Search

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method. [ABSTRACT FROM AUTHOR]

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity. [ABSTRACT FROM AUTHOR]

In this article, a time‐fractional nonlinear moving boundary problem is considered to express the mathematical model of diffusion of a solvent into a glassy polymeric device. A numerical method, including an iterative process based on a non‐standard implicit finite difference method with variable time‐step is developed to solve this problem. The purpose of solving this problem is to study and predict the solvent diffusion behavior in the polymer and the position of the moving boundary. We analyze the monotonicity and positivity of the numerical solution. To show the effectiveness and robustness of the numerical method, two test problems are considered and the numerical results are reported for different values of model parameters. [ABSTRACT FROM AUTHOR]

This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or illposed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applied to obtain stable and accurate results. Finally, to demonstrate the accuracy and effectiveness of our scheme, two benchmark test problems have been considered, and its good working with different noise levels. [ABSTRACT FROM AUTHOR]

The new fractional-order Genesio-Tesi system is introduced, and its boundedness, stability of the equilibrium points, Lyapunov stability, uniqueness of the solution, and bifurcation are all discussed in this paper. Using the efficient predictor-corrector approach, we statistically analyze the Genesio-Tesi system in fractional order. The results effectively conceptualize and visualize the novel fractional order Genesio-Tesi systems that are suggested. When the system's order shifts from integer to fractional, the revolution around the fixed point increases. The chaotic character of the modified Genesio-Tesi system is comparable to that of the original Genesio system. The major changes were made to the Geensio-Tesi system by including the trigonometric functions, keeping the initial conditions and parameter values intact. The system is fractionalised with the help of Caputo fractional operator. In particular, the modified system's nature is more complex, which may aid in signal processing and secure communication. Future research on the modified Genesio-Tesi system can now proceed in light of this finding. This article offers a fresh approach to utilizing and thoroughly researching the Genesio-Tesi systems that have been provided. [ABSTRACT FROM AUTHOR]

A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen–Cahn equation (TFACE). Initially, Caputo's formula is used to discretize the time-fractional derivative, while a new ExCBS is used for the spatial derivative's discretization. Convergence analysis is carried out and the stability of the proposed method is also analyzed. The scheme's applicability and feasibility are demonstrated through numerical analysis. [ABSTRACT FROM AUTHOR]

This article, published in the International Journal of Applied Mechanics, discusses a new theory of fractional-order rate-dependent piezoelectric thermoelasticity and its application in analyzing the structural transient response of smart piezoelectric composite laminates. The authors develop unified forms of fractional-order strain and heat conduction by adopting different types of fractional derivatives. They establish a fractional-order rate-dependent piezoelectric thermoelasticity theory and derive constitutive and governing equations using an extended thermodynamics framework. The theory is then applied to investigate the dynamic thermo-electromechanical responses of smart piezoelectric composite laminates with imperfect interfacial conditions. The influences of different fractional derivatives, imperfect interfacial conditions, and materials constants ratios on wave propagations and structural thermo-electromechanical responses are evaluated and discussed. [Extracted from the article]

In this article, two dimensional inverse problem of determining convolution kernel in the fractional diffusion equation with the time-fractional Caputo derivative is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation with Riemann–Liouville derivative is constructed. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function, which was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. The local existence and global uniqueness results are proven by the aid of fixed point argument in suitable functional classes. Also the stability estimate is obtained. [ABSTRACT FROM AUTHOR]

The paper aims to utilize an integral transform, specifically the Khalouta transform, an abstraction of various integral transforms, to address fractional differential equations using both Riemann-Liouville and Caputo fractional derivative. We discuss some results and the existence of this integral transform. In addition, we also discuss the duality between Shehu transform and Khalouta transform. The numerical examples are provided to confirm the applicability and correctness of the proposed method for solving fractional differential equations. [ABSTRACT FROM AUTHOR]

The central-symmetric time-fractional heat conduction equation with heat absorption is investigated in a solid with a spherical hole under time-harmonic heat flux at the boundary. The problem is solved using the auxiliary function, for which the Robin-type boundary condition with a prescribed value of a linear combination of a function and its normal derivative is fulfilled. The Laplace and Fourier sine–cosine integral transformations are employed. Graphical representations of numerical simulation results are given for typical values of the parameters. [ABSTRACT FROM AUTHOR]

