On the determination of the impulsive Sturm–Liouville operator with the eigenparameter‐dependent boundary conditions

Abstract

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter‐dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h1,h2, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H1,H2, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.

Inverse fractional Sturm‐Liouville problem with eigenparameter in the boundary conditions

“…The concept of fractional differential equation is an extension of the classical differential equation which gives the chance of taking the $$$ \alpha $$$‐order derivative of a function, where $$$ 0<\alpha \le 1 $$$. This extension has been proved to be very effective as it gives many chances of calculating and computing on some phenomenon that can't be done by the classical differentiation and integration as described in many works 13–43 . Rashid et al 36 derived some generalizations that captured novel results under investigation by fractional operators.…”

Parameter identification in multidimensional hyperbolic partial differential equations using wavelet collocation method

“…Numerical methods for parameter identification problem of parabolic type have been developed by few authors (see previous studies 7–11 and references therein). Source identification problem for the generalized time fractional diffusion equation based on regularized hyper Bessel operator has been studied by Luc et al 12 Inverse problem for the impulsive Sturm–Liouville equations with eigenparameter‐dependent boundary conditions are studied by Khalili et al, 13 whereas Triet et al 14 studied inverse nonlinear sideways heat equation based on filter method.…”

Convergence Analysis of Implicit Euler Method for a Class of Nonlinear Impulsive Fractional Differential Equations

“…e research of impulsive fractional differential equations can be found in literatures [10][11][12][13][14][15][16][17][18][19][20] and monograph [21]. However, there are few literatures on numerical methods for impulsive fractional differential equations.…”