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This study aims to develop a first-order Chebyshev neural network-based technique for solving ordinary and partial distributed-order fractional differential equations. The neural network is used as a trial solution to construct the loss function. The loss function is utilized to train the neural network via an extreme learning machine and obtain the solution. The novelty of this work is developing and implementing a neural network-based framework for distributed-order fractional differential equations via an extreme learning machine. The proposed method is validated on several test problems. The error metrics utilized in the study include the absolute error and the L 2 error. A comparison with other previously available approaches is presented. Also, we provide the computation time of the method. [ABSTRACT FROM AUTHOR]

The global impact of high unemployment rates has significant economic and social consequences. To overcome this, various skill development programs are initiated by governments of developing countries. But the problem of unemployment is still increasing day by day. So, there is a pressing necessity to revise the current policies and models. Therefore, this research proposes a fractional‐order mathematical model that examines the impact of various skill development programs for youths. The proposed model incorporates fractional‐order differential equations to capture the complex dynamics of unemployment. The main objective of this research is to examine the impact of training programs aimed at enhancing the abilities of unemployed individuals, with the ultimate goal of reducing the overall unemployment rate. The reproduction number is calculated using the next‐generation matrix approach, which is crucial for both the existence and stability analysis of the equilibria. When the reproduction number is less than 1, the employment‐free equilibrium is locally and globally asymptotically stable. The employment‐persistence equilibrium point emerges only when the reproduction number exceeds one. We also explore the possibility of transcritical bifurcation and investigate the impact of skill development on the unemployment rate. We conduct numerical simulations to validate our analytical findings, further supporting our qualitative conclusions. These simulations help illustrate the unemployment dynamics and confirm the stability and behavior of the equilibrium points predicted by the mathematical model. [ABSTRACT FROM AUTHOR]

In this work, the time-fractional Tricomi equation is investigated via two efficient computational techniques. This equation is used to explain the nearly sonic speed gas dynamics phenomenon. The Homotopy perturbation transform technique, which is a combination of Laplace transform and a semi-analytical technique, and Homotopy analysis method are used to solve the time-fractional Tricomi equation. The existence and uniqueness of the solution is analyzed by using two different fixed-point theorems. Finally, the effectiveness of the proposed techniques is illustrated through two test examples by comparing the absolute error of proposed techniques with the existing techniques and the result achieved in this paper benefits (but not limited to) the gas flow dynamics. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) is a serious disease that threatens and affects capital stock, population composition, and economic growth. This research paper aims to study the mathematical modeling and disease dynamics of HIV/acquired immunodeficiency syndrome (AIDS) with memory effect. We propose two fractional models in the Caputo sense for HIV/AIDS with and without migration. First, we prove the existence and positivity of both models and calculate the basic reproduction number R0$$ {\mathcal{R}}_0 $$ using the next generation method. Then, we study the local and global stability of the obtained equilibria. In addition, numerical simulations are provided for different values of the fractional order ρ$$ \rho $$ using the Adams–Bashforth–Moulton fractional scheme, to verify the theoretical results. Moreover, a sensitivity analysis of the parameters for the model with migration is carried out. [ABSTRACT FROM AUTHOR]

