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This paper presents the numerical method utilized for approximate solution of fast and slow diffusion models having fractional order derivative in time. The proposed method approximates the time derivative of the model using L1-formula and applies the integrated Lucas and Fibonacci polynomial functions for approximating the unknown functions and their derivatives. Then, with the help of spatial discretization, the approximated model is converted into system of linear equations. The performance of the proposed algorithm was studied by calculating the solutions of one- and two-dimensional fast and slow diffusion models and these solutions were compared with the existing methods. The comparison illustrates that the proposed method is robust and highly efficient.

In this paper, we employed a novel technique with the help of the Daftardar-Gejji and Jafari method (DGJM) to solve the generalized time-fractional Cattaneo model (GTFCM) via Caputo derivative with order $1<\alpha <2$. The existence and uniqueness of solutions are examined using Banach’s fixed-point theorem, and stability is evaluated under Ulam–Hyers conditions. We compared our results with the exact solution and existing techniques, demonstrating our approach’s efficiency.

In this paper, we investigate a novel model problem involving a viscoelastic, frictionless contact model characterized by history-dependent operators. The constitutive relation is formulated based on a time-fractional Kelvin–Voigt model. We represent the contact by incorporating normal compliance within the framework of a time-fractional derivative. In addressing the contact problem, we begin by formulating its weakness and subsequently verify the existence of a solution within this framework. Finally, we study the solution’s behavior concerning the integral term and its relation to the solution, while also presenting convergence results.

In this paper, we study the replicator equation for the hawk–dove (HD) and Prisoner's Dilemma (PD) games and generalized it to the fractional differential equation.

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is $0<\beta \le 1$ and for the time domain is $0<\gamma \le 2$. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

We start by presenting a brief summary of fractional quantum mechanics, as means to convey a motivation towards fractional quantum cosmology. Subsequently, such application is made concrete with the assistance of a case study. Specifically, we investigate and then discuss a model of stiff matter in a spatially flat homogeneous and isotropic universe. A new quantum cosmological solution, where fractional calculus implications are explicit, is presented and then contrasted with the corresponding standard quantum cosmology setting.

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G${}^{\prime }$)-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L₂ and L${}_{\infty }$ error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables.

The main goal of this study is to find solutions for the generalized coupled Ramani equation with the fractional order using the fractional natural decomposition method (FNDM). Four distinct cases are chosen to illustrate and validate the effectiveness of the considered method. The simulations in terms of numeric have been illustrated to confirm the reliability and proficiency of the projected scheme. Moreover, the behavior of the obtained results is captured for distinct fractional order. The comparison study is illustrated to verify the accuracy of the projected procedure. The achieved results exemplify that the projected solution procedure offers a simple algorithm and is also very efficient to analyze the nature of the coupled differential equations with arbitrary order situated in associated areas of Science and Engineering.

In this paper, we will first present the TFMIADM with its adequate Dirichlet constraints. Right after that, we will review the formation of that model under the terms and assumptions of the RKHSM computational approach. The solutions and modeling of the utilized model will be discussed based on Caputo’s connotation of the partial time derivative. We will present the scores required to construct the appropriate spaces for the method and we will present several theories such as solutions representations, convergence restriction, and order of error. With the use of the Fourier functions expansion rule, the numeric–analytic solutions are expressed by collection sets of orthonormal functions system in $𝔄({\mathscr{H}})$ and $𝔅({\mathscr{H}})$ spaces. Right after that, we will solve this model in both time and space domains using the algorithms of the method used. Indeed, several drawings and tables that expound on the effectiveness and strength of the approach and its adaptation to the issue reviewed are utilized. In the end, some points of view and highlights are presented side by side with the most important modern references used.

This work proposes a semi-analytical new hybrid approach, so-called differential $𝕁$-transform method (D$𝕁$TM), to evaluate the behavior of n-space dimensional fractional-nonlinear hyperbolic-like wave equations, where time-fractional derivative is considered in Caputo format. The D$𝕁$TM is the hybrid method in which projected differential transform is implemented after imposing the recently introduced integral transform, i.e., so-called $𝕁$ transform [W. Zhao and S. Maitama, J. Appl. Anal. Comput. 10, 1223 (2020)]. The efficiency and applicability of the proposed D$𝕁$TM had been tested by considering three different test examples of the Caputo time-fractional nonlinear hyperbolic-like wave equations in terms of absolute error norms, and the different order D$𝕁$TM solutions are compared with exact solution behaviors and the existing results, for the large time level $\tau \in [0,10]$. In addition, the convergence analysis of D$𝕁$TM is studied theoretically and verified it numerically as well as graphically, which confirms that the numerical experiments via D$𝕁$TM for distinct fractional orders support the theoretical findings excellently, and the presented D$𝕁$TM results converge to their exact solution behavior, very fast. The evaluated series approximations are expressed in the compact form of Mittag-Leffler functions.

Hyper-chaotic systems have useful applications in engineering applications due to their complex dynamics and high sensitivity. This research is supposed to introduce and analyze a new noninteger hyper-chaotic system. To design its fractional model, we consider the Caputo fractional operator. To obtain the approximate solutions of the extracted system under the considered fractional-order derivative, we employ an accurate nonstandard finite difference (NSFD) algorithm. Moreover, the existence and uniqueness of the solutions are provided using the theory of fixed-point. Also, to see the performance of the utilized numerical scheme, we choose different values of fractional orders along with various amounts of the initial conditions (ICs). Graphs of solutions for each case are provided.

Within the context of fractional calculus, we investigate novel mathematical possibilities. In this context, using the fractional dispersion relations for the fractional wave equation, we explore a class of the generalized fractional wave equation numerically. Some important classes of differential equations in the theory of wave studies are Drinfeld–Sokolov–Wilson and Shallow Water equations. In this effort, the natural transform decomposition technique has been implemented to investigate the explicit result of fractional-order coupled schemes of Drinfeld–Sokolov–Wilson and Shallow Water coupled systems. The proposed method is obtained by coupling the Natural transform with the Adomian decomposition process. The current technique significantly works to find the approximate solution without any discretization or constraining parameter assumptions. The obtained numerical and graphical outcomes by the devised technique are compared with the available exact result to verify the convergence of the method. For mathematical calculations, the Mathematica software package is used.

The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as two couple space-time FDDEs. Furthermore, some properties of the analytical solutions are illustrated by graphs.

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

This paper examines the variable-order fractional Lorenz system with distinct types of delays. The application of the Arzela–Ascoli theorem is made to prove the existence of solutions for the given problem, while the Banach fixed point theorem is employed to derive the uniqueness results. The utilization of the Adams–Bashforth–Moulton technique facilitates the exploration and resolution of the approximation solution, complemented by a thorough error analysis of particular approaches. The computational simulations showing chaotic behaviors in various delayed systems with different variable orders demonstrate the effectiveness of the approach.

A fractional-order proportional-derivative controller is designed to address bifurcation issues in a dual-time-delay fractional-order predator–prey model, achieving anti-control of Hopf bifurcation at equilibrium of the system. By selecting different delays as bifurcation parameters, the stability and Hopf bifurcation conditions of the controlled system are derived. The results show that the fractional-order, delays and control parameters play an important role in determining the stability and Hopf bifurcation of the system. By selecting reasonable system parameters (fractional-order, delays, and control parameters), effective system control strategies can be devised. Finally, the key findings of this study are verified through numerical examples. The research in this paper has potential application value for the management of natural ecosystems.

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, lower semicontinuous energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology.

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using $q$-homotopy analysis transform method ($q$-HATM). The considered scheme is a unification of $q$-homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered.