Related papers: A Stochastic Fractional Calculus with Applications…
This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…
This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or…
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential…
Stochastic mechanics is regarded as a physical theory to explain quantum mechanics with classical terms such that some of the quantum mechanics paradoxes can be avoided. Here we propose a new variational principle to uncover more insights…
A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional…
Viscoelasticity and related phenomena are of great importance in the study of mechanical properties of material especially, biological materials. Certain materials show some complex effects in mechanical tests, which cannot be described by…
The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
The general idea of a stochastic gauge representation is introduced and compared with more traditional phase-space expansions, like the Wigner expansion. Stochastic gauges can be used to obtain an infinite class of positive-definite…
We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term…
Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in…
In this paper, we state with a variational method a general theorem providing the existence of a weak solution $u$ for fractional Euler-Lagrange equations of the type: $$ \dfrac{\partial L}{\partial x} (u,D^\alpha_- u,t) + D^\alpha_+…
We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the…
The purpose of this article is to review the developments related to the notion of local fractional derivative introduced in 1996. We consider its definition, properties, implications and possible applications. This involves the local…
We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving…
The classical relativistic wave equations are presented as partial difference equations in the arena of covariant discrete phase space. These equations are also expressed as difference-differential equations in discrete phase space and…
We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these…
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any…
In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \alpha \leq 1$ and the Riemann- Liouville fractional integral of order $\beta >0$ under certain constraints. A…
Based on the requirement of covariance, we propose a new approach for generalizing fractional calculus in multi-dimensional space. As a first application we calculate an approximation for the ground state energy of the fractional…