On Some Fractional-Order Equations of Evolution
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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...
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.
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