Linear Fractional Models Research Articles

Optimizing UAV performance with IoT and fuzzy linear fractional transportation models

Unmanned aerial vehicles (UAVs) have become indispensable in replacing human pilots for high-risk, high-intensity operations, including search-and-rescue missions, infrastructure inspection, and environmental monitoring. As a key enabler of these operations, UAV task planning systems require advanced trajectory optimization to operate effectively in dynamic, complex environments. This paper presents an innovative model integrating fully fuzzy linear fractional transportation programming (FTMP) with Internet of Things (IoT) technologies to enhance UAV performance. Unlike conventional path-planning methods, the proposed model leverages bio-inspired artificial intelligence (AI) algorithms and mixed-integer programming to solve multi-objective optimization problems under uncertainty. A critical feature of this platform is its ability to continuously adjust flight altitude based on real-time data, accounting for varying terrain morphology, fuel constraints, flight restrictions, and threats, thus ensuring optimal 3D trajectory planning. This makes the platform particularly effective in emergency scenarios such as natural disasters, explosions, or fires, where it supports rapid and accurate search-and-rescue efforts. The UAV system not only minimizes human involvement but also reduces the response time by transmitting real-time video, audio, and GPS data to responders over long distances. The proposed solution offers several advantages over traditional aerial platforms, including: (a)Precision tracking and path optimization with a demonstrated accuracy exceeding 88 % across recall, precision, and F-measure metrics. (b)Cost-effectiveness: The UAV is compact, lightweight, and affordable, significantly reducing operational expenses compared to helicopters. (c)Adaptability: Remote sensing allows for seamless control in rugged and obstructed environments. (d)Scalability: IoT integration supports multi-drone coordination for continuous real-time data collection and transmission to smartphones or command centers. This pragmatic UAV model offers transformative potential for emergency response, environmental monitoring, and smart infrastructure management, validating its reliability and efficiency through comprehensive performance metrics.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

Implementation of non-linear mixed effects models defined by fractional differential equations.

Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis-Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.

On the Solution of the Variable Order Time Fractional Schrödinger Equation

In the present paper, we used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional Schrödinger equations. Twelve illustrative applications have been presented. When the fractional order is unity, the results are the same as those obtained by the standard integer order derivative. Some of the obtained results are discussed graphically. Some of the obtained results 1st appear in the literature. This technique can be applied to other linear or nonlinear fractional differential models.
 HIGHLIGHTS
 
 We study a class of time variable-order fractional nonlinear Schrödinger equations (tVOFNLSEs)
 We introduce a new procedure for solving the tVOFNLSE and tVOFSE
 We used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional equations
 The calculations are performed in the frame of Caputo sense
 
 GRAPHICAL ABSTRACT

A new approach to modeling and simulation of the nonlinear, fractional behavior of Li-ion battery cells

Two nonlinear infinite dimensional models of a Lithium-ion battery are introduced. The proposed models interpolate linear fractional models which are obtained from a widely spread equivalent circuit model. The linear models are parameterized by means of so-called dynamic impedance measurements, i.e. impedance measurements in several operating points. In each operating point, the fractional input output behavior is recovered by an infinite dimensional state space description. For numerical implementation the introduced models require for an appropriate approximation. Therefore, a Krylov subspace method and a finite element approach are proposed. The nonlinear lumped parameter models are validated on the basis of experimental data.

Iterative Method for Approximate-Analytical Solutions of Linear Schrödinger Equation

In this study, the modified Picard Iterative Method (MPIM) is used to provide analytic and numerical solutions to linear Schrödinger Equations. These approximate-analytical solutions for the examples under consideration are easily computed. The suggested method is employed without any transformation, discretization, linearization, or limiting assumptions. The obtained results are similar to their exact forms. As a result, the approach is highly suggested for both linear and non-linear time-space fractional partial differential models with applications in various applied disciplines.

A novel approach for solving decision-making problems with stochastic linear-fractional models

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

Numerical Simulation of Fractional Zakharov–Kuznetsov Equation for Description of Temporal Discontinuity Using Projected Differential Transform Method

The core aim of this study is to propose a novel computational procedure, namely, Elzaki transform iterative method to work out two‐dimensional nonlinear time‐fractional Zakharov–Kuznetsov equation numerically. We execute the suggested iterative procedure on two models and results are presented graphically in the form of surface plot and absolute error is compared with the VIM and HPM to show that the method is more powerful than VIM and HPM and deduce that the offered numerical pattern is more efficient in simulating linear and nonlinear fractional order models.

Identification of fractional-order models for condition monitoring of solid-oxide fuel cell systems

With rising market deployment the condition monitoring of solid oxide fuel cell systems is gaining particular importance. The conventional approaches mainly use electrochemical impedance spectroscopy based on the repeated sinusoidal perturbation over a range of frequencies. One of the notable weaknesses of the approach is excessively long perturbation time needed to properly evaluate the impedance curve. In this paper, we propose a time-efficient approach in which, a short, persistently exciting and small-amplitude perturbation is used to excite all the relevant system eigenmodes. A model structure from a class of linear fractional order models is selected to describe the perturbed dynamics and to account for anomalous diffusion processes in the cells. Then, the model parameters are estimated directly from measured input and output records. The paper presents a computationally efficient parameter estimation procedure in which the numerical issues of differentiation of noisy signals are alleviated by using modulating functions. In practice, that means a combination of filtering and application of conventional least squares. The approach is applied on a case of health assessment of solid oxide fuel cells.

One Model of a Linear-Fractional Three-Index Transportation Problem

In specialized literature and in practice some methods for production-transportation planning using the models of three-index transportation problem are well-known. Different linear-fractional models for production-transportation planning are applied as well. In the present paper the authors aim to combine elements of the three-index transportation problem and elements of the linear-fractional optimization, thus offering a model of three-index transportation problem with a linear-fractional function which combines the advantages of both methods and gives better opportunities for determining an optimal solution and making the post-optimal analysis.

