Parabolic Transform and Some Ill-posed Problems

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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.

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