Calculation of proportional derivative on time scales

Fractional calculus has a very important place in mathematical analysis and applications. In the first part of this study, the proportional derivative that has unit operator, and the importance of the time scale in the literature are given. In the second part, some important concepts and definitions on the time scale are considered for the concept and characteristics of proportional derivative, proportional second order linear equations, self-adjoint equations and the Sturm-Liouville equation for proportional derivation. In the third part, the proportional delta derivatives and properties of the time scale, proportional dynamic equations are examined. In the fourth part, linear homogeneous proportional dynamic equations and solution methods from the second order are taken into account. In the fifth chapter is studied self-adjoint dynamic equations for proportional nabla derivative and some of the key concepts in spectral theory. In the last section, the Sturm-Liouville equation and spectral properties of the proportional delta derivative are examined on time scale, and the method of variation of parameters for third order linear proportional dynamic equations is obtained. Keywords: Time scales, Proportional Delta Derivative, Dynamic Equation, Self-Adjoint Equation