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Analysis and Applications of Composed Forms of Caputo Fractional Derivatives

Błaszczyk T., Kotela E., Hall M. R., Leszczyński J.

Acta Mechanica et Automatica

2011

Vol. 5, no. 2

11-14

In this paper we consider two ordinary fractional differential equations with composition of the left and the right Caputo derivatives. Analytical solution of this type of equations is known for particular cases, having a complex form, and therefore is difficult in practical calculations. Here, we present two numerical schemes being dependent on a fractional order of equation. The results of numerical calculations are compared with analytical solutions and then we illustrate convergence of our schemes. Finally, we show an application of the considered equation.

Positive and bounded below solutions for certain nonlinear fractional differential equations

Klimek M., Błasik M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2010

Vol. 9, nr 2

91-102

Two types of one-term nonlinear fractional differential equations are considered and the existence of solutions in the space of continuous, positive and bounded below functions is proved. We transform an equation containing the left- or right-sided Caputo derivative into a fixed point condition and apply the Banach theorem and extended Bielecki method of equivalent norms.

On differentiable solutions for one-term nonlinear fractional differential equations

Klimek M., Lupa M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2010

Vol. 9, nr 2

103-113

Two one-term nonlinear fractional differential equations with the left- or rightsided Caputo derivative are discussed. The existence and uniqueness of solutions, generated by the respective stationary function, is proved in the space of continuously differentiable function. The proof, based on the Banach theorem, includes the extension of the Bielecki method of equivalent norms.