Search

Zika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional‐order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals are x1(t), infected individuals are x2(t), x3(t) shows susceptible mosquitos, and x4(t) shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana–Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional‐order model has been made and verify the non‐negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional‐order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we propose an anisotropic diffusion equation based on fractional‐order tensor diffusion applied on noisy images. We start by investigating the existence and uniqueness results of the proposed model. We also provide a discretization scheme based on the finite difference method. In order to prove the efficacy of our proposed approach, we deliver some simulations, which show robustness and remarkable results against noise while preserving edges and corners, compared to some well‐known models in literature. For a fair comparison, we manage to obtain the optimal parameters of every model thanks to PSNR and SSIM. We demonstrate the robustness efficacy of our model against high level of noise. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev-type hemivariational differential systems of order r ∈ (1, 2) with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed-point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study. [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]

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ of the model is computed using the method of next generation matrix, and we prove that if R0<1$$ {\mathcal{R}}_0<1 $$, the healthy equilibrium is locally asymptotically stable, and when R0>1$$ {\mathcal{R}}_0>1 $$, the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive L∞0,T;H01(Ω)$$ {L}^{\infty}\left(0,T;{H}_0^1\left(\Omega \right)\right) $$ bound as well as L2(0,T;H2(Ω))$$ {L}^2\left(0,T;{H}^2\left(\Omega \right)\right) $$ bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of O(h+k2−α)$$ O\left(h+{k}^{2-\alpha}\right) $$, where α(0<α<1)$$ \alpha \kern0.1em \left(0<\alpha <1\right) $$ is the fractional derivative exponent and h$$ h $$ and k$$ k $$ are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 σ$$ {}_{\sigma } $$ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the new corresponding convolution, inverse and Plancherel formulas, and Hausdorff–Young type inequality. We show that this generalized Fourier transform is useful in the study of several fractional differential equations (both ordinary and partial differential equations). [ABSTRACT FROM AUTHOR]

Mathematical modeling of biological systems is an important issue having significant effect on human beings. In this direction, the description of immune systems is an attractive topic as a result of its ability to detect and eradicate abnormal cells. Therefore, this manuscript aims to investigate the asymptotic behavior of immunogenic tumor dynamics based on a new fractional model constructed by the concept of general fractional operators. We discuss the stability and equilibrium points corresponding to the new model; then we modify the predictor–corrector method in general sense to implement the model and compare the associated numerical results with some real experimental data. As an achievement, the new model provides a degree of flexibility enabling us to adjust the complex dynamics of biological system under study. Consequently, the new general model and its solution method presented in this paper for the immunogenic tumor dynamics are new and comprise quite different information than the other kinds of classical and fractional equations. In addition to these, we implement a tracking control method in order to decrease the development of tumor‐cell population. The satisfaction of control purpose is confirmed by some simulation results since the controlled variables track the tumor‐free steady state in the whole realistic cases. [ABSTRACT FROM AUTHOR]

This paper concerns the existence and uniqueness of the strong solutions to the initial‐boundary value problems of fractional analog of Voigt model in the planar case. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of solutions for a coupled system of nonlinear multi‐term fractional differential equations with anti‐periodic type coupled nonlocal boundary conditions. The existence of a unique solution is proved via Banach contraction mapping principle, while two existence results are proved by using Krasnosel'skiic̆'s fixed point theorem and Leray‐Schauder alternative. We also discuss a variant of the proposed problem involving fractional differential types nonlinearities. The obtained results are well illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper proposes a general theoretical framework to establish the time‐space fractional derivative models for non‐Fourier heat conduction. Similar to viscoelastic modeling in mechanics, some new heat conduction elements are proposed instead of mechanical ones. With different combinations of the thermal elements, the time‐space fractional generalizations, such as the Kelvin‐Voigt model, the Maxwell model, and the Zener model, are presented. In addition, it is possible to obtain the fractional derivative heat conduction equations from the corresponding transport equations. While incorporating the existing non‐Fourier heat conduction models into the proposed theoretical framework, some new time‐space fractional derivative models are presented. Finally, the nonlocal models considering the long‐range heat fluxes are investigated and confirmed to be consistent with the proposed framework. [ABSTRACT FROM AUTHOR]

