Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives

Liouville and Bogoliubov Equations with Fractional Derivatives

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fracti...

Fractional Derivatives, Fractional Integrals, and Fractional Differential Equations in Matlab

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a letter to L’Hospital in 1695 Leibniz raised the following question (Miller and Ross...

Fractional differential equations with Legendre polynomials

Relativistic wave equations with fractional derivatives and pseudo-differential operators

The class of the free relativistic covariant equations generated by the fractional powers of the D’Alambertian operator ( ) is studied. Meanwhile the equations corresponding to n = 1 and 2 (Klein-Gordon and Dirac equations) are local in their nature, the multicomponent equations for arbitrary n > 2 are non-local. It is shown, how the representation of generalized algebra of Pauli and Dirac matr...

Calculus of variations with fractional derivatives and fractional integrals

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.