Search

Awareness of virus spreading is an important issue for building various defence strategies and protecting personal computers, smart devices, network devices, etc. In this research work, we develop epidemiological models to address this problem and introduce certain modified epidemiological fractional SAIR model, where we consider the fractional derivative in Caputo sense. We utilize the residual power series method to construct approximate solutions for the governing system. To show the efficiency and suitability of the proposed technique, we introduce a comparative between the obtained solutions and those solutions that are constructed using the fourth-order Runge-Kutta method. We derive some numerical results by considering specific values for the parameters in the governing model, and then we depict some of these results into two and three dimensions. [ABSTRACT FROM AUTHOR]

The Burger equation has been widely used to study nonlinear acoustic plane waves in gas-filled tubes, waves in shallow water, and shock waves in gas. Recently, a more comprehensive version of the equation, known as the fractional-order Burger equation, has emerged. However, finding a closed-form for approximate analytic solution using analytical methods for this type of equation is challenging. This paper focuses on researching the multidimensional fractional-order time and space Burger equation based on the Caputo-Katugampola derivative. An approximate analytic solution is obtained using the generalized Laplace homotopy perturbation method. The coefficients of the approximate analytic solution have a recurrence relation similar to the Catalan number in number theory, and the closed form of the approximate analytic solution can be obtained using number theory knowledge. It is worth noting that the Caputo-Katugampola derivative can be reduced to the Caputo derivative, and hence, the closed form for the approximate analytic solution of the multidimensional fractional-order time and space Burger equation based on the Caputo derivative is also obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the system of switched coupled implicit fractional Langevin equations of nonlinear form with anti‐periodic boundary conditions. In the first step, problem's equivalence and the corresponding integral equation by applying fractional calculus tools are established, and a fixed point problem is defined. For mixed monotone mappings, we have used coupled fixed point theorems to achieve the existence and uniqueness of solutions of these equations. In the next step, by using Banach's fixed point theorem, Ulam–Hyers, Ulam–Hyers–Rassias, generalized Ulam–Hyers and generalized Ulam–Hyers–Rassias stabilities of our considered model are discussed. An example is given at the end for the verification of our results. [ABSTRACT FROM AUTHOR]

In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order α ∈ ( 0 , 1 ) . The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near t = 0. Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme. [ABSTRACT FROM AUTHOR]

The quasi‐projective synchronization problems for Caputo inertial neural networks (INNs) with delays are analyzed in this article. The considered model is transformed into two subsystems by selecting a suitable variable substitution. To realize synchronization, a novel fractional differential inequality is established, and the new lemma is derived via Laplace transform. Applying the given lemmas and inequality techniques, the synchronization conditions are inferred through different nonlinear feedback controllers. From the derivation of the theorems, it can be seen that the designed controllers are more extensive and efficacious. Furthermore, the error bound is validly estimated, and the validity of obtained theorems is illustrated by the numerical simulations. [ABSTRACT FROM AUTHOR]

This paper aims to retrieve the initial value for a non-local fractional Sobolev–Galpern problem. The given data are subject to noise by the discrete random model. We show that the solution to the problem is ill-posed in the sense of Hadamard. In this paper, we applied the Fourier truncation method to construct the regularized solution. We estimate the convergence between the solution and the regularized solution. In addition, the numerical example is also proposed to assess the efficiency of the theory. [ABSTRACT FROM AUTHOR]

In this typescript, we study system of nonlinear implicit coupled differential equations of arbitrary (non–integer) order having nonlocal boundary conditions on closed interval [0, 1] with Caputo fractional derivative. We establish sufficient conditions for the existence, at least one and a unique solution of the proposed coupled system with the help of Krasnoselskii's fixed point theorem and Banach contraction principle. Moreover, we scrutinize the Hyers–Ulam stability for the considered problem. We present examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

