fractional derivatives
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Lagrangians which depend on higher-order derivatives appear frequently in many areas of physics. In this paper, we reformulate Podolsky's Lagrangian in fractional form using left-right Riemann-Liouville fractional derivatives. The equations of motion are obtained using the fractional Euler Lagrange equation. In addition, the energy stress tensor and the Hamiltonian are obtained in fractional form from the Lagrangian density. The resulting equations are very similar to those found in classical field theory.

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivative is  employed for constructing some results related to Mittag-Leffler functions and established a number of important relationships between the Mittag-Leffler functions and Wright function.

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature.

Abstract In this note, the main emphasis is to study two kinds of high-order fractional p-Laplacian differential equations with mixed derivatives, which include Caputo type and Riemann–Liouville type fractional derivative. Based on fixed point theorems on the cone, the existence-uniqueness of positive solutions for equations and two iterative schemes to uniformly approximate the unique solutions are discussed theoretically. What’s more, the sufficient conditions for stability of the equations are given. Some exact examples are further provided to verify the analytical results at the end of the article.

<abstract><p>In this paper, we consider a two-times nonlinear fractional differential inequality involving both Hadamard-Caputo and Caputo fractional derivatives of different orders, with a singular potential term. We obtain sufficient criteria depending on the parameters of the problem, for which a global solution does not exist. Some examples are provided to support our main results.</p></abstract>