Terms Of Fractional Derivatives Research Articles

Existence of positive solutions for boundary value problems of nonlinear fractional functional integro-differential equations

In this paper, we study the existence of positive solutions for a class of fractional functional integro-differential equations with two fractional derivative terms. First, we transform the boundary value problem into an equivalent integral equation, establish the operator T and prove its full continuity, then the existence theorems of positive solutions of boundary value problems is established by using the fixed point theorems of cone extension and cone compression.

Stochastic bifurcation analysis of a bistable Duffing oscillator with fractional damping under multiplicative noise excitation

The stochastic P-bifurcation behavior of bi-stability in a Duffing oscillator with fractional damping under multiplicative noise excitation is investigated. Firstly, in order to consider the influence of Duffing term, the non-linear stiffness can be equivalent to a linear stiffness which is a function of the system amplitude, and then, using the principle of minimal mean square error, the fractional derivative term can be equivalent to a linear combination of damping and restoring forces, thus, the original system is simplified to an equivalent integer order Duffing system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging, and then according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis, and the method used in this paper can directly guide the design of the fractional order controller to adjust the behaviors of the system.

On the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations

<abstract> In this paper, we investigate the numerical solutions of coupled nonlinear time-fractional reaction-diffusion equations obtained by applying a procedure that combines the Lie symmetry analysis with the numerical methods. By Lie symmetries, the model, governed by two fractional differential equations defined in terms of the Riemann-Liouville fractional derivatives, is reduced into nonlinear fractional ordinary differential equations that, by introducing the Caputo derivative, are numerically solved by the implicit trapezoidal method. The solutions of the original model are computed by the inverse transformations. Numerical examples are performed in order to show the efficiency and the reliability of the proposed approach applied for solving a wide class of fractional models. </abstract>

On numerical approximation of Atangana-Baleanu-Caputo fractional integro-differential equations under uncertainty in Hilbert Space

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations. Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

Abstract The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 &amp;lt; β &amp;lt; 1) and $_{ - \infty }D_t^\alpha x$ (1 &amp;lt; α &amp;lt; 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 &amp;lt; β &amp;lt; 1) and α (1 &amp;lt; α &amp;lt; 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.

Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

Abstract In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the L ⁢ 2 - 1 σ {L2-1_{\sigma}} format on non-uniform meshes, with σ = 1 - α 2 {\sigma=1-\frac{\alpha}{2}} , while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering k = 3 , 4 , 5 {k=3,4,5} , we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders O ⁢ ( N - min ⁡ { k ⁢ α , 2 } ) {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms

The paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.

Analysis of Legendre pseudospectral approximations for nonlinear time fractional diffusion-wave equations

A finite difference/pseudospectral scheme is developed for solving nonlinear time fractional parabolic equations with Caputo fractional derivative of order . The boundedness and unique solvability of numerical solution are given. Then we prove rigorously the unconditional stability and convergence of the fully discrete scheme, where the optimal error estimate in norm is obtained. Furthermore, an improved scheme by adding correction terms is proposed to deal with the weak singularity, which makes the approximations of fractional derivative and nonlinear term exact or sufficiently accurate for the weak singular parts of solutions. Numerical experiments are provided to show the sharpness of the error analysis.

Fractional convection-dispersion equation with conformable derivative approach

Abstract In this present work, a well-structured and limit-based derivative definition of fractional derivative term, known as conformable derivative, is employed to develop a local generalized form of time variable fractional convection-dispersion equation (FCDE). The fractional models of convection-dispersion equation have been widely established as more authentic mode to characterize the pollutant transport in geological structures. In this work, models are formulated corresponding to spatial dependency of velocity and dispersion coefficient and also for the temporally varying decay rate, that dealt with more realistic phenomenon of the pollutant transport in groundwater reservoir. Due to the non-linearity of the problem, homotopy analysis method (HAM) is adopted to investigate the complex solutions of FCDE. As the FCDE comprises the classical convection-dispersion equation (CDE) as a special case corresponding to the fractional order α = 1 , so the obtained solutions are validated by the corresponding numerical solution and exact analytical solution for α = 1 and solutions are also verified for different fractional values of α . The effect of conformable derivative order α is properly visible over concentration strength. Hence, this investigation helps to interpret the accurate description of time dependent behaviour of contaminant transport in porous structure.

Time-dependent performance of damaged marble and corresponding fractional order creep constitutive model

The time-dependent behaviour of a rock mass is crucial to the long-term stability of underground engineering projects, such as large tunnels, caverns and roadways. This paper focused on the experimental and theoretical studies of marble creep behaviour considering its initial damage due to excavation damage. Based on the onsite investigation of the creep performance of Jinping marble, detailed creep tests were carried out on damaged marble under different confining stresses, which indicated that damaged marble exhibited more notable creep properties than the intact marble. A new time-dependent creep constitutive model was proposed to characterize the time-dependent mechanical behaviour of damaged marble in terms of fractional derivatives and damage variables by replacing the Newtonian dashpot with the Abel dashpot. Furthermore, this one-dimensional creep equation was extended to a three-dimensional numerical model for analyzing the long-term deformation of hydraulic caverns. The simulation results indicated that the proposed model provided a precise description for the full creep process of marble containing initial damage.

Fractional Abstract Differential Equations and Applications

Boundary value problems for fractional elliptic equations with parameter in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for elliptic equations in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operators generated by these equations are positive and also are generators of the analytic semigroups. Moreover, maximal regularity properties of the fractional abstract parabolic equation are established. As an application, the parameter-dependent anisotropic fractional differential equations and the system of fractional differential equations are studied.

Why Fractional Derivatives with Nonsingular Kernels Should Not Be Used

Abstract In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

Nonlocal Fractional Differential Equations and Applications

The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional differential equations are studied.

