Abstract:
We consider an inverse problem on determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative. The right-hand side of the equation has the form f(x)g(t) and the unknown is the function f(x). The condition u(x,t0)=ψ(x) is taken as the over-determination condition, where t0 is some interior point of the considered domain and ψ(x) is a given function. By the Fourier method we show that under certain conditions on the functions g(t) and ψ(x) the solution of the inverse problem exists and is unique. We provide an example showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions g(t). For such functions g(t) we find necessary and sufficient conditions on the initial function and on the function from the over-determination condition, which ensure the existence of a solution to the inverse problem.
Keywords:
subdiffusion equation, forward and inverse problems, the Caputo derivatives, Fourier method.
Citation:
R. R. Ashurov, M. D. Shakarova, “Inverse problem for subdiffusion equation with fractional Caputo derivative”, Ufa Math. J., 16:1 (2024), 112–126