Numerical Analysis of a Fractional Variational Problem.

LIC. BARRIOS MELANI; DRA. REYERO GABRIELA; DR. LOMBARDI ARIEL

Bilbao

Otro; Fractional and Other Nonlocal Models.; 2018

The realm of numerical methods in scientific fields is vastly growing due to the very fast progresses in computational sciences and technologies. Nevertheless, the intrinsic complexity of fractional calculus, caused partially by non-local properties of fractional derivatives and integrals (\cite {Die}), makes it rather difficult to find efficient numerical methods in this field.\\Fractional derivatives and integrals were introduced more than three centuries ago, but only recently have they gained more attention due to its applications. Motivated by numerous applications in physics and other scientific areas, fractional calculus of variations finds itself in fast development. In this work we consider variational problems of the form:\[J(y)=\int_{a}^{b}L(x,y,y',\,_{a}^{C}D_{x}^{\alpha }y)\,dx\]where in the Lagrangian function $L$ appears $\,_{a}^{C}D_{x}^{\alpha }y$, which is the left Caputo fractional derivative of order $\alpha>0$ (\cite{Agr2},\cite{AlTo},\cite{OdMaTo}). Also considered is the fractional Euler-Lagrange equation that depends only on the derivative of Caputo (\cite {BaRe}, \cite {LaTo}). \\A new numerical method based on the method L1 (\cite{Li}) is presented here to obtain approximations to the solutions of these types of problems and even though it is more complicated than in the classical case, it still inherits some sort of simplicity and an ease of implementation (\cite{PoAlTo}).\\Two test examples were considered: \begin{center}$min \rightarrow J(y)=\int_{0}^{1}\left[\,_{0}^{C}D_{x}^{\alpha}[y](x)-(y'(x))^2\right] dx$\\$y(0)=0$ \hspace{0,25cm} $y(1)=1$.\\\end{center}And\begin{center}$min \rightarrow J(y)=\int_{0}^{1}\left[y(x)(y'(x)+1)+ \,_{0}^{C}D_{x}^{\alpha}[y]^{2}(x)\right] dx$\\$y(0)=0$ \hspace{0,25cm} $y(1)=0$.\\\end{center}Both were solved by implementing the method through the Matlab software.