A representation for solutions of the Sturm–Liouville equation

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

“…This result generalizes what was done for the Sturm-Liouville equation (1.2) pu + qu = λru in [31] (see also [32,33]). …”

“…As was shown in a number of recent publications the SPPS representation provides an efficient and accurate method for solving initial value, boundary value and spectral problems (see [7], [8], [16], [21], [23], [22], [24], [25], [31], [32], [33], [35], [47]). In this paper we demonstrate this fact in application to equation (1.1).…”

Bloch Solutions of Periodic Dirac Equations in SPPS Form

“…Notice that the solution u of the equation (6) for λ = 0 can be obtained as follows u(x) = e − Φ(x)dx and u(x) is a nodeless periodic function with the period T if Φ(x) ∈ C 1 and T 0 Φ(x)dx = 0. Once having the function u(x) the solutions f 1 (x, λ) and f 2 (x, λ) of (6), (2) for all values of the parameter λ can be given using the SPPS method [1] .…”

Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

“…The completeness of the system is with respect to the uniform norm in the closed rectangle Ω. The system of solutions is shown to be useful for uniform approximation of solutions of initial boundary value problems for (1) as well as for explicit solution of the noncharacteristic Cauchy problem (see [1]) for (1) in terms of the formal powers arising in the spectral parameter power series (SPPS) method (see [2,3]). In the paper, the Dirichlet boundary conditions are considered; however the proposed method can be easily extended onto other standard boundary conditions.…”

“…The formal powers arise in the spectral parameter power series (SPPS) representation for solutions of the onedimensional Schrödinger equation (see [2,3]). …”