Direct and some inverse problems for a generalized diffusion equation with variable coefficients

Direct and two inverse problems for a Legendre equation involving integral convolution in time are studied. The inverse problems are ill-posed in the sense of Hadamard. The analytical series solutions of the problems are constructed by using method of variable separation. The determination of only u(x, y) is studied in the direct problem, the recovery of a pair of functions, i.e., \(\{u(x,y), f(x)\}\) $\{u(x,y),f(x)\}$ with appropriate addition data at some T is investigated in the 1st inverse problem while the identification of a pair of functions, i.e., \(\{u(x,y), q(y)\}\) $\{u(x,y),q(y)\}$ with an integral type data is considered in the 2nd inverse problem. By imposing certain regularity conditions, the unique existence of series solutions is developed. We provided some numerical examples to illustrate our results for the inverse problems.