Parabolic Transform and Some Ill-posed Problems | Chapter 09 | Advances in Mathematics and Computer Science Vol. 3
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.
Author(s) Details
Professor Mahmoud M.
El-Borai
Department of Mathematics
and Computer Science, Faculty of Science, Alexandria University, Alexandria,
Egypt.
Professor Khairia El-said
El-Nadi
Department of Mathematics
and Computer Science, Faculty of Science, Alexandria University, Alexandria,
Egypt.
View Volume: https://doi.org/10.9734/bpi/amacs/v3
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