Geodesic
Well-Posedness of the Cauchy Problem for Inhomogeneous Time-Fractional Pseudo-Differential Equations
Fractional calculus and applied analysis, Tome 9 (2006) no. 1, pp. 01-16

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In the present paper the Cauchy problem for partial inhomogeneous pseudo-differential equations of fractional order is analyzed. The solvability theorem for the Cauchy problem in the space ΨG,2(R^n) of functions in L2(R^n) whose Fourier transforms are compactly supported in a domain G ⊆ R^n is proved. The representation of the solution in terms of pseudo-differential operators is given. The solvability theorem in the Sobolev spaces H^s,2(R^n), s ∈ R^1 is also established.
Keywords: Pseudo-Differential Equations, Cauchy Problem, Caputo Fractional Derivative, Mittag-Leffler Function, Inhomogeneous Equation, Time-Fractional Equation, 26A33, 45K05, 35A05, 35S10, 35S15, 33E12 
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Saydamatov, Erkin. Well-Posedness of the Cauchy Problem for Inhomogeneous Time-Fractional Pseudo-Differential Equations. Fractional calculus and applied analysis, Tome 9 (2006) no. 1, pp. 01-16. http://geodesic.mathdoc.fr/item/FCAA_2006_9_1_a0/