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It is well known that investigation on exact solutions of nonlinear fractional partial differential equations (PDEs) is a very difficult work compared with integer‐order nonlinear PDEs. In this paper, based on the separation method of semifixed variables and integral bifurcation method, a combinational method is proposed. By using this new method, a class of generalized time‐fractional thin‐film equations are studied. Under two kinds of definitions of fractional derivatives, exact solutions of two generalized time‐fractional thin‐film equations are investigated respectively. Different kinds of exact solutions are obtained and their dynamic properties are discussed. Compared to the results in the existing references, the types of solutions obtained in this paper are abundant and very different from those in the existing references. Investigation shows that the solutions of the model defined by Riemann–Liouville differential operator converge faster than those defined by Caputo differential operator. It is also found that the profiles of some solutions are very similar to solitons, but they are not true soliton solutions. In order to visually show the dynamic properties of these solutions, the profiles of some representative exact solutions are illustrated by 3D graphs. [ABSTRACT FROM AUTHOR]

In the paper mentioned in the title, we studied the semi‐Markovian random walk processes with jumps and delaying screen in zero. More precisely, we obtained a mathematical modeling of the semi‐Markov random walk processes with a delaying screen in zero, given in general form by means of fractional differential equation. In this note, we provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper. [ABSTRACT FROM AUTHOR]

In this paper, a class of fractional stochastic differential equations (SFDEs) with impulses is considered. By virtue of Mönch's fixed point theorem and Banach contraction principle, we explore the existence and uniqueness of solutions to the addressed system. Furthermore, with the aid of the Jensen inequality, Hölder inequality, Burkholder–Davis–Gundy inequality, Grönwall–Bellman inequality, and some novel assumptions, the averaging principle of our considered system is obtained. At the end of this paper, an example is provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of multiple positive solutions and stability results for nonlinear fractional delay differential equations involving p$$ p $$‐Laplacian operator. We derived adequate conditions to ensure that at least three nonnegative solutions exist by applying the conditions of the Leggett–Williams fixed‐point theory and some Green function properties. Due to a small change in time‐delay, we analyzed the Hyers–Ulam stability‐type of the equation. We used Riemann–Liouville fractional differential definition, and we assumed that nonzero delay ϑ>0$$ \vartheta >0 $$. In addition, for application purpose, comprehensive examples are given to ensure the effectiveness and feasibility of the results in this paper. Our proposed equation generalize some literature in the system. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding the order 0<α<1$$ 0<\alpha <1 $$ in the fractional discrete system: C∇αun=Aun,n≥1,u0=x0,$$ \left\{\begin{array}{cll}{\kern0.1em }_C{\nabla}^{\alpha }{u}^n& =& A{u}^n,\kern0.30em n\ge 1,\\ {}{u}^0& =& {x}_0,\end{array}\right. $$where A$$ A $$ is a closed linear operator defined in a Banach space X,x0∈X$$ X,{x}_0\in X $$, and C∇αun$$ {\kern0.1em }_C{\nabla}^{\alpha }{u}^n $$ is the discrete Caputo fractional derivative of a given vector‐valued sequence (un)n∈ℕ0$$ {\left({u}^n\right)}_{n\in {\mathrm{\mathbb{N}}}_0} $$. [ABSTRACT FROM AUTHOR]

