An approximation of the analytical solution of the fractional Euler-Lagrange equation

Abstract

Abstract. In this paper the fractional Euler-Lagrange equation of order α ∈ (0, 1] in the finite time interval is considered. This equation is transformed to the integral form by the use of the fractional integral operators. Next, the numerical approximation of the analytical solution is presented. Finally, some examples of numerical solutions are presented.

Fractional oscillator equation – Transformation into integral equation and numerical solution

“…One of the new directions in fractional calculus and its applications is to investigate the numerical solutions of fractional differential equations, containing composition of the left and the right derivatives [2,3,5,[7][8][9][10]13,20,27,31]. This problem is an important area of investigations in fractional differential equations theory.…”

Fractional heat conduction in a sphere under mathematical and physical Robin conditions

“…The properties of fractional derivatives and different analytical methods to solve fractional differential equations are presented in (Atanacković et al, 2014;Klimek, 2009;Leszczyński, 2011;Magin, 2006;Mainardi, 2010;Povstenko, 2015). Approximate numerical methods were applied to solving fractional initial-boundary problems in numerous papers, for example in (Blaszczyk and Ciesielski, 2017;Ciesielski and Błaszczyk, 2013;Dimitrov, 2014).…”

Fractional oscillator equation: analytical solution and algorithm for its approximate computation

“…In our previous work (Ciesielski and Blaszczyk, 2013) we investigated the solution of the fractional oscillator equation dealing with the approximation of the analytical solution of this equation that is based on the numerical evaluation of fractional integrals. In this work we use a similar method (we applied more accurate numerical schemes) and extend our results.…”