This work is dedicated to the investigation of a regularization method for the problem of determining Caputo fractional derivatives of a function in the Banach space L ∞ [ 0 , T ] . This regularization method is based on the approximation of the first-order derivative of the function by the solution of a well-posed problem depending on a regularization parameter. We then discuss the Hölder type stability results for the method according to two choice rules for the regularization parameter, which are an a priori parameter choice rule and an a posteriori parameter choice rule. Some numerical examples are provided. [ABSTRACT FROM AUTHOR]

Arrival of a new disease marks a yearlong destruction of human lives and economy in general, and if the disease turns out to be a pandemic the loss is frightening. COVID-19 is one such pandemic that has claimed millions of lives till date. There is a suffering throughout the world due to various factors associated with the pandemic, be it loss of livelihoods because of sudden shutdown of companies and lockdown, or loss of lives due to lack of medical aid and inadequate vaccination supplies. In this study, we develop a six-compartmental epidemiological model incorporating vaccination. The motivation behind the study is to analyze the significance of higher vaccination efficacy and higher rate of population getting vaccinated in controlling the rise in infectives and thereby the untimely demise of various individuals. The work begins with an ordinary differential equation model followed by stability analysis of the same, after which a fractional-order derivative model of the same is formulated and the existence of uniformly stable solution for the system is proved. In addition to this, we present the stability of the equilibria in general for the fractional model framed. The sensitivity analysis of the basic reproduction number along with its correlation with various parameters is presented. In addition to this, sensitivity of certain state variables in the fractional model with respect to different fractional orders as well with respect to different infection rate is exhibited in this work. Factors related to lockdown and usage of face shields are incorporated in the entire study, and importance of these is highlighted in the study as well. The major takeaway from the study is that mere vaccination will not suffice in eradication of the virus. The vaccine efficacy plays a major role along with other intervention included in the model. The numerical simulations are carried out in MATLAB software using ode45 and fde12. [ABSTRACT FROM AUTHOR]

Nano-batteries have been widely used in electric vehicles, new energy, and aerospace engineering since their high energy density, low manufacturing cost, and long cycle life. In recent years, there have been many papers that contributed to investigate the diffusion-mechanical coupling problems under non-uniform molar concentration environments (e.g., rapid charging, etc.). Nevertheless, the memory dependence of strain relaxation and mass transfer has not been considered yet. This paper aims to construct a unified fractional-order non-Fick mechanical-diffusion coupling model by introducing the fractional derivatives of the Caputo (C), Caputo–Fabrizio (CF), Atangana–Baleanu (AB), and Tempered-Caputo (TC) types. The proposed theoretical model is applied to investigate structural transient dynamic responses of multilayered composite laminates with imperfect interfacial conditions by Laplace transformation approach. The influences of different fractional derivatives, imperfect interfacial conditions, and materials constants ratios on the wave propagations and dynamic mechanical-diffusion responses are evaluated and discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we extend fractional-order derivative for the shifted Vieta-Lucas polynomial to generalized-fractional integro-differential equations involving non-local boundary conditions using Galerkin method as transformation technique and obtained N - ⌈δ⌉ + 1 system of linear algebraic equations with λi, i = 0, . . ., N unknowns, together with ⌈δ⌉ non-local boundary conditions, we obtained (N + 1)- linear equations. The accuracy and effectiveness of the scheme was tested on some selected problems from the literature. Judging from the table of results and figures, we observed that the approximate solution corresponding to the problem that has exact solution in polynomial form gives a closed form solution while problem with non-polynomial exact solution gives better accuracy compared to the existing results. [ABSTRACT FROM AUTHOR]

To extend the existing results of differential systems of integer order, we have considered a class of Caputo fractional linear differential systems with multiple delays. By applying the multinomial theorem for nonpermutable matrices without a commutativity assumption on the matrix coefficients, the representations of exact solutions for the studied systems were obtained. The results show that Hyers-Ulam stability in finite time holds for this class of systems. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution. [ABSTRACT FROM AUTHOR]

