Marc Jornet
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View article: Mikusiński Algebraic Calculus and Multivariate Fractional Operators for Representing Solutions of Linear Fractional Systems and Functionals
Mikusiński Algebraic Calculus and Multivariate Fractional Operators for Representing Solutions of Linear Fractional Systems and Functionals Open
The aim of this paper is to advance in the field of fractional calculus, in the topics of Mikusiński algebraic calculus and multivariate operators. On the one hand, we show how the Mikusiński method can be used for finding closed‐form solu…
View article: Patterns of dengue and SARS-CoV-2 coinfection in the light of deterministic and stochastic models
Patterns of dengue and SARS-CoV-2 coinfection in the light of deterministic and stochastic models Open
View article: A Mathematical Analysis for the Dynamics of Multiple Languages
A Mathematical Analysis for the Dynamics of Multiple Languages Open
View article: On partial Caputo fractional models
On partial Caputo fractional models Open
A partial Caputo fractional model mixes, in the same formulation, classical and Riemann–Liouville terms, in the realm of epidemiology and sociology. The Riemann–Liouville part introduces non-Markovian behavior and makes the hazard risk fun…
View article: A general Volterra integral problem of Riemann–Liouville type
A general Volterra integral problem of Riemann–Liouville type Open
We study a Volterra convolution equation of the second kind, based on a combination of Riemann–Liouville integrals. The problem can be reformulated involving the Caputo fractional derivative, hence the equation becomes of differintegral ty…
View article: Fractional Bateman equations in the Atangana-Baleanu sense
Fractional Bateman equations in the Atangana-Baleanu sense Open
Considering experimental evidence of fractional behavior for alpha-type radioactive decay, this paper is placed in the context of the fractional Bateman equations with several memory orders. Recent papers studied these equations for Caputo…
View article: Theory on New Fractional Operators Using Normalization and Probability Tools
Theory on New Fractional Operators Using Normalization and Probability Tools Open
We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel cl…
View article: Closed-form solution for a mathematical extension of the multi-term fractional Bateman equations via Mikusiński operational method
Closed-form solution for a mathematical extension of the multi-term fractional Bateman equations via Mikusiński operational method Open
We give a closed-form solution, in terms of multivariate Mittag–Leffler functions, for a lower triangular linear fractional system consisting of Riemann–Liouville derivatives. For such a task, we use Mikusiński algebraic calculus, while so…
View article: Representation and inequalities involving continuous linear functionals and fractional derivatives
Representation and inequalities involving continuous linear functionals and fractional derivatives Open
We investigate how continuous linear functionals can be represented in terms of generic operators and certain kernels (Peano kernels), and we study lower bounds for the operators as a consequence, in the space of square-integrable function…
View article: Inverse uncertainty quantification for stochastic systems by resampling. Applications to modeling of alcohol consumption and infection by HIV
Inverse uncertainty quantification for stochastic systems by resampling. Applications to modeling of alcohol consumption and infection by HIV Open
View article: On a nonlinear stochastic fractional differential equation of fluid dynamics
On a nonlinear stochastic fractional differential equation of fluid dynamics Open
View article: Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus
Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus Open
View article: A new lower bound for the $$\textrm{L}^2$$-norm of the Caputo fractional derivative
A new lower bound for the $$\textrm{L}^2$$-norm of the Caputo fractional derivative Open
We prove a novel, tight lower bound for the norm in $$\textrm{L}^2[0,T]$$ of the Caputo fractional derivative. It is based on continuous linear functionals, Peano kernels, and the Gaussian hypergeometric function.
View article: On the interpretation of Caputo fractional compartmental models
On the interpretation of Caputo fractional compartmental models Open
In the last decade, hundreds of papers have been published on fractional modeling. The approach is often
\nsimilar, with the motivation of incorporating a ‘‘memory effect’’: take the integer-order differential-equation
\nmodel, and replace…
View article: Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function
Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function Open
The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a d…
View article: Generalized polynomial chaos expansions for the random fractional Bateman equations
Generalized polynomial chaos expansions for the random fractional Bateman equations Open
View article: Power-series solution of the L-fractional logistic equation
Power-series solution of the L-fractional logistic equation Open
View article: On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations
On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations Open
We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a…
View article: Theory on new fractional operators using normalization and probability tools
Theory on new fractional operators using normalization and probability tools Open
We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel cl…
View article: Power-series solutions of fractional-order compartmental models
Power-series solutions of fractional-order compartmental models Open
Compartmental models based on coupled differential equations of fractional order have been widely employed in the literature for modeling. An abstraction of these models is given by a system with polynomial vector field. We investigate the…
View article: A dynamical mathematical model for crime evolution based on a compartmental system with interactions
A dynamical mathematical model for crime evolution based on a compartmental system with interactions Open
We use data on imprisonment in Spain to fit a system of three ordinary differential equations that describes the temporal evolution of three different groups in the country: offenders that are not in prison, offenders that are in prison, a…
View article: PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE
PROPERTIES OF A NEW GENERALIZED CAPUTO-FABRIZIO FRACTIONAL DERIVATIVE Open
View article: Finite-dimensional probability distributions in the random Burgers-Riemann problem
Finite-dimensional probability distributions in the random Burgers-Riemann problem Open
We fully quantify the uncertainty for the Burgers' partial differential equation with random Riemann initial states, by obtaining the generalized probability density functions associated to the finite-dimensional distributions of the respo…
View article: Derivative of Certain Stochastic Integrals with Anticipating Integrands
Derivative of Certain Stochastic Integrals with Anticipating Integrands Open
We investigate whether an analogue of the fundamental theorem of calculus holds for the Ayed–Kuo stochastic integral. This integral was defined for anticipating processes $$\phi =\{\phi _t:t\in [a,b]\}$$ of the form $$\ph…
View article: On the mean‐square solution to the Legendre differential equation with random input data
On the mean‐square solution to the Legendre differential equation with random input data Open
In this short note, we investigate a linear stochastic differential equation from mathematical physics, driven by parametric uncertainty. Given the Legendre differential equation with random inputs, the goal is to give a proof of a conject…
View article: On the random fractional Bateman equations
On the random fractional Bateman equations Open
We study the random fractional Bateman equations for a radioactive decay chain, in the Caputo-derivative sense. On the one hand, a fractional order entails memory effects in the system, and on the other hand, randomness accounts for uncert…
View article: Two-dimensional probability distribution of the solution to the random Burgers–Riemann problem
Two-dimensional probability distribution of the solution to the random Burgers–Riemann problem Open
We investigate Burgers' partial differential equation with random Riemann initial states and random-field solution. Previous works obtained the generalized probability density function in the one-dimensional case, by using different method…
View article: A Stratonovich integral for anticipating processes
A Stratonovich integral for anticipating processes Open
A stochastic integral for anticipating integrands was introduced by Ayed and Kuo in 2008. Riemann–Stieltjes sums were considered, where the adapted part of the integrand was evaluated at the left endpoints of the subintervals, while the in…
View article: Spatial modeling of crime dynamics: Patch and reaction–diffusion compartmental systems
Spatial modeling of crime dynamics: Patch and reaction–diffusion compartmental systems Open
We study the dynamics of abstract models for crime evolution. The population is divided into three compartments, taking into account the participation in crime and incarceration. Individuals transit between the three segments, assuming tha…
View article: Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions
Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions Open