Spectral identification of nonlinear multi-degree-of-freedom structural systems with fractional derivative terms based on incomplete non-stationary data

A novel spectral identification technique is developed for determining the parameters of nonlinear and time-variant multi-degree-of-freedom (MDOF) structural systems based on available input-output (excitation-response) realizations. A significant advantage of the technique relates to the fact that it can readily account for the presence of fractional derivative terms in the system governing equations, as well as for the cases of non-stationary, incomplete and/or noise corrupted data. Specifically, the technique relies on recasting the governing equations as a set of multiple-input/multiple-output systems in the wavelet domain. Next, an l1-norm minimization procedure based on compressive sampling theory is employed for determining the wavelet coefficients of the available incomplete non-stationary input-output data. Finally, these wavelet coefficients are utilized to reconstruct the non-stationary incomplete signals, and consequently, to determine system related time- and frequency-dependent wavelet-based frequency response functions and associated parameters. Two illustrative MDOF systems are considered in the numerical examples for demonstrating the reliability of technique. The first refers to a nonlinear time-variant system with fractional derivative terms, while the second addresses a nonlinear offshore structural system subjected to flow-induced forces. It is worth noting that for the offshore system, a novel recently proposed evolutionary version of the widely used JONSWAP spectrum is employed for modeling the non-stationary free-surface elevation in cases of time-dependent sea states.

On the application of fractional calculus for the formulation of viscoelastic Reddy beam

The focus of the current work is to present the bending analysis of visco-elastic beams based on Reddy’s third-order shear deformation theory. Fractional calculus is taken into account for dealing with the fractional derivative terms, able to better describe the damping behaviour of any visco-elastic material. Numerical analyses of beams with different boundary conditions have been proposed and discussed following two different approaches, namely the finite element method and the Galerkin method. An assessment of the proposed approach is presented by comparing the computed solutions with those obtained with the classical and first-order shear deformation theories available in the literature.

Reflection properties of zeta related functions in terms of fractional derivatives

Abstract We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ζ(n, a) of integer first argument can be extended to the more general case of ζ(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ζ(s, a) with ℜ(s) &gt; 2 is essentially the Weyl (s − 2)-derivative of ζ(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch’s transcendent.

Duhem histeresisin Riemann-Liouville kesirli türevi ile modellenmesi

In the view of memory effect of hysteresis, this work aims to interpret hysteresis nonlinearities in terms of Riemann-Liouville fractional derivative which is a singular operator with memory and hereditary properties. For this purpose, Duhem hysteresis, a model defined by a first order differential equation, is considered and adapted to a fractional order differential equation. Since the fractional order Duhem hysteresis cannot be solved by an analytical scheme, Grünwald-Letnikov approximation is used to obtain numerical solutions. Thus, the effect of fractional order derivative to Duhem hysteresis is demonstrated with graphics obtained by this approximation and plotting using MATLAB. As a result, it is observed that the fractional order model exhibits hysteresis behavior for the orders that are smaller than 1.

A new model of dengue fever in terms of fractional derivative.

It is eminent that the epidemiological patterns of dengue are threatening for both the global economy and human health. The experts in the field are always in search to have better mathematician models in order to understand the transmission dynamics of epidemics models and to suggest possible control or the minimization of the infection from the community. In this research, we construct a new fractional-order system for dengue infection with carrier and partially immune classes to visualize the intricate dynamics of dengue. By using the basics of fractional theory, we determine the fundamental results of the proposed fractional-order dengue model. We obtain the basic reproduction number $R_0$ by next generation method and present the results based on it. The stability results are established for the infection-free state of the system. Moreover, sensitivity of $R_0$ is analyzed through partial rank correlation coefficient(PRCC) method to show the importance of different parameters in $R_0$. The influence of different input factors is shown on the output of basic reproduction number $R_0$ numerically. Our result showed that the threshold parameter $R_0$ can be decreased by increasing vaccination and treatment in the system. Finally, we illustrate the solution of the suggested dengue system through a numerical scheme to notice the influence of the fractional-order $\vartheta$ on the system. We observed that the fractional-order dynamics can explain the complex system of dengue infection more precisely and accurately rather than the integer-order dynamics. In addition, we noticed that the index of memory and biting rate of vector can play a significant part in the prevention of the infection.

&lt;i&gt;S&lt;/i&gt;-function associated with fractional derivative and double Dirichlet average

The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.

Finite difference method for a nonlinear fractional Schrödinger equation with Neumann condition

Abstract In this paper, a special case of nonlinear fractional Schrödinger equation with Neumann boundary condition is considered. Finite difference method is implemented to solve the nonlinear fractional Schrödinger problem with Neumann boundary condition. Previous theoretical results for the abstract form of the nonlinear fractional Schrödinger equation are revisited to derive new applications of these theorems on the nonlinear fractional Schrödinger problems with Neumann boundary condition. Consequently, first and second order of accuracy difference schemes are constructed for the nonlinear fractional Schrödinger problem with Neumann boundary condition. Stability analysis show that the constructed difference schemes are stable. Stability theorems for the stability of the nonlinear fractional Schrödinger problem with Neumann boundary condition are presented. Additionally, applications of the new theoretical results are presented on a one dimensional nonlinear fractional Schrödinger problem and a multidimensional nonlinear fractional Schrödinger problem with Neumann boundary conditions. Numerical results are presented on one and multidimensional nonlinear fractional Schrödinger problems with Neumann boundary conditions and different orders of derivatives in fractional derivative term. Numerical results support the validity and applicability of the theoretical results. Numerical results present the convergence rates are appropriate with the theoretical findings and construction of the difference schemes for the nonlinear fractional Schrödinger problem with Neumann boundary condition.