A Non-Local Problem for the Benney–Luke Type Fractional Order Equation

In the paper, we consider the problem of finding a solution to a fractional order equation of the form $$D^{\alpha}_{t}u(t)+A\left(D^{\alpha}_{t}u(t)\right)+A^{2}\left(D^{\alpha}_{t}u(t)\right)+Au(t)=f$$ , $$0<\alpha<1$$ , and $$0<t<T$$ , satisfying the non-local condition $$u(T)=au(+0)+\varphi$$ . Here $$a$$ and $$T$$ are given numbers, $$A:H\rightarrow H$$ be a self-adjoint, unbounded, and positive operator defined on a separable Hilbert space $$H$$ . In this work, we examine the role of the parameter $$a$$ in determining the existence and uniqueness of solutions to the associated problem. Furthermore, we consider the inverse problem of reconstructing the right-hand side of the equation based on additional information about the solution.

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