Numerical solution of composite left and right fractional Caputo derivative models for granular heat flow

European Journal of Mechanics - A/Solids

Liebold³

2015

In this paper the classical EULER-BERNOULLI beam (CEBB) theory is reformulated utilising fractional calculus. Such generalisation is called fractional Euler-Bernoulli beams (FEBB) and results in non-local spatial description. The parameters of the model are identified based on AFM experiments concerning bending rigidities of micro-beams made of the polymer SU-8. In experiments both force as well as deflection data were recorded revealing significant size effect with respect to outer dimensions of the specimens. Special attention is also focused on the proper numerical solution of obtained fractional differential equation.

Section: Introductionmentioning

confidence: 99%

Fractional Euler–Bernoulli beams: Theory, numerical study and experimental validation

Sumelka

European Journal of Mechanics - A/Solids

Liebold³

2015

“…The fractional differential equations, both ordinary and partial ones, are very useful tools for modelling many phenomena in physics, mechanics, control theory, biochemistry, bioengineering and economics [10,12,21,22,24,38]. Therefore, the theory of fractional differential equations is an area that has developed extensively over the last decades.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of fractional Sturm-Liouville equation in integral form

Ciesielski

2014

fcaa

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In this paper a fractional differential equation of the Euler-Lagrange/ Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.MSC 2010 : Primary 26A33; Secondary 34A08, 65L10

“…This is also a fundamental problem in finding solution of variational differential equations [3], [17], [21]. Recently many researchers have focused their attention on the numerical solution of equations containing integral or differential operators [5], [6], [8], [9], [7], [10], [11], [12], [13], [14], [27].…”

Section: Introductionmentioning

confidence: 99%

Application of the Rayleigh-Ritz method to solve a class of fractional variational problem

2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR)

2015

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The aim of this paper is to investigate a direct numerical method for solving a class of a fractional variational problem (FVP). The fractional derivative in the FVP is in the Caputo sense. The Rayleigh-Ritz method is introduced for the numerical solution of the FVP. The corresponding Euler-Lagrange equation contain the left and right Caputo fractional derivatives. We approximate the solution of the FVP by using the trigonometric function. Finally, the illustrative example is presented.