In this paper, we develop and compare four kinds of fully discrete continuous Galerkin finite element methods for a fractional heat transport equation describing the deep underground unsteady flow. We adopt continuous Galerkin finite element methods for the spatial discretization and the backward Euler together with linear interpolation (L1), second‐order backward differentiation formula (BDF2), weighted and shifted Grünwald difference (WSGD), and Crank‐Nicolson (CN) time‐stepping methods for time discretization. We derive the stability estimates and a priori error estimates for the semidiscrete and fully discrete finite element schemes. Finally, numerical comparisons of the four families of fully discrete finite element methods are presented to illustrate the effectiveness of the studied numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Galerkin finite element method for the Cahn‐Hilliard‐Cook equation involving the Caputo‐type fractional derivative, which can be used to describe the interface phenomena for modeling the phase transitions with the random effects and the properties of shape memory. The regularity properties of mild solution to the given problem are presented, and a result concerning the convergence error estimate of the corresponding semidiscrete scheme is established. Finally, we construct the fully discrete scheme based on the approximations of the Mittag‐Leffler function, and the strong convergence error estimate of the proposed scheme is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents analytical‐approximate solutions of the time‐fractional Cahn‐Hilliard (TFCH) equations of fourth and sixth order using the new iterative method (NIM) and q‐homotopy analysis method (q‐HAM). We obtained convergent series solutions using these two iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate with a time fractional‐order derivative in a three‐species predator‐prey model with the presence of prey social behavior. A new approximation for predator‐prey interaction in the presence of prey social behavior has been considered. For the model analysis, the study has been divided into two principal parts. First of all, we study the local stability of the equilibria and the existence of Hopf bifurcation. Then, for the numerical analysis, the Caputo fractional derivative operator is utilized to approximate the numerical solution of the model. An excellent agreement is seen between the numerical results and the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a modification of a compartmental model for Ebola disease previously introduced in the literature. For this initial model, the basic reproduction number is obtained. Moreover, a new dimensionless model is introduced. For this new model, we analyze the local stability and obtain the basic reproduction number. Finally, a fractional analogue of the dimensionless model is presented. Some numerical experiments are also included. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this present paper, we discuss some properties of fractional operators with fixed memory length (Riemann–Liouville fractional integral, Riemann–Liouville and Caputo fractional derivatives). Some observations and examples are discussed during the article, in order to make the results well defined and clear. Furthermore, we consider the fundamental theorem of calculus for fractional operators with fixed memory length. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of pseudoparabolic equations with the nonlocal condition and the Caputo derivative. Two cases of problems (1–2) will be studied, which are linear case and nonlinear case. For the first case, we establish the existence, the uniqueness, and some regularity results by using some estimates technique and Sobolev embeddings. Second, the Banach fixed‐point theorem will be applied to the nonlinear case to prove the existence and the uniqueness of the mild solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. By using these definitions, we obtain an equiaffine Frenet formula and then construct an analogue of the fundamental theorem. The plane curves of constant equiaffine curvature with fractional order are classified. Several examples are also illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we study a fractional nonlinear Rayleigh–Stokes problem with nonlocal conditions. Existence, uniqueness, and regularity of the mild solution of the problem are investigated in some appropriate spaces. Furthermore, when the parameter goes to zero, we consider the convergence of the mild solution. Our analysis relies on Banach's fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro-differential systems with distributed delays in control. The results are obtained by using the Mittag- Leffler functions and the Schauder fixed-point theorem. Examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

Fractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differential-difference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The main objective is to apply the concept of newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function in fluid dynamics. In this article, a newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function and the nonsingular kernel has been applied to study viscous fluid flow in the presence of an applied magnetic field. The flow is considered over an infinite vertical plate moving with arbitrary velocity. The modeled problem is transformed into a nondimensional form via dimensionless analysis, and then the Laplace transform method is applied for the solution of the problem. Due to the complexity in Laplace inversion, a strong numerical inversion procedure, namely, Zakian's algorithm, has been used, and the results are computed in various plots and tables. The corresponding discussion of results is included in detail. It is concluded that the generalized fractional‐order derivative is accurate and efficient for describing general fractional‐order dynamics in complex and power law phenomena. [ABSTRACT FROM AUTHOR]

An inverse source problem (ISP) of recovering space‐dependent source term along with diffusion concentration for a two‐dimensional diffusion equation involving integral convolution of arbitrary memory kernel in time is considered. The unique existence of the solution is proved using a bi‐orthogonal system of functions obtained from the associated non‐self‐adjoint spectral problem and its adjoint problem which form Riesz basis in L2[(0,1)×(0,1)]$$ {L}&#x0005E;2\left[\left(0,1\right)\times \left(0,1\right)\right] $$. In addition, some particular cases of ISP are described as special cases of our results. [ABSTRACT FROM AUTHOR]

The goal of this study is to analyze and obtain a new numerical approach to an important mathematical model called polio, which is one of the highly infectious and dangerous diseases challenging many lives in most developing nations, most especially in Africa, Latin America, and Asia. A number of research outputs have proven beyond doubt that modeling with noninteger‐order derivative is much more accurate and reliable when compared with the integer‐order counterparts. In the present case, an extension is given to the polio model by replacing the classical time derivative with the newly defined operator known as the Atangana–Baleanu fractional derivative which has its formulation based on the noble Mittag–Leffler kernel. This derivative has been tested and applied in number of ways to model a range of physical phenomena in science and engineering. The Picard–Lindelöf theorem is applied to determine the condition under which the proposed model has a solution, also to show that the solution exists and unique. Local stability analysis of the disease‐free equilibrium and endemic state is also discussed. A novel approximation based on the Adams–Bashforth method is formulated to numerically approximate the fractional derivative operator. To justify the theoretical findings, some numerical results obtained for different instances of fractional order are presented. [ABSTRACT FROM AUTHOR]

In this work, we study the stability of a one‐dimensional Timoshenko system with localized internal fractional Kelvin–Voigt damping in a bounded domain. First, we reformulate the system into an augmented model and using a general criteria of Arendt–Batty we prove the strong stability. Next, we investigate three cases: The first one when the damping is localized in the bending moment, the second case when the damping is localized in the shear stress, we prove that the energy of the system decays polynomially with rate t−1$$ {t}^{-1} $$ in both cases. In the third case, the fractional Kelvin–Voigt is acting on the shear stress and the bending moment simultaneously. We show that the system is polynomially stable with energy decay rate of type t−42−α$$ {t}^{\frac{-4}{2-\alpha }} $$, provided that the two dampings are acting in the same subinterval. The method is based on the frequency domain approach combined with multiplier technique. [ABSTRACT FROM AUTHOR]

Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order). [ABSTRACT FROM AUTHOR]

In this article, we formulate fractional differential inclusion of pantograph type (IFDIP), incorporating impulsive behavior of the solution. The boundary conditions taken into account are nonlocal in nature. We will consider the convex problem and prove the Filippov–Wazewski‐type theorem. Moreover, existence of solution, uniqueness of a solution, and the topological properties of the solution's set will be examined for the problem under consideration. In the second part, the study will be confined to the second‐order impulsive fractional differential equation of pantograph type. For certain geometric characteristics of the solution's set, Aronszajn–Browder–Gupta‐type results will be explored for the newly introduced differential equation. Also, it will prove the existence of solution for the first‐order fractional differential equation of pantograph type having impulsive behavior of the solution. [ABSTRACT FROM AUTHOR]

