Parabolic Transform and Some Ill-posed Problems

A new transform is constructed, which is called parabolic. By using this transform, existence and stability
results can be obtained for singular integro-partial differential equations and also for stochastic ill-posed
problems. It is well known that the cauchy problem for elliptic partial differential equations is ill-posed. The
question, which arises, how a priori knowledge about solutions and the set of initial conditions can bring about
stability? With the help of the parabolic tranform, we can study, not only elliptic partial differential equations,
but also a general stochastic partial differential equations and singular integro-partial differential equations
without any restrictions on the charachtrestic forms of the partial differential operators. The cauchy problem of
fractional general partial differential equations can be considered as a special case from the obtained results. In
addition, Hilfer fractional differential equations can be solved also without any restrictions on the charachtrestic
forms. Many physical and engineering problems in areas like biology, seismology, and geophysics require the
solutions of ill-posed stochastic problems and general singular integro-partial differential equations.