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Jan Freundlich - One of the best experts on this subject based on the ideXlab platform.
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Vibrations of a Simply Supported Beam with a Fractional Viscoelastic Material Model – Supports Movement Excitation
Shock and Vibration, 2013Co-Authors: Jan FreundlichAbstract:The paper presents vibration analysis of a simply supported beam with a Fractional order viscoelastic material model. The Bernoulli-Euler beam model is considered. The beam is excited by the supports movement. The Riemann – Liouville Fractional Derivative of order 0 α ⩽ 1 is applied. In the first stage, the steady-state vibrations of the beam are analyzed and therefore the Riemann – Liouville Fractional Derivative with lower terminal at −∞ is assumed. This assumption simplifies solution of the Fractional differential equations and enables us to directly obtain amplitude-frequency characteristics of the examined system. The characteristics are obtained for various values of Fractional Derivative of order α and values of the Voigt material model parameters. The studies show that the selection of appropriate damping coefficients and Fractional Derivative order of damping model enables us to fit more accurately dynamic characteristic of the beam in comparison with using integer order Derivative damping model.
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Vibrations of a simply supported beam with a Fractional Derivative order viscoelastic material model - Supports movement excitation
2012Co-Authors: Jan FreundlichAbstract:The paper presents vibration analysis of a simply supported beam with a Fractional order viscoelastic material model. The Bernoulli-Euler beam model is considered. The beam is excited by the supports movement. The Riemann –Liouville Fractional Derivative of order 0 < α ≤ 1 is applied. In the first stage, the steady-state vibrations of the beam are analyzed and therefore the Riemann –Liouville Fractional Derivative with lower terminal at -∞ is assumed. This assumption simplifies solution of the Fractional differential equations and enables us to directly obtain amplitude-frequency characteristics of the examined system. The characteristics are performed for various values of Fractional Derivative of order a and values of the Voigt material model parameters.
Hans J. Haubold - One of the best experts on this subject based on the ideXlab platform.
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space time Fractional reaction diffusion equations associated with a generalized riemann liouville Fractional Derivative
arXiv: Mathematical Physics, 2014Co-Authors: R K Saxena, Arak M. Mathai, Hans J. HauboldAbstract:Office for Outer Space Affairs, United Nations, P.O. Box 500, Vienna International Centre,Vienna 1400, Austria* Author to whom correspondence should be addressed; E-Mail: hans.haubold@gmail.com;Tel.: +43-676-425-2050; Fax: +43-1-260-605-830.Received: 24 March 2014; in revised form: 8 July 2014 / Accepted: 22 July 2014 /Published: 4 August 2014Abstract: This paper deals with the investigation of the computational solutions of aunified Fractional reaction-diffusion equation, which is obtained from the standard diffusionequation by replacing the time Derivative of first order by the generalized Riemann–LiouvilleFractional Derivative defined by others and the space Derivative of second order by theRiesz–Feller Fractional Derivative and adding a function ˚(x;t). The solution is derivedby the application of the Laplace and Fourier transforms in a compact and closed formin terms of Mittag–Leffler functions. The main result obtained in this paper provides anelegant extension of the fundamental solution for the space-time Fractional diffusion equationobtained by others and the result very recently given by others. At the end, extensions of thederived results, associated with a finite number of Riesz–Feller space Fractional Derivatives,are also investigated.Keywords: Fractional operators; Fractional reaction-diffusion; Riemann-Liouville FractionalDerivative; Riesz-Feller Fractional Derivative; Mittag-Leffler function
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Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann-Liouville Fractional Derivative
Axioms, 2014Co-Authors: Ram K. Saxena, Arak M. Mathai, Hans J. HauboldAbstract:This paper deals with the investigation of the computational solutions of a unified Fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time Derivative of first order by the generalized Riemann–Liouville Fractional Derivative defined by others and the space Derivative of second order by the Riesz–Feller Fractional Derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time Fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space Fractional Derivatives, are also investigated.
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Further solutions of Fractional reaction-diffusion equations in terms of the H-function
Journal of Computational and Applied Mathematics, 2010Co-Authors: Hans J. Haubold, Arak M. Mathai, Ram K. SaxenaAbstract:This paper is in continuation of our earlier paper in which we have derived the solution of a unified Fractional reaction-diffusion equation associated with the Caputo Derivative as the time-Derivative and Riesz-Feller Fractional Derivative as the space-Derivative. In this paper, we consider a unified reaction-diffusion equation with the Riemann-Liouville Fractional Derivative as the time-Derivative and Riesz-Feller Derivative as the space-Derivative. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The results derived are of general character and include the results investigated earlier in [7,8]. The main result is given in the form of a theorem. A number of interesting special cases of the theorem are also given as corollaries.
Ozkan Guner - One of the best experts on this subject based on the ideXlab platform.
