section 8 presents some illustrations of its numerical performance on a test problem admitting an exact solution. In Appendix A we prove error and decay rate estimates of the NSBF representation.
2 Preliminaries on the Liouville transformation
2.1 Definition of the Liouville transformation
Consider the Sturm-Liouville differential equation
with being a finite interval. Let , and be such that , and for all . Define the mapping by
where . Denote . Then (2.1) is related to the one-dimensional Schrödinger differential equation
and the coefficient is given by the relation
The Liouville transformation can be considered as an operator acting according to the rule
Let us introduce the following notations for the differential expressions
The following proposition summarizes the main properties of the operator .
The inverse operator is defined by .
The uniform norms of the operators and are and .
The operator equality
is valid on .
2.2 Transformation of formal powers
Let be a non-vanishing solution of the equation
On the existence of a non-vanishing see Remark 2.5 below. Note that in general complex valued solutions are considered even when the coefficient is real valued. Denote .
The families of functions and constructed according to the rules
are called systems of formal powers associated with equation (2.3).
Analogously, let us introduce a system of formal powers corresponding to equation (2.1).
Let be a solution of the equation
such that for all (see Remark 2.5). Then the following two families of auxiliary functions are well defined
The existence of a non-vanishing solution of (2.6) for complex valued and such that , and are continuous on was proved in [14, Remark 5] (see also ). Moreover, the only reason for the requirement of the absence of zeros of the functions and is to make sure that the formal powers be well defined. As was shown in  this is true even when and/or have zeros, though in that case corresponding formulas are slightly more complicated.
Theorem 2.6 ().
In order that the equality be fulfilled, must be chosen so that
We will assume this initial condition to be satisfied.
It is easy to see that for one obtains
3 Solution of the one-dimensional Schrödinger equation
Let , and , denote the solutions of (2.3) satisfying the following initial conditions in the origin
where is an arbitrary complex number.
Theorem 3.1 ().
The solutions and of the equation
admit the following representations
where stands for the spherical Bessel function of order , the functions are defined as follows
where is the coefficient of in the Legendre polynomial of order . The series in (3.2) and (3.3) converge uniformly with respect to on and converge uniformly with respect to on any compact subset of the complex plane of the variable . Moreover, for the functions
the following uniform estimates hold
for any , , and
for any , belonging to the strip , . Here is a function satisfying as . These estimates are slight refinements for those presented in  using the ideas from . Some estimates for are presented in Appendix A.
Inequalities (3.7) and (3.8) reveal an interesting feature of the representations (3.2) and (3.3): the accuracy of approximation of the exact solutions by the partial sums (3.5) and (3.6) does not depend on meanwhile it belongs to a strip in the complex plane. This feature was tested and confirmed numerically in . An analogous result is valid for the derivatives of the solutions. We formulate it in the following statement.
Theorem 3.2 ().
The derivatives of the solutions and with respect to admit the following representations
where are defined as follows
Moreover, for the approximations
the following inequalities are valid
for any , , and
for any , belonging to the strip , .
Remark 3.3 ().
The functions also can be constructed as solutions of the recurrent equations
with the first functions given by and and initial conditions where .
The functions can be calculated from the equalities
One can use as well.
4 Solution of the Sturm-Liouville equation
Let the functions , and satisfy the conditions from Subsection 2.1 and be a solution of (2.6) satisfying (2.7) and such that for all . Then two linearly independent solutions and of equation (2.1) for can be written in the form
with defined by (2.2), the coefficients being defined by the equalities
where are from Definition 2.4. The solutions and satisfy the following initial conditions
Moreover, for the functions
the following estimates hold
for any , , and
for any , belonging to the strip , , where is a function from Theorem 3.1.
are then solutions of (2.1) with . In these representations all the magnitudes except the coefficients can be defined with no reference to equation (2.3). Let us show that the same observation is applicable to the functions
for any function and any . Then
Inequalities (4.8) and (4.9) show that the representations (4.1) and (4.2) preserve the important property of the representations (3.2) and (3.3): the accuracy of approximation of the exact solutions by the partial sums (4.6) and (4.7) does not depend on meanwhile it belongs to a strip in the complex plane.
Similarly to the coefficients (see Remark 3.3) the functions can be constructed as solutions of the recurrent equations
The initial conditions satisfied by have the form
This sequence of equations for leads to a stable recursive integration procedure which is proposed in Section 6.
5 Representation of the derivatives of the solutions and
and the functions and have the form (cf. [11, Remark 4.7])
Hence, due to Theorem 2.6,
This is a direct formula for calculating in terms of the functions and . A recurrent formula for can be obtained from (3.13). We have
One may also use