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid-saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin-Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi-Kober integral. In this way, we obtain a nonlinear model in the framework of time-fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time-fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we will use the definitions of the Caputo–Fabrizio fractional derivative and the Riemann–Liouville fractional q‐integral to look into the existence, uniqueness, and continuous dependence of solutions for a coupled system of fractional q‐integro‐differential equations. We provide an overview of the finite‐trapezoidal method. Finally, we present some numerical examples to demonstrate the method's effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: −(D0+αx)(t)=f(t,x(t),D0+α−1x(t)),n−1<α≤n,t∈[0,1],x(0)=x′(0)=...=x(n−2)(0)=0,x(k)(1)=∫01x(k)(t)dA(t),$$ {\displaystyle \begin{array}{cc}\hfill -\left({D}_{0+}^{\alpha }x\right)(t)=& f\left(t,x(t),{D}_{0+}^{\alpha -1}x(t)\right),n-1<\alpha \le n,t\in \left[0,1\right],\hfill \\ {}\hfill x(0)=& {x}^{\prime }(0)=\dots ={x}^{\left(n-2\right)}(0)=0,{x}^{(k)}(1)=\int_0^1{x}^{(k)}(t) dA(t),\hfill \end{array}} $$by using Mawhin's coincidence degree theory. Here, D0+α$$ {D}_{0+}^{\alpha } $$ is the standard Riemann–Liouville fractional derivative of order α,f:[0,1]×ℝ2→ℝ$$ \alpha, f:\left[0,1\right]\times {\mathbb{R}}^2\to \mathbb{R} $$, and ∫01x(k)(t)dA(t)$$ {\int}_0^1{x}^{(k)}(t) dA(t) $$ is the Riemann–Stieltjes integral of x(k)$$ {x}^{(k)} $$ with respect to A$$ A $$. Our choice of k$$ k $$ in the boundary condition can be any integer between 0 and n−1$$ n-1 $$, which supplements many boundary conditions assumed in the literature. Several examples are given to strengthen our result. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and the uniqueness of solutions for a class of nonlinear fractional differential equations with a mixed fractional boundary value, by using Banach fixed‐point theorem. Moreover, we compare obtained results with another two works considering similar problem. [ABSTRACT FROM AUTHOR]

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples. [ABSTRACT FROM AUTHOR]

Our main objectives in the paper are to investigate the existence and uniqueness of solution for a class of implicit fractional‐order boundary value problem with Riemann–Liouville derivative via generalized boundary conditions. Furthermore, different kinds of Ulam stability are discussed. By giving a few examples, the key findings are demonstrated. [ABSTRACT FROM AUTHOR]

This paper first finishes off the exact solutions of Riemann‐Liouville fractional differential time‐delay oscillatory system of order ρ∈(1,2)$$ \rho \in \left(1,2\right) $$ by using two newly defined delayed perturbations of Mittag‐Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved. [ABSTRACT FROM AUTHOR]

This paper aims to explore and apply differential quadrature based on different test functions to find an efficient numerical solution of fractional order Cauchy reaction‐diffusion equations (CRDEs). The governing system is discretized through time and space via novel techniques of differential quadrature method and the fractional operator of Caputo kind. Two problems are offered to explain the accuracy of the numerical algorithms. To verify the reliability, accuracy, efficiency, and speed of these methods, computed results are compared numerically and graphically with the exact and semi‐exact solutions. Then mainly, we deal with absolute errors and L∞ errors to study the convergence of the presented methods. For each technique, MATLAB Code is designed to solve these problems with the error reaching ≤1 × 10−5. In addition, a parametric analysis is presented to discuss influence of fractional order derivative on results. The achieved solutions prove the viability of the presented methods and demonstrate that these methods are easy to implement, effective, high accurate, and appropriate for studying fractional partial differential equations emerging in fields related to science and engineering. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with the existence of extremal solutions for a novel class of nonlinear fractional p‐Laplacian differential equation in terms of ψ‐Caputo fractional derivative equipped with a new class of nonlinear boundary conditions. Initially, we focus on the linear problem and we give an explicit form of the solutions, from which we derive new comparison results which is crucial for the proof of the main theorem of this paper. Secondly, with the help of the monotone iterative method along with its accompanying upper and lower solutions, the existence of extremal solutions for the aforementioned problem is investigated. Then, based on the proposed method, we can construct two types of successive approximations which converge uniformly monotonically to minimal and maximal solutions of the given problem. Finally, we illustrate our results with an example. [ABSTRACT FROM AUTHOR]

The paper aims to analyze the fractional Bloch equation with Caputo and Atangana–Baleanu–Caputo (ABC) derivative having a nonsingular kernel. The fixed point theorem is used to prove the existence and uniqueness of the proposed equation. Furthermore, the approximate‐analytical solution of the proposed equation is obtained by Aboodh transform iterative method (ATIM) in the form of a convergent series. The technique (ATIM) combines the new iterative method with the Aboodh transform. The convergence analysis of the approximate solution is discussed using the Cauchy convergence theorem. The validity of the ATIM is shown by numerical simulation and graphs. Also, the preference for nonsingular kernel over singular kernel is shown with the help of tables and graphs. For integer order, the obtained solutions from Caputo and ABC derivative are compared with the exact solutions and published work. [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]

One out of eight people in the United Arab Emirates (UAE) are estimated to have diabetes. Modeling the relationship between developing this health condition and complications is an important step towards understanding this disease. In this paper, we investigate a system of fractional differential equations to estimate the size of diabetic population with and without complication in the UAE. We provide the well‐posedness analysis of the model in the sense of Hadamard. We also provide the numerical approximations of the solution as a means of prediction of the future size of UAE diabetic population and the economic burden of the country that may be used by the government to take necessary preventative measure. [ABSTRACT FROM AUTHOR]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

Uncertain fractional differential equation (UFDE) is a useful tool for studying complex systems in uncertain environments. The mathematical characteristics of solution of an UFDE are also widely used in various fields. In this paper, we give the extreme value theorems of solution of Caputo–Hadamard UFDE and applications. A numerical algorithm for obtaining the inverse uncertainty distributions (IUDs) for extreme values of solution of Caputo–Hadamard UFDE is presented; the stability and feasibility of the proposed algorithm are validated by numerical experiments. As an application of extreme value theorems in uncertain financial market, the pricing formulas of the American option based on the new uncertain stock model are given. Besides, the algorithms for computing the price of the American option without explicit pricing formulas based on the Simpson's rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments. [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]

This paper investigates a new class of p‐Laplacian boundary value problems of Hadamard type fractional differential equations. We emphasize that few works on Hadamard fractional boundary value problems include p‐Laplacian Hadamard type fractional differential equations with nonlinear terms involving Hadamard fractional derivatives of unknown functions. By means of the five functional fixed point theorem, multiple positive solutions are established for the p‐Laplacian Hadamard fractional boundary value problem. An example is constructed for illustration of the main result. [ABSTRACT FROM AUTHOR]

In the current paper, we intend to check the existence aspects of solutions for a category of the multi‐point boundary value problem (BVP) involving a p$$ p $$‐Laplacian differential operator within the generalized fractional derivatives depending on another function. Based on two fixed point results attributed to Schaefer and Banach, the desired results are verified. Further, the continuity of solutions in terms of inputs (fractional orders, associated parameters, and appropriate function) is extensively discussed. A simulative example is prepared to demonstrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

The Ornstein–Uhlenbeck process is a fundamental tool in various applications ranging from finance where it is known under Vasiček process name to fractional diffusion in the context, for instance, Klein–Kramers dynamics. In this paper, we define the Ornstein–Uhlenbeck process driven by a symmetric tempered α‐stable process. We investigate its properties in the non‐stationary case and its long time limit. Moreover, we provide an explicit representation of its characteristic function in terms of special functions, that is, generalized hypergeometric function and its special case hypergeometric Gauss function. Thus, using probabilistic arguments, we provide a new identity between those two functions. Our calculations are also confirmed by numerical simulations of this new process. We discuss the Fokker–Planck type equation describing the probability density function of our model as a solution of the fractional differential equation with the so‐called tempered fractional derivative. [ABSTRACT FROM AUTHOR]

