Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

Abstract

We study matrix representations of Sturm-Liouville problems with coupled eigenparameter-dependent boundary conditions and transmission conditions. Meanwhile, given any matrix eigenvalue problem with coupled eigenparameter-dependent boundary conditions and transmission conditions, we construct a class of Sturm-Liouville problems with given boundary conditions and transmission conditions such that they have the same eigenvalues.

Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity

“…Assume that v i (i = 1, M) is defined in (21). Then, using (26), and utilizing induction on (27), (22) holds. Moreover, ( 23) and ( 24) can be similarly obtained.…”

“…Ao et al generalized this problem to various differential operators, for example, ASLPs with interface conditions, ASLPs with eigenparameters contained in boundary conditions, higher-order differential operators, etc. [21][22][23][24][25][26]. They discussed the existence of a finite spectrum and gave the corresponding matrix representation.…”

Eigenvalues of a Class of Eigenparameter Dependent Third-Order Differential Operators

“…Various applications in physics and other fields such as the vibration of loaded strings, diffusion processes in probability theory and so on yield such problems [1]. A large number of literature have devoted to the study of such problems for Sturm-Liouville (S-L) problems and fourth-order beam equations, and numerous significance results are obtained (see, for example [2][3][4][5][6][7][8][9][10][11][12]).…”

Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

“…The differential boundary value problems (see [12][13][14]) with eigenparameter dependent boundary conditions have been widely investigated by many authors such as [3,4,6,8,9,11,17,20,24,26] because of the important application in probability theory and physics and so on. In the present paper we consider Sturm-Liouville problems with two singular endpoints and one eigenparameter dependent boundary condition.…”