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Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition. (English) Zbl 1218.35245
Summary: Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data are continuous, the initial data are continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.
MSC:
35R11Fractional partial differential equations
45D05Volterra integral equations
35C15Integral representations of solutions of PDE
35D30Weak solutions of PDE