Compartmental models based on coupled differential equations of fractional order have been widely employed in the literature for modeling. An abstraction of these models is given by a system with polynomial vector field. We investigate the use of power series for solving generic polynomial differential equations in any dimension, with Caputo fractional derivative. As is well known, power series convert a continuous formulation into a discrete system of difference equations, which are easily solved by recursion. The novelty of this paper is that we rigorously prove that the series converge on a neighborhood of the initial instant, which is an analogue of the Cauchy–Kovalevskaya theorem. Besides, these series are proved to be continuous with respect to the fractional index. For applications, a general-purpose symbolic implementation of truncated power series is developed, and its execution is illustrated for the fractional SIR epidemiological model. [ABSTRACT FROM AUTHOR]

We present how the invariant subspace method of differential equations can be extended to scalar and coupled fractional differential-difference equations, and illustrate its applicability through fractional discrete Burger's equation, fractional discrete heat equation, fractional Toda lattice system, coupled fractional discrete Volterra equation. Then, we show how to find the partially invariant subspaces for the nonlinear difference operator that possesses at least one linear difference factor and illustrate its applicability through the fractional Lotka Volterra equation, fractional discrete KdV equation, and fractional Toda lattice equation. Also, we extend the generalized separation of variables technique to time-fractional differential-difference equation. We demonstrate its efficiency through the fractional discrete KdV equation, fractional discrete Modified KdV equation, fractional discrete Burgers equation, fractional Lotka Volterra equation, and fractional discrete heat equation. Finally, we generalize the invariant subspace method to a time-space fractional differential-difference equation, and its usefulness is illustrated through examples. The above-proposed methods provide efficient analytical techniques to derive the exact solution of time-fractional differential-difference equations in the Caputo sense or Riemann Liouville sense. [ABSTRACT FROM AUTHOR]

This study examines a fractional-order prey-predator model incorporating fear effects and harvesting impacts on prey dynamics, employing both continuous and discretized frameworks with the Monod–Haldane functional response. The existence, uniqueness, and boundedness of the system's solutions, along with their non-negativity, are established through rigorous analysis. The system is further evaluated for potential equilibrium points, with their stability conditions meticulously assessed. It is revealed that the model possesses three locally stable equilibrium points, provided certain conditions are met. In the context of the discretized model, an optimal harvesting strategy is formulated, guided by Pontryagin’s Maximum Principle, to ensure maximum economic yield. Numerical simulations complement the analytical findings, offering insights into the system's dynamic behavior under both continuous and discrete scenarios. Moreover, the optimality problem associated with harvesting strategies is resolved. The study concludes by summarizing the significant outcomes and their implications for ecological management. [ABSTRACT FROM AUTHOR]

The non-existence of solutions is discussed for a system of fractional differential equations including two types of fractional derivatives: the Caputo fractional derivative (CFD) and the Riemann–Liouville fractional derivative (RLFD). The nonlinear sources are nonlocal in time. The system we consider is more general than those previously discussed in the literature. Our results are obtained by using several properties of fractional derivatives, the test-function method and by applying some integral inequalities. Finally, we provide some examples to illustrate our findings. [ABSTRACT FROM AUTHOR]

In this study, we proposed a numerical technique to solve a class of variable‐order time‐fractional advection–diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional‐order Lagrange polynomials (FOLPs). The variable‐order fractional derivative is assumed to be Caputo's derivative. Using the operational matrix and collocation method, the advection–diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton's iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples. [ABSTRACT FROM AUTHOR]

