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

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

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

In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

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

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

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

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

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

This paper 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]

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]

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]

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]

In this manuscript, the mathematical model of COVID‐19 is considered with eight different classes under the fractional‐order derivative in Caputo sense. A couple of results regarding the existence and uniqueness of the solution for the proposed model is presented. Furthermore, the fractional‐order Taylor's method is used for the approximation of the solution of the concerned problem. Finally, we simulate the results for 50 days with the help of some available data for fractional differential order to display the excellency of the proposed model. [ABSTRACT FROM AUTHOR]

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID‐19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID‐19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID‐19 model with time‐ arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary‐order problem into algebraic equations. The corrector scheme is also used for solving the COVID‐19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary‐order COVID‐19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet‐based results. [ABSTRACT FROM AUTHOR]

In this article, we studied the time‐fractional Burgers–Huxley equation using the natural transform decomposition method. The fractional operator is treated in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu senses. We employed the natural transform with the Adomian decomposition process on time‐fractional Burgers–Huxley equation to obtain the solution. To establish the uniqueness and convergence of the accomplished solution, the Banach's fixed point theorem is used. The obtained findings are visually shown in two‐ and three‐dimensional graphs for various fractional orders. To illustrate the efficacy of the method under discussion, numerical simulations are provided. The proposed solution captures the behavior of the reported findings for various fractional orders. A comparative study was conducted to ascertain the proposed method's correctness. The findings of this study establish that the technique investigated is both efficient and accurate for solving nonlinear fractional differential equations that arise in science and technology. [ABSTRACT FROM AUTHOR]

This paper proposes the shifted Legendre Gauss–Lobatto collocation (SL‐GLC) scheme to solve two‐dimensional space‐fractional coupled reaction–diffusion equations (SFCRDEs). The proposed method is implemented by expressing the function and its spatial fractional derivatives as a finite expansion of shifted Legendre polynomials. Then the expansion coefficients are determined by reducing the SFCRDEs with their initial and boundary conditions to a system of ordinary differential equations for these coefficients. Subsequently, we applied the proposed method to discretize the temporal and spatial variables to convert the two‐dimensional spacetime fractional coupled reaction–diffusion equations (STFCRDEs) to a system of algebraic equations. Some results regarding the error estimation are obtained. Several examples are discussed to validate the capability and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

Since December 2019, the whole world has been facing the big challenge of Covid‐19 or 2019‐nCoV. Some nations have controlled or are controlling the spread of this virus strongly, but some countries are in big trouble because of their poor control strategies. Nowadays, mathematical models are very effective tools to simulate outbreaks of this virus. In this research, we analyze a fractional‐order model of Covid‐19 in terms of the Caputo fractional derivative. First, we generalize an integer‐order model to a fractional sense, and then, we check the stability of equilibrium points. To check the dynamics of Covid‐19, we plot several graphs on the time scale of daily and monthly cases. The main goal of this content is to show the effectiveness of fractional‐order models as compared to integer‐order dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness theorem (EUT) of Pantograph fractional stochastic differential equations (PFSDE) using the Banach fixed point theorem (BFPT). We show the Ulam–Hyers stability (UHS) of PFSDE by the generalized Gronwall inequalities (GGI). We illustrate our results by two examples. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Cauchy problem for a fractional diffusion equation and the time‐fractional derivative is taken in the Caputo type sense. We give a representation of solutions under Fourier series and analyze initial value problems for the semi‐linear fractional diffusion equation with a memory term. We also discuss the stability of the fractional derivative order for the time under some assumptions on the input data. Our key idea is to use Mittag‐Leffler functions, the Banach fixed point theorem, and some Sobolev embeddings. [ABSTRACT FROM AUTHOR]

