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System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and…
The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization…
We present a theoretical framework and numerical methods for predicting the large-scale properties of solutions of partial differential equations that are too complex to be properly resolved. We assume that prior statistical information…
A heat equation with uncertain domains is thoroughly investigated. Statistical moments of the solution is approximated by the counterparts of the shape derivative. A rigorous proof for the existence of the shape derivative is presented.…
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…
Standard methods in non-linear analysis are used to show that there exists a parabolic branching of solutions of the Lichnerowicz-York equation with an unscaled source. We also apply these methods to the extended conformal thin sandwich…
We investigate evolution equations for anomalous diffusion employing fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. Fractional diffusion equations…
Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are…
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
Line integration of generalized functions is studied. Second order partial differential equations with piecewise continuous and generalized variable coefficients over Cayley-Dickson algebras are investigated. Formulas for integrations of…
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
Aim of this work is the study of differential equations governing non--dissipative non--linear oscillators; these arise in different physical models such as the treatment of relativistic oscillators, up to generalizations to Duffing's…
We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…
We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…
A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using…
In the present paper the classical point symmetry analysis is extended from partial differential to functional differential equations with functional derivatives. In order to perform the group analysis and deal with the functional…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…