In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials ...

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by fractional-order derivatives. This article considers a nonlocal inverse problem and shows that the exponents of the fractional time and space derivatives are determined uniquely by the data u(t, 0) = g(t), 0 < t < T . The uniqueness result is a theoretical background for determining experimental...

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...

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve different...

The use of non-local operators, defining Riemann–Liouville or Caputo derivatives, is a very useful tool to study problems involving non-conventional diffusion problems. case electric circuits, ruled by non-integer derivatives capacitors with fractional dielectric permittivity, fairly natural frame relevant applications. We techniques, generalized exponential obtain suitable solutions for this t...

Because of the nonlocal and nonsingular properties fractional derivatives, they are more suitable for modelling complex processes than integer derivatives. In this paper, we use a factor to investigate Hamilton’s canonical equations Poisson theorem mechanical systems. Firstly, derivative integral with presented, multivariable differential calculus is given. Secondly, obtained under new definiti...

This study presents an efficient method that is suitable for differential equations, both with integer-order and fractional derivatives. examines the construction of solutions equations are associated varying delay proportional to independent variable using a hybrid Sumudu transform method. considers Caputo derivatives orders their applications in Nuclear Physics. The application indicates have...

A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems chaotic solutions. The Adams-Bashforth involving Lagrange interpolation the fundamental theorem of fractional calculus. We provide existence uniqueness solutions, also convergence result state...

We present a toy model for extending the Friedmann equations of relativistic cosmology using fractional derivatives. do this by replacing integer derivatives, in few well-known cosmological results with derivatives leaving their order as free parameter. All intention to explain current observed acceleration Universe. apply Last Step Modification technique calculus construct some useful cosmolog...