N-Fractional Calculus Operator Method to the Euler Equation

Abstract

We can obtain the explicit solutions of the Euler equation by using the fractional calculus methods. So, we apply the N operator method in the fractional calculus to solve this equation in this paper. We take advantage of some results of previous studies related to the fractional calculus.

Discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations

“…where n ∈  [28]. Lemma 5 (Index law) [20]. Suppose that ( ) t ϕ is an analytic and single-valued function.…”

“…A similar theory was started for discrete fractional calculus and the definition and properties of fractional sums and differences theory were developed. Many article related to this topic have seemed lately [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Finding exact solutions to non-linear PDE defining the evolution of localized waveforms is an important subject in non-linear science [21][22][23].…”

On discrete fractional solutions of the hydrogen atom type equations

“…this part takes a centripetal and Colomb part, the usual singularities of the nuclear problem [17]. Yilmazer [18] acquired fractional solutions of eq. (2) by using the Nishimoto operator.…”

An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation

“…It is possible to see many works based on the Sonine-Letnikov fractional derivative, although it is often known as N-fractional calculus operator. These works include the solutions of the Gauss equation [12], solutions of modified Whittaker equations [13], an almost free damping vibration equation [14], differential operators and integral operators in univalent function theory [15], geometric univalent function theory [16], power and logarithmic functions, Weber's equation, Gauss hypergeometric equations and some double infinite, finite and mixed sums [17], products of some power functions and some doubly infinite sums [18], some composite functions [19], some algebraic functions [20], some functions which include a root sign [21], a modified hydrogen atom equation [22], some second order homogeneous Euler's equation [23], some logarithmic functions and some identities [24], fractional solutions of homogeneous and nonhomogeneous Chebyshev's equations [25,26], explicit solutions of Gegenbauer equation [27], fractional solutions of Bessel equation [28], fractional solutions of the radial part in the fractional Schrödinger equation [29] and some singular differential equations [30].…”