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# Article

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Keywords:
inverse source problem; space fractional diffusion equation; weak solution theory; adjoint problem; Lipschitz continuity
Summary:
We consider the problem of determining the unknown source term $f=f(x,t)$ in a space fractional diffusion equation from the measured data at the final time $u(x,T) = \psi (x)$. In this way, a methodology involving minimization of the cost functional $J(f) = \int _0^l ( u(x,t;f )|_{t = T} - \psi (x)) ^2 {\rm d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence $\{J'(f^{(n)}) \}$, where $f^{(n)}$ is the $n$th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.
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