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Analysis and Applications of Composed Forms of Caputo Fractional Derivatives

100%

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

tom 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.

Using the Euler's method to solve ordinary differential equations of higher order with a mixture of integer and Caputo derivatives

100%

Błaszczyk T. , Leszczyński J. S.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2007

tom Vol. 6, nr 1

31-40

In this paper we present an application of the Euler's method to the numerical solution of fractional ordinary differential equations. These equations include both a classical differential operator of integer order and the fractional one defined in the Caputo sense. Our previous work was limited to the order of fractional derivative α ∈ (0,1) . This study considers numerical schemes for higher orders of a fractional derivative. We then compare our schemes with analytical solutions in order to show their good numerical precision.

Positive and bounded below solutions for certain nonlinear fractional differential equations

88%

Klimek M. , Błasik M.

tom 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

88%

Klimek M. , Lupa M.

tom 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.