This article deals with the approximate controllability problem for fractional evolution equations involving noninstantaneous impulses and state‐dependent delay. In order to derive sufficient conditions for the approximate controllability of our problem, we first consider the linear‐regulator problem and find the optimal control in the feedback form. By using this optimal control, we develop the approximate controllability of the linear fractional control system. Further, we obtain sufficient conditions for the approximate controllability of the nonlinear problem. In the end, we provide a concrete example to support the applicability of the derived results. [ABSTRACT FROM AUTHOR]

A backward problem for composite fractional relaxation equations is considered with Caputo's fractional derivative, which covers as particular case of Basset problem that concerns the unsteady motion of a particle accelerating in a viscous fluid in fluid dynamics. Based on a spectral problem, the representation of solutions is established. Next, we show the maximal regularity for the corresponding initial value problem. Due to the mildly ill‐posedness of current backward problem, the fractional Landweber regularization method will be applied to discuss convergence analysis and error estimates. [ABSTRACT FROM AUTHOR]

The main motivation of our conversation is the approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1 < r < 2. Using principles and ideas from stochastic analysis, the theory of cosine family, fractional calculus, and Banach fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions and then demonstrate the approximate controllability of the fractional stochastic control equation. We then apply our findings to the theory of nonlocal conditions. At last, an application is established for drawing the theory of the main results. [ABSTRACT FROM AUTHOR]

In this article, a novel collocation method is developed based on Gegenbauer wavelets together with the quasi‐linearization technique to facilitate the solution of population growth model of fractional order in a closed system. The operational matrices of fractional order integration are obtained via block‐pulse functions. The obtained matrices are employed to transform the given time‐fractional population growth model into a non‐linear system of algebraic equations. Then, the quasi‐linearization technique is invoked to convert the underlying equations to a linear system of equations. The performance and accuracy of the proposed method is elucidated by a presenting a comparison with some numerical methods existing in the open literature. The numerical outcomes shows that the present method is more efficient than the existing ones. [ABSTRACT FROM AUTHOR]

This study deals with the inverse problem of retrieving the initial status for a fractional diffusion system from the measured final and source data. Here, the mentioned system is taken as a nonhomogeneous time fractional diffusion problem in a general bounded domain Ω⊂Rn. A regularized sought solution is obtained by the regularization scheme, namely, modified quasi‐boundary value method. Further, the convergence estimates between the exact and regularized solution are derived based on the choice of strategies of the regularization parameter. Eventually, numerical examples are illustrated to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we focus on a class of singular linear systems of fractional differential equations with given nonconsistent initial conditions (IC). Because the nonconsistency of the IC can not lead to a unique solution for the singular system, we use two optimization techniques to provide an optimal solution for the system. We use two optimization techniques to provide the optimal solution for the system because a unique solution for the singular system cannot be obtained due to the non‐consistency of the IC. These two optimization techniques involve perturbations to the non‐consistent IC, specifically, an l2 perturbation (which seeks an optimal solution for the system in terms of least squares), and a second‐order optimization technique at an l1 minimum perturbation, (which includes an appropriate smoothing). Numerical examples are given to justify our theory. We use the Caputo (C) fractional derivative and two recently defined alternative versions of this derivative, the Caputo‐Fabrizio (CF) and the Atangana‐Baleanu (AB) fractional derivative. [ABSTRACT FROM AUTHOR]

In this work, we study a space–time Petrov–Galerkin method for third‐ and fifth‐order time fractional Korteweg–de Vries–Burgers equations. The method is based on the framework of Legendre and Jacobi polynomials. The basis functions of the fractional part are constructed by the generalized Jacobi functions, which contained the singularity of weak solutions. The numerical schemes of the problems are transformed into the nonlinear schemes constructed by matrices. Based on the orthogonality of ideal basis functions, we get the optimal estimation under the specific weighted Sobolev spaces. Numerical experiments confirm the expected convergence. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence and uniqueness of solutions for nonlinear fractional integro‐differential equation on an unbounded domain in a weighted Banach space. New results are showed owing to the fixed point theorems of Banach and Krasnoselskii. We illustrate our results by examples to ending our theoretical study. [ABSTRACT FROM AUTHOR]