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the g g expansion method using modified riemann liouville Derivative for some space time Fractional differential equations
Ain Shams Engineering Journal, 2014Co-Authors: Ahmet Bekir, Ozkan GunerAbstract:Abstract In this paper, the Fractional partial differential equations are defined by modified Riemann–Liouville Fractional Derivative. With the help of Fractional Derivative and traveling wave transformation, these equations can be converted into the nonlinear nonFractional ordinary differential equations. Then G ′ G -expansion method is applied to obtain exact solutions of the space-time Fractional Burgers equation, the space-time Fractional KdV-Burgers equation and the space-time Fractional coupled Burgers’ equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. These results reveal that the proposed method is very effective and simple in performing a solution to the Fractional partial differential equation.
Ram K. Saxena - One of the best experts on this subject based on the ideXlab platform.
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Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann-Liouville Fractional Derivative
Axioms, 2014Co-Authors: Ram K. Saxena, Arak M. Mathai, Hans J. HauboldAbstract:This paper deals with the investigation of the computational solutions of a unified Fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time Derivative of first order by the generalized Riemann–Liouville Fractional Derivative defined by others and the space Derivative of second order by the Riesz–Feller Fractional Derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time Fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space Fractional Derivatives, are also investigated.
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Fractional Helmholtz and Fractional Wave Equations with Riesz-Feller and Generalized Riemann-Liouville Fractional Derivatives
European Journal of Pure and Applied Mathematics, 2014Co-Authors: Ram K. Saxena, Trifce Sandev, Saints CyrilAbstract:The objective of this paper is to derive analytical solutions of Fractional order Laplace, Pois- son and Helmholtz equations in two variables derived from the corresponding standard equations in two dimensions by replacing the integer order partial Derivatives with Fractional Riesz-Feller deriva- tive and generalized Riemann-Liouville Fractional Derivative recently defined by Hilfer. The Fourier- Laplace transform method is employed to obtain the solutions in terms of Mittag-Leffler functions, Fox H-function and an integral operator containing a Mittag-Leffler function in the kernel. Results for Fractional wave equation are presented as well. Some interesting special cases of these equations are considered. Asymptotic behavior and series representation of solutions are analyzed in detail. Many previously obtained results can be derived as special cases of those presented in this paper. 2010 Mathematics Subject Classifications: 26A33, 33E12, 33C60, 76R50, 44A10, 42A38.
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Fractional Helmholtz and Fractional wave equations with Riesz-Feller and generalized Riemann-Liouville Fractional Derivatives
arXiv: Mathematical Physics, 2014Co-Authors: Ram K. Saxena, Zivorad Tomovski, Trifce SandevAbstract:The objective of this paper is to derive analytical solutions of Fractional order Laplace, Poisson and Helmholtz equations in two variables derived from the corresponding standard equations in two dimensions by replacing the integer order partial Derivatives with Fractional Riesz-Feller Derivative and generalized Riemann-Liouville Fractional Derivative recently defined by Hilfer. The Fourier-Laplace transform method is employed to obtain the solutions in terms of Mittag-Leffler functions, Fox $H$-function and an integral operator containing a Mittag-Leffler function in the kernel. Results for Fractional wave equation are presented as well. Some interesting special cases of these equations are considered. Asymptotic behavior and series representation of solutions are analyzed in detail. Many previously obtained results can be derived as special cases of those presented in this paper.
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Further solutions of Fractional reaction-diffusion equations in terms of the H-function
Journal of Computational and Applied Mathematics, 2010Co-Authors: Hans J. Haubold, Arak M. Mathai, Ram K. SaxenaAbstract:This paper is in continuation of our earlier paper in which we have derived the solution of a unified Fractional reaction-diffusion equation associated with the Caputo Derivative as the time-Derivative and Riesz-Feller Fractional Derivative as the space-Derivative. In this paper, we consider a unified reaction-diffusion equation with the Riemann-Liouville Fractional Derivative as the time-Derivative and Riesz-Feller Derivative as the space-Derivative. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The results derived are of general character and include the results investigated earlier in [7,8]. The main result is given in the form of a theorem. A number of interesting special cases of the theorem are also given as corollaries.
Franco Tomarelli - One of the best experts on this subject based on the ideXlab platform.
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Fractional Sobolev Spaces and Functions of Bounded Variation of One Variable
Fractional Calculus & Applied Analysis, 2017Co-Authors: Maitine Bergounioux, Antonio Leaci, Giacomo Nardi, Franco TomarelliAbstract:We investigate the 1D Riemann-Liouville Fractional Derivative focusing on the connections with Fractional Sobolev spaces, the space BV of functions of bounded variation, whose Derivatives are not functions but measures and the space SBV, say the space of bounded variation functions whose Derivative has no Cantor part. We prove that SBV is included in W^{s,1} $ for every s \in (0,1) while the result remains open for BV. We study examples and address open questions.
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Fractional Sobolev Spaces and Functions of Bounded Variation
arXiv: Optimization and Control, 2016Co-Authors: Maitine Bergounioux, Antonio Leaci, Giacomo Nardi, Franco TomarelliAbstract:We investigate the 1D Riemann-Liouville Fractional Derivative focusing on the connections with Fractional Sobolev spaces, the space BV of functions of bounded variation, whose Derivatives are not functions but measures and the space SBV, say the space of bounded variation functions whose Derivative has no Cantor part. We prove that SBV is included in W^{s,1} $ for every s \in (0,1) while the result remains open for BV. We study examples and address open questions.
