Conservation laws for systems of non-standard Birkhoffians with fractional derivatives

Conservation laws for systems of exponential, power-law and logarithmic non-standard Birkhof-fian with fractional derivatives are investigated respectively, and their relationships with Noether symmetries are revealed. Corresponding to the three forms of non-standard Birkhof-fians, the Pfaff-Birkhoff principles are proposed and non-standard Birkhoff's equations with fractional derivatives are derived. The variation of Pfaff action is investigated in depth, and the formulas of total variation are deduced, and on this basis the criterion equations of Noether symmetry are set up. Then, Noether's theorems for systems of non-standard Birkhoffian with fractional derivatives are proved, which reveals the relationship between symmetry and conserved quantity. Finally, three examples are given to illustrate how to calculate the symmetry and find out the conserved quantity, and the correctness of the obtained conserved quantities is shown by numerical calculation.