We study the question of positivity of the fundamental solution for fractional diffusion and wave equations of the form, which may be of fractional order both in space and time. We give a complete characterization for the positivity of the fundamental solution in terms of the order of the time derivative α∈(0,2), the order of the spatial derivative β∈(0,2], and the spatial dimension d. It turns out that the fundamental solution fails to be positive for all α∈(1,2) and either β∈(0,2] and d ≥ 2 or β<α and d=1, whereas in the other cases, it remains positive. The proof is based on delicate properties of the Fox H‐functions and the Mittag‐Leffler functions. [ABSTRACT FROM AUTHOR]

A widescale description of various phenomena in science and engineering such as physics, chemical, acoustics, control theory, finance, economics, mechanical engineering, civil engineering, and social sciences is well described by nonlinear fractional differential equations (NLFDEs). In turbulence, fluid dynamics, and nonlinear biological systems, applications of NLFDEs can also be found. NLFDEs are believed to be powerful tools to describe real‐world problems more precisely than the differential equation of the integer‐order. In this research, we have used the fractional reduced differential transform method (FRDTM) to find the solution of the time‐fractional Sawada‐Kotera‐Ito seventh‐order equation. The novelty of the FRDTM is that it does not require any discretization, transformation, perturbation, or any restrictive conditions. In addition, compared to other methods, this approach needs less calculation. For special cases of an integer and noninteger orders, computed results are compared with existing results. Present results are in good agreement with the existing solutions. Here, the fractional derivatives are considered in the Caputo sense. Convergence analysis of the results has also been studied with the increasing number of terms of the solution. [ABSTRACT FROM AUTHOR]

Fractional Brusselator reaction‐diffusion system (BRDS) is used for modeling of specific chemical reaction‐diffusion processes. It may be noted that numerous models in nonlinear science are defined by fractional differential equations (FDEs) in which an unknown function appears under the operation of a fractional‐order derivative. Even though many researchers have studied the applicability and practicality of this model, the analytical approach of this model is rarely found in the literature. In this investigation, a novel semi‐analytical technique called fractional reduced differential transform method (FRDTM) has been applied to solve the present model, which is characterized by the time‐fractional derivative (FD). Obtained outcomes are compared with the solution of other existing methods for a particular case. Also, the convergence analysis of this model has been studied here. [ABSTRACT FROM AUTHOR]

We consider the time‐fractional Keller‐Segel system of order α∈(0,1). Interesting properties of solutions are highlighted, like regularity and large time behavior in Lebesgue spaces, which depend on the fractional exponent α. [ABSTRACT FROM AUTHOR]

We investigate a backward problem for the Rayleigh‐Stokes problem, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well‐known to be ill‐posed because of the rapid decay of the forward process. We construct a regularized solution using the filter regularization method in the Gaussian random noise. Under some a priori assumptions on the exact solution, we establish the expectation between the exact solution and the regularized solution in the L2 and Hm norms. [ABSTRACT FROM AUTHOR]

This work presents a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier-Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case. [ABSTRACT FROM AUTHOR]

The well posedness of the evolutive problem for visco-plastic materials represented by two different fractional constitutive equations is proved. We show that, for these materials, we can observe permanent deformations. So that, as it is usual in plasticity, when the stress goes to zero, then the strain assumes a constant nonzero behavior. Moreover, we prove the compatibility of our models with the classical laws of thermodynamics. For the second model, described through a fractional derivative with an exponential kernel, we obtain the exponential decay of the solutions by means of the semigroup theory. [ABSTRACT FROM AUTHOR]

We study the nonexistence of global solutions to the Cauchy problem for systems of time fractional parabolic-hyperbolic and time fractional hyperbolic thermo-elasticity equations in [ABSTRACT FROM AUTHOR]

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L2[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]