In this study, we present convergence analysis along with an error estimate for time‐fractional biological population equation in terms of the Caputo derivative using a new technique called the fractional decomposition method (FDM). Further, we present exact solutions to four test problems of nonlinear time‐fractional biological population models to show the accuracy and efficiency of the FDM. This method based on constructing series solutions in a form of rapidly convergent series with easily computable components and without the need of linearization, discretization and perturbations. The results prove that the FDM is very effective and simple for solving fractional partial differential equations in multi‐dimensional spaces, special cases of which we have described in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we utilize a time‐fractional anisotropic diffusion equation for image denoising. Theoretical results are provided thanks to Schauder fixed‐point theorem. A discretization scheme by finite difference is also presented. Numerical experiments show a great performance in deleting the noise while preserving important features. The model robustness is tested, which is clear visually and quantitatively. All the obtained results conclude that our model surpasses the competitive models, such as Perona–Malik and Weickert. [ABSTRACT FROM AUTHOR]

In this article, a powerful modification of the optimized decomposition method is presented for dealing with nonlinear fractional differential equations, where the fractional derivatives are described in the Caputo sense. The new modification, the Legendre‐based approach of the optimized decomposition method, is suggested by employing the Legendre orthogonal polynomials to construct an optimal decomposition of the series solutions. An illustrative comparison between the Legendre‐based approaches of the Adomian decomposition method and the optimized decomposition method is performed to highlight the features and the computational efficiency of the suggested approach. The resulting numerical approximations demonstrate that the approximate solutions of the proposed approach have better accuracy and convergence. [ABSTRACT FROM AUTHOR]

In this article, we focused on solving numerically the coupled nonlinear Schrödinger equations (NLFSEs) with Caputo fractional derivative in time. The discrete schemes are constructed by using the trigonometric B‐spline collocation method and the non‐polynomial B‐spline method for space discretization respectively, while the L1‐formula is applied in time discretization. The Von Neumann approach is applied to examine the stability of the proposed methods. For validating the accuracy and efficiency of the presented schemes, numerical tests are compared with the exact solution. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to employ fractional natural decomposition method (FNDM) to find the solution for a nonlinear system arising in thermoelasticity. The considered coupled system is generalised many physical phenomena associated with the material with elastic characters and its temperature and also which is called a Cauchy problem. We consider the coupled system by incorporating the Caputo fractional operator and investigate three distinct cases for different initial values to illustrate the applicability and efficiency of the FNDM. With respect to fractional order, we capture the behaviour of the achieved solution cited in three different cases and exemplified with the aid of 2D and 3D plots for the particular value of the parameters in the model. Moreover, some interesting behaviours of the projected model are confirms the prominence of the employed fractional operator while analysing the nonlinear coupled equations exemplifying real‐world problems and also shows the capability of the considered algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we have studied the seventh‐order time‐fractional Sawada Kotera Ito equation via natural transform decomposition method (NTDM) with singular and nonsingular kernel derivatives. The fractional derivative considered in Caputo, Caputo‐Fabrizio (CF), and Atangana‐Baleanu‐Caputo (ABC). We employed the natural transform with the Adomian decomposition process on time‐fractional Sawada Kotera Ito equation to obtain the solution. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient numerical method is developed for solving a class of nonlinear fractional differential equations. The main idea is to transform the nonlinear fractional differential equations into a system of integral equations involved the generalized Mittag‐Leffler functions, and then to discretize the integral equations by the technique of exponential integrators and collocation methods with the polynomials of Lagrange basis. The convergence of this method is proven. The linear stability analysis of this method is carried out, and the stability region is derived. Finally, numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we propose a method to solve the time‐fractional hyperbolic heat equation. We first formulate a boundary value problem for the standard hyperbolic heat equation in a finite domain and provide an analytical solution by means of separation of variables and Fourier series. Then, we consider the same boundary value problem for the fractional hyperbolic heat equation. The fractional problem is solved using three different definitions of the fractional derivative: the Caputo fractional derivative and two recently defined alternative versions of this derivative, the Caputo–Fabrizio and the Atangana–Baleanu. A closed form of the solution is provided for each case. Finally, we compare the solutions of the fractional and the standard problem and show numerically that the solution of the standard hyperbolic heat equation can be retrieved from the solution of the fractional equation in the limit γ→2, where γ represents the exponent of the fractional derivative. [ABSTRACT FROM AUTHOR]

