Fractional Derivative Equation Research Articles

Cancer modeling by fractional derivative equation and chemotherapy stabilizing

This paper discusses the theme of cancer modeling and the problem of control by treatments. We model the cancer cells growth by fractional derivative equation and we stabilize their concentration by chemotherapy. The considered model will be converted into simple partial derivative equation and associated with a viability problem. We use the set-valued analysis to make viable the converted model by regulation law of regulation map. Applied chemotherapy concentration on cancer cells concentration follows the regulation law.

A Unified Phenomenological Model Captures Water Equilibrium and Kinetic Processes in Soil

AbstractSoil water sustains life on Earth, and how to quantify water equilibrium and kinetics in soil remains a challenge for over a century despite significant efforts. For example, various models were proposed to interpret non‐Darcian flow in saturated soils, but none of them can capture the full range of non‐Darcian flow. To unify the different models into one overall framework and improve them if needed, this technical note proposes a theory based on the tempered stable density (TSD) assumption for the soil‐hydraulic property distribution, recognizing that the underlying hydrologic processes all occur in the same, albeit very complex and not measurable at all the relevant scales, soil‐water system. The TSD assumption forms a unified fractional‐derivative equation (FDE) using subordination. Preliminary applications show that simplified FDEs, with proposed hydrological interpretations and TSD distributed properties, effectively capture core equilibrium and kinetic water processes, spanning non‐Darcian flow, water retention, moisture movement, infiltration, and wetting/drying, in the soil‐water system with various degrees and scales of system heterogeneity. Model comparisons and evaluations suggest that the TSD may serve as a unified density for the properties of a broad range of soil‐water systems, driving multi‐rate mass, momentum, and energy equilibrium/kinetic processes often oversimplified by classical models as single‐rate processes.

Nonlocal theory on plane waves in a higher-order thermo-diffusive semiconducting medium under the gravitational field

This article explores the reflection of elastic waves in nonlocally generalized thermo-diffusive semiconducting elastic solids under the influence of gravity. The problem is framed within the higher-order fractional derivative heat equation. The frequency dispersion relation reveals the presence of five coupled waves. The wave speeds are graphed against angular frequency for the nonlocal parameter under the influence of gravity, and the cut-off frequency of the waves is visually represented. The incident wave at the free surface of the solid is the coupled longitudinal P-wave, and its reflection coefficients are computed. The study investigates the impact of gravity, fractional order, and nonlocal parameters on propagation speed and amplitude ratios, revealing their significant influence. The findings are substantiated in the context of energy conservation. This research provides valuable insights for scientists engaged in geophysics and various branches of mechanics.

Flow and convection heat of spatial fractional derivative non-Newtonian fluids in fractal main channels

This paper mainly investigates the convective diffusion of non-Newtonian fluids in fractal main channels. The spatial fractional derivative constitutive equations of non-Newtonian fluids are coupled with the flow equations to derive the governing equations for fluid flow and heat transfer. The shifted Grünwald– Letnikov formula is used to obtain numerical solutions for the model. The effects of fractional derivatives, Reynolds number, and Prandtl number on fluid flow and heat transfer are analyzed.

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

A Fractional-order dual-continuum model to capture non-Fickian solute transport in a regional-scale fractured aquifer

Contaminant transport in fractured media exhibits complex dynamics, including multiple peaks in breakthrough curves (BTCs) and non-Fickian diffusion, thereby posing significant challenges to the application of traditional transport models. Here we undertook a detailed study of a natural-gradient tracer test conducted in a regional-scale fractured carbonate aquifer situated in southwestern Germany, where the observed BTCs contained both dual peaks and positive skewness. These BTCs were used to optimize parameters and interpret their physical meanings for several transport models, including the dual-continuum model (DCM) and the fractional derivative equation (FDE) model. Tracer concentration distributions were simulated in both single- and dual-continuum media employing the DCM and FDE models. Our results demonstrated that while the DCM model could reasonably replicate the bimodal BTC, the FDE (which accounts for solute retention) outperformed in capturing the heavy-tailed BTC. This was attributed to the limitations of grid-based numerical models that assume Fickian diffusion and fail to map small-scale medium heterogeneity exhaustively. In contrast, a parsimonious model like the FDE, with upscaled parameters, was found to be more effective in capturing regional-scale non-Fickian transport. To further characterize the multiple BTC peaks the standard FDE missed, we proposed a fractional derivative dual-continuum model (fDCM). This model was found to be adept at capturing both the multi-peak and late-time heavy tail in the BTC. Our study thus opens an alternate pathway for modeling solute transport in regional-scale fractured to partially karstified aquifers.

