arXiv subjects

Tomasz Blaszczyk

Publications and source records attributed to Tomasz Blaszczyk.

3 recordsLinked to original sources

arXiv2015

Numerical solution of fractional Sturm-Liouville equation in integral form

In this paper a fractional differential equation of the Euler-Lagrange / Sturm-Liouville type is considered. The fractional equation with derivatives of order $α\in \left( 0,1 \right]$ 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.

math.NA↗

arXiv2015

Fractional Euler-Bernoulli beams: theory, numerical study and experimental validation

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.

math-ph↗

arXiv2007

A novel numerical technique used in the solution of ordinary differential equations with a mixture of integer and fractional derivatives

Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some algorithms where four discrete forms of the Caputo derivative and three different numerical techniques of solving ordinary differential equations are proposed. We then illustrate how to introduce classical initial conditions into equations where the Riemann-Liouville derivative is included.

math.NA↗