New Fractional Analytical Study of Three-Dimensional Evolution Equation Equipped With Three Memory Indices

Abstract Herein, analytical solutions of three-dimensional (3D) diffusion, telegraph, and Burgers' models that are equipped with three memory indices are derived by using an innovative fractional generalization of the traditional differential transform method (DTM), namely, the threefold-fractional differential transform method (threefold-FDTM). This extends the applicability of DTM to comprise initial value problems in higher fractal spaces. The obtained solutions are expressed in the form of a γ¯-fractional power series which is a fractional adaptation of the classical Taylor series in several variables. Furthermore, the projection of these solutions into the integer space corresponds with the solutions of the classical copies for these models. The results detect that the suggested method is easy to implement, accurate, and very efficient in (non)linear fractional models. Thus, research on this trend is worth tracking.

Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave equation, is studied. The explicit form of fundamental solution is calculated, and wave propagation speed, arising from solution's support, is found to be connected with the material properties at initial time instant. Existence and uniqueness of the fundamental solutions to the fractional wave equations corresponding to four thermodynamically acceptable classes of linear fractional constitutive models, as well as to power type distributed order model, are established and explicit forms of the corresponding fundamental solutions are obtained.

PRM88 - Estimating Patient Utility in Migraine: Longitudinal Analysis of Erenumab Clinical Trial Data

Migraine is a debilitating disease with substantial impact on patients’ health-related quality of life (HRQoL). Cost-utility analysis of migraine preventives requires the estimation of patient utility values and how these relate to monthly migraine days (MMD). The objective of this analysis is to model utility values in a longitudinal framework, allowing more accurate estimates of expected utility values. Data from two pivotal erenumab studies in episodic migraine (EM; NCT02456740) and chronic migraine (CM; NCT02066415) were pooled to analyze HRQoL outcomes. Migraine Specific Questionnaire (MSQ) version 2.1 and Headache Impact Test (HIT-6) condition-specific instruments were mapped to the EQ-5D using published mapping algorithms (Gillard et al. 2012) to generate generic, preference-based utility estimates, suitable for use in economic models. Longitudinal non-linear beta regressions were fitted between mapped EQ-5D utility values, MMD and treatment group (erenumab 140mg and placebo), adjusting for age, sex, race and baseline MMD in the various time periods considered. Expected utility values by MMD were calculated using the Delta method. Multilevel linear models and fractional response models with probit and logit links were also assessed. The analysis sample included data from 1,113 patients; 83.7% females with a mean age of 41.6 years. The longitudinal beta regression models closely approximated the observed utility values. Patients with 14 MMD were predicted to have MSQ mapped mean utility values of 0.608 (95% CI: 0.595–0.621) for patients treated with erenumab 140mg and 0.590 (0.577–0.602) for patients receiving placebo. The respective predicted HIT-6 mapped mean utility values were 0.698 (0.654–0.741) and 0.659 (0.629–0.689). Mapped utility values for erenumab patients were consistently higher than placebo patients with the same number of MMD. Linking patient QoL to MMD allows utility estimates for different levels of MMD to be predicted, for use in economic evaluations of preventive therapies.

Linear-fractional model for global warming

A unique linear-fractional model is designed to describe the global warming in terms of the ratio of atmospheric carbon dioxide (CO2) concentration to the pre-industrial level. As an application, doubled CO2 concentration is expected to raise surface temperatures by about 2°C in average (above pre-industrial records). Prior to that, an algebraic greenhouse function is provided to correlate the global average temperature to the atmospheric emissivity, for which a parameterised collection of linear-fractional models involving CO2 ratio is proposed using a practical characterisation of such models.

Linear-fractional model for global warming

A unique linear-fractional model is designed to describe the global warming in terms of the ratio of atmospheric carbon dioxide (CO2) concentration to the pre-industrial level. As an application, doubled CO2 concentration is expected to raise surface temperatures by about 2°C in average (above pre-industrial records). Prior to that, an algebraic greenhouse function is provided to correlate the global average temperature to the atmospheric emissivity, for which a parameterised collection of linear-fractional models involving CO2 ratio is proposed using a practical characterisation of such models.

Artificial neural network approximations of linear fractional neutron models

In this paper the artificial neural network (ANN) approximation for fractional neutron point kinetics (FNPK) model and fractional reduced-order model (F-ROM) is presented. The input-output data of the step response generated using the closed-loop stable fractional-order models was used for the training the ANN models. The ANN topology with various layers and different number of neutrons was tried to obtain the best approximation for the fractional-order model. The results confirm that the designed ANNs provide a good approximation to linear FNPK and F-ROM models. It was also observed that the convergence and learning of ANN is greatly affected by the type of model (FNPK or F-ROM), value of anomalous diffusion coefficient (fractional-order derivative) and the relaxation time.

Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Approaches to Determining Box Consistent Parameter Sets for Polytopic Output Constraints on Linear Fractional Models Using Structured Singular Values

This technical note considers the problem of maximizing the volume of a box of parameters that satisfy polytopic output constraints for linear fractional models using the structured singular value $\mu$ . In particular, four kinds of boxes are considered: 1) a hypercube centered at a specified location; 2) a freely located hypercube; 3) a freely located box of free shape; and 4) a freely located box of free shape and rotation. It is found that the problem for 1) can be solved by computing $\mu$ for the number of scalar constraints, while problems for 2)–4) can be reformulated as a constant-matrix $\mu$ -synthesis problem, which can be approached by $D,G-K$ iteration.

A nonlinear fractional model to describe the population dynamics of two interacting species

In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.