A novel hybrid variation iteration method and eigenvalues of fractional order singular eigenvalue problems

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues ( \(\mu \) $\mu$ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, \(\begin{aligned} D^{2\alpha } y(t) + \frac{k}{t^\alpha } D^\alpha y(t) + \mu y (t) =0,~~ 0< t<1,~~ 0< \alpha \le 1, \end{aligned}\) $$\begin{array}{c}\hfill D^{2\alpha }y(t)+\frac{k}{t^{\alpha }}{D}^{\alpha }y(t)+\mu y(t)=0,0<t<1,0<\alpha \le 1,\end{array}$$ with boundary condition, \(y'(0)=0, ~~~~y(1)=0,\) $$y^{\prime }(0)=0,y(1)=0,$$ where \(D^\alpha , D^{2\alpha }\) $D^{\alpha },D^{2\alpha }$ represents the Caputo fractional derivative, \(k \ge 1\) $k\ge 1$ . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective.