On the method for calculating the trace formulas of the fractional Sturm-Liouville operator

INTERNATIONAL CONFERENCE ON ANALYSIS OF MATHEMATICS AND EXACT SCIENCES Volume 02, Issue 03, 2025 34 INTERNATIONAL CONFERENCE ON ANALYSIS OF MATHEMATICS AND EXACT SCIENCES universalconference.us On the method for calculating the trace formulas of the fractional SturmLiouville operator Fayzullayev Bexzod Shuhrat ogʻli Urganch davlat universiteti doktoranti oli[email protected] The trace formulas of linear differential operators deeply reveal the structure of the spectrum of differential operators and have important applications in the numerical calculation of eigenvalues, the theory of inverse problems, and the theory of solitons. For the first time in work [1], a regularized trace for the Sturm-Liouville operator is calculated. This result was generalized by many authors, and methods were developed for generalizing this formula to the case of other operators [2-4]. The trace formulas for the conformable fractional Sturm-Liouville problem was studied in [5]. In this paper, we present a new approach for obtaining the trace formula for the conformable fractional Sturm-Liouville problem. We consider the conformable fractional Sturm-Liouville problem : ( ) (0) (0) 0 ( ) ( ) 0 xx x x l y D D y q x y y D y hy D y y       = − + = −=  −=  0x   where λ is a spectral parameter and ()qx is a real continuous function. In (1), x Dy  denotes the fractional derivative defined as follows: Definition. Let :[0, )fR→ be a given function. Then, the conformable fractional derivative of f of order  is defined by: 1 0 ( ) ( ) ( ) lim , h f x hx f x D f x h   − → +− = (1) for all x > 0, (0,1]   . For more information about conformable fractional derivative see [6-7]. Let consider the following (1) Dirichlet boundary value problem Theorem. The following trace formula is hold 222 00 0 (0) ( ) 1 1 ( ) ( ) 4 2 2 n n qq n c c h H h H       =  + − − = − − + − +    Where INTERNATIONAL CONFERENCE ON ANALYSIS OF MATHEMATICS AND EXACT SCIENCES Volume 02, Issue 03, 2025 35 INTERNATIONAL CONFERENCE ON ANALYSIS OF MATHEMATICS AND EXACT SCIENCES universalconference.us 0 0 ()c q t d t      = References 1. Gelfand I.M., Levitan B.M. Trace formulas and conservation laws for nonlinear evolution equations // Rev. Math. Phys. 6, 51–95 (1994). 2. Sadovnichy V.A., Podol’skii V.E. Traces of operators // UMN, 2006, v. 61, issue 5(371), pp. 89-156. 3. Gesztesy, H. Holden, B. Simon, and Z. Zhao. Higher order trace relations for Schrodinger operators // Rev. Math. Phys. 7, 893-922 (1995). 4. F. Gesztesy and H. Holden. Trace formulas and conservation laws for nonlinear evolution equations // Rev. Math. Phys. 6, 51–95 (1994). 5. Mortazaasl H., Jodayree Akbarfam A. Trace formula and inverse nodal problem for a conformable fractional Sturm-Liouville problem. // Inverse Problems in Science and Engineering, 28(4), 524–555 (2019). 6. Khalil R., Al Horani M., Yousef A. et al. A new definition of fractional derivative. // J Comput Appl Math. 2014, vol.264, pp. 65–70. 7. Chung W.S. Fractional Newton mechanics with conformable fractional derivative.// J Comput Appl Math. 2015,vol.290, pp.150–158.