Mohamed Herzallah

On Some Fractional-order Equations of Evolution Mohamed Herzallah

In this thesis we study, under certain conditions, the existence of a unique solution of the nonhomogeneous fractional order evolution equation D^? u(t)=Au(t)+f(t),u(0)=u_o,t?J=[0, T],??(0,1), the nonhomogeneous fractional order evolutionary integral equation D^? u(t)=f(t)+?_0^t?? h(t-s) Au(s)ds,u(0)=u_o,??(0,1),t?J=[0, T] and the nonhomogeneous fractional order evolutionary integro-differential equation D^? u(t)=?Au(t)+?_0^t?? k(t-s) Au(s)ds+f(t), u(0)=x,u'(0)=y,??(1,2),??0, where A is a closed linear operator with dense domain D(A)=X_A in the Banach space X. Also we prove the continuation properties of the solution u_? (t) and its fractional derivative D^? u_? (t) in the first two problems as ??1^- and in the third problem we prove the continuation properties of the solution u_? (t) and its fractional drerivative D^? u_? (t) as ??1^+ and as ??2^-. Finally we prove the maximal regularity property of the solution of each problem and give some examples of the three problems.