<p>This study addresses the inverse problem of identifying the source term in a nonlinear subdiffusion equation governed by the Caputo time-fractional derivative. The nonlinearity arises from the right-hand side’s dependence on the solution itself, in addition to time and spatial variables. The goal is to reconstruct the source coefficient, which varies in both time and space, using measurements in integral form. Unlike prior works that considered source terms dependent solely on time or space, this paper tackles the more general case of combined dependence. Employing the Galerkin method, we seek a weak solution and derive a priori estimates to establish the existence and uniqueness of the solution to the inverse problem. Notably, our results extend to standard diffusion equations, offering novel contributions to the field.</p>

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Source identification problem for a nonlinear subdiffusion equation

  • Ravshan Ashurov,
  • Oqila Mukhiddinova

摘要

This study addresses the inverse problem of identifying the source term in a nonlinear subdiffusion equation governed by the Caputo time-fractional derivative. The nonlinearity arises from the right-hand side’s dependence on the solution itself, in addition to time and spatial variables. The goal is to reconstruct the source coefficient, which varies in both time and space, using measurements in integral form. Unlike prior works that considered source terms dependent solely on time or space, this paper tackles the more general case of combined dependence. Employing the Galerkin method, we seek a weak solution and derive a priori estimates to establish the existence and uniqueness of the solution to the inverse problem. Notably, our results extend to standard diffusion equations, offering novel contributions to the field.