Weakly perturbed linear boundary-value problem for system of fractional differential equations with Caputo derivative

Abstract

We consider a perturbed linear boundary-value problem for a system of fractional differential equations with Caputo derivative. The boundary-value problem is specified by a linear vector functional, the number of components of which does not coincide with the dimension of the system of differential equations. This formulation of the problem is being considered for the first time and includes both underdetermined and overdetermined boundary-value problems. Under the condition that the solution of the homogeneous generating boundary-value problem is not unique and that the inhomogeneous generating boundary-value problem is unsolvable, the conditions for the bifurcation of solutions of this problem are determined. An iterative procedure for constructing a family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter ɛ with singularity at the point ɛ=0 is proposed. The results obtained by us generalize the known results of perturbation theory for boundary-value problems for ordinary differential equations.

Keywords

• Boundary-value problem
• Fractional differential equation
• Caputo derivative
• Pseudoinverse Moore–Penrose matrix
• Bifurcation