In recent years, fuzzy and fractional calculus are utilized for simulating complex models with uncertainty and memory effects. This study is focused on fuzzy-fractional modeling of (2+1)-dimensional Wu–Zhang (WZ) system. Caputo-type time-fractional derivative and triangular fuzzy numbers are employed in the model to observe uncertainties in the presence of non-local and memory effects. The extended He–Mohand algorithm is proposed for the solution and analysis of the current model. This approach is based on homotopy perturbation method along with Mohand transformation. Effectiveness of proposed methodology at upper and lower bounds is confirmed through residual errors. The theoretical convergence of proposed algorithm is proved alongside numerical computations. Existence and uniqueness of solution are also theoretically proved in the given paper. Current investigation considers three types of fuzzifications i.e. fuzzified equations, fuzzified conditions, and finally fuzzification in both model and conditions. Different physical aspects of WZ system profiles are analyzed through 2D and 3D illustrations at upper and lower bounds. The obtained results highlight the impact of uncertainty on WZ system in fuzzy-fractional space. Hence, the proposed methodology can be used for other fuzzy-fractional systems for better accuracy with lesser computational cost. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this article, we construct the series solution of the time-fractional Korteveg de Vries (K-dV) equation through a computational approach named as Laplace residual power series (LRPS) that combines the Laplace transform with the residual power series method (RPS). Time-fractional K-dV equation is used to modeled various real life phenomena like propagation of waves in elastic rods, dispersion effects in shallow coastal regions, anomalous diffusion observed in financial markets. The Caputo fractional derivative is used in the formulation of time-fractional K-dV equation. LRPS method is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity without any perturbation, discretization and linearization. To assess the effectiveness of proposed computational strategy, we perform a comparative analysis among the fractional residual power series method, the Adomian decomposition method, and the RPS method. Additionally, we examine the convergence of the fractional series solution across different α values and assess the solution's behavior as the time domain increased. The efficiency and authenticity of the LRPS method is shown by computing the absolute error, relative error and residual error. This work is supported by 2D and 3D graphical representations made in accordance with Maple and MATLAB. [ABSTRACT FROM AUTHOR]

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system's state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. [ABSTRACT FROM AUTHOR]

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian's and He's polynomials, thereby offering a more efficient way of solving nonlinear problems. [ABSTRACT FROM AUTHOR]

In this paper, we present the design of a fractional-order sliding mode observer (FO-SMO) for actuator fault estimation in a quadrotor unmanned aerial vehicle (QUAV) system. Actuator faults can significantly compromise the stability and performance of QUAV systems; therefore, early detection and compensation are crucial. Sliding mode observers (SMOs) have recently demonstrated their accuracy in estimating faults in QUAV systems under matched uncertainties. However, existing SMOs encounter difficulties associated with chattering and sensitivity to initial conditions and noise. These challenges significantly impact the precision of fault estimation and may even render fault estimation impossible depending on the magnitude of the fault. To address these challenges, we propose a new fractional-order SMO structure based on the Caputo derivative definition. To demonstrate the effectiveness of the proposed FO-SMO in overcoming the limitations associated with classical SMOs, we assess the robustness of the FO-SMO under three distinct scenarios. First, we examined its performance in estimating actuator faults under varying initial conditions. Second, we evaluated its ability to handle significant chattering phenomena during fault estimation. Finally, we analyzed its performance in fault estimation under noisy conditions. For comparison purposes, we assess the performance of both observers using the Normalized Root-Mean-Square Error (NRMSE) criterion. The results demonstrate that our approach enables more accurate actuator fault estimation, particularly in scenarios involving chattering phenomena and noise. In contrast, the performance of classical (non-fractional) SMO suffers significantly under these conditions. We concluded that our FO-SMO is more robust to initial conditions, chattering phenomena, and noise than the classical SMO. [ABSTRACT FROM AUTHOR]

In this paper, a generalized equal width wave equation involving space fractal derivatives and time Caputo fractional derivatives is studied and its approximate analytical solution is presented by the Adomian decomposition method. An example shows that the method is efficient to solve fractal non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

The present work investigates Bloch equations arising in magnetic resonance with Caputo and Caputo Fabrizio derivatives. Banach's fixed point approach is used to construct the existence theory for the model's solution. Also, the stability of the solution is established by Ulam–Hyers conditions. A novel 3-step iterative method is used for the considered model's numerical simulation with Caputo and Caputo Fabrizio derivative. This iterative method is formulated by combining Lagrange's interpolation with the fundamental theorem of integral calculus. The proposed method's error estimate is provided. The simulation results are displayed in tabulated and graphical form for distinct values of fractional order. The results demonstrate how the proposed method is accurate and appropriate for analysing fractional Bloch model. Further, this technique can also approximate the solution of other equations arising in engineering physics and quantum fields. [ABSTRACT FROM AUTHOR]

This paper is concerned with solving the problem of identifying the control vector problem for a fractional multi-order system of nonlinear ordinary differential equations (ODEs). We describe a quasilinearization approach, based on minimization of a quadratic functional, to compute the values of the unknown parameter vector. Numerical algorithm combining the method with appropriate fractional derivative approximation on graded mesh is applied to SIS and SEIR problems to illustrate the efficiency and accuracy. Tikhonov regularization is implemented to improve the convergence. Results from computations, both with noisy-free and noisy data, are provided and discussed. Simulations with real data are also performed. [ABSTRACT FROM AUTHOR]

