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In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of  L   p  -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation