Variational Formulations for Fractional Integral and Differential Equations

Given a fractional-order linear equation $L^{\alpha }u=f$, we define an appropriate symmetric bilinear form so that the fractional operator $L^{\alpha }$ is symmetric with respect to that bilinear form. Using the bilinear form, we then define a functional of the fractional calculus of variations proving that the solutions of the given fractional-order equation are critical points of the fractional variational functional. In the case of fractional integral equations, the provided bilinear form is non-degenerate, and all critical points are solutions of the given equation. In the case of fractional differential equations, a relation with the least-squares method is obtained.