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Solving second-order singularly perturbed ODE by the collocation method based on energetic Robin boundary functions

Chein-Shan Liu;  Botong Li — 2019

Applications of Mathematics

For a second-order singularly perturbed ordinary differential equation (ODE) under the Robin type boundary conditions, we develop an energetic Robin boundary functions method (ERBFM) to find the solution, which automatically satisfies the Robin boundary conditions. For the non-singular ODE the Robin boundary functions consist of polynomials, while the normalized exponential trial functions are used for the singularly perturbed ODE. The ERBFM is also designed to preserve the energy, which can quickly...

The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu;  Botong Li — 2024

Applications of Mathematics

The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n -dimensional matrix eigenvalue problem is derived with a special matrix 𝐀 : = [ a i j ] , that is, a i j = 0 if i + j is odd.Based on the product formula, an integration method with a fictitious time, namely...

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