Geodesic
Initial-value problem for a~linear ordinary differential equation of noninteger order
Sbornik. Mathematics, Tome 202 (2011) no. 4, pp. 571-582

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An initial-value problem for a linear ordinary differential equation of noninteger order with Riemann-Liouville derivatives is stated and solved. The initial conditions of the problem ensure that (by contrast with the Cauchy problem) it is uniquely solvable for an arbitrary set of parameters specifying the orders of the derivatives involved in the equation; these conditions are necessary for the equation under consideration. The problem is reduced to an integral equation; an explicit representation of the solution in terms of the Wright function is constructed. As a consequence of these results, necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. Bibliography: 7 titles.
Keywords: fractional order derivative, Cauchy problem, differential equation of fractional order, Wright function, Hille-Tamarkin formula. 
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     author = {A. V. Pskhu},
     title = {Initial-value problem for a~linear ordinary differential equation of noninteger order},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {202},
     number = {4},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_4_a4/}
}
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A. V. Pskhu. Initial-value problem for a~linear ordinary differential equation of noninteger order. Sbornik. Mathematics, Tome 202 (2011) no. 4, pp. 571-582. http://geodesic.mathdoc.fr/item/SM_2011_202_4_a4/