In this study, we explore a mathematical model of human immunodeficiency virus type 1 (HIV-1) infection in CD4+ T-cells, considering fractional-order dynamics in the Caputo sense. Fractional models are of great significance due to their capacity to predict disease outbreaks while incorporating memory and non-local characteristics. To establish the existence and uniqueness of solutions for the fractional model, a fixed-point theorem and an iterative technique are employed. Furthermore, it is confirmed that the solutions of the model are both positive and bounded. The basic reproduction number, denoted as R 0 , is calculated using the next-generation matrix approach, and the equilibrium points of the model, including disease-free and endemic equilibrium points, are presented. A sensitivity analysis of R 0 is performed by varying different parameters. To analyze the local stability of equilibrium points, the Routh-Hurwitz technique is employed. The global stability of equilibrium points is demonstrated using the Lyapunov function and LaSalle's principle, as well as through UlamHyers (UH) stability and the generalized UH stability conditions. The proposed numerical scheme Adams-Bashforth predictor-corrector is validated by comparing it with the fourth-order Runge-Kutta (RK4) technique. We examine the influence of the fractional order (ξ) by conducting numerical simulations for different values of ξ using the Adams-Bashforth predictor-corrector approach. Furthermore, graphical presentations illustrate how different crucial model parameters impact disease dynamics. The results obtained from this study demonstrate that the adopted strategies significantly improve prediction accuracy. [ABSTRACT FROM AUTHOR]

A full-discrete scheme is presented for a Langevin equation involving Caputo fractional derivative and additive white noise. Based on a spectral truncation of white noise, the fractional Langevin equation is converted to an approximate equation with random parameters and a finite difference scheme is constructed. Consistency of the approximate equation as well as error estimate of the finite difference scheme are obtained. It is proved that when spectral truncation level and the step size are inversely proportional, the convergence order of the finite difference scheme is 1.5 in mean-square sense and independent of the value of fractional derivative order. Numerical examples verify the theoretical analysis. Moreover, to fulfill long-time simulation, a scheme based on piecewise spectral approximation of white noise is further developed. [ABSTRACT FROM AUTHOR]

In this paper, the inverse cumulative grey time power model with a Caputo fractional derivative is established, and the solution to the whitening equation is given by the Laplace transform. To improve the prediction accuracy of the model, the linear-tangent-function transformation is used to improve the smoothness of the data sequence, and a grey time power model is obtained, which has higher accuracy than the sinusoidal-function transformation, negative-exponential-function transformation and logarithmic-function transformation. The form and application range of the model are generalized. [ABSTRACT FROM AUTHOR]

This study carried out the free convective non‐Newtonian fluid of Brinkman type flow near an upright plate moving with velocity f(t). A fractional order model for non‐Newtonian fluid of Brinkman type flow is proposed. The time derivative in the proposed fractional flow model is considered by using the two types of fractional derivatives namely Caputo fractional derivative and Atangana–Baleanu fractional derivative. The system of conjugated fractional partial differential equations for the temperature θ, velocity u and concentration C are worked out by applying optimal homotopy asymptotic technique. The effectuate of tangible and fractional variables on the domains of velocity u, temperature θ and concentration C are envisioned graphically. The rate of heat and mass transfer in the form of Nu and Sh is also calculated for both fractional derivatives. The numerical results reveal the efficiency, reliability, significant features, and simple in computation with high accuracy of consider method for non‐Newtonian fractional order fluid of Brinkman type flow. We ascertained that our results are in excellent agreement with the exact results. [ABSTRACT FROM AUTHOR]

In this article, we examine the Kuramoto–Sivashinsky equation, a nonlinear model that modifies a variety of physical and chemical scenarios. The main goal of this article is to find the analytical solution to the fuzzy fractional Kuramoto–Sivashinsky equations (FFKSEs). We employ the fractional reduced differential transform method (FRDTM) for dealing with the fractional Caputo operator. By using three examples, we tested the developed technique. The upper and lower parts of the fuzzy solutions for all three challenges were obtained using two different fractional-order simulations between 0 and 1. The attained series-type result was identified in the current investigation. The results show that the suggested strategy leads to accurate solutions at the integer level. The outcomes of applying the suggested methodology demonstrate the effectiveness and simplicity of our approach for analysing the behaviour of nonlinear models used in science and technology. [ABSTRACT FROM AUTHOR]

Of concern is a fractional differential problem of order between zero and one. The model generalizes an existing well-known problem in heat conduction theory with memory. First, we justify the replacement of the first order derivative by a fractional one. Then, we establish a Mittag-Leffler stability result for a class of heat flux relaxation functions. We will combine the energy method with some properties from fractional calculus. [ABSTRACT FROM AUTHOR]