Asymptotics Solutions of a Singularly Perturbed Integro-differential Fractional Order Derivative Equation with Rapidly Oscillating Coefficients

In this paper, the regularization method of S.A. Lomov is generalized to singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients. The main purpose of the study is to reveal the influence of the integral term and rapidly oscillating coefficients on the asymptotics of the solution of the original problem. To study the influence of rapidly oscillating coefficients on the leading term of the asymptotics of solutions, we consider a simple case, i.e. the case of no resonance (when an entire linear combination of frequencies of a rapidly oscillating cosine does not coincide with the frequency of the spectrum of the limit operator).

Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales

Recent studies have emphasized the importance of nonlocal models in characterizing bedload transport in natural rivers, particularly in mixed-size gravel beds or steep hillslopes. Nonlocality denotes that a quantity (flux) at a specific location x is dependent on the conditions in the surrounding area, as opposed to solely at the location itself. This concept applies in bedload transport even in planar flumes, where particles are entrained at an upstream position and travel a finite distance, ultimately contributing nonlocally to the sediment flux. However, existing bedload transport models, such as the advection–diffusion equation (ADE) or the fractional derivative equation (FDE) models, are inadequate in characterizing the nonlocal transport behavior of bedload at a regional scale. Large errors may arise from the lack of an accurate description of the nonlocal bedload transport processes at regional scales. This study proposes a regional-based nonlocal bedload transport model, which is conceptualized from the probabilistic Exner-based equations and the peridynamic (PD) differential operator. The PD model encapsulates the nonlocal motion of bedload sediments on the basis of the PD differential operator, by utilizing a pre-defined weight function and influence domain. Comparisons demonstrate that the PD model serves as a generalized tool connecting the local and the global models with different PD functions and influence domains. Its variability on kernel function and influence domain, enable it conveniently describe sub-, super-, and normal diffusion behaviors of bedload transport.

Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate.

The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.

The non-uniform L1-type scheme coupling the finite volume method for the time–space fractional diffusion equation with variable coefficients

In this paper, we consider the time–space fractional diffusion equation on a bounded interval with the Caputo time fractional derivative and the two-sided Riemann–Liouville spatial fractional derivative. |The diffusion coefficients are variable with respect to x and t. The temporal non-uniform L1-type discretization and the finite volume method based on the nodal basis functions are adopted to discrete this fractional partial derivative equation. We strictly prove that our scheme is conditionally stable with convergence rate Oh2+N−min{2−α,rα}, where h is the spatial step and N,α,r are the temporal nodal numbers, the fractional parameter and the graded parameter respectively. Finally, some numerical results are presented to validate the theoretical analysis.

A New Approach for Solving a New Class of Nonlinear Optimal Control Problems Generated by Atangana-Baleanu- Caputo Variable Order Fractional Derivative and Fractional Volterra-Fredholm Integro-Differential Equations

In the sequel, the numerical solution of a new class of nonlinear optimal control problems (OCPs) generated by Atangana-Baleanu-Caputo (ABC) variable order (V-O) fractional derivative (FD) and fractional Volterra-Fredholm integro-differential equations (FVFIDEs) is found by Bezier curve method (BCM). The main idea behind this work is the use of the BCM. In this technique, the solution is found in the form of a rapid convergent series. Using this method, it is possible to obtain BCM solution of the general form of multipoint boundary value problems. To shown the efficiency of the developed method, numerical results are stated as the main results in this study.

Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD model.

The response of a cell population is often delayed relative to drug injection, and individual cells in a population of cells have a specific age distribution. The application of transit compartment models (TCMs) is a common approach for describing this delay. In this paper, we propose a TCM in which damaged cells caused by a drug are given by a single fractional derivative equation. This model describes the delay as a single equation composed of fractional and ordinary derivatives, instead of a system of ODEs expressed in multiple compartments, applicable to the use of the PK concentration in the model. This model tunes the number of compartments in the existing model and expresses the delay in detail by estimating an appropriate fractional order. We perform model robustness, sensitivity analysis, and change of parameters based on the amount of data. Additionally, we resolve the difficulty in parameter estimation and model simulation using a semigroup property, consisting of a system with a mixture of fractional and ordinary derivatives. This model provides an alternative way to express the delays by estimating an appropriate fractional order without determining the pre-specified number of compartments.

