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

entropy Article A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow Jagdev Singh 1,*, Devendra Kumar 2and Juan J. Nieto 3,4 1Department of Mathematics, Jagan Nath University, Jaipur 303901, India 2Department of Mathematics, JECRC University, Jaipur 303905, India; [email protected] 3 Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain; [email protected] 4Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia *Correspondence: [email protected]; Tel.: +91-946-090-5224 Academic Editor: Carlo Cattani Received: 13 April 2016; Accepted: 23 May 2016; Published: 25 May 2016 Abstract: The pivotal proposal of this work is to present a reliable algorithm based on the local fractional homotopy perturbation Sumudu transform technique for solving a local fractional Tricomi equation occurring in fractal transonic flow. The proposed technique provides the results without any transformation of the equation into discrete counterparts or imposing restrictive assumptions and is completely free of round-off errors. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate. Keywords: Tricomi equation; fractal transonic flow; local fractional derivative; homotopy perturbation method; local fractional Sumudu transform method 1. 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, yuxx `uyy “ 0, because of Tricomi, who found this mathematical model, for the function u“upx , yq of two variables xand 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 yof the second order partial differential coefficient of the required function u“upx , yq with respect to xchanges sign. This mathematical equation is also parabolic at the points where yvanishes. 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 – 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 Entropy 2016,18, 206; doi:10.3390/e18060206 www.mdpi.com/journal/entropy Entropy 2016,18, 206 2 of 8 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] yβ Γp1`βq B2βupx,yq Bx2β`B2βupx,yq By2β“0, (1) where the size upx , yq is the non-differentiable function, and the local fractional operator indicates [ 10 ] Bβupx,tq Btβ“∆βpupx,tq ´ upx,t0qq pt´t0q, (2) where ∆βpupx,tq ´ upx,t0qq – Γp1`βq rupx,tq ´ upx,t0qs (3) The Tricomi equation finds its application in modeling transonic flow [ 21 – 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 – 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. 2. Local Fractional Integrals and Derivatives In this section, we review the basic theory of local fractional calculus, which is applied in this research article. Definition 1 ([ 10 – 20 ]) . For the relation |x´x0| ă δ , when ε , δą 0 and εąR , we permit the function fpxq P Cβpa,bq, while ˇˇˇfpxq ´ fpx0qˇˇˇăεβ, 0 ăβď1, (4) exists. Definition 2 ([ 10 – 20 ]) . Consider the interval [a, b] and ptj , tj`1q , j“ 0, . . . , N´ 1, t0“a , and tN“b with ∆tj“tj`1´tj , ∆t“max t∆t0,∆t1,∆t2, . . .u a partition of this interval. Then, the local fractional integral of fpxqis explained as aIbpβq fpxq“1 Γp1`βq b ż a fptqpdtqβ“1 Γp1`βqlim ∆tÑ0 j“N´1 ÿ j“0 fptjqp∆tjqβ, (5) Definition 3 ([ 10 – 20 ]) . Suppose the function fpxq fulfill state in Equation (4), then the inverse formula of Equation (5) is defined as follows: dβfpx0q dxβ“Dpβq xfpx0q “ ∆βpfpxq ´ fpx0qq px´x0qβ, (6) where ∆βpfpxq ´ fpx0qq – Γp1`βqr fpxq ´ fpx0qs. (7) Entropy 2016,18, 206 3 of 8 The formula of the local fractional derivative, employed in this paper, is given as follows [10]: dβ dxβ xnβ Γp1`nβq“xpn´1qβ Γp1` pn´1qβq,nPN. (8) 3. 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 fpxqis first introduced and defined by Srivastava et al. [36] in the following manner: LFSβtfpxqu “ Fβpzq “1 Γp1`βq 8 ş 0 Eβp´z´βxβqfpxq zβpdxqβ, 0 ăβď1(9) and the inverse formula is expressed as follows LFS´1 β Fβpzq(“fpxq, 0 ăβď1. (10) 4. 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 Lβupx,tq ` Rβupx,tq “ hpx,tq, (11) where Lβ denotes the linear local fractional differential operator, Rβ is the remaining linear operator and hpx,tqis a source term. Using the local Sumudu transform on Equation (11) yields Uβpx,zq “ upx, 0q ` zβuβpx, 0q ` z2βu2βpx, 0q ` . . . `zpk´1qβupk´1qβpx, 0q ´ zkβLFSβ“Rβupx,tq‰`zkβLFSβrhpx,tqs .