A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow

Jagdev Singh 1,*, Devendra Kumar 2 and Juan J. Nieto 3,4 1 Department of Mathematics, Jagan Nath University, Jaipur 303901, India 2 Department of Mathematics, JECRC University, Jaipur 303905, India; devendra.maths@gmail.com 3 Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain; juanjose.nieto.roig@usc.es 4 Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia * Correspondence: jagdevsinghrathore@gmail.com; Tel.: +91-946-090-5224


Introduction
Partial differential equations of mixed type with boundary conditions have played an important role in describing real world problems such as in elementary research conducted by Tricomi [1].Mixed type partial differential equations [2,3] are used to investigate transonic flow and they produce special boundary value problems, known as Tricomi and Frankl problems [1,4].Transonic flows include a change from the subsonic to the hypersonic region [4,5] via the sonic curve.Consequently, transonic flows are very attractive phenomena occurring in aeronautics and hydraulics.The familiar mixed type partial differential equation is known as a Tricomi equation, yu xx `uyy " 0, because of Tricomi, who found this mathematical model, for the function u " upx, yq of two variables x and y.It acts as a basis for the mathematical modelling of the transonic flows, since it is of elliptic and hyperbolic type, where the coefficient y of the second order partial differential coefficient of the required function u " upx, yq with respect to x changes sign.This mathematical equation is also parabolic at the points where y vanishes.
The Tricomi equation [1] is a mixed type of linear partial differential equation of the second order, which has been used to narrate the theory of plane transonic flow [6][7][8][9].The Tricomi equation was used to recount differentiable problems for the theory of plane transonic flow.However, for the fractal theory of plane transonic flow with non-differentiable expressions, the Tricomi equation is not registered to report them.Recently, local fractional calculus [10] was tried for non-differentiable problems, for instance fractal heat conduction [10,11], damped and dissipative wave equations in fractal strings [12], local fractional Laplace equations [13], the Helmholtz equation associated with local fractional derivatives [14], the wave equation on Cantor sets [15], Navier-Stokes equations on Cantor sets [16], local fractional Schrödinger equations [17], Korteweg-de Vries equations involving local fractional derivative [18], etc.In recent times, the local fractional model of the Tricomi equation in fractal transonic flows was recommended in the form [19,20] where the size upx, yq is the non-differentiable function, and the local fractional operator indicates [10] where ∆ β pupx, tq ´upx, t 0 qq -Γp1 `βq rupx, tq ´upx, t 0 qs The Tricomi equation finds its application in modeling transonic flow [21][22][23].Inspired and motivated by the ongoing research in this area and wide applications of local fractional differential equations, we propose the local fractional homotopy perturbation Sumudu transform method (LFHPSTM) to solve the local fractional model of the Tricomi equation appearing in fractal transonic flow pertaining to the local fractional derivative boundary value conditions.The LFHPSTM is a conjunction of the classical homotopy perturbation method (HPM) [24][25][26] and the local fractional Sumudu transform technique.The formation of this article is as developed.In Section 2, the local fractional integrals and derivatives are initiated.In Section 3, the local fractional homotopy perturbation Sumudu transform method is proposed.In Section 4, the non-differentiable numerical solutions for local fractional Tricomi equation along the local fractional derivative boundary value conditions are specified.Finally, in Section 5, the conclusions are discussed.

Local Fractional Integrals and Derivatives
In this section, we review the basic theory of local fractional calculus, which is applied in this research article.
The formula of the local fractional derivative, employed in this paper, is given as follows [10]: , n P N. (8)

Local Fractional Sumudu Transform
The Sumudu transform was initially proposed and developed by Watugala [27] and some of its important properties were discovered and investigated by Belgacem et al. [28] and Belgacem and Karaballi [29].Katatbeh and Belgacem [30] employed the Sumudu transform to solve fractional differential equations.Gupta et al. [31] used the Sumudu transform to solve generalized fractional kinetic equations.Belgacem [32] investigated the applications of the Sumudu transform to Bessel functions and equations.Belgacem [33] introduced and analyzed deeper Sumudu properties.Bulut et al. [34] obtained the analytical solutions of some fractional ordinary differential equations by using the Sumudu transform technique.The Sumudu transform method is also coupled with HPM to investigate the fractional biological population model [35].The local fractional Sumudu transform of a function f pxq is first introduced and defined by Srivastava et al. [36] in the following manner: and the inverse formula is expressed as follows

Local Fractional Homotopy Perturbation Sumudu Transform Method
In order to establish the basic idea of the LFHPSTM, we assume the following linear differential equation with a local fractional derivative L β upx, tq `Rβ upx, tq " hpx, tq, where L β denotes the linear local fractional differential operator, R β is the remaining linear operator and hpx, tq is a source term.

`tpk´1qβ
Γp1`pk´1qβq u pk´1qβ px, 0q ´pLFS ´1 which is a mixture of the local fractional Sumudu transform technique and HPM.Comparing the coefficients of like powers of p, we get Therefore, the solution of Equation ( 11) is given by upx, tq " lim

Nondifferential Solutions for the Local Fractional Tricomi Equation
In this section, we present the nondifferential solutions for the Tricomi equation pertaining to the local fractional derivative occurring in fractal transonic flow with local fractional derivative boundary value conditions.

Conclusions
In the present paper, the local fractional Tricomi equation with its applications in fractal transonic flow is discussed by using the local fractional homotopy perturbation Sumudu transform technique.We obtain the solution with non-differential terms by applying this approach.The results show that the proposed technique is very efficient and can be used to solve various kinds of local fractional differential equations.Hence, the introduced method is a powerful tool for solving local fractional linear equations of physical importance.