1,128 research outputs found
A fractional variational approach to the fractional basset-type equation
No full textIn this paper we discuss an application of fractional variational calculus to the Basset-type fractional equations. It is well known that the unsteady motion of a sphere immersed in a Stokes fluid is described by an integro-differential equation involving derivative of real order. Here we study the inverse problem, i.e. We consider the problem from a Lagrangian point of view in the framework of fractional variational calculus. In this way we find an application of fractional variational methods to a classical physical model, finding a Basset-type fractional equation starting from a Lagrangian depending on derivatives of fractional order. © 2013 Polish Scientific Publishers
Fractional Unit-Root Tests Allowing for a Fractional Frequency Flexible Fourier Form Trend: Predictability of Covid-19
No full textBaleanu, Dumitru/0000-0002-0286-7244; Omay, Tolga/0000-0003-0263-2258In this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unit-root test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its small-sample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid-19 pandemic
Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators
No full textA new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense
On Fractional <i>schrodinger</I> Equation in Α-Dimensional Fractional Space
No full textBaleanu, Dumitru/0000-0002-0286-7244The Schrodinger equation is solved in a-dimensional fractional space with a Coulomb potential proportional to 1/r(beta-2), 2 <= beta <= 4. The wave functions are studied in terms of spatial dimensionality alpha and beta and the results for beta = 3 are compared with those obtained in the literature. (C) 2008 Elsevier Ltd. All rights reserved.Scientific and Technical Research Council of TurkeyThis work is partially supported by the Scientific and Technical Research Council of Turkey
Numerical simulation of a fractional mathematical model for epidermal wound healing
No full textA number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider numerical simulation of fractional model based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in advection and diffusion terms belong to the intervals (0; 1) or (1; 2], respectively. Some numerical techniques will be used. \ud
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Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. \ud
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Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of the Riemann-Liouville and Gr¨unwald-Letnikov fractional derivative definitions. \ud
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Thirdly, its stability and convergence are discussed, respectively. \ud
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Finally, some numerical results are given to demonstrate the theoretical analysis
Analytical solutions for the two- and three-dimensional time-fractional telegraph equations by the method of separating variables
No full textIn this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) in two and three dimensions. We discuss and derive the analytical solution of the TFTE in two and three dimensions with nonhomogeneous Dirichlet boundary condition. This method can be extended to other kinds of the boundary conditions
The Galerkin finite element approximation of the fractional cable equation
No full textThe cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed.\ud
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The main contribution of this work is outlined as follow:\ud
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• A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt).\ud
• A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2).\ud
• Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated.\ud
• Finally, some numerical results are given to demonstrate the theoretical analysis.\u
A freely damped oscillating fractional dynamic system modeled by fractional Euler-Lagrange equations
No full textBaleanu, Dumitru/0000-0002-0286-7244The behaviors of some vibrating dynamic systems cannot be modeled precisely by means of integer representation models. Fractional representation looks like it is more accurate to model such systems. In this study, the fractional Euler-Lagrange equations model is introduced to model a fractional damped oscillating system. In this model, the fractional inertia force and the fractional damping force are proportional to the fractional derivative of the displacement. The fractional derivative orders in both forces are considered to be variable fractional orders. A numerical approximation technique is utilized to obtain the system responses. The discretization of the Coimbra fractional derivative and the finite difference technique are used to accomplish this approximation. The response of the system is verified by a comparison to a classical integer representation and is obtained based on different values of system parameters
Numerical treatment of a two-dimensional variable-order fractional nonlinear reaction-diffusion model
No full textA two-dimensional variable-order fractional nonlinear reaction-diffusion model is considered. A second-order spatial accurate semi-implicit alternating direction method for a two-dimensional variable-order fractional nonlinear reaction-diffusion model is proposed. Stability and convergence of the semi-implicit alternating direct method are established. Finally, some numerical examples are given to support our theoretical analysis. These numerical techniques can be used to simulate a two-dimensional variable order fractional FitzHugh-Nagumo model in a rectangular domain. This type of model can be used to describe how electrical currents flow through the heart, controlling its contractions, and are used to ascertain the effects of certain drugs designed to treat arrhythmia
The Dynamics of Shear-Type Frames Equipped with Chain-Based Nonlinear Braces
No full textIn recent years a number of bracing devices have been proposed,
analyzed, and applied to real cases, since in engineering applications the construction
of frames equipped with braces is a widespread practice. In the present
contribution, a nonlinear bracing system is introduced and applied to the case of
shear-type moment-resistant frames. The frame is considered here as the primary
structure and is assumed to have linear elastic behavior and the bracing system is
considered as a secondary, additional structure. The bracing system is made of two
chains, each of them constructed as the assemblage of two axial elements (springs)
undergoing axial force, only. The springs that are assumed to have linear elastic
behavior are connected to each other in the chain and to the frame through hinges.
The global behavior of the system is nonlinear, since the restoring force of the
bracing system is a piecewise-defined function. In order to asses the performance of
the whole nonlinear system, its behavior is compared with that of the linear primary
structure alone, through a suitable concise descriptor
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