We consider a general class of linear integro-differential equations with Caputo fractional derivatives and weakly singular kernels. First, the underlying initial value problem is reformulated as an integral equation and the possible singular behavior of its exact solution is determined. After that, using a suitable smoothing transformation and spline collocation techniques, the numerical solution of the problem is discussed. Optimal convergence estimates are derived and a superconvergence result of the proposed method is established. The obtained theoretical results are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when α → 1 − . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when α → 1 − . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a spectral collocation method for the numerical solutions of fractional multipantograph delay differential equations. The fractional derivatives are described in the Caputo sense. We present that some suitable variable transformations can convert the equations to a Volterra integral equation defined on the standard interval [−1, 1]. Then the Jacobi–Gauss points are used as collocation nodes, and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence analysis of the proposed method is investigated in the infinity norm and weighted L2 norm. To perform the numerical simulations, some test examples are investigated, and numerical results are presented. Further, we provide the comparative study of the proposed method with some existing numerical methods. [ABSTRACT FROM AUTHOR]

The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton's law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically. [ABSTRACT FROM AUTHOR]

This work presents a highly accurate method for the numerical solution of the advection–diffusion equation of fractional order. In our proposed method, we apply the Laplace transform to handle the time-fractional derivative and utilize the Chebyshev spectral collocation method for spatial discretization. The primary motivation for using the Laplace transform is its ability to avoid the classical time-stepping scheme and overcome the adverse effects of time steps on numerical accuracy and stability. Our method comprises three primary steps: (i) reducing the time-dependent equation to a time-independent equation via the Laplace transform, (ii) employing the Chebyshev spectral collocation method to approximate the solution of the transformed equation, and (iii) numerically inverting the Laplace transform. We discuss the convergence and stability of the method and assess its accuracy and efficiency by solving various problems in two dimensions. [ABSTRACT FROM AUTHOR]

Rubella is a viral disease that can lead to severe health complications, especially in pregnant women and their unborn babies. Understanding the dynamics of the Rubella disease model is crucial for developing effective strategies to control its spread. This paper introduces a major innovation by employing a novel piecewise approach that incorporates two different kernels. This innovative approach significantly enhances the accuracy of modeling Rubella disease dynamics. In the first interval, the Caputo operator is employed to address initial conditions, while the Atangana–Baleanu derivative is utilized in the second interval to account for anomalous diffusion processes. A thorough theoretical analysis of the piecewise derivative for the problem is provided, discussing mathematical properties, stability, and convergence. To solve the proposed problem effectively, the piecewise numerical Newton polynomial technique is employed and the numerical scheme for both kernels is established. Through extensive numerical simulations with various fractional orders, the paper demonstrates the approach's effectiveness and flexibility in modeling the spread of the Rubella virus. Furthermore, to validate the findings, the simulated results are compared with real data obtained from Rubella outbreaks in Uganda and Tanzania, confirming the practical relevance and accuracy of this innovative model. [ABSTRACT FROM AUTHOR]

Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to ς -nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are illustrated by an example of the pantograph-type model from electrodynamics. [ABSTRACT FROM AUTHOR]

In this research, we study the Caputo fractional and constant proportional derivative numerical approximation of electrical symmetric circuits. It has been assumed that the derivative is in the order 0 ≤ σ ≤ 1. For the fractional electrical symmetric circuits, the RC, LC, and RLC solutions are obtained by using the Sumudu transformation. We also compare the numerical simulation of each equation to its classical equivalent. We use a highly efficient integral transform to examine the impact of the power-law kernel. In our upcoming works, we will apply this to electrical circuits that are more intricate. [ABSTRACT FROM AUTHOR]

