Lie Point Symmetries and Conservation Law to Fractional $$\zeta (t)$$-KdV Equation

Abstract In this present paper we apply the Lie group theory associated fractional calculus to obtain the symmetries of the $$\zeta (t)$$ ζ ( t ) -KdV fractional partial differential equation, which is given in terms of the $$\zeta (t)$$ ζ ( t ) -Riemann-Liouville time fractional partial derivative, in which a particular case of the $$\zeta (t)$$ ζ ( t ) -Hilfer fractional partial derivative is obtained. The calculus of symmetries consider the explicit formula of this infinitesimal extension of the fractional operator. The fractional partial equation is reduced to a fractional differential ordinary equation, and an analytical solution is proposed in the form of a power series. We obtain a nonlinear recurrence for the coefficients of the series. We discuss the linearized case for the fractional KdV equation, obtaining the Mainardi function as the solution, and the results are interpreted graphically. We then present the conservation law theorem for fractional $$\zeta (t)$$ ζ ( t ) operators, and we applied the law to find the law associated with each symmetry.