The first case of COVID‐19 in India detected on January 30, 2020, after its emergence in Wuhan, China, in December 2019. The lockdown was imposed as anemergency measure by the Indian government to prevent the spread of COVID‐19 but gradually eased out due to its vast economic consequences. Just 15 days after the relaxation of lockdown restrictions, Delhi became India's worst city in terms of COVID‐19 cases. In this paper, we propose a variable‐order fractional SIR (susceptible, infected, removed) model at state‐level scale. We introduce a algorithm that uses the differential evolution algorithm in combination with Adam–Bashforth–Moulton method to learn the parameters in a system of variable‐order fractional SIR model. The model can predict the confirm COVID‐19 cases in India considering the effects of nationwide lockdown and the possible estimate of the number of infliction inactive cases after the removal of lockdown on June 1, 2020. A new parameter p is introduced in the classical SIR model representing the fraction of infected people that get tested and are thereby quarantined. The COVID‐19 trajectory in Delhi, as per our model, predicts the slowing down of the spread between January and February 2021, touching a peak of around 5 lakh confirmed cases. [ABSTRACT FROM AUTHOR]

The main objective of the paper is to study the non‐local problem for a pseudo‐parabolic equation with fractional time and space. The derivative of time is understood in the sense of the time derivative of the Caputo fraction of the order α, 0 < α < 1. The first result is an investigation of the existence and uniformity of the solution; the formula for mild solution and the regularity properties will be given. The proofs are based on a number of sophisticated techniques using the Sobolev embedding and also on the construction of the Mittag–Lefler operator. In the second part, we investigate the convergence of the mild solution for non‐local problem to the solution of the local problem when two non‐local parameters reach 0. Finally, we present some numerical examples to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

Novel coronavirus (COVID‐19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID‐19 SEIR epidemic model via Caputo fractional derivatives using a predictor–corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag–Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper aims to develop a numerical method for the solutions of Hadamard fractional differential equations. We introduced Hadamard fractional Legendre functions by modifying classical Legendre polynomials and used them to approximate the numerical solutions of linear and nonlinear Hadamard fractional differential equations. The solution is approached by approximating the appropriate terms in Hadamard fractional differential equations using Hadamard fractional Legendre functions and converting the problem to a system of algebraic equations. Quasilinearization technique is employed to linearize the nonlinear Hadamard fractional differential equations. An upper bound for approximation error is derived. This method provides reasonably accurate results for a relatively smaller order of Hadamard fractional Legendre functions. [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]

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]

This paper is in concern with the study of differential problems involving the ψ$$ \psi $$‐shifted fractional derivatives, where ψ$$ \psi $$ is a scaling function. Such operators can be thought of as a generalization of several fractional derivatives such as the classical Riemann–Liouville and Caputo operators, the Hadamard operators, the generalized fractional operators, and the Erdélyi–Kober operators, among others. Two types of boundary conditions are considered: the Cauchy problems and the integral boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we put forward numerical algorithms to simulate fractional HIV infected CD4+T cell model and fractional Schrodinger equation. First, we put forward numerical algorithm to solve fractional differential systems with trigonometric neural network. Then trigonometric neural network method is used to simulate the fractional HIV infected CD4+T cell model. It is found that the infection equilibrium has the restriction condition for the number of virus particles released by each infected cell. Under this condition, the fractional system has a unique positive equilibrium. Second, we put forward numerical algorithm to solve fractional PDE with trigonometric neural network. Then this numerical method is used to simulate fractional Schrodinger equation, and some important results in fractional quantum mechanics are concluded by the corresponding numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary stabilization for a class of neural networks with non‐identical nodes described by the fractional Orr‐Sommerfeld (OS) equation cascaded with the fractional Squire equation and fractional ordinary differential equations (ODEs), in which the disturbances flow to the control end of both the OS equation and the Squire equation. By applying the Backstepping transform, the diffusion subsystems in the considered neural networks are decoupled, while only the ODEs are still cascaded with the diffusion terms through the Neumann interconnection. By the active disturbance rejection control (ADRC) approach, two auxiliary systems are constructed to compensate each disturbance by an estimator without high gain. Finally, a boundary feedback control law is designed to achieve the Mittag‐Leffler stability of the neural networks in the closed‐loop. [ABSTRACT FROM AUTHOR]