In recent decades, numerous and varied numerical methods have been proposed and studied to approximate solutions for various classes of fractional differential equations, primarily those involving single-term or multipleorder equations. However, equations incorporating fractional iterated derivatives have not received widespread attention. In this work we describe a reliable strategy to approximate the solution of higher-order fractional differential equations where both the fractional derivative and the iterated derivatives are described in the Caputo sense. Specifically, we propose a fractional variational iteration method (FVIM) where the Lagrange multiplier associated with the correction term is explicitly determined by means of the Laplace transform. For the second-order case, we give a sufficient condition -involving the coefficients of the equation and the fractional order of the Caputo derivative-which guarantees the convergence of the sequence generated by the FVIM. Furthermore, this convergence is independent of the initial function considered for the iteration. Finally, some examples are presented in order to illustrate the applicability of the method and the reliability of the theoretical results obtained. In particular, for most of them we observe that the FVIM leads to the exact solution which shows the power of the method in practice. [ABSTRACT FROM AUTHOR]

This paper deals with the theoretical results for solutions of a nonlocal fractional neutral integro-differential problem with boundary integral conditions. We prove the existence and uniqueness results using the Krasnoselskii’s and Banach fixed point theorems. Based on the results obtained, sufficient conditions are provided that ensure the generalized results. Finally, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

The goal of this research is to develop a new, simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations (PDEs) with variable coefficient. ARA-transform is a robust and highly flexible generalization that unifies several existing transforms. The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor's expansion. The process of finding approximations for dynamical fractional-order PDEs is challenging, but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts. This approach is effective and useful for solving a massive class of fractional-order PDEs. Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients. Additionally, several visualizations are drawn for different fractional-order values. Besides that, the estimated findings by the proposed technique are in close agreement with the exact outcomes. Finally, statistical analyses further validate the efficacy, dependability and steady interconnectivity of the suggested ARA-residue power series approach. [ABSTRACT FROM AUTHOR]

Substantial advancements have been made in applying fractional calculus to understanding the epidemiology of viral diseases. However, it remains troubling that over 2,000,000 children, particularly those under the age of 5 and elderly individuals over 65, succumb to pneumonia annually in developing nations. This paper uses a deterministic SVEIR (susceptible, vaccinated, exposed, infected, and recovered) model to investigate pneumonia disease dynamics from a mathematical perspective. The governing model has been generalized from Caputo fractional derivatives, and the generalized Euler’s method has been used to calculate the approximate solution of the governing model. The dynamical systems theory examines the model’s stability analysis, basic reproduction number, and equilibrium points. The linearization method for spatial equilibrium and the Lyapunov functional method demonstrate that the model is locally asymptotic stable for disease-free equilibrium. Graphical results are showcased through MATLAB21 to illustrate the resolution of the problem. The research findings suggest that the disease will eventually disappear from the population if vaccination coverage rises over the necessary vaccination threshold. According to sensitivity analysis, the most sensitive factors are the transmission rate and the pace at which exposed individuals become contagious. According to the study, we must strengthen treatment effectiveness to prevent disease spreading.

Abstract:: In this paper, a hybridized Newton-Raphson method with convergence order (2+1) is coupled with modified Laplace Adomian decomposition technique (LADT) to obtain approximate soliton of Time Fractional Klein Gordon Equation (TFKGE) in Caputo environment and associated with relativistic fractional quantum mechanics. Our approach involves utilising an effective Laplacian and inverse Laplacian method to approximate the soliton of the proposed equation. The comprehensive investigation of convergence and stability for the current system improves the theoretical notion. The corresponding graphs of the exact and approximate solutions, as well as the error norms L2 and L∞ for each test problem, are provided to validate the numerical results of the problem. The present result is compared with the analytical solution and existing solutions in the literature to illustrate the effectiveness and applicability of our proposed method.

