A strong solution of a high-order mixed type partial differential equation with integral conditions

An Integral Two Space-Variables Condition for Parabolic Equations

“…Boundary-value problems for parabolic equations with integral boundary condition are investigated by Batten (1963); Bouziani and Benouar (1998); Cannon (1963);(1984); Cannon et al (1987); Ionkin (1977); Kamynin (1964); Field and Komkov (1992); Shi (1993); Marhoune and Bouzit (2005); Marhoune and Hameida (2008); Denche et al (1994); Denche and Marhoune (2001); Marhoune and Latrous (2008); Yurchuk (1986) and many references therein. The problem with integral one space-variable (respectively two space-variables) condition is studied in Fairweather and Saylor (1991) and Denche and Marhoune (2000) (respectively in Marhoune (2007) and Marhoune and Lakhal (2009)).…”

On a Class of Caputo Time Fractional Problems with Boundary Integral Conditions

“…For example the case m = 1 in [1] and [8]. Other authors studied the integer order case ( [2], [4] and [5]). We started by studying the time fractional problem in the Caputo sense applying the energy inequality method which is a powerful functional analysis method for showing problems wellposedness, it is mainly based on an a priori estimate to prove uniqueness of the strong solution, meanwhile the range density of the operator generated by the problem ensures its existence.…”