Neural network has good self‐learning and adaptive capabilities. In this paper, a wavelet neural network is proposed to be used to solve the value problem of fractional differential equations (FDE). We construct a wavelet neural network (WNN) with the structure 1 ×N× 1 based on the wavelet function and give the conditions for the convergence of the given algorithm. This method uses the truncated power series of the solution function to transform the original differential equation into an approximate solution, then, using WNN, update the parameters, and finally get the FDE solution. Simulation results prove the validity of WNN. [ABSTRACT FROM AUTHOR]

In the paper, we consider some fractional integral formulas in terms of the Riemann–Liouville, Erdélyi–Kober type, and Weyl fractional integral operators and present the general fractional kinetic model involving the hypergeometric superhyperbolic sine function via the Gauss hypergeometric series. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss basic results of boundary value problems of fractional differential equations (BVP‐FDEs) via the concept of Caputo fractional derivative with respect to another function with the order α∈(1,2). The existence and uniqueness results of a solution for BVP‐FDEs are discussed by utilizing Banach fixed point theorem and Schaefer's fixed point theorem. We also provide new sufficient conditions to guarantee the Hyers‐Ulam stability and the Hyers–Ulam–Rassias stability of BVP‐FDEs. Furthermore, some concrete examples to consolidate the obtained results are also considered. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear fractional Schrödinger integro‐differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one‐dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish global fractional Halanay‐type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag–Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag–Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality. [ABSTRACT FROM AUTHOR]

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional parabolic‐elliptic Keller–Segel system with time‐space dependent logistic source on the whole space 0.1ut+(−Δ)su+χ∇·(u∇v)=u(a(t,x)−b(t,x)u),t>0,x∈ℝN,0=Δv−λv+μu,t>0,x∈ℝN,$$ \left\{\begin{array}{cc}{u}_t&amp;#x0002B;{\left(-\Delta \right)}&amp;#x0005E;su&amp;#x0002B;\chi \nabla \cdotp \left(u\nabla v\right)&amp;#x0003D;u\left(a\left(t,x\right)-b\left(t,x\right)u\right),\kern0.30em &amp; t&gt;0,x\in {\mathbb{R}}&amp;#x0005E;N,\\ {}0&amp;#x0003D;\Delta v-\lambda v&amp;#x0002B;\mu u,\kern0.30em &amp; t&gt;0,x\in {\mathbb{R}}&amp;#x0005E;N,\end{array}\right. $$where s∈(0,1),χ,λ$$ s\in \left(0,1\right),\chi, \lambda $$ and μ$$ \mu $$ are positive constants and the functions a(t,x)$$ a\left(t,x\right) $$ and b(t,x)$$ b\left(t,x\right) $$ are positive and bounded. When the order of fractional diffusion s∈1/2,1$$ s\in \left(1/2,1\right) $$ and the functions a(t,x)$$ a\left(t,x\right) $$ and b(t,x)$$ b\left(t,x\right) $$ satisfy some conditions, we first prove the local existence and uniqueness of nonnegative classical solutions (u(t,x;t0,u0),v(t,x;t0,u0))$$ \left(u\left(t,x;{t}_0,{u}_0\right),v\left(t,x;{t}_0,{u}_0\right)\right) $$ with u(t0,x;t0,u0)=u0$$ u\left({t}_0,x;{t}_0,{u}_0\right)&amp;#x0003D;{u}_0 $$ for every t0∈ℝ$$ {t}_0\in \mathbb{R} $$ and every nonnegative bounded and uniformly continuous initial function u0$$ {u}_0 $$. Next, under some parameter conditions, we prove the global existence and boundedness of classical solutions (u(t,x;t0,u0),v(t,x;t0,u0))$$ \left(u\left(t,x;{t}_0,{u}_0\right),v\left(t,x;{t}_0,{u}_0\right)\right) $$ for given initial function u0$$ {u}_0 $$. Finally, we obtain the pointwise persistence phenomena of solutions; that is, for any strictly positive initial function u0$$ {u}_0 $$, there are 00$$ T\left({u}_0\right)&gt;0 $$ such that m≤u(t,x;t0,u0)≤M,∀t≥T(u0),x∈ℝN.$$ m\le u\left(t,x;{t}_0,{u}_0\right)\le M,\forall t\ge T\left({u}_0\right),x\in {\mathbb{R}}&amp;#x0005E;N. $$In summary, these results are extension to the case of fractional parabolic‐elliptic Keller–Segel system. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo–Hadamard fractional differential equations on a semi‐infinite domain. Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo–Hadamard fractional delay differential equations. Findings: We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration, and Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo–Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo–Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method. We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo–Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate. Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present wavelets method has never been introduced and implemented for Caputo fractional and Caputo–Hadamard fractional differential equations. [ABSTRACT FROM AUTHOR]