The citrus epidemic huanglongbing (HLB), allied with an uncultured bacterial pathogen, is blusterous the industry of citrus worldwide. In this research work, we analysed a fractional huanglongbing model to study the transmission dynamics of the disease. We considered Caputo and new generalized form of Caputo type fractional derivatives to solve the proposed HLB model using two different methods. We exemplified the all necessary graphical observations by the call of real numerical data to show the nature of the given model classes. The analysis regarding to existence of the solution are mentioned with the help of theorems. We observed that the given numerical techniques are strong and worked well to show the dynamics of the model against the time variable. [ABSTRACT FROM AUTHOR]

The current study presents a detailed analysis of two crucial real‐world problems under the Caputo fractional derivative in order to deliver some desired results for the ecosystem. In view of the fact that memory effect plays a vital role in the application, we utilize an advantageous non‐local fractional operator to investigate and analyze a mathematical model of the planktonic ecosystem and biological system for the ecosystem on Planet GLIA‐2. On the other hand, theoretical and numerical results are given for the model created for phytoplankton, which is of great importance in preventing global warming, and the biological model. Existence and uniqueness are discussed for the solutions of both models with the help of the fixed‐point theorem under the Caputo operator. Additionally, the first‐order convergent numerical technique which is accurate, conditionally stable, and convergent in obtaining the solution of fractional‐order nonlinear systems of ordinary differential equations is utilized to simulate the two governing models. Numerical simulations including different values of arbitrary order ρ indicate the righteousness of the conditions for phytoplankton, Jancor, Murrot, and Vekton populations to develop. [ABSTRACT FROM AUTHOR]

In this paper, we consider switched coupled system of nonlinear implicit impulsive Langevin equations with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam–Hyers–Rassias stability of our proposed model, with the help of Generalized Diaz–Margolis's fixed point approach, over generalized complete metric space. We give an example which supports our main result. [ABSTRACT FROM AUTHOR]

This study explores the fractional damped generalized regularized long‐wave equation in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio fractional derivatives. With the aid of fixed‐point theorem in the Atangana‐Baleanu fractional derivative with Mittag‐Leffler–type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long‐wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio (in the Riemann sense) operators is used in securing the approximate‐analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of ρ, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures. [ABSTRACT FROM AUTHOR]

In this paper, we find the solution for an interesting family of coupled Schrödinger equations exemplifying simulating consequences. The fractional Schrödinger–Boussinesq (FSB) and fractional generalized Zakharov (FGZ) equations are analyzed in the present framework using fractional natural decomposition method (FNDM). The hired technique is graceful amalgamations of natural transform technique with Adomian decomposition scheme. Three different cases are considered, one with FSB equations and two cases are associated with FGZ equations with different initial conditions. To validate and illustrate the proficiency of the projected solution procedure, we analyzed the projected model in terms of fractional order. Further, we captured the nature of FNDM results for different values of fractional order in terms of the plots. The considered scheme is highly effective and structured while examining nonlinear models and which can be observed and confirmed from the obtained results. Moreover, the plots show that the hired fractional operator and algorithm can help to exemplify the more fascinating properties of the complex system associated with real‐world problems. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical solution of the variable coefficient multiterm time fractional advection–diffusion equation via exponential B‐splines. We discretize the temporal part by using the Crank–Nicolson method and spatial part by the exponential B‐splines. The unconditional stability is obtained by the Von‐Neumann method. The convergence rates are also studied. Numerical simulations confirm the theoretically expected accuracy in both time and space directions. A comparative analysis with the other methods shows the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This work is concerned with the development of a stable finite difference method (SFDM) for time‐fractional singularly perturbed convection–diffusion problems with a delay in time. The fractional derivative is considered in the Caputo sense. The SFDM is constructed based on the stability of the analytical solution. Unlike the other classical numerical methods for singular perturbation problems, this method works nicely on a uniform mesh. The method is easily adaptable on Shishkin mesh and also on any graded mesh in space. Error estimates are presented to show the convergence of the numerical scheme. To support the theory, numerical results are presented in tables for different values of the fractional derivative parameter and perturbation parameter. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel indirect convergent Jacobi spectral collocation method for fractional optimal control problems governed by a dynamical system including both classical derivative and Caputo fractional derivative. First, we present some necessary optimality conditions. Then we suggest a new Jacobi spectral collocation method to discretize the obtained conditions. By the proposed method, we get a system of algebraic equations by solving of which we can approximate the optimal solution of the main problem. Finally, we present a convergence analysis for our method and solve three numerical examples to show the efficiency and capability of the method. [ABSTRACT FROM AUTHOR]

