Papers by Marco Antonio Taneco-Hernández
Linear Electrical Circuits Described by a Novel Constant Proportional Caputo Hybrid Operator
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 2025
In this work, we obtain analytical solutions via Laplace transform of fractional electrical circu... more In this work, we obtain analytical solutions via Laplace transform of fractional electrical circuits by using the proportional Caputo derivative with power law. Numerical simulations were obtained to see the impact of the memory concept represented by the fractional parameter order. Also, we collect a lot of experimental data obtained in our laboratory for an RC circuit, and for each case, we calculate the value of the fractional-order by using a particle swarm optimization approach. The results obtained reveals that the experimental data deviate slightly from that obtained in the integer-order behavior and the application of the proportional Caputo derivative best fits the experimental data.
Physica Scripta, 2024
Fractional Partial Differential equations (FPDEs) are essential for modeling complex systems acro... more Fractional Partial Differential equations (FPDEs) are essential for modeling complex systems across various scientific and engineering areas. However, efficiently solving FPDEs presents significant computational challenges due to their inherent memory effects, often leading to increased execution times for numerical solutions. This study proposes a highly parallelizable hybrid computational approach that combines the Finite Element Method (FEM) for spatial discretization with Numerical Inversion of the Laplace Transform (NILT) for time-domain solutions, optimized for execution on Graphics Processing Units (GPUs). The NILT method's high parallelizability, stemming from the independence of its series terms, combined with the robust spatial discretization provided by FEM, enables the efficient and accurate solution of FPDEs on GPUs, demonstrating substantial performance improvements over traditional CPU-based implementations. We observe a generalized pattern in execution time behavior that accounts for both the number of nodes and the number of NILT terms. Specifically, execution time scales quadratically with the number of nodes, while showing only a logarithmic marginal increase with the number of NILT terms These behaviors not only enables consistent performance assessment but also highlights potential areas for algorithm optimization. Validation against exact solutions of fractional diffusion and wave equations, employing Caputo, modified Caputo-Fabrizio, and modified Atangana-Baleanu derivatives, demonstrates the accuracy and convergence of the hybrid FEM-NILT method. Notably, the exact solutions of wave equation are novel in literature. The results highlight the method's potential for enabling high-precision, largescale simulations in fractional calculus applications, thereby advancing computational capabilities and efficiency in the field.
Initial-boundary value and interface problems on the real half line for the fractional advection-diffusion type equation
Mathematical Methods in the Applied Sciences, 2024
We use the unified transform method (UTM) to solve initial-boundary value problems for the fracti... more We use the unified transform method (UTM) to solve initial-boundary value problems for the fractional advection-diffusion type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana-Baleanu fractional derivative of order α∈ (0, 1] in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non-singular kernel. In addition, an interface problem is stated and solved in the adjacent domains IR+ and IR−, where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection-diffusion equations with integer-order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.
Representaciones gráficas de funciones complejas
Tlamati, 2015
Assessment of the performance of the hyperbolic-NILT method to solve fractional differential equations
Mathematics and Computers in Simulation
The performance of the hyperbolic–numerical inverse Laplace transform (hyperbolic-NILT) method is... more The performance of the hyperbolic–numerical inverse Laplace transform (hyperbolic-NILT) method is evaluated when it is used to solve time-fractional ordinary and partial differential equations. With this purpose, the formalistic fractionalization approach of Gompertz and diffusion equations are used as model problems, i.e., in the Gompertz and diffusion equations the integer-order time derivative is replaced by the Caputo or Atangana–Baleanu fractional derivative or the Caputo–Fabrizio non-integer order operator. The accuracy, stability and convergence of the numerical solutions are analyzed by comparing the numerical and exact solutions. From our analysis, we obtain an independent formula of the fractional order, which together with the initial condition is used to optimize the parameter of the hyperbolic-NILT method. This expression can be implemented in linear fractional differential equations with non-homogeneous initial condition. Finally, we show that the value of the parameter is transferred throughout the time domain with the certainty that the accuracy of the inverted solution remains between certain orders of magnitude. In fact, everything indicates that this conclusion fits well with model problems that are similar (fractional linear differential equations) to those studied here and for which we highly recommended the hyperbolic-NILT method to solve them.
Exact solutions to fractional pharmacokinetic models using multivariate Mittag-Leffler functions
Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena., 2023
The aim of this paper is to provide a mathematical study of the amount of drug administered as a ... more The aim of this paper is to provide a mathematical study of the amount of drug administered as a continuous intravenous infusion or oral dose. For this purpose, we consider fractional-order mammillary-type models describing the anomalous dynamics of exchange of concentrations between compartments at, constant input rates, power-law type, and in the form of oral doses at given (discrete) times. We have developed a general analysis strategy for these models, in which we have found closed-form analytical solutions written in terms of the multivariate Mittag-Leffler function. Numerical simulations have been performed using our formulas, with parameters from the literature.
Mathematical Methods in the Applied Sciences, 2022
We consider the equations of motion of a bar, with a given density, infinite in both directions, ... more We consider the equations of motion of a bar, with a given density, infinite in both directions, subject to longitudinal vibrations under the action of an external load, and a stress-strain relation represented by a fractional order operator. Using three types of fractional operators, the initial-boundary value problems associated with the described phenomenon are posed and solved. Through the bivariate Mittag-Leffer function, which has been recently introduced, we find the fundamental solution to these problems and calculate their moments.
Journal of Porous Media, 2020
Modeling of fluid flow considering radially symmetric reservoirs is common in groundwater science... more Modeling of fluid flow considering radially symmetric reservoirs is common in groundwater science and petroleum engineering. The Hankel transform is suitable for solving boundary value problems, considering this flow geometry. However, there are few applications of this transform for reservoirs with a finite wellbore radius, although there are formulas of the finite Hankel transform for homogeneous boundary conditions. In this work, we refer to them as the Cinelli formulas, which are used to obtain novel solutions for transient fluid flow in bounded naturally fractured reservoirs with time-varying influx at the outer boundary, i.e., a technique to incorporate inhomogeneous boundary conditions based on the Cinelli formulas is developed. An analysis shows that the results of the solutions are highly oscillating and slowly convergent. Nevertheless, we show that this problem is largely overcome when the long-time solution is expressed as a closed relationship. Accordingly, we present the characteristic drawdown pressure curves and its Bourdet derivatives for a double-porosity reservoir with influx recharge. These curves allow us to distinguish between the pressure drops of a single-porosity reservoir with influx recharge from that of a double-porosity closed reservoir, which have been stated in the literature to resemble one another. Similarly, double-and triple-porosity reservoirs are analyzed.
Results in Physics, 2021
In this paper will use fractional calculus to analyse the model that describes a biofluid equippe... more In this paper will use fractional calculus to analyse the model that describes a biofluid equipped with charged particles influenced by a magnetic field. For this purpose, the Atangana-Baleanu fractional operator in the Riemann-Liouville sense was used to solve the initial-boundary value problem. The fluid flow through a circular cylinder is influenced by a magnetic field which is perpendicular to the circular tube and an oscillating pressure gradient. Integral transforms are used to find solutions for the velocity potentials of the blood flow and its magnetic particles. Finally the effect of physical variables (H_a, R ,G) on the dynamics of fluid and magnetic parameters are highlighted graphically.
Results in Physics, 2021
Analytical solutions of the fractional wave equation via Caputo-Fabrizio fractional derivative ar... more Analytical solutions of the fractional wave equation via Caputo-Fabrizio fractional derivative are presented in this paper. For this analysis, three cases are considered, the classical, the damped and the damped with a source term defined by fractional wave equations. We show that these solutions are special cases of the time fractional equations with exponential law. Illustrative examples are presented.
Stability and Lyapunov functions for systems with Atangana-Baleanu Caputo derivative: An HIV/AIDS epidemic model
Chaos, Solitons and Fractals, 2020
In this paper, we derive extensions of classical Lyapunov and Chetaev instability theorems and La... more In this paper, we derive extensions of classical Lyapunov and Chetaev instability theorems and LaSalle's invariance principle to the case of Atangana-Baleanu derivative in the Caputo sence. Moreover, we get some results to estimates fractional derivatives of quadratic and Volterra-type Lyapunov functions when γ ∈ (0, 1). Finally, we present rigorous proofs about the complete classification for global dynamics of an HIV/AIDS epidemic model with Atangana-Baleanu Caputo derivative (Chaos, Solitons and Fractals 126 (2019) 41-49).
Communications in Nonlinear Science and Numerical Simulation, 2020
We obtain the exact analytic solutions of a fluid flow model that includes the Caputo–Fabrizio op... more We obtain the exact analytic solutions of a fluid flow model that includes the Caputo–Fabrizio operator and new constitutive equations in its definition. Formulas are obtained for a slightly compressible fluid in an infinite single-porosity reservoir with the inner boundary having a constant pressure. The flow equation is given by $$ ^{CFC}_{\ \ \ \ 0}{\mathcal{D}}_t^{\alpha}p + \tau \; ^{CFC}_{\ \ \ \ 0}{\mathcal{D}}_t^{\beta}p = \frac{1}{r}\frac{\partial}{\partial r} \left( r \frac{\partial p}{\partial r} \right), \begin{array}{lll} && 0<\alpha, \beta \leq 1 \mbox{ or }\\ && 1\leq\alpha, \beta \leq 2,$$ , where $p$ is the pressure, which depends on the position $r$ and the time $t$, τ is the relaxation time, and $^{CFC}_{\ \ \ \ 0}{\mathcal{D}}_t^{\alpha}$ is the Caputo-Fabrizio operator of order α. We show that this equation is local and does not involve the flow properties associated with a fractal medium. Consequently, the mean square displacement at both short and long time scales exhibits normal behaviour when 0 < α, β ≤ 1, while it shows a combination of ballistic and normal behaviours when 1 ≤ α, β ≤ 2. Indeed, in general, the discretized models with the CF operator exhibit the absence of a memory process, with the potential of speeding up the computations. On the other hand, we show that the solutions have a finite propagation velocity only when both α and β (or either of them) are equal to 2. In addition, the known solutions of the Helmholtz, diffusion, wave, and Cattaneo equations are recovered by our generalized solutions given the appropriate integer orders of α and β.
