A Fractional Analog of the Duhamel Principle

Mathematics Subject Classification: 35CXX, 26A33, 35S10

Fractional Calculus of P-transforms

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99

Hyperbolic Fibrations and PDE

2010 Mathematics Subject Classification: 35L10, 35L90.

Finite time extinction for nonlinear fractional evolution equations and related properties

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

Finite time extinction for nonlinear fractional evolution equations and related properties.

The finite time extinction phenomenon (the solution reaches an equilibrium after a finite time) is peculiar to certain nonlinear problems whose solutions exhibit an asymptotic behavior entirely different from the typical behavior of solutions associated to linear problems. The main goal of this work is twofold. Firstly, we extend some of the results known in the literature to the case in which the ordinary time derivative is considered jointly with a fractional time differentiation. Secondly, we consider the limit case when only the fractional derivative remains. The latter is the most extraordinary case, since we prove that the finite time extinction phenomenon still appears, even with a non-smooth profile near the extinction time. Some concrete examples of quasi-linear partial differential operators are proposed. Our results can also be applied in the framework of suitable nonlinear Volterra integro-differential equations.

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p = 1 , in any dimension d ∈ N , by including a pointwise state-constraint. More precisely, given a x ‾ ( ⋅ ) ∈ W p , 1 ( [ a , b ] , R d ) solving the convexified p-th order differential inclusion L p x ‾ ( t ) ∈ co { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e., consider the general problem consisting in finding bang-bang solutions (i.e. L p x ˆ ( t ) ∈ { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e.) under the same boundary-data, x ˆ ( k ) ( a ) = x ‾ ( k ) ( a ) & x ˆ ( k ) ( b ) = x ‾ ( k ) ( b ) ( k = 0 , 1 , … , p − 1 ); but restricted, moreover, by a pointwise state constraint of the type 〈 x ˆ ( t ) , ω 〉 ≤ 〈 x ‾ ( t ) , ω 〉 ∀ t ∈ [ a , b ] (e.g. ω = ( 1 , 0 , … , 0 ) yielding x ˆ 1 ( t ) ≤ x ‾ 1 ( t ) ). Previous results in the scalar d = 1 case were the pioneering Amar & Cellina paper (dealing with L p x ( ⋅ ) = x ′ ( ⋅ ) ), followed by Cerf & Mariconda results, who solved the general case of linear differential operators L p of order p ≥ 2 with C 0 ( [ a , b ] ) -coefficients. This paper is dedicated to: focus on the missing case p = 1 , i.e. using L p x ( ⋅ ) = x ′ ( ⋅ ) + A ( ⋅ ) x ( ⋅ ) ; generalize the dimension of x ( ⋅ ) , from the scalar case d = 1 to the vectorial d ∈ N case; weaken the coefficients, from continuous to integrable, so that A ( ⋅ ) now becomes a d × d -integrable matrix; and allow the directional vector ω to become a moving AC function ω ( ⋅ ) . Previous vectorial results had constant ω, no matrix (i.e. A ( ⋅ ) ≡ 0 ) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).

Pseudo-almost periodic solutions for abstract partial functional differential equations

In this work we study the existence and uniqueness of pseudo-almost periodic solutions for a first-order abstract functional differential equation with a linear part dominated by a Hille-Yosida type operator with a non-dense domain. (C) 2009 Published by Elsevier Ltd

Differential evolution based classification with pool of distances and aggregation operators

The objective of this thesis is to develop and generalize further the differential evolution based data classification method. For many years, evolutionary algorithms have been successfully applied to many classification tasks. Evolution algorithms are population based, stochastic search algorithms that mimic natural selection and genetics. Differential evolution is an evolutionary algorithm that has gained popularity because of its simplicity and good observed performance. In this thesis a differential evolution classifier with pool of distances is proposed, demonstrated and initially evaluated. The differential evolution classifier is a nearest prototype vector based classifier that applies a global optimization algorithm, differential evolution, to determine the optimal values for all free parameters of the classifier model during the training phase of the classifier. The differential evolution classifier applies the individually optimized distance measure for each new data set to be classified is generalized to cover a pool of distances. Instead of optimizing a single distance measure for the given data set, the selection of the optimal distance measure from a predefined pool of alternative measures is attempted systematically and automatically. Furthermore, instead of only selecting the optimal distance measure from a set of alternatives, an attempt is made to optimize the values of the possible control parameters related with the selected distance measure. Specifically, a pool of alternative distance measures is first created and then the differential evolution algorithm is applied to select the optimal distance measure that yields the highest classification accuracy with the current data. After determining the optimal distance measures for the given data set together with their optimal parameters, all determined distance measures are aggregated to form a single total distance measure. The total distance measure is applied to the final classification decisions. The actual classification process is still based on the nearest prototype vector principle; a sample belongs to the class represented by the nearest prototype vector when measured with the optimized total distance measure. During the training process the differential evolution algorithm determines the optimal class vectors, selects optimal distance metrics, and determines the optimal values for the free parameters of each selected distance measure. The results obtained with the above method confirm that the choice of distance measure is one of the most crucial factors for obtaining higher classification accuracy. The results also demonstrate that it is possible to build a classifier that is able to select the optimal distance measure for the given data set automatically and systematically. After finding optimal distance measures together with optimal parameters from the particular distance measure results are then aggregated to form a total distance, which will be used to form the deviation between the class vectors and samples and thus classify the samples. This thesis also discusses two types of aggregation operators, namely, ordered weighted averaging (OWA) based multi-distances and generalized ordered weighted averaging (GOWA). These aggregation operators were applied in this work to the aggregation of the normalized distance values. The results demonstrate that a proper combination of aggregation operator and weight generation scheme play an important role in obtaining good classification accuracy. The main outcomes of the work are the six new generalized versions of previous method called differential evolution classifier. All these DE classifier demonstrated good results in the classification tasks.

Femtosecond pulse shaping using differential evolutionary algorithm and wavelet operators

Evolutionary meta-algorithms for pulse shaping of broadband femtosecond duration laser pulses are proposed. The genetic algorithm searching the evolutionary landscape for desired pulse shapes consists of a population of waveforms (genes), each made from two concatenated vectors, specifying phases and magnitudes, respectively, over a range of frequencies. Frequency domain operators such as mutation, two-point crossover average crossover, polynomial phase mutation, creep and three-point smoothing as well as a time-domain crossover are combined to produce fitter offsprings at each iteration step. The algorithm applies roulette wheel selection; elitists and linear fitness scaling to the gene population. A differential evolution (DE) operator that provides a source of directed mutation and new wavelet operators are proposed. Using properly tuned parameters for DE, the meta-algorithm is used to solve a waveform matching problem. Tuning allows either a greedy directed search near the best known solution or a robust search across the entire parameter space.

EXISTENCE RESULTS FOR NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

In this paper we discuss the existence of mild, strict and classical solutions for a class of abstract integro-differential equations in Banach spaces. Some applications to ordinary and partial integro-differential equations are considered.

Existence Results for Abstract Partial Neutral Integro-differential Equation with Unbounded Delay

In this paper we study the existence and regularity of mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay.

GLOBAL SOLUTIONS FOR ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper we study the existence of global solutions for a class of abstract functional differential equation with nonlocal conditions. An application is considered.