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the Ψ$$ \Psi $$‐fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the Ψ$$ \Psi $$‐tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann–Liouville‐type and Caputo‐type cases of these operators. Furthermore, we study some non‐linear fractional differential equations involving Ψ$$ \Psi $$‐tempered derivatives, proving existence‐uniqueness theorems by using the Banach contraction principle and proving stability results by using Grönwall type inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we present a definition of the generalized fractional local derivative, which contains as a particular case several of those reported in the literature. In this framework, we apply this operator to the study of the solutions of some fractional differential equations and we focus mainly on a Gompertz model applied to the study of tuberculosis in Mexico. We also show the effectiveness of this model by fitting it to real data and by comparing it with other models using other well‐known definitions of fractional derivative. [ABSTRACT FROM AUTHOR]

This work presents the generalized nonlinear multi‐terms fractional variable‐order delay differential equation with periodic and anti‐periodic conditions. In this paper, a novel shifted Jacobi operational matrix technique is introduced to solve a class of these equations mentioned, so that the main problem becomes a system of algebraic equations that we can solve numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical tests are provided to demonstrate the generality, efficiency, accuracy of presented scheme and the flexibility of this technique. The numerical experiments compared it with the true solution, indicating the validity and efficiency of this scheme. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated. [ABSTRACT FROM AUTHOR]

This paper is devoted to the implementation of fractional differential transform method (FDTM) for the evaluation of new approximations for the fractional models of the system of multi‐pantograph differential equations. Some new properties of FDTM have also been proposed for the fractional integro‐differential equations. It is noteworthy that the integro‐differential equations are studied for integers in the literature, and we have proposed the solutions of proportional delayed integro‐differential equations for fractional order. The evaluated results are verified numerically by considering some test examples of fractional integro‐differential equations and fractional models of the system of multi‐pantograph differential equations. The outcomes conclude that evaluated series solutions via FDTM promise more appropriate approximations possess very negligible error and represent the convergent approximations. Thus, evaluated series solutions are comparatively more accurate and require less computational time. [ABSTRACT FROM AUTHOR]

In this paper, we construct an analytical study of the epidemic pine wilt disease model in a Caputo–Fabrizio type fractional order; the basic properties of the fractional differential equation and the model are presented; we give the basic properties of the model. Applying the homotopy analysis transform method to obtain numerical solutions to the model, the numerical result represented in graphs is presented. [ABSTRACT FROM AUTHOR]

In this paper, stability analysis which ensures the convergence of a part of the solutions towards a ball of a class of fractional‐order nonlinear systems is described. Using the Lyapunov‐like functions, such nonlinear systems depending on a small parameter is studied, and such practical stability is ensured. Numerical examples are given to illustrate the validity of the proposed theoretical results as well as a real application to a class of cobweb models. [ABSTRACT FROM AUTHOR]

For the first time, the result on finite‐time stability (FTS) for fractional delay differential equations with non‐instantaneous impulses (NI‐FDDEs) involving the generalized Caputo fractional derivative is presented. Based on an extensive estimation of the fractional integral inequality provided in this paper, a sufficient condition is proposed to the FTS of NI‐FDDEs. Some examples are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]