Along with the developing technology, obesity and diabetes are increasing rapidly among people. The identification of diabetes diseases, modeling, predicting their behavior, conducting simulations, studying control and treatment methods using mathematical methods has become of great importance. In this paper, we have obtained numerical solutions by considering the glucose-insulin fractional model. This model consists of three compartments: the blood glucose concentration (G), the blood insulin concentration (I) and the ready-to-absorb glucose concentration (D) in the small intestine. The fractional derivative is used in the sense of Caputo. By performing mathematical analyzes for the Glucose-Insulin fractional mathematical model, numerical results were obtained with the help of the Euler method and graphs were drawn. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative. [ABSTRACT FROM AUTHOR]

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering. [ABSTRACT FROM AUTHOR]

This paper presents the computational analysis of fractional differential equations of variable‐order delay systems. To the proposed problem, the existence of solutions is derived using Arzela‐Ascoli theorem, and the Banach fixed point theorem is used for uniqueness results. To investigate and address the computational solutions, Adams‐Bashforth‐Moulton technique is established. To demonstrate the method's efficiency, computational simulations of chaotic behaviors in several one‐dimensional delayed systems with distinct variable orders are employed. The numerical solution of the proposed problem gives high precision approximations. [ABSTRACT FROM AUTHOR]

In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme. [ABSTRACT FROM AUTHOR]

The creation of an explicit finite difference scheme with the express purpose of resolving initial boundary value issues with linear and semi-linear variable-order temporal fractional properties is presented in this study. The rationale behind the utilization of the Caputo derivative in this scheme stems from its known importance in fractional calculus, an area of study that has attracted significant interest in the mathematical sciences and physics. Because of its special capacity to accurately represent physical memory and inheritance, the Caputo derivative is a relevant and appropriate option for representing the fractional features present in the issues this study attempts to address. Moreover, a detailed Fourier analysis of the explicit finite difference scheme's stability is shown, demonstrating its conditional stability. Finally, certain numerical example solutions are reviewed and MATLAB-based graphic presentations are made. [ABSTRACT FROM AUTHOR]

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. [ABSTRACT FROM AUTHOR]

This article presents an efficient implicit spline-based numerical technique to solve the time-fractional generalized coupled Burgers' equation. The time-fractional derivative is considered in the Caputo sense. The time discretization of the fractional derivative is discussed using the quadrature formula. The quasilinearization process is used to linearize this non-linear problem. In this work, the formulation of the numerical scheme is broadly discussed using cubic B-spline functions. The stability of the proposed method is proved theoretically through Von-Neumann analysis. The reliability and efficiency are demonstrated by numerical experiments that validate theoretical results via tables and plots. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

The primary goal of the current work is to use a novel technique known as the natural transform decomposition method to approximate an analytical solution for the fractional smoking epidemic model. In the proposed method, fractional derivatives are considered in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo senses. An epidemic model is proposed to explain the dynamics of drug use among adults. Smoking is a serious issue everywhere in the world. Notwithstanding the overwhelming evidence against smoking, it is nonetheless a harmful habit that is widespread and accepted in society. The considered nonlinear mathematical model has been successfully used to explain how smoking has changed among people and its effects on public health in a community. The two states of being endemic and disease-free, which are when the disease dies out or persists in a population, have been compared using sensitivity analysis. The proposed technique has been used to solve the model, which consists of five compartmental agents representing various smokers identified, such as potential smokers V, occasional smokers G, smokers T, temporarily quitters O, and permanently quitters W. The results of the suggested method are contrasted with those of existing numerical methods. Finally, some numerical findings that illustrate the tables and figures are shown. The outcomes show that the proposed method is efficient and effective. [ABSTRACT FROM AUTHOR]

In this work, the susceptible-infectious-removed (SIR) dynamics are considered in relation to the effects on the health system. With the help of the Caputo derivative fractional-order method, the SIR epidemic model for childhood diseases is designed. Subsequenly, a set of sufficient conditions ensuring the existence and uniqueness of the addressed model by choosing proper fuzzy approximation methods. In particular, the fuzzy Laplace method along with the Adomian decomposition transform were employed to better understand the dynamical structures of childhood diseases. This leads to the development of an efficient methodology for solving fuzzy fractional differential equations using Laplace transforms and their inverses, specifically with the Caputo sense derivative. This innovative approach facilitates the numerical resolution of the problem and numerical simulations are executed for considering parameter values. [ABSTRACT FROM AUTHOR]

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

The mathematical efficiency of fractional-order differential equations in modeling real systems has been established. The first-order and second-order time derivatives are substituted in integer-order problems by a fractional derivative of order 0 < ω ≤ 1 , resulting in time-fractional heat and wave problems with variable coefficients. In this research, we analyze fractional-order wave and heat problems with variable coefficients within the framework of a Caputo derivative (CD) using the Elzaki residual power series method (ERPSM), which is a coupling of the residual power series method (RPSM) and the Elzaki transform (E-T). It relies on a novel form of fractional power series (FPS), which provides a convergent series as a solution. The accuracy and convergence rates have been proven by the relative, absolute, and recurrence error analyses, demonstrating the validity of the recommended approach. By employing the simple limit principle at zero, the ERPSM excels at calculating the coefficients of terms in a FPS, but other well-known approaches such as Adomian decomposition, variational iteration, and homotopy perturbation need integration, while the RPSM needs the derivative, both of which are challenging in fractional contexts. ERPSM is also more effective than various series solution methods due to the avoidance of Adomian's and He's polynomials to solve nonlinear problems. The results obtained using the ERPSM show excellent agreement with the natural transform decomposition method and homotopy analysis transform method, demonstrating that the ERPSM is an effective approach for obtaining the approximate and closed-form solutions of fractional models. We established that our approach for fractional models is accurate and straightforward and researcher can use this approach to solve various problems. [ABSTRACT FROM AUTHOR]

We solve the fractional Liouville equation by using Riemann–Liouville and Caputo derivatives for systems exhibiting noninteger power laws in their Hamiltonians. Based on the fractional Liouville equation, we calculate the density function (DF) of a classical ideal gas. If the Riemann–Liouville derivative is used, the DF is a function depending on both the momentum and the coordinate , but if the derivative in the Caputo sense is used, the DF is a constant independent of and . We also study a gas consisting of fractional oscillators in one-dimensional space and obtain that the DF of the system depends on the type of the derivative. [ABSTRACT FROM AUTHOR]

The second order Burger's equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo's formula, while the CBS technique with the help of a θ -weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique. [ABSTRACT FROM AUTHOR]

The initial value problem for the Caputo type time‐fractional Sobolev equation with a nonlinear exponential source function is investigated in this work. We establish the existence and uniqueness of mild solutions corresponding to two different initial data assumptions. We derive global results of a unique mild solution with small initial data using some Sobolev/Sobolev‐Orlicz embeddings, a weighted Banach space, and the fixed point theorem. In the absence of any smallness assumptions, the Cauchy iteration method demonstrates that the mild solution blows up at a finite time or exists globally in time. Finally, we consider some illustrated examples to test the results obtained in theory. [ABSTRACT FROM AUTHOR]

The memory effect in vesicle dynamics refers to the persistence of shape changes in lipid vesicles, a type of lipid bilayer membrane that mimics some features of real cells, particularly red blood cells (RBCs). To study this effect, a fractional rigid sphere model in five dimensions has been investigated, which maps the dynamics of a vesicle using the Caputo operator. This model provides new insights into the mechanisms behind the memory effect as well as the emergence of two other motions, namely, tank‐treading with underdamped and tank‐treading with overdamped vesicle oscillations, in addition to the tank‐treading mode where the vesicle rotates about its axis while maintaining a constant shape. Overall, this work represents a significant advance in our understanding of vesicle dynamics and has important implications for understanding the behavior of RBCs. [ABSTRACT FROM AUTHOR]

This paper gives a highly efficient technique to analyse the fractional BBM-Burger equation and fractional Diffusion-Wave equation. These equations are used to model various real-life phenomena like acoustic gravity waves, diffusion theory, anomalous diffusive systems, and wave propagation phenomena. A modified technique, which is the combination of the Homotopy perturbation method and Laplace transform, is used for getting the numerical solution. The Lyapunov function is used to investigate asymptotic stability, and the maximum absolute error for the proposed technique is also examined. The efficiency of the proposed technique is shown by computing the root mean square (RMS), L 2 , and L ∞ errors and comparing the results with the other techniques. [ABSTRACT FROM AUTHOR]

The reproducing kernel Hilbert space method (RK-HS method) is used in this research for solving some important nonlinear systems of fractional ordinary differential equations, such as the fractional Susceptible-Infected-Recovered (SIR) model. Nonlinear systems are widely used across various disciplines, including medicine, biology, technology, and numerous other fields. To evaluate the RK-HS method's accuracy and applicability, we compare its numerical solutions with those obtained via Hermite interpolation, the Adomian decomposition method, and the residual power series method. To further support the reliability of the RK-HS method, the convergence analysis is discussed. [ABSTRACT FROM AUTHOR]

In this work, the main goal is to implement Homotopy perturbation transform method (HPTM) involving Katugampola fractional operator. As an example, a fractional order Hepatitis model is considered to analyze the solutions. At first, the integer order model is converted to fractional order model in Caputo sense. Then, the new operator Katugampola fractional derivative is used to present the model. The new such kind of operator is illustrated in Caputo sense. HPTM is described to get the solution of the proposed model using the new kind of operator. Also, there are some analyses about the new kind of operator to prove the efficiency of the operator. [ABSTRACT FROM AUTHOR]

In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix's spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform. [ABSTRACT FROM AUTHOR]

This article looks for a reliable numerical technique to solve the Allen–Cahn equation using the Caputo time-fractional derivative. The fractional derivative semi-discretization approach using finite differences of the second order is shown first. The cubic B-spline collocation method is used to get a full discretization. We prove the conditional stability and convergence of the suggested approach. The technique's effectiveness is demonstrated with numerical examples using two test problems. Numerical analysis confirms the approach's efficiency and the method's continued correctness. [ABSTRACT FROM AUTHOR]

Most nutrient uptake problems are modeled by the convection–diffusion equation (CDE) abiding by Fick's law. Because nutrients needed by plants exist in the soil solution as a form of ions and the soil is a typical fractal structure of heterogeneity, it makes the solute transport appear anomalous diffusion in soil. Taking anomalous diffusion as a transport process, we propose time and space fractional nutrient uptake models based on the classic Nye–Tinker–Barber model. There does not appear apparent sub-diffusion of nitrate in the time fractional model until four months and the time fractional models are appropriate for describing long-term dynamics and slow sorption reaction; the space fractional model can capture super-diffusion in short term and it is suitable for describing nonlocal phenomena and daily variations driven by transpiration and metabolism; the anomalous diffusion more apparently appears near the root surface in the modeling simulation. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper focuses on development of an efficient numerical technique for approximation of a fractional neutron diffusion equation with delayed neutrons for neutron flux profile in a nuclear reactor. The L 1 method is used to approximate the Caputo time-fractional derivatives in the governing equation. A collocation technique based on quintic B-spline (QBS) basis function is employed for discretization of space derivative. The proposed numerical scheme is applied to solve the considered fractional neutron diffusion equation in order to illustrate its applicability and efficiency. The influences of fractional order derivative (α), radioactive decay constant (λ) and relaxation time (τ) on the neutron flux profile are investigated. The numerical results of fractional neutron diffusion equation are compared with those of classical neutron diffusion equation. Further, the results obtained by the proposed approach are compared with the results obtained by the finite difference method to justify the superiority of our method. The CPU time (the execution time of the computer program) is given in order to demonstrate the method's computational efficiency. [ABSTRACT FROM AUTHOR]

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α + 1 where 0 < α < 1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L 2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider a solution u to an initial boundary value problem in a bounded domain Ω over a time interval (0 , T) for a time-fractional wave equation where the order of time derivative is between 1 and 2. We prove that if u | ω × (0 , T) = 0 for arbitrarily chosen subdomain ω ⊂ Ω , then u = 0 in Ω × (0 , T). [ABSTRACT FROM AUTHOR]