This paper is devoted to the study of a Wright-type hypergeometric function (Virchenko, Kalla and Al-Zamel in Integral Transforms Spec. Funct. 12(1):89-100, 2001) by using a Riemann-Liouville type fractional integral, a differential operator and Lebesgue measurable real or complex-valued functions. The results obtained are useful in the theory of special functions where the Wright function occurs naturally. MSC: 33C20; 33E20; 26A33; 26A99

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In this paper, we have solved the non-linear Korteweg-de Vries equation by considering it in time-fraction Caputo sense and offered intrinsic properties of solitary waves. The fractional residual power series method is used to obtain the approximate solution of the aforesaid equation and compared the obtained results with Adomian Decomposition Method. Obtained results are efficient, reliable, and simple to execute on most of the non-linear fractional partial differential equations, which arise in various dynamical systems.