Mathematical-Chemistry Research Lab

The group’s work in the field of Fractional Calculus officially began at UNIFAL-MG in 2018, when we supervised our first M.Sc. projects centered on this topic, with the graduate students Camila A. Tavares and Taináh M. R. Santos. Although our interest in this research line emerged several years earlier, our first publication on the subject, entitled A speculative study of non-linear Arrhenius plot by using fractional calculus, dates back to 2015. This speculative work generalized the Van’t Hoff equation to fractional order and, from it, derived a generalized Arrhenius equation. Also in 2015, we generalized an epidemiological model to fractional order in order to model bovine babesiosis and tick populations in the work A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations.

In 2016, a differential equation of non-integer order was employed to model an anomalous luminescence decay process. This was one of the works previously highlighted, entitled A generalized Mittag-Leffler function to describe nonexponential chemical effects. Although this process is, in principle, an exponential decay process, recent data indicate that this is not the case for longer observation times.

In 2020, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black-body radiation problem in the study entitled Improving a Tikhonov regularization method with a fractional-order differential operator for the inverse black body radiation problem. The project was developed within the M.Sc. dissertation of the graduate student Taináh M. R. Santos.

In 2021, the Hopfield neural network (HNN), one of the most widely used neural network architectures today, was generalized to fractional order in time (FHNN). In this work, developed by the M.Sc. student Camila A. Tavares, three prototype problems for examining the FHNN model were presented and discussed. In all cases, the solution obtained with the FHNN model was achieved in less time when compared to the classical HNN model. This work, Solving ill-posed problems faster using fractional-order Hopfield neural network, was also among the articles previously highlighted. In the same year, we supervised our first undergraduate final project in this research line, involving a study of Newton’s cooling law using the Caputo fractional derivative. This work was conducted by the undergraduate student Higor V. M. Ferreira, who is currently pursuing his M.Sc. degree within the group. Higor also carried out an undergraduate research project exploring the use of Fractional Calculus in epidemiological models.

In 2022, we supervised our first undergraduate scientific initiation project in this research line, exploring the use of Fractional Calculus in the description of heat conduction in a linear bar, conducted by the undergraduate student Ivan Assunção Murad Ramos.

In 2023, the group developed a modified Tikhonov method to remove random noise from experimental data. The proposed method incorporates the Euclidean norm of the fractional-order derivative of the solution as an additional criterion in the Tikhonov functional. The method was applied both to simulated data and to surface-enhanced Raman scattering (SERS) spectra of crystal violet dye in colloidal dispersions of silver nanoparticles. This was another work previously highlighted: Smoothing and Differentiation of Data by Tikhonov and Fractional Derivative Tools, Applied to Surface-enhanced Raman Scattering (SERS) Spectra of Crystal Violet Dye.

At the beginning of 2024, I participated in the 15th Summer Workshop of the Graduate Program in Mathematics, held from March 11 to 15 at the Federal University of Juiz de Fora (UFJF), in Juiz de Fora, Minas Gerais, Brazil. On that occasion, I delivered the lecture Applications of Fractional Calculus in Chemistry. Also in 2024, I presented a seminar on the YouTube channel Fractional Calculus Seminar, entitled An overview of the fractional descent gradient method and its applications.

Still in 2024, the group participated in the III Symposium on Fractional Calculus with the works Fractional Kinetic Modeling of Adsorption and Desorption Processes from Experimental QCM Curves and The Relationship Between Fractional Order and the Complexity of the Cooling Process.

In 2025, the graduate student Higor V. M. Ferreira will defend his M.Sc. dissertation in this area, with two recently submitted works: Fractional kinetic modelling of the adsorption and desorption process from experimental SPR curves and An overview of the fractional-order gradient descent method and its applications, both currently available on the arXiv repository.

A steadily increasing activity of the group in the field of Fractional Calculus can therefore be observed.

About Nelson H. T. Lemes

Curious chemist and hardworking mathematician. I was born in Três Corações, Brazil. I studied at UFMG during both undergraduate and graduate studies, earning my M.Sc. and Ph.D. degrees in Physical Chemistry. I have been teaching for 27 years (as of 2025), having taught at all educational levels, from middle school classes to graduate courses. Currently (in 2025), I have a 9-year-old student who is curious about everything.

I have been working at UNIFAL since 2009 and became a Full Professor in 2025. During this period, I served as Chemistry area coordinator for PIBID (Institutional Scholarship Program for Teaching Initiation). At the undergraduate level, I teach General Chemistry courses in several programs, as well as elective courses such as Chemical Kinetics, Chemometrics, Mathematics for Chemists, and Practical Projects in Scientific Programming.

In the Graduate Program in Chemistry, I supervise M.Sc. and Ph.D. students and teach courses in Advanced Physical Chemistry, Quantum Mechanics, and other subjects related to my research area.

My research field is Mathematical Chemistry. Mathematical Chemistry is a research area that employs non-routine mathematical methodologies to address chemically important problems requiring new approaches. In other words, Mathematical Chemistry focuses on new mathematical ideas and concepts, adapting and developing them for use within the context of Chemistry.

My interests include cinema, comics, electronics (Arduino®), programming (App Inventor® and MATLAB®), Mathematics (Fractional Calculus), and Chemistry (Solution Thermodynamics, Chemical Kinetics, Statistical Thermodynamics, and Quantum Chemistry).

I also served as coordinator of the Bachelor’s Degree Program in Chemistry from 2020 to 2024.

Research Interests

Research Area: Mathematical Chemistry

Description: Mathematical Chemistry is a research field that employs non-routine mathematical methodologies to address chemically important problems requiring new approaches. In other words, Mathematical Chemistry focuses on new mathematical ideas and concepts, adapting and developing these concepts for use within the context of Chemistry.

Mathematical Chemistry is a truly interdisciplinary field encompassing all areas of chemistry and mathematics, with overlap among theory, numerical methods, computation, and experimental chemistry. It is currently an area of increasing importance due to the growing complexity of chemical problems.

It should be emphasized that Mathematical Chemistry is distinct from Computational Chemistry (which may be viewed as an experimental chemistry whose experimental apparatus is the computer and whose focus is data extraction), Chemometrics (with a strong emphasis on statistical treatment), and Theoretical Chemistry (where the connection with physics rather than mathematics is emphasized). Finally, it should be stressed that Mathematical Chemistry differs from the routine use of already consolidated mathematical tools in solving chemical problems.

The field of Mathematical Chemistry is relatively recent and emerged as an attempt to better characterize the interests of a group of researchers whose work does not fit strictly within Chemometrics, Theoretical Chemistry, or Computational Chemistry.

Research Line 1: Applications of Generalized-Order Calculus in the Context of Chemistry

Description: The differential equations describing the rates of many chemical and physical processes have recently undergone extensive reformulation under the framework known as Fractional Calculus. Generalized-order calculus (Fractional Calculus) is the theory that unifies and generalizes the concepts of integer-order derivatives and n-th order integrals, with integer n, to non-integer orders.

Although its history dates back to 1695, significant interest in the subject has emerged only recently. Despite the growing activity in the area, several fundamental questions remain open, such as the mathematical, physical, or geometrical interpretation of a generalized-order derivative or integral.

In this research line, we explore the application of these tools to different problems and techniques relevant to Chemistry, seeking deeper insight into these fundamental questions.

Selected Works in This Research Line:

A Fractional Variational Approach to Spectral Filtering Using the Fourier Transform, Nelson H. T. Lemes, José Claudinei Ferreira, Higor V. M. Ferreira. https://doi.org/10.48550/arXiv.2511.20675
Fractional kinetic modelling of the adsorption and desorption process from experimental SPR curves, Higor V. M. Ferreira, Nelson H. T. Lemes, Yara L. Coelho, Luciano S. Virtuoso, Ana C. dos Santos Pires, Luis H. M. da Silva. https://doi.org/10.48550/arXiv.2503.00003

Part of this work was presented at the III Brazilian Symposium on Fractional Calculus under the title: Fractional Calculus Modeling of Adsorption and Desorption Kinetics from Experimental QCM Curves.

An overview of the fractional-order gradient descent method and its applications, Higor V. M. Ferreira, Camila A. Tavares, Nelson H. T. Lemes and José P. C. dos Santos. (Submitted)

Part of this work was presented on the YouTube channel Fractional Calculus Seminar under the title: An Overview of the Fractional Descent Method.

CARVALHO DOS SANTOS, JOSÉ PAULO; MONTEIRO, EVANDRO; FERREIRA, JOSÉ CLAUDINEI; TEIXEIRA LEMES, NELSON HENRIQUE; RODRIGUES, DIEGO SAMUEL. Well-posedness and qualitative analysis of a SEIR model with spatial diffusion for COVID-19 spreading. BIOMATH, v. 12, p. 2307207, 2023.
Smoothing and differentiation of data by Tikhonov and fractional derivative tools, applied to surface-enhanced Raman scattering (SERS) spectra of crystal violet dye. JOURNAL OF CHEMOMETRICS, v. e3507, 2023.
Solving ill-posed problems faster using fractional-order Hopfield neural network. Journal of Computational and Applied Mathematics, Volume 381, 2021, 112984.
Improving a Tikhonov regularization method with a fractional-order differential operator for the inverse black body radiation problem. Inverse Problems in Science and Engineering.
A generalized Mittag-Leffler function to describe nonexponential chemical effects. Applied Mathematical Modelling, Volume 40, Issues 17–18, 2016, pp. 7971–7976.
A speculative study of non-linear Arrhenius plot by using fractional calculus.
A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations. Abstract and Applied Analysis, Volume 2015, Article ID 729894.

Research Line 2: Mathematical Modeling, Optimization, Inverse Problems, and Artificial Intelligence in the Context of Chemistry

Description: Inverse problems are optimization problems involving weakly sensitive parameters in mathematical models and input data containing experimental errors, which render the problem ill-posed. In such cases, conventional optimization methods are generally inadequate. To overcome this difficulty, regularization methods such as Tikhonov regularization and truncated singular value decomposition are employed. Artificial Intelligence techniques, including Hopfield neural networks, are also important tools in this context.

Recent Works in This Research Line:

Functional sensitivity analysis approach to retrieve the potential energy function from the quantum second virial coefficient. PHYSICA A: Statistical Mechanics and its Applications, v. 536, p. 122539, 2019.
Accurate potential energy curve for helium dimer retrieved from viscosity coefficient data at very low temperatures. PHYSICA A: Statistical Mechanics and its Applications, v. 487, p. 32–39, 2017.

Research Line 3: Development of Electronic Devices Using Arduino® Boards and Educational Android® Applications Using App Inventor®

Description: In this research line, we are interested in exploring Arduino® boards for the development of electronic devices (controllers and sensors) aimed at teaching chemical concepts or acquiring data in physical chemistry laboratory experiments. The App Inventor® software has been employed as a resource for teaching logic and programming languages, as well as for developing educational Android® applications addressing Chemistry content in an interactive and playful way.

Recent Works in This Research Line:

Study of Newton’s cooling law using the Caputo fractional derivative. Higor Vinicius Monteiro Ferreira. Undergraduate Final Project (B.Sc. in Chemistry), 2021.

Part of this work was recently presented at the III Brazilian Symposium on Fractional Calculus.

Construction of an Arduino-based electronic system for heat conduction data acquisition in a metal bar and modeling using Fractional Calculus. Ivan Assunção Murad Ramos. Undergraduate Research Project, 2021.
Development of an electronic nose for sensory analysis of coffee. LETÍCIA DE PAIVA BARBOSA, 2024.