A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

Modeling some real phenomena by fractional differential equations

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

A Discretization of the Hadamard fractional derivative

We present a new discretization for the Hadamard fractional derivative, that simplifies the computations. We then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the Hadamard fractional derivative.

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method.

Development of variable displacement oil pump for internal combustion engines

In the last years the need to develop more environmentally friendly and efficient cars as led to the development of several technologies to improve the performance of internal combustion engines, a large part of the innovations are focused in the auxiliary systems of the engine, including, the oil pump, this is an element of great importance in the dynamics of the engine as well a considerable energy consumer. Most solutions for oil pumps to this day are fixed displacement, for medium and high speeds, the pump flow rate is higher than the needs of the engine, this excess flow leads to the need for recirculation of the fluid which represents a waste of energy. Recently, technological advances in this area have led to the creation of variable displacement oil pumps, these have become a 'must have' due to the numerous advantages they bring, although the working principle of vane or piston pumps is relatively well known, the application of this technology for the automotive industry is new and brings new challenges. The focus of this dissertation is to develop a new concept of variable displacement system for automotive oil pumps. The main objective is to obtain a concept that is totally adaptable to existing solutions on the market (engines), both dimensionally as in performance specifications, having at the same time an innovative mechanical system for obtaining variable displacement. The developed design is a vane pump with variable displacement going in line with existing commercial solutions, however, the variation of the eccentricity commonly used to provide an variable displacement delivery is not used, the variable displacement is achieved without varying the eccentricity of the system but with a variation of the length of the pumping chamber. The principle of operation of the pump is different to existing solutions while maintaining the ability to integrate standard parts such as control valves and mechanical safety valves, the pump is compatible with commercial solutions in terms of interfaces for connection between engine systems and pump. A concept prototype of the product was obtained in order to better evaluate the validity of the concept. The developed concept represents an innovation in oil pumps design, being unique in its mechanical system for variable displacement delivery.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.