In this work, we study a dynamic model of social media addiction. Addiction to social media is a behavioral addiction that is very complex in nature. To study such kinds of complex behavior, the fractional-order operators are very useful due to their memory nature. Considering all these facts, we consider the fractional and fractal-fractional operators with a singular kernel to investigate the social media addiction model. This paper's main aim is to demonstrate the importance of non-classical-order derivatives in the investigation of the social media addiction model. First, we propose the model using Caputo fractional derivative and present some basic mathematical computations. The existence condition for a solution of the model system is presented via fixed-point theory. Furthermore, the behavior of the social media addiction model is examined using the fractal-fractional concept with the Caputo operator, which, due to its memory effect, is very effective in modeling. The existence and uniqueness of the solution are investigated using fixed point theory. Ulam–Hyers stability is demonstrated using nonlinear functional analysis. We demonstrate the simulated numerical results graphically for the proposed models via numerical methods based on Lagrange polynomial interpolation. The results are plotted for choices of arbitrary-order parameter values. Based on our findings, we can say the fractal-fractional operators provide more realistic information about the complexity of the dynamics of the social media addiction model. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new technique known as Tempered Fractional -Transform (TF T). This scheme can be applied to study numerous linear and nonlinear dynamical systems in tempered fractional (TF) calculus in both Riemann–Liouville and Caputo and sense. Some new theories, properties, and applications of the above-mentioned -transform are calculated in detail. The proofs of some important theorems on TF Riemann–Liouville and Caputo derivatives are proved based on TF T. For validation, accuracy and efficiency, the general TF equations as well as TF linear and nonlinear Klein–Gordon equations are studied by using the proposed transform with the numerical illustrations. It is observed that the proposed technique is fast convergent and the results are the first precise confirmations of TF T in tempered calculus for nonlinear systems. This work can be studied as a substitute to present mathematical methods and will have extensive applications in physical sciences. [ABSTRACT FROM AUTHOR]

The generalized time‐fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high‐order numerical scheme for the generalized time‐fractional Fisher's equation. The time‐fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of Ok2−α+h4$$ O\left({k}^{2-\alpha }+{h}^4\right) $$, where h$$ h $$ and k$$ k $$ are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem. [ABSTRACT FROM AUTHOR]

The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved. [ABSTRACT FROM AUTHOR]

In this paper. the application of the natural transform decomposition method (NTDM) and variational iteration transform method (VITM) is applied to obtain the solution for the one-dimensional groundwater recharge through porous media in a fractional order with a singular kernel derivative. The mathematical formulation of the one-dimensional groundwater recharge problem has been derived with assuming that the average coefficient of diffusivity remains constant across the entire moisture content range, while the permeability is allowed to vary. The convergence and uniqueness of the solutions by the proposed methods have been analysed and discussed. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations. The obtained results demonstrate that the proposed methods are straightforward, highly accurate, and efficient. [ABSTRACT FROM AUTHOR]

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville's fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method's effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. [ABSTRACT FROM AUTHOR]

This study investigates the effects of magnetic induction, ion slip and Hall current on the flow of linear viscous fluids in a rectangular buoyant channel. In a hydro-magnetic flow scenario with permeable and conducting walls, one wall has a temperature variation that changes over time, while the other wall keeps a constant temperature; the research focuses on this situation. Asymmetric wall heating and suction/injection effects are also examined in the study. Using the Laplace transform, analytical solutions in the Laplace domain for temperature, velocity and induced magnetic field have been determined. The Stehfest approach has been used to find numerical solutions in the real domain by reversing Laplace transforms. The generalized thermal process makes use of an original fractional constitutive equation, in which the thermal flux is influenced by the history of temperature gradients, which has an impact on both the thermal process and the fluid's hydro-magnetic behavior. The influence of thermal memory on heat transfer, fluid movement and magnetic induction was highlighted by comparing the solutions of the fractional model with the classic one based on Fourier's law. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we first established a high-accuracy difference scheme for the time-fractional Schrödinger equation (TFSE), where the factional term is described in the Caputo derivative. We used the L1-2-3 formula to approximate the Caputo derivative, and the fourth-order compact finite difference scheme is utilized for discretizing the spatial term. The unconditional stability and convergence of the scheme in the maximum norm are proved. Finally, we verified the theoretical result with a numerical test. [ABSTRACT FROM AUTHOR]

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. [ABSTRACT FROM AUTHOR]

In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using q -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function χ ρ = 1 , ρ and ρ 2 to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme. [ABSTRACT FROM AUTHOR]

