Constructing Solutions to Multi-Term Cauchy–Euler Equations with Arbitrary Fractional Derivatives

Abstract

We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations ∑i=1mdixαiDαiu(x)+μu(x)=0,αi>0, with the derivatives in Caputo or Riemann–Liouville sense. Unlike the existing works, we consider multi-term equations without any restrictions on the order of fractional derivatives. The results are based on the characteristic equations which generate the solutions. Depending on the roots of the characteristic equations (real, multiple, or complex), we construct the corresponding solutions and prove their linear independence.

Keywords

• fractional derivative
• fractional Cauchy–Euler equation
• characteristic equation