The Cauchy problem for a fourth-order equation involving a fractional derivative in the Caputo sense

In this article, the Cauchy problem in a half-plane is studied for a fourth-order inhomogeneous equation with a fractional derivative in the Caputo sense. The uniqueness of the solution is demonstrated using the Laplace transform. In constructing the solution, partial solutions expressed in terms of Wright functions are first found. Green’s functions are then constructed using these partial solutions. The solution is constructed explicitly using the Green function. An explicit form of the fundamental solution is also obtained.

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