The fractional d'Alembert's formulas (Q2329292)

The authors present generalized d'Alembert formulas for the solutions of the abstract fractional integro-differential equations \[u(t)=\phi+\displaystyle\frac{t^{\alpha/2}}{\Gamma\left(1+\frac{\alpha}{2}\right)}\psi+\displaystyle\frac{1}{\Gamma(\alpha)}\displaystyle\int_0^t (t-s)^{\alpha-1}A^2u(s)\,ds,\,\,\,t>0,\] and \[u(t)=\phi+\displaystyle\frac{t^{\beta}}{\Gamma(1+\beta)}\psi+\displaystyle\frac{1}{\Gamma(\alpha)}\displaystyle\int_0^t (t-s)^{\alpha-1}A^2u(s)\,ds,\,\,\,t>0,\] where \(A\) is a closed linear densely defined operator on the Banach space \(X\), \(1<\alpha\le 2\) and \(\frac{\alpha}{2}\le\beta\le \alpha\). D'Alembert formula for the sequential fractional telegraph equation with initial conditions \[\left\{\begin{array}{l} D_t^{\alpha/2}D_t^{\alpha/2}u(t)+2h D_t^{\alpha/2}u(t)=Au(t),\,\,\,t>0,\\ u(0)=\phi,\,\,\,D_t^{\alpha/2}u(0)=\psi, \end{array}\right.\] where \(1<\alpha\le 2\) and \(D_t^{\alpha}u\) is the \(\alpha\)-order Caputo fractional derivative of function \(u\), is also provided. Some examples which illustrate the obtained results are finally given.