Computational and Mathematical Methods in Medicine, 2019
The estimation of parameters in biomathematical models is useful to characterize quantitatively t... more The estimation of parameters in biomathematical models is useful to characterize quantitatively the dynamics of biological processes. In this paper, we consider some systems of ordinary differential equations (ODEs) modelling the viral dynamics in a cell culture. These models incorporate the loss of viral particles due to the absorption into target cells. We estimated the parameters of models by least-squares minimization between numerical solution of the system and experimental data of cell cultures. We derived a first integral or conserved quantity, and we proved the use of experimental data in order to test the conservation law. The systems have nonhyperbolic equilibrium points, and the conditions for their stability are obtained by using a Lyapunov function. We complemented these theoretical results with some numerical simulations.
Physica A, 2019
The presence of a complex spatio-temporal behavior in spatially extended systems (SES) as a resu... more The presence of a complex spatio-temporal behavior in spatially extended systems (SES) as a result of several mechanisms that interact non-linearly with other nearby has attracted a lot of attention in recent decades. A well-known example of SES is the Kuramoto-Sivashinsky (KS) equation. In the search for a broader perspective of some unusual irregularities observed in the context of phase turbulence in the reaction-diffusion systems, the propagation of the wrinkled flame front and the unstable drift waves driven by the collision of electrons in a Tokamak, we explored the possibility of extending the analysis of the KS equation with three perturbation levels using the conceptions of fractional differentiation with non-local and non-singular kernel. We use the fractional order operator of Atangana-Baleanu in the sense of Liouville-Caputo (ABC) to set the fractional model of KS that we analyze in this article. We prove existence and uniqueness of a continuous solution to fractional KS model, using the fixed-point theorem and the Picard-Lindelöf approach, provided conditions on the perturbation parameters and the order of the fractional operator. In addion, using the theory of fixed point, we present the stability of the iterative method generated by the Picard-Lindelöf approach. Finally, we presented the approximate analytical solution of the fractional KS equation using the homotopy perturbation transform method. Some numerical simulations are carried out for illustrating the results obtained.
Discrete and Continuous Dynamical System, Serie S, 2020
This paper presents the solution for a fractional Bergman’s minimal blood glucose-insulin model e... more This paper presents the solution for a fractional Bergman’s minimal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo fractional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homotopy polynomial we obtain analytical approximate solutions for both derivatives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homotopy analysis transform method are simple and straightforward.
Physica A, 2019
The purpose of this paper is to offer a discussion on the fundamental solutions for the vibratio... more The purpose of this paper is to offer a discussion on the fundamental solutions for the vibration fractional equation of rods semi-infinite (also called Fresnel equation). For this end, using the fractional derivative of Riemann-Liouville of order γ ∈ (1, 2] with respect to the spatial variable, we solve boundary-value problems associated to the mentioned equation showing the connection with Brownian behavior and the heat equation. We obtain solutions for the fractional Fresnel equation using the unified transform method. In the intermediate of the investigation, we have solved generalized Cauchy problems associated with linear fractional Schrödinger equations with order γ .
Revista Mexicana de Física, 2019
In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fraction... more In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order α. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense, and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the mod-eling of anomalous diffusion, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.
This paper presents the analytical solutions of fractional linear electrical systems by using the... more This paper presents the analytical solutions of fractional linear electrical systems by using the Caputo-Fabrizio fractional-order operator in Liouville-Caputo sense. This novel operator involves an exponential kernel without singularities. The fractional equations were solved analytically by using the properties of Laplace transform operator, as well as the convolution theorem. To validate the analytical solutions, numerical simulations were carried out.
In this paper, we give analytical solutions of a fractional-time wave equation with memory effect... more In this paper, we give analytical solutions of a fractional-time wave equation with memory effect and frictional memory kernel of Mittag–Leffler type via the Atangana–Baleanu fractional order derivative. The method of separation of variables and the Laplace transform has been used to obtain the exact solutions for the fractional order wave equations. Additionally, we present analytical solutions considering the Caputo–Fabrizio fractional derivative with exponential kernel. We showed that the solutions obtained via Caputo–Fabrizio fractional order derivative were a particular case of the solutions obtained with the new fractional derivative based in the Mittag–Leffler law.
In this paper, we analyze the fractional modeling of the giving up the smoking using the definiti... more In this paper, we analyze the fractional modeling of the giving up the smoking using the definitions of Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives. Applying the homotopy analysis method and the Laplace transform with polynomial homotopy, the analytical solution of the smoking dynamics has obtained. Furthermore, using an iterative scheme by the Laplace transform, and the Atangana-Baleanu fractional integral, special solutions of the model are obtained. Uniqueness and existence of the solutions by the fixed-point theorem and Picard-Lindelof approach are studied. Finally, some numerical simulations are carried out for illustrating the results obtained.
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Papers by Marco Antonio Taneco-Hernández