Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem

Two classes of conformable fractional Sturm‐Liouville problems: Theory and applications

“…Proposition 6.8. Fractional diffusion Equation (86) has a unique real-valued, weak solution, continuous in [0, ] × (0, ∞) obeying initial condition (87) and boundary conditions (88). The solution is given as series…”

“…Proof. According to the separation of variables method, we expand the solution with regard to the eigenfunctions basis n , u(x, t) = ∑ ∞ n=1 T n (t) (x, n ), where n and n are the eigenvalues and eigenfunctions of the operator on the right-hand side of (86). By using the orthonormality of eigenvectors, we arrive at the following set of fractional differential equations for variable coefficients T n :…”

“…with solutions proportional to the Mittag-Leffler function T n (t) = c n E (− n t ), where constants c n can be easily derived from initial condition (87). From theorem 3.12 of Mortazaasl and Jodayree Akbarfam, 86 we have…”

“…[77][78][79] In recent years, many researchers have focused their attention on certain generalizations of Sturm-Liouville problems. [81][82][83][84][85][86][87][88] Existence and uniqueness theorems for fractional differential equations of conformable type are investigated in Gökdogan et al 83 In particular, a regular conformable fractional Sturm-Liouville eigenvalue problem was considered in Al-Rifae and Abdeljawad. 84 Mortazaasl and Jodayree Akbarfam in a previous study 86 presented the spectral theory for a conformable fractional Sturm-Liouville problem (CFSLP) and formulated a self-adjoint conformable fractional Sturm-Liouville operator (CFSLO) in a Hilbert space.…”

Partial inverse nodal problems for differential pencils on a star‐shaped graph

“…It is worth mentioning that Theorem 2.2 in Guo and Wei 5 shows us that not only the potential up to its mean value and coefficients of boundary conditions can be uniquely determined by a twin‐dense subset on ( a , b ) with 1/2∈( a , b ) under some conditions but also the length b − a of subinterval ( a , b ) can be arbitrarily small. Because we cannot possibly describe the recent developments in details in this paper, one may refer to other studies, 1,3‐23 and the references therein, which lead the interested reader into a variety of directions. Meanwhile, the inverse nodal problem for differential pencils was also studied in Buterin and Shieh and Guo and Wei, 24‐26 respectively.…”