Ricardo Almeida

Dina Tavares, Ricardo Almeida, Delfim F. M. Torres

We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains fractional derivatives into a classical problem in which only derivatives of integer order are present. Corresponding approximations provide useful numerical tools to compute fractional derivatives of functions. Application of such approximations to fractional differential equations and fractional problems of the calculus of variations are discussed. Illustrative examples show the advantages and disadvantages of each approximation.

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres

Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grunwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points.

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations and fractional problems of the calculus of variations, is illustrated.

Ricardo Almeida

The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo--Hadamard fractional derivatives, that are dependent on a real parameter ro. Sufficient and necessary conditions of the first and second order are presented. The cases of integral and holonomic constraints are also considered.

Ricardo Almeida, Nuno R. O. Bastos

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.

Hassan Khosravian-Arab, Ricardo Almeida

We exhibit a numerical method to solve fractional variational problems, applying a decomposition formula based on Jacobi polynomials. Formulas for the fractional derivative and fractional integral of the Jacobi polynomials are proven. By some examples, we show the convergence of such procedure, comparing the exact solution with numerical approximations.

Ricardo Almeida, Agnieszka B. Malinowska, M. Luísa Morgado et al.

The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method.

Ricardo Almeida, Nuno R. O. Bastos

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.

Ricardo Almeida

The Herglotz problem is a generalization of the fundamental problem of the calculus of variations. In this paper, we consider a class of non-differentiable functions, where the dynamics is described by a scale derivative. Necessary conditions are derived to determine the optimal solution for the problem. Some other problems are considered, like transversality conditions, the multi-dimensional case, higher-order derivatives and for several independent variables.

Rodrigo Tertulino, Ricardo Almeida, Laercio Alencar

The digitization of healthcare has generated massive volumes of Electronic Health Records (EHRs), offering unprecedented opportunities for training Artificial Intelligence (AI) models. However, stringent privacy regulations such as GDPR and HIPAA have created data silos that prevent centralized training. Federated Learning (FL) has emerged as a promising solution that enables collaborative model training without sharing raw patient data. Despite its potential, FL remains vulnerable to poisoning and Sybil attacks, in which malicious participants corrupt the global model or infiltrate the network using fake identities. While recent approaches integrate Blockchain technology for auditability, they predominantly rely on probabilistic reputation systems rather than robust cryptographic identity verification. This paper proposes a Trustworthy Blockchain-based Federated Learning (TBFL) framework integrating Self-Sovereign Identity (SSI) standards. By leveraging Decentralized Identifiers (DIDs) and Verifiable Credentials (VCs), our architecture ensures only authenticated healthcare entities contribute to the global model. Through comprehensive evaluation using the MIMIC-IV dataset, we demonstrate that anchoring trust in cryptographic identity verification rather than behavioral patterns significantly mitigates security risks while maintaining clinical utility. Our results show the framework successfully neutralizes 100% of Sybil attacks, achieves robust predictive performance (AUC = 0.954, Recall = 0.890), and introduces negligible computational overhead (<0.12%). The approach provides a secure, scalable, and economically viable ecosystem for inter-institutional health data collaboration, with total operational costs of approximately $18 for 100 training rounds across multiple institutions.

Rodrigo Tertulino, Ricardo Almeida

Identifying the factors that influence student performance in basic education is a central challenge for formulating effective public policies in Brazil. This study introduces a multi-level machine learning approach to classify the proficiency of 9th-grade and high school students using microdata from the System of Assessment of Basic Education (SAEB). Our model uniquely integrates four data sources: student socioeconomic characteristics, teacher professional profiles, school indicators, and principal management profiles. A comparative analysis of four ensemble algorithms confirmed the superiority of a Random Forest model, which achieved 90.2% accuracy and an Area Under the Curve (AUC) of 96.7%. To move beyond prediction, we applied Explainable AI (XAI) using SHAP, which revealed that the school's average socioeconomic level is the most dominant predictor, demonstrating that systemic factors have a greater impact than individual characteristics in isolation. The primary conclusion is that academic performance is a systemic phenomenon deeply tied to the school's ecosystem. This study provides a data-driven, interpretable tool to inform policies aimed at promoting educational equity by addressing disparities between schools.

Rodrigo Tertulino, Ricardo Almeida

The increasing digitalization of education presents unprecedented opportunities for data-driven personalization, but it also introduces significant challenges to student data privacy. Conventional recommender systems rely on centralized data, a paradigm often incompatible with modern data protection regulations. A novel privacy-preserving recommender system is proposed and evaluated to address this critical issue using Federated Learning (FL). The approach utilizes a Deep Neural Network (DNN) with rich, engineered features from the large-scale ASSISTments educational dataset. A rigorous comparative analysis of federated aggregation strategies was conducted, identifying FedProx as a significantly more stable and effective method for handling heterogeneous student data than the standard FedAvg baseline. The optimized federated model achieves a high-performance F1-Score of 76.28%, corresponding to 92% of the performance of a powerful, centralized XGBoost model. These findings validate that a federated approach can provide highly effective content recommendations without centralizing sensitive student data. Consequently, our work presents a viable and robust solution to the personalization-privacy dilemma in modern educational platforms.

Rodrigo Tertulino, Ricardo Almeida

This study proposes and validates a Federated Learning (FL) framework to proactively identify at-risk students while preserving data privacy. Persistently high dropout rates in distance education remain a pressing institutional challenge. Using the large-scale OULAD dataset, we simulate a privacy-centric scenario where models are trained on early academic performance and digital engagement patterns. Our work investigates the practical trade-offs between model complexity (Logistic Regression vs. a Deep Neural Network) and the impact of local data balancing. The resulting federated model achieves strong predictive power (ROC AUC approximately 85%), demonstrating that FL is a practical and scalable solution for early-warning systems that inherently respects student data sovereignty.

Ricardo Almeida

In this paper we present three types of Caputo-Hadamard derivatives of variable fractional order, and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained, and an estimation for the error is given. At the end we compare the exact fractional derivative of a concrete example with some numerical approximations.