Nonlocal magneto-thermoelastic infinite half-space due to a periodically varying heat flow under Caputo–Fabrizio fractional derivative heat equation

Abstract This article deals with a new modified heat conduction model with fractional order that includes the Caputo–Fabrizio differential operator (CF) and the thermal relaxation time. This new approach to the CF fractional derivative has attracted many researchers because it includes a nonsingular kernel. The nonlocal theory proposed by Eringen has also been applied to demonstrate the effect of scale-dependent thermoelastic materials. The problem of thermal isotropic semi-infinite space is addressed as an application of the presented model. The medium is exposed to regularly changing heat sources and is initially placed in a continuous external magnetic field. The system of governing equations was expressed in the field of the Laplace transform, and the problem in this field was solved by the state-space operation. The inverse of the transformed expressions of physical quantities is found numerically using Zakian’s algorithm. The effects of the nonlocal parameter, the fractal order parameter, and the magnetic field were graphically presented and analyzed in detail. Some of the previous investigations were extracted in some special cases.

Operator & Fractional Order Based Nonlinear Robust Control for a Spiral Counter-Flow Heat Exchanger with Uncertainties and Disturbances

This paper studies operator and fractional order nonlinear robust control for a spiral counter-flow heat exchanger with uncertainties and disturbances. First, preliminary concepts are presented concerning fractional order derivative and calculus, fractional order operator theory. Then, the problem statement about nonlinear fractional order derivative equation with uncertainties is described. Third, the design of an operator fractional order controller and fractional order PID controller and determination of several related parameters is described. Simulations were performed to verify tracking and anti-disturbance performance by comparison to different control cases; verification is described and concluding remarks provided.

Numerical implementation and error analysis of nonlinear coupled fractional viscoelastic fluid model with variable heat flux

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

FORCED TRANSVERSE VIBRATIONS OF FRACTIONAL VISCOELASTIC EULER-BERNOULLI BEAM USING THE MODAL ANALYSIS METHOD

In the paper, the forced transverse vibration of fractional viscoelastic Euler-Bernoulli is studied. Based on the fractional relationship of stress and strain, the partial differential equation describing transverse vibration of Euler-Bernoulli viscoelastic beam is considered. The Riemann-Liouville fractional derivative of the order and is used. Using the Ritz-Galerkin method, the fractional partial derivative equation describing the vibration of the beam is transformed into a system of differential equations containing fractional derivatives. The dynamic response of a simply supported fractional viscoelastic beam to a harmonic concentrated force is calculated in detail. The forced vibration solution of the beam is determined using the harmonic balancing method. The solution to the vibration equations is determined analytically, while dynamic responses are calculated numerically. The effects of fractional–order parameters on the vibration amplitude-time curves are investigated. From the calculation results, we can see that the lower the parameter is, the larger the vibration amplitude. This is consistent with our logic thinking. A comparison between the approximately analytical solution and the numerical one shows a good agreement, and the correctness of the obtained results is therefore verified.

Asymptotics solutions of a singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.

Fractal Characterization of Heat and Mass Transfer Processes in Nanostructured Materials

The relationship between the fractal self-organization of structural-phase inhomogeneity in nanostructured materials and the processes of heat and mass transfer in them is considered. It is characterized by a non-Euclidean relationship between the rate of increase in the number of the considered elements of the medium and an increase in the scale of their consideration and is described by the transfer equations in fractional derivatives with respect to coordinate and time.

An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

Thermomagnetic behavior of a nonlocal finite elastic rod heated by a moving heat source via a fractional derivative heat equation with a non-singular kernel

The use of fractional derivatives has been very useful in recent decades for modeling many problems of engineering sciences. The literature includes several different concepts of a fractional derivative. In this regard, most fundamental models are based on the Riemann- Liouville and Caputo concepts that include singular kernels. Based on the Caputo–Fabrizio fractional derivative, the current investigation deals with providing a new mathematical heat conduction model involving a non-singular kernel. Also, Eringen's nonlocal elasticity theory has been applied to study the size dependent effect. The suggested model is then used to analyze the transient reactions of a finite thermoelastic rod when exposed to a heat resource that moves to the right axial direction. The rod is posed in a constant primary magnetic field and its ends are assumed to be thermally insulated and fixed. In solving the governing equations, the Laplace transform technique has been applied and the inversions have been calculated numerically using an appropriate numerical procedure. The effect of the coefficient of nonlocality, fractional order parameter and the velocity of the moving heat supply has been studied in separate cases. The findings indicate that these factors significantly affect the scale of the studied physical variables.