This endeavor proposes an effective implementation of a hybrid technique for computing the approximate solution of fractional order Helmholtz equation, with Dirichlet boundary conditions. The novel scheme is an amalgamation of the traditional finite difference method with the bat optimization algorithm (BOA). This nature‐inspired optimization technique simulates the echolocation behavior of foraging bats, which ascertain the surroundings through the echo of their emitted sound pulse. Systematically, the deliberated fractional order system is altered into an integer order partial differential equation by virtue of linearized expansion of Laplace transformation. Subsequently, the attained system is processed through the proposed innovative finite difference optimization technique (FDOT) the numerical discussions. Furthermore, some experiments are carried out in order to expound the effective execution and application of the technique. In addition, the convergence and accuracy of the scheme are also delineated numerically, via statistical inference. The significant outcomes of the analysis reveal the advantageousness of the proposed numerical scheme, which can efficiently handle the complexities of the fractional order 2‐D Helmholtz equation. [ABSTRACT FROM AUTHOR]

In the current paper, we develop a new method based on the finite difference formulation and meshless method to solve 2D time fractional stochastic Tricomi‐type equation with Caputo derivative in temporal direction and Neumann boundary condition defined on irregular domain, numerically. In this method, first the fractional order derivative is discretized by a finite difference scheme and then meshless method based on radial basis functions is employed to approximate the functions in the spatial directions. So, the solution of considered problem is transformed to the solution of a linear system of algebraic equations which will be solved at each time step. The most advantage of this method is that it has no triangular, quadrangular, or other type of meshes and therefore is very applicable to solve various high dimensional problems. Finally, some test problems with different domains have been solved via present method to verify the accuracy and efficiency of the newly developed meshless formulation. Numerical results confirm the present technique is very effective to approximate the solution of 2D fractional stochastic Tricomi‐type equation with Neumann boundary condition. [ABSTRACT FROM AUTHOR]

The Lotka‐Volterra model is a very famous model and frequently used to describe the dynamics of ecological systems in which two species interact, one a predator and one its prey. In this present work, a comparative study is presented for solving Lotka‐Volterra model which has an important role in Biological sciences. The Lotka‐Volterra equations are solved by spectral collocation method (SCM) by using shifted Chebyshev polynomials of first kind. Then, a numerical example with two different cases is presented to demonstrate the accuracy and effectiveness of the numerical schemes. Using spectral method, we get a system of nonlinear algebraic equations which are solved by Newton's iteration method. Also the achieved solutions are compared with another solutions obtained by the Hermite wavelets method (HWM). Further, this work reflects that the integer order Lotka‐Volterra model and the fractional order one have close relationship where integer order model is a special case of fractional order. The behaviors study of predator and prey populations due to presence of fractional derivative are the key features of the present work. Our numerical and graphical results indicate that the proposed methods are simple, powerful and fully compatible for treatment of the Lotka‐Volterra model of fractional order. [ABSTRACT FROM AUTHOR]