(12) Applying the inverse of the local fractional Sumudu transform on Equation (12), we have the following result upx,tq “ upx, 0q ` tβ Γp1`βquβpx, 0q ` t2β Γp1`2βqu2βpx, 0q ` . . . `tpk´1qβ Γp1`pk´1qβqupk´1qβpx, 0q ´ LFS´1 β”zkβLFSβ“Rβupx,tq‰ı `LFS´1 β”zkβLFSβrhpx,tqsı. (13) Now we use the HPM [24–26] upx,tq “ 8 ÿ n“0 pnunpx,tq. (14) Entropy 2016,18, 206 4 of 8 Substituting Equation (14) in Equation (13), we get the following result: 8 ř n“0 pnunpx,tq “ upx, 0q ` tβ Γp1`βquβpx, 0q ` t2β Γp1`2βqu2βpx, 0q ` . . . `tpk´1qβ Γp1`pk´1qβqupk´1qβpx, 0q ´ pLFS´1 β„zkβLFSβ„Rβ 8 ř n“0 pnunpx,tq `LFS´1 β”zkβLFSβrhpx,tqsı. (15) which is a mixture of the local fractional Sumudu transform technique and HPM. Comparing the coefficients of like powers of p, we get p0:u0px,tq “ upx, 0q ` tβ Γp1`βquβpx, 0q ` t2β Γp1`2βqu2βpx, 0q ` . . . `tpk´1qβ Γp1`pk´1qβqupk´1qβupx, 0q ` LFS´1 β”zkβLFSβrhpx,tqsı, p1:u1px,tq“´LFS´1 α”zkβLFSβ“Rβu0px,tq‰ı, p2:u2px,tq“´LFS´1 β”zkβLFSβ“Rβu1px,tq‰ı, . . . (16) Therefore, the solution of Equation (11) is given by upx,tq “ lim NÑ8 N ÿ n“0 unpx,tq(17) 5. 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. Example 1. Firstly, we investigate the following local fractional Tricomi equation yβ Γp1`βq B2βupx,yq Bx2β`B2βupx,yq By2β“0 (18) subject to the initial-boundary value conditions upx, 0q “ 0, uph,lq “ 0, upx, 0q “ x2β Γp1`2βq, Bβupx,0q Byβ“0. (19) Applying the local fractional Sumudu transform on Equation (18), we get Uβpx,zq “ upx, 0q ` zβuβpx, 0q ` z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β(20) which implies Uβpx,zq “ x2β Γp1`2βq`z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β(21) Entropy 2016,18, 206 5 of 8 Applying the inverse local fractional Sumudu transform to Equation (21) gives upx,yq “ x2β Γp1`2βq`LFS´1 β„z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β (22) Now using HPM [24–26] we get 8 ÿ n“0 pnunpx,yq “ x2β Γp1`2βq`pLFS´1 β » — — – z2βLFSβ» — — – ´yβ Γp1`βq B2βp8 ř n“0 pnunpx,yqq Bx2β fi ffi ffi fl fi ffi ffi fl (23) Comparing the like powers of p, we get the following components of the series solution p0:u0px,yq “ x2β Γp1`2βq, p1:u1px,yq“´y3β Γp1`3βq, p2:u2px,yq “ 0, . . . (24) Finally, we get the exact solution of Equation (18) with the local fractional derivative boundary value conditions Equation (19), namely upx,yq “ lim NÑ8 N ř n“0 unpx,yq “x2β Γp1`2βq´y3β Γp1`3βq (25) Example 2. Next, we study the local fractional Tricomi equation of the form yβ Γp1`βq B2βupx,yq Bx2β`B2βupx,yq By2β“0 (26) and the initial conditions are presented as upx, 0q “ xβ Γp1`βq, Bβupx,0q Byβ“xβ Γp1`βq, (27) Applying the local fractional Sumudu transform to Equation (26), we get Uβpx,zq “ upx, 0q ` zβuβpx, 0q ` z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β(28) which gives Uβpx,zq “ xβ Γp1`βq`zβxβ Γp1`βq`z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β(29) Applying the inverse local fractional Sumudu transform to Equation (29), we have upx,yq “ xβ Γp1`βq„1`yβ Γp1`βq`LFS´1 β„z2βLFSβ„´yβ Γp1`βq B2βupx,yq Bx2β (30) Entropy 2016,18, 206 6 of 8 Now using HPM [24–26], we get 8 ř n“0 pnunpx,yq “ xβ Γp1`βq”1`yβ Γp1`βqı` pLFS´1 β» — –z2βLFSβ» — –´yβ Γp1`βq B2βp8 ř n“0 pnunpx,yqq Bx2βfi ffi flfi ffi fl (31) Comparing the coefficients of like powers of pprovides p0:u0px,yq “ xβ Γp1`βq”1`yβ Γp1`βqı, p1:u1px,yq “ 0, p2:u2px,yq “ 0, . . . (32) Hence, we get the exact solution with non-differential term, as follows: upx,yq “ lim NÑ8 N ÿ n“0 unpx,yq “ xβ Γp1`βq„1`yβ Γp1`βq(33) 6. 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. Acknowledgments: The authors are very grateful to the referees for their invaluable suggestions and comments for the improvement of this paper. Author Contributions: Jagdev Singh, Devendra Kumar and Juan J. Nieto contributed equally to this work. 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