The paper is devoted to present considerations concerning the solvability on a bounded interval of a system of nonlinear integral equations, which create the generalization of a fractional-order neuron model under the electromagnetic field expressed in terms of Caputo fractional derivatives. Our approach depends on replacing the neuron models in question by a system of nonlinear integral equations of the Volterra–Stieltjes type. That enables us to apply suitable tools of nonlinear analysis and to obtain a transparent and easy-to-apply result. The established criteria significantly extend and refine the solvability theory for this class of generalized fractional neuron models.

In this paper, we study the transient thermal behavior of a constant-area longitudinal fin composed of a functionally graded material. The longitudinal variation in thermal conductivity along the fin is modeled using four types: linear, quadratic, power, and exponential. The proposed problem is studied using a Caputo-type fractional-order model. In materials with spatially varying properties, the thermal response often exhibits nonlocal behavior that motivates the use of fractional derivatives. We derive the numerical solution to the model using the L1 approximation of the Caputo derivative, while the spatial operator is treated using the Chebyshev collocation method based on the fourth kind Chebyshev polynomials. The numerical results are plotted for fins with linear, quadratic, power-law, and exponential grading at various fractional orders. Also, the functionally graded longitudinal fin efficiency and the influence of inhomogeneity parameters on the fin's temperature distribution are shown through graphical simulations. The numerical results reveal that the transient thermal response is strongly influenced by both the fractional order and the thermal-conductivity grading profile. The fin efficiency increases with dimensionless time and asymptotically approaches a steady-state value. Increasing the convective parameter reduces the fin efficiency. The effects of linear, quadratic, power-law, and exponential thermal conductivity gradings on the thermal performance of the functionally graded fin are examined and compared.

this paper, we propose an efficient hybrid numerical approach to obtain approximate solutions to the two-dimensional time-fractional cable equation of distributed-order involving Riesz space-fractional operators. This combined numerical approach includes two numerical approaches in time and space directions. The weighted and shifted Gr & uuml;nwald difference numerical method is used to approximate the fractional problem in the time direction, and the fractional compact numerical method is applied in the direction of the space variable. The stability and convergence analyses are studied for the proposed numerical approach. To show the effectiveness of the presented numerical method, some numerical examples are given, and the numerical results for these examples are plotted. Also, this numerical approach is compared with other methods.

. This work investigates a fractional boundary value problem in the sense of Riemann-Liouville derivative and integral. We derive some novel results for the necessary and sufficient conditions for the existence and uniqueness of the positive solution. In this regard, some fixed-point theorems on cones are used. Also, a convergent successive sequence to find the solution to the problem is introduced. We derive the numerical scheme for the proposed problems. The correctness of the proposed results is verified with some illustrative examples.

In this paper, we propose a novel fractional-order neutral-type delay neural network (FNDNN) considering two delay variables in terms of the Caputo fractional derivatives. We prove the existence of a unique solution within the given time domain. We analyse the bifurcation with respect to both delay parameters and the initial state’s stability of the FNDNN. We derive the numerical solution of the proposed FNDNN using a recently proposed algorithm. We provide the necessary graphical simulations to justify the correctness of our theoretical proofs. We investigate how both delay parameters affect stability and induce bifurcations in the FNDNN. Also, we check the influence of fractional orders on the dynamical behaviour of the FNDNN. We find that, in comparison with the integer-order case, the proposed FNDNN has faster convergence performance.

We propose a fractional-order improved FitzHugh-Nagumo(FHN)neuron model in terms of a generalized Caputo fractional derivative.Following the existence of a unique solution for the proposed model,we derive the numerical solu-tion using a recently proposed L1 predictor-corrector method.The given method is based on the L1-type discretization algorithm and the spline interpolation scheme.We perform the error and stability analyses for the given method.We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns,chaotic patterns,and quasi-periodic patterns.The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better un-derstanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics,which are inherent to many biological systems.

In this paper, we propose a fractional-order bicyclic crossed neural network (NN) with multiple time delays consisting of two sharing neurons between rings. The given fractional-order NN is defined in terms of the Caputo fractional derivatives. We prove boundedness and the existence of a unique solution for the proposed NN. We do the stability and the onset of Hopf bifurcation analyses by converting the proposed multiple-delayed NN into a single-delay NN. Later, we numerically solve the proposed NN with the help of the L1 predictor-corrector algorithm and justify the theoretical results with graphical simulations. We explore that the time delay and the order of the derivative both influence the stability and bifurcation of the fractional-order NN. The proposed fractional-order NN is a unique multi-delayed bicyclic crossover NN that has two sharing neurons between rings. Such ring structure appropriately mimics the information transmission process within intricate NNs.

In this research article, we investigate a COVID-19 model of fractional-order defined in terms of functional shape with square root susceptible-infected interaction. Firstly, we simulate the positivity and boundedness of the solution and then calculate the nature of equilibria. For exploring the dynamics of investigated fractional-order model, we use the Hurwitz criterion and then a graph theoretical method for the derivation of a Lyapunov function. For the given model, a unique solution exists under the results of the fixed-point theory. We use the Harr wavelets method to derive the numerical solution of the investigated model. As a result, some graphical illustrations are used to ensure the theoretical results, which indicates the good agreement between numerical illustrations and theoretical findings. The motivation of this article is to show how the given square root susceptible-infected interaction model effectively explores the outbreaks of COVID-19 at various fractional-order values. The inclusion of the Caputo fractional derivative incorporates the memory effects in the proposed model.

This article considers the Cauchy reaction-diffusion equations and derives the numerical solutions using the fractional natural decomposition method (FNDM). The projected solution approach works without conversion or perturbation. The examples confirm the method’s accuracy and reliability, allowing for fractional order studies in real-world problems. Plots and tables validate the accuracy of the proposed scheme. This research reveals the influences of temporal history in the fractional Cauchy reaction-diffusion equations, which is the novelty of this work.

Nowadays, different real-life phenomena are being modelled using fractional-order operators. In this paper, a Caputo-type fractional-order mathematical model is proposed for defining the dynamics of ending student strikes at a university by taking per-year constant admissions. We analyse the possible strategies to control the strikes on the university campus. We prove the existence of a unique global solution for the given fractional-order model using a new characteristic of the well-known Mittag–Leffler function and fixed-point theory. We derive the numerical solution of the proposed model via the Haar wavelet method, which is one of the efficient numerical algorithms. A number of plots are performed, taking different cases, for a good understanding of the proposed problem. The aim of this study is to understand how fractional derivatives are useful to capture memory effects in such problems. All results are given with supporting arguments.

This paper proposes a novel L1-based predictor–corrector method for the fractional differential equations involving generalized-Caputo type derivative. A decomposition scheme is used to obtain the three-point predictor–corrector formula. The error and stability of the proposed method are given in detail. A computer virus and a five-dimensional Hopfield neural network models are solved using the proposed approach.

In this article, we define a nonlinear model for exploring the strategy of combating the atmospheric level of carbon dioxide (CO _2 ) considering development activities in terms of variable-order Liouville–Caputo fractional derivatives. There are two types of variable-order Liouville–Caputo fractional derivatives used to derive the proposed model. We prove the existence and uniqueness of the solution for the given model using fixed-point theory. The numerical solution is derived by using a recently proposed predictor-corrector scheme. We perform several graphical simulations to describe the outcomes of the given model. The outputs performed at various fractional-order values provide novel findings to understand how to combat atmospheric CO _2 . A novel variable-order fractional model that captures memory effects in the proposed dynamics, along with a recent numerical methodology, are the key features of this study. The simulation analysis shows that the leafy tree plantation on the excess land will be efficient against atmospheric CO _2 .

This paper deals with the oscillatory behavior of the Psi-Hilfer generalized proportional fractional initial value problem. Using the Volterra integral equation and Young's inequality, we establish sufficient conditions for each solution of the problem to oscillate. For the appropriate choice of the kernel Psi$$ \Psi $$, our obtained results generalize and recover some existing results in the literature. Additionally, we present some examples to emphasize the importance of our results.

This paper focuses on the distributed-order time-fractional diffusion-wave equations with the Riesz space fractional derivatives. A combined method based on the midpoint quadrature rule, linear B-spline interpolation, and the Galerkin finite element method is proposed to obtain the approximate solution. Two steps are used to calculate the approximate solution to this type of equation. The first step approximates the temporal direction by combining a midpoint quadrature rule and linear B-spline interpolation. In the second step, a Galerkin finite element method in the space direction is applied to compute a full-discrete method. Furthermore, the error estimate has been displayed to demonstrate unconditional stability and convergence. Finally, two numerical examples are reported to show the simplicity and efficiency of the proposed method.

In this paper, we develop a novel numerical scheme, namely 'NPCM-PCDE,' to integrate fractional ordinary differential equations with proportional Caputo derivatives of the type (pc)D(alpha)u(t) = f(1)(t, u(t)), t >= 0, 0 < alpha < 1 involving a non-linear operator f(1). A new method is developed using a natural discretization of the proportional Caputo derivative and the decomposition method to decompose the non-linear operator f(1). The error and stability analyses for the proposed method are provided. Some illustrated examples are given to compare the solution curves graphically with the exact solution and to prove the utility and efficiency of the method. The proposed NPCM-PCDE is found to be efficient, easy to implement, convergent, and stable.

In this paper, we solved a model of a well-known infectious disease called dengue fever via fractional natural decomposition and modified Predictor–Corrector (PC) methods. A study of the dengue epidemic in the Cape Verde Islands off the coast of West Africa in 2009 has been resumed here for a better understanding of the results. The results are obtained using Liouville–Caputo and new generalized Caputo-type fractional derivatives. The numerical simulations are presented for various orders of given derivatives. Existence and uniqueness analysis of the given problem are also performed in the new generalized Caputo sense. The explored results are verified using figures. The main target of this paper is to explore the different dynamics of the given dengue fever model via two types of fractional numerical algorithms.

In this article, we propose an effective scheme based on a combination of the Tau method and fractional-order Gegenbauer wavelets for solving the multi-term time-fractional differential equations of distributed order. First, we define fractional order Gegenbauer wavelets and then obtain operational matrices of these orthogonal functions. Applying the Legendre–Gauss quadrature for the integral term, we use function approximations obtained by the presented wavelets and the Tau method for the solution of the distributed-order multi-term time-fractional telegraph equation. The proposed method reduces the numerical solution of multi order time-fractional equations to a system of algebraic equations. Then, the convergence analysis and error bounds of the proposed scheme are studied. Three illustrative examples are solved to justify the effectiveness of the proposed method compared with some previously published results.

This study developed and examined a new operational matrix approach utilizing the Vieta-Fibonacci polynomial for the numerical solution of generalized Caputo-type differential equations with fractal-fractional terms. Based on the proposed approach, the fractal-fractional differential equations with generalized Caputo-type derivatives were reduced into a system of algebraic equations, which was further solved to obtain the unknown solution. The convergence and error bounds are theoretically calculated. The results are quantitatively confirmed in various cases. To demonstrate the correctness and computational efficacy of this proposed technique, it is compared to other well-known methods.