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Korteweg–de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey–Dodd–Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error’s analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approach is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

This work investigates the exact and accurate approximate series solutions of the strongly nonlinear wave-like differential equations of fractional order with variable coefficients utilizing the Laplace residual power series technique in the 2D, and 3D-space. The posed model is a notable model in mathematical physics, engineering physics, and biophysics, to characterize the variations in laser light intensity, and fluid particle velocity distributions in turbulent flows. In the meaning of Caputo fractional derivative, the posed models are solved analytically via simulation of the generalized Taylor formula in the Laplace space. Different from the other analytic methods, an easy recurrence formula for finding the Laplace power series coefficients is derived. To validate the proposed approach, it is applied to three interesting problems. The approximate analytical solutions of studied fractional problems are expressed in fast-converging Caputo-fractional Maclaurin formulas. The results are numerically and graphically analyzed to demonstrate the efficiency and applicability of the technique, as well as to examine the effects of partial arrangement on the behavior of the solutions. Ultimately, the results and numerical simulations indicate that the used technique is an effective approach and accurate tool for solving fractional evolution models and other fractional models arising in nonlinear phenomena.

Hepatitis B virus (HBV) is a much more threatening and prolonged disease than hepatitis A. It can start as an acute disease, but in roughly 5 to 10% of situations, it can progress to a chronic condition that permanently damages the liver. After getting or coming into contact with the Hepatitis B virus (HBV), the said symptoms normally last from 10 days to 6 months. In the present paper, to have a firm understanding of the dynamics of HBV, we derived a new fractional model in the Caputo sense with asymptomatic carriers, vaccination, and treatment classes. We obtain the basic reproduction number R0 for the diagnosis and recurrence of the disease. We determine the stability of the proposed model locally and globally for R0 values less than and greater than 1. We use the Lyapunov function theory in fractional environments to establish the global stability of the fractional Hepatitis B virus (HBV) model. We demonstrate the existence and uniqueness of the fractional-order model. To look at the fractional order model’s solution graphically, we used a useful numerical strategy to get more detailed numerical results for the factors that have a big effect on getting rid of diseases. Additionally, the inclusion of the treatment class in the population allows for the determination of the impact of alternative medicines used to treat infected populations. Numerical simulation of the proposed fractional order (FO) Caputo model is performed, allowing for the analysis of graphical representations and the significance of fractional order derivatives to illustrate the impact of our theoretical findings.

Shigellosis, a highly contagious bacterial infection causing diarrhea, fever, and abdominal pain, necessitates a deep understanding of its transmission dynamics to devise effective control measures. Our study takes a novel approach, employing a fractional order framework to explore the influence of memory and control measures on Shigellosis transmission dynamics, thereby making a unique contribution to the field. The model is presented as a system of Caputo fractional differential equations capturing time constant controls. The Caputo derivatives are chosen for their inherent benefits. The qualitative features of the model, such as the solutions' existence and uniqueness, positivity, and boundedness, are thoroughly investigated. Moreover, the equilibria of the model are derived and analyzed for their stability using suitable theorems. In particular, local stability was proved through Routh's criteria, while global stability results were established in the Ulam-Hyers sense. The model is then solved numerically with the help of the predict-evaluate-correct-evaluate method of Adams-Bashforth-Moulton. The numerical results underscore the significant impact of memory on disease evolution, highlighting the novelty of integrating memory-related aspects into the meticulous planning of effective disease control strategies. Using fractional-order derivatives is more beneficial for understanding the dynamics of Shigellosis transmission than integral-order models. The advantage of fractional derivatives is their ability to provide numerous degrees of freedom, allowing for a broader range of analysis of the system's dynamic behaviour, including nonlocal solutions. Also, an investigation on the impacts of control measures via parameter variation is done; the findings show that applying treatment and sanitation minimizes disease eruption.