Background: Tuberculosis (TB), caused by Mycobacterium tuberculosis, is a contagious infectious disease that primarily targets the lungs but can also impact other critical systems such as the bones, joints, and neurological system. Despite significant efforts to combat TB, it remains a major global health concern. To address this challenge, this study aims to explore and evaluate various tuberculosis control approaches using a mathematical modeling framework. Results: The study utilized a novel SEITR mathematical model to investigate the impact of treatment on physical limitations in tuberculosis. The model underwent qualitative analysis to validate key aspects, including positivity, existence, uniqueness, and boundedness. Disease-free and endemic equilibria were identified, and both local and global stability of the model was thoroughly examined using the derived reproduction number. To estimate the impact of each parameter on each compartment, sensitivity analysis was conducted, and numerical simulations were performed using Maple 18 software with the homotopy perturbation method. The obtained results are promising and highlight the potential of the proposed interventions to significantly reduce tuberculosis virus prevalence. The findings emphasize the significance of fractional-order analysis in understanding the effectiveness of treatment strategies for mitigating tuberculosis prevalence. The study suggests that the time fractional dynamics of TB treatment correspond to the treatment's efficacy, as the conceptual results showed that non-local interactions between the disease and the treatment may lead to more accurate ways of eradicating tuberculosis in real-world scenarios. These insights contribute to a better understanding of effective treatment strategies and their potential impact on tuberculosis control and public health. Conclusions: In conclusion, scientists, researchers, and healthcare personnel are urged to take action and utilize the discoveries from this research to facilitate the eradication of the hazardous tuberculosis bacteria. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson's equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson's equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique's precision. [ABSTRACT FROM AUTHOR]

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. [ABSTRACT FROM AUTHOR]

Richards' equation is a classical differential equation describing water transport in unsaturated porous media, in which the moisture content and the soil matrix depend on the spatial derivative of hydraulic conductivity and hydraulic potential. This paper proposes a nonlocal model and the peridynamic formulation replace the temporal and spatial derivative terms. Peridynamic formulation utilizes a spatial integration to describe the path-dependency, so the fast diffusion process of water transport in unsaturated porous media can be captured, while the Caputo derivative accurately describes the sub-diffusion phenomenon caused by the fractal nature of heterogeneous media. A one-dimensional water transport problem with a constant permeability coefficient is first addressed. Convergence studies on the nonlocal parameters are carried out. The excellent agreement between the numerical and analytical solutions validates the proposed model for its accuracy and parameter stability. Subsequently, the wetting process in two porous building materials is simulated. The comparison of the numerical results with experimental observations further demonstrates the capability of the proposed model in describing water transport phenomena in unsaturated porous media. [ABSTRACT FROM AUTHOR]

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time t=0$$ t=0 $$. The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This work investigates a fractional Susceptible Infected Recovered (SIR) model to study childhood disease. We analyzed the proposed model by applying Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivatives. Here, we use the fractional Adams–Bashforth method to solve the childhood disease model with nonlocal operator. The proposed numerical technique is developed by combining the fundamental theorem of integral calculus with Lagrange's interpolation. This numerical approach is more efficient than the other existing numerical techniques and is easily computed using Matlab as a programming language. The existence and uniqueness of this fractional model are studied with the fixed‐point theorem. These fractional derivatives show different asymptotic behaviors for the distinct values of the fractional order. The numerical results are presented graphically as well as in tabulated form for some value of fractional order. [ABSTRACT FROM AUTHOR]

In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional‐order (FO) heat‐like equations. To show this, we first establish the Elzaki transform in the context of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat‐like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer‐order solutions. EHPTM is considered to be an appropriate and convenient approach for solving FO time‐dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis. [ABSTRACT FROM AUTHOR]

A numerical scheme is developed to solve the time-fractional linear Kuramoto-Sivahinsky equation in this work. The time-fractional derivative (of order γ) is taken in the Caputo sense. The scheme comprises the backward Euler formula in the temporal direction and the quintic B-spline collocation approach in the spatial direction. Through rigorous analysis, the proposed method is shown to be unconditionally stable and convergent of order 2 − γ and two in the temporal and spatial directions, respectively. Two test problems are solved numerically to demonstrate the convergence and accuracy of the method. [ABSTRACT FROM AUTHOR]

Extensive research was conducted on the transmission dynamics of tuberculosis epidemics during its reemergence from the 1980s to the early 1990s, but this global problem of investigating tuberculosis spread dynamics remains of paramount importance. Our study utilized a fractional-order delay differential model to study tuberculosis transmission, where the time delay in the model was attributed to the disease's latent period. What is more, this model accounts for endogenous reactivation, exogenous reinfection, and treatment of tuberculosis. The model qualitative properties and the basic reproduction number were analyzed. The primary goal of the study was to recover the important dynamic parameters of tuberculosis. Our understanding of these complex processes leverages the efficacy of efforts for controlling the disease, forecasting future dynamics, and applying further appropriate strategies to prevent its spread.The calibration itself was carried out via minimization of a quadratic cost functional. Computational simulations demonstrated that the algorithm is capable of working with noisy real data. [ABSTRACT FROM AUTHOR]

We study a nonlinear fractional boundary value problem (BVP) subject to non-local multipoint boundary conditions. By introducing an appropriate parametrization technique we reduce the original problem to an equivalent one with already two-point restrictions. Using a notion of Chebyshev nodes and Lagrange polynomials we construct a successive iteration scheme, that converges to the exact solution of the non-local problem for particular values of the unknown parameters, which are calculated numerically. [ABSTRACT FROM AUTHOR]

In this article, we study fractional stochastic Burgers equations perturbed by fractional noise. Existence and uniqueness of a mild solution is given by a fixed point argument. Then, we explore Hölder regularity of the mild solution in C'([0, T*]; LP (Ω; Ḣγ) for some stopping time T*. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional order Coronavirus (COVID-19) model incorporating non-pharmaceutical interventions and vaccine hesitancy. The proposed model was calibrated with data from literature and validated with reported daily cases of COVID-19 from Wuhan, China. We derived the reproduction number and demonstrated that it is an important threshold parameter for disease persistence and extinction. We examined the relationship between the reproduction number and model parameters. Our findings underscore the importance of awareness and vaccine uptake on mitigating the spread of COVID-19. [ABSTRACT FROM AUTHOR]

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented. [ABSTRACT FROM AUTHOR]

The focus of this work is to construct a pseudo-operational Jacobi collocation scheme for numerically solving the Caputo variable-order time-fractional diffusion-type equations with applications in applied sciences. Modeling scientific phenomena in the context of fluid flow problems, curing reactions of thermosetting systems, solid oxide fuel cells, and solvent diffusion into heavy oils led to the appearance of these equations. For this reason, the numerical solution of these equations has attracted a lot of attention. More precisely, using pseudo-operational matrices and appropriate approximations based on bivariate Jacobi polynomials, the approximate solutions of the variable-order time-fractional diffusion-type equations in the Caputo sense with high accuracy are formally retrieved. Based on orthogonal bivariate Jacobi polynomials and their operational matrices, a sparse algebraic system is generated which makes implementing the proposed approach easy. An error bound is computed for the residual function by proving some theorems. To illustrate the accuracy and efficiency of the scheme, several illustrative examples are considered. The results demonstrate the efficiency of the present method compared to those achieved by the Legendre and Lucas multi-wavelet methods and the Crank-Nicolson compact method. [ABSTRACT FROM AUTHOR]

Numerical methods for fractional differential equations have specific properties with respect to the ones for ordinary differential equations. The paper discusses Euler methods for Caputo differential equation initial value problem. The common properties of the methods are stated and demonstrated by several numerical experiments. Python codes are available to researchers for numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the harmonic balance method and its variants are used to analyze the response of Mathieu–Duffing oscillator with Caputo derivative. First, the exact and approximate expressions of the Caputo derivatives of trigonometric function and composite function are derived. Next, using the approximate expression of the Caputo derivative of the composite function, the resonance of Duffing oscillator with Caputo derivative is analyzed by the harmonic balance method. Finally, Mathieu–Duffing oscillator with Caputo derivative is approximated by three kinds of methods, i.e., the harmonic balance method, the residue harmonic balance method and the improved harmonic balance method. The corresponding numerical simulations are given to illustrate the performance of these methods as well. The results show that the residue harmonic balance method is more precise than the harmonic balance method and the improved harmonic balance method in analyzing the dynamic response of Mathieu–Duffing oscillator with Caputo derivative. [ABSTRACT FROM AUTHOR]