This paper exhibits a numerical method for solving general fractional optimal control problems involving a dynamical system described by a nonlinear Caputo fractional differential equation, associated with a fractional Bolza cost composed as the aggregate of a standard Mayer cost and a fractional Lagrange cost given by a Riemann–Liouville fractional integral. By using the Lagrange multiplier within the calculus of variations and by applying integration by part formula, the necessary optimality conditions are derived in terms of a nonlinear two‐point fractional‐order boundary value problem. An operational matrix of fractional order right Riemann–Liouville integration is proposed and by utilizing it, the obtained two‐point fractional‐order boundary value problem is reduced into the solution of an algebraic system. An L2‐error estimate in the approximation of unknown variable by Legendre wavelet is derived and in the last, illustrative examples are included to demonstrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a pseudo‐parabolic equation with the Caputo fractional derivative. We study the existence and uniqueness of a class of mild solutions of these equations. For a nonlinear problem, we first investigate the global solution under the initial data u0 ∈ L2. In the case of initial data u0 ∈ Lq,  q ≠ 2, we obtain the local existence result. Our main tool here is using fundamental tools, namely, Banach fixed point theorem and Sobolev embeddings. In addition, we give an example to illustrate the effectiveness of the method has been proposed. [ABSTRACT FROM AUTHOR]

In this manuscript, the main objective is to introduce the derivatives of fractional‐order into a delayed dynamical model of oncolytic virotherapy. The system consists of populations of infected cells, uninfected cells, and virus particles. The local asymptotic stability of all the equilibrium points is discussed by analyzing the corresponding characteristic polynomials. The existence of Hopf bifurcation is shown due to the effect of delay. The fractional‐order dynamical system is compared with the integer order counterpart. Numerical simulations are also carried out to verify how the fractional‐order model is more stable than its integer‐order counterpart and fractional‐order parameter can be used to pacify the effect of delay. [ABSTRACT FROM AUTHOR]

In this paper, we develop two fast implicit difference schemes for solving a class of variable‐coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M‐matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum‐of‐exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from O(MN3+M2N) by a direct solver to O(MN(logN+Nexp)) per preconditioned Krylov subspace iteration and a memory requirement from 𝒪(MN2) to 𝒪(NNexp), where N and (Nexp ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods. [ABSTRACT FROM AUTHOR]

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present work is to find the numerical solution for fractional Benney–Lin equation by using two efficient methods, called q‐homotopy analysis transform method and fractional natural decomposition method. The considered equation exemplifies the long waves on the liquid films. Projected methods are distinct with solution procedure and they are modified with different transform algorithms. To illustrate the reliability and applicability of the considered solution procedures we consider eight special cases with different initial conditions. The fractional operator is considered in Caputo sense. The achieved results are drowned through two and three‐dimensional plots for different Brownian motions and classical order. The numerical simulations are presented to ensure the efficiency of considered techniques. The behavior of the obtained results for distinct fractional order is captured in the present framework. The outcomes of the present investigation show that, the considered schemes are efficient and powerful to solve nonlinear differential equations arise in science and technology. [ABSTRACT FROM AUTHOR]

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro‐differential equation of Lane–Emden type with nonlocal multi‐point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results. [ABSTRACT FROM AUTHOR]

This paper focuses on the finite element method for Caputo‐type parabolic equation with spectral fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0,1) and the spatial derivative is the spectral fractional Laplacian. The time discretization is based on the Hadamard finite‐part integral (or the finite‐part integral in the sense of Hadamard), where the piecewise linear interpolation polynomials are used. The spatial fractional Laplacian is lifted to the local spacial derivative by using the Caffarelli–Silvestre extension, where the finite element method is used. Full‐discretization scheme is constructed. The convergence and error estimates are obtained. Finally, numerical experiments are presented which support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results. [ABSTRACT FROM AUTHOR]