This study stems from the growing need for effective mathematical tools to tackle nonlinear fractional evolution equations, which find wide applications in physics and engineering. Traditional methods often struggle to provide exact closed-form solutions for such equations, leading to the development of novel approaches. In this context, the combination of the homotopy analysis method with the Laplace transform presents a promising avenue for obtaining exact solutions to these complex equations. By exploring this hybrid approach, the study aims to address the existing challenges in solving nonlinear fractional models and provide a reliable method for obtaining closed-form solutions. In this way, this study focuses on rendering the coupling between the homotopy analysis approach and the Laplace transform in the search for exact closed-form solutions for the class of fractional evolution and heat-typed equations. In fact, this class of equations features some interesting nonlinear Caputo fractional partial differential equations that model dissimilar physical processes in physics and engineering. A generalized recurrent scheme is derived, which is applicable to all the members of the governing class, and further analyzed the application of this very scheme on some test fractional initial-value problems of much concern. Eventually, the method has been found to be advantageous, with various advantages, including convergence to available exact solutions. Thus, of the advantage of the current study is innovative combination of mathematical methods, the applicability of these methods to a special class of fractional equations, the development of a generalized scheme, the successful solution of prototype problems, and the practical advantages of the method.

This article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis–Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics, utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model.

This article talks about a model of fractional differential equations to describe how COVID-19 is spread in the world in general and Iraq in particular. The model contains five fractional differential equations. Moreover, we have proven the existence and uniqueness of the solution of the model, found the equilibrium points of the model and checked its stability. Then we solved it using a fractional linear multi-step method. When we compare the results with the data documented by the World Health Organisation (WHO), we find that the total number of active cases in the world are equal to 584498294 on 3/8/2022, and it is similar to what was done with the system. The number of those who have recovered and died from the disease has also been calculated. For Iraq, the total number of active cases is 2448484 and the total number of active cases calculated by the model is 2628000. In comparison, the calculated number is slightly higher than what is given in the data. This is quite normal because not all infected patients go to the health centres and it is difficult to record them as active cases.

In this paper, a biophysical fractional diffusive cancer model with virotherapy is thoroughly analyzed and analytically simulated. The goal of this biophysical model is to represent both the dynamics of cancer development and the results of virotherapy, which uses viruses to target and destroy cancer cells. The Caputo sense is applied to the fractional derivatives. We look at the governing model's existence and uniqueness. For analytical solutions, the Laplace residual power series approach is used. The study investigates the model's dynamic behavior, shedding light on the development of cancer and the effects of virotherapy. The research advances our knowledge of cancer modeling and treatment options. Numerical simulations show the agreement between the analytical results and the related numerical solutions, proving the usefulness of the analytical solution.

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order q,0

Abstract Background Following a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method. Methodology Foremost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates. Recommendations We propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.

A review on Hermite-Hadamard type inequalities connected with a different classes of convexities and fractional differential operators is presented. In the various classes of convexities it includes, classical convex functions, quasi-convex functions, p-convex functions, strongly-m-convex functions, strongly-(θ,m)-convex functions, (s, m)-convex functions, (θ, h- m)-convex functions, strongly (θ, h- m)-convex functions, (h, m)-convex functions of the second type, m-convex functions, h-convex functions, (h,m)-convex functions, relative-convex functions, exponentially (θ, h- m)-convex functions, harmonically h-convex functions and geometric-arithmetically s-convex functions. In the fractional differential operators it includes, Caputo fractional derivative, k-Caputo fractional derivative and Hilfer fractional derivative.

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

Under uncertainty, the analytical behaviour of fractional partial differential equations is frequently puzzling and challenging to predict. Therefore, in order to address these problems, it is essential to create a proper, extensive, and highly effective theory. The theory of fuzzy fractional partial differential equations is a relatively new idea with applications in applied mathematics and engineering. In this article, we examined the different types of fuzzy fractional-order Korteweg-de-Vries (KdV) equations along with their analytical fuzzy solutions. To construct a series-type solution under a fuzzy notion for the relevant research, we utilise the fractional reduced differential transform method (FRDTM) in the Caputo sense. The solutions of various kinds of fuzzy fractional KdV equations that have been developed are more broadly applicable. The fuzzy concept is used to overrule the uncertainty in physical models based on the parametric form of the fuzzy number. The numerical and graphical presentation demonstrates the symmetry between the lower and upper cut representations of the fuzzy solutions and may be helpful in improving understanding of automatic control models, measure theory, physics, biology, computer science, quantum optics, economics, artificial intelligence, and other domains, as well as non-financial analysis.

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields.