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Research ArticleLocal Fractional Derivative Boundary Value Problems forTricomi Equation Arising in Fractal Transonic Flow
Xiao-Feng Niu1 Cai-Li Zhang1 Zheng-Biao Li2 and Yang Zhao3
1 College of Materials Science and Engineering Taiyuan University of Technology Taiyuan 030024 China2 College of Mathematics and Information Science Qujing Normal University Qujing Yunnan 655011 China3Department of Electronic and Information Technology Jiangmen Polytechnic Jiangmen 529090 China
Correspondence should be addressed to Yang Zhao zhaoyang19781023gmailcom
Received 19 June 2014 Accepted 26 June 2014 Published 13 July 2014
Academic Editor Xiao-Jun Yang
Copyright copy 2014 Xiao-Feng Niu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The local fractional decomposition method is applied to obtain the nondifferentiable numerical solutions for the local fractionalTricomi equation arising in fractal transonic flow with the local fractional derivative boundary value conditions
1 Introduction
The Tricomi equation [1] is the second-order linear partialdifferential equations of mixed type which had been appliedto describe the theory of plane transonic flow [2ndash7] TheTricomi equation was used to describe the differentiableproblems for the theory of plane transonic flow Howeverfor the fractal theory of plane transonic flow with nondif-ferentiable terms the Tricomi equation is not applied todescribe them Recently the local fractional calculus [8] wasapplied to describe the nondifferentiable problems such asthe fractal heat conduction [8 9] the damped and dissipativewave equations in fractal strings [10] the local fractionalSchrodinger equation [11] the wave equation on Cantor sets[12] the Navier-Stokes equations on Cantor sets [13] andothers [14ndash19] Recently the local fractional Tricomi equationarising in fractal transonic flowwas suggested in the form [19]
119910120572
Γ (1 + 120572)
1205972120572
119906 (119909 119910)
1205971199092120572+
1205972120572
119906 (119909 119910)
1205971199102120572= 0 (1)
where the quantity 119906(119909 119910) is the nondifferentiable functionand the local fractional operator denotes [8]
120597120572
119906 (119909 119905)
120597119905120572=
Δ120572
(119906 (119909 119905) minus 119906 (119909 1199050))
(119905 minus 1199050)120572
(2)
where
Δ120572
(119906 (119909 119905) minus 119906 (119909 1199050)) cong Γ (1 + 120572) [119906 (119909 119905) minus 119906 (119909 119905
0)]
(3)
The local fractional decomposition method [12] was used tosolve the diffusion equation on Cantor time-space The aimof this paper is to use the local fractional decompositionmethod to solve the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions The structure of this paper isas follows In Section 2 the local fractional integrals andderivatives are introduced In Section 3 the local fractionaldecomposition method is suggested In Section 4 the non-differentiable numerical solutions for local fractional Tricomiequation with the local fractional derivative boundary valueconditions are given Finally the conclusions are shown inSection 5
2 Local Fractional Integrals and Derivatives
In this section we introduce the basic theory of the localfractional integrals and derivatives [8ndash19] which are appliedin the paper
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 872318 5 pageshttpdxdoiorg1011552014872318
2 Abstract and Applied Analysis
Definition 1 (see [8ndash19]) For |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and
120576 isin 119877 we give the function 119891(119909) isin 119862120572(119886 119887) when
1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572
0 lt 120572 le 1 (4)
is valid
Definition 2 (see [8ndash19]) Let (119905119895 119905119895+1
) 119895 = 0 119873minus1 1199050= 119886
and 119905119873
= 119887withΔ119905119895= 119905119895+1
minus119905119895andΔ119905 = maxΔ119905
0 Δ1199051 be
a partition of the interval [119886 119887] The local fractional integralof 119891(119909) in the interval [119886 119887] is defined as
119886119868119887
(120572)
119891 (119909) =1
Γ (1 + 120572)int
119887
119886
119891 (119905) (119889119905)120572
=1
Γ (1 + 120572)limΔ119905rarr0
119895=119873minus1
sum
119895=0
119891 (119905119895) (Δ119905119895)120572
(5)
As the inverse operator of (6) local fractional derivative of119891(119909) of the order 120572 in the interval (119886 119887) is presented as [8ndash19]
119889120572
119891 (1199090)
119889119909120572= 119863119909
(120572)
119891 (1199090) =
Δ120572
(119891 (119909) minus 119891 (1199090))
(119909 minus 1199090)120572
(6)
where
Δ120572
(119891 (119909) minus 119891 (1199090)) cong Γ (1 + 120572) [119891 (119909) minus 119891 (119909
0)] (7)
The formulas of local fractional derivative and integral whichappear in the paper are valid [8]
119889120572
119889119909120572
119909119899120572
Γ (1 + 119899120572)=
119909(119899minus1)120572
Γ (1 + (119899 minus 1) 120572) 119899 isin 119873
119889120572
119889119909120572119864120572(119909120572
) = 119864120572(119909120572
)
119889120572
119889119909120572sin120572(119909120572
) = cos120572(119909120572
)
119889120572
119889119909120572cos120572(119909120572
) = minus sin120572(119909120572
)
0119868119909
(120572)119909(119899minus1)120572
Γ (1 + (119899 minus 1) 120572)=
119909119899120572
Γ (1 + 119899120572) 119899 isin 119873
0119868119909
(120572)cos120572(119909120572
) = sin120572(119909120572
)
(8)
3 Analysis of the Method
In this section we give the local fractional decompositionmethod [12] We consider the following local fractionaloperator equation in the form
119871(2)
120572119906 + 119877120572119906 = 0 (9)
where 119871(2)120572
is linear local fractional operators of the order 2120572with respect to119909 and119877
120572is the linear local fractional operators
of order less than 2120572 We write (9) as
119871(2120572)
119909119909119906 + 119877120572119906 = 0 (10)
where the 2120572-th local fractional differential operator denotes
119871(119899)
120572= 119871(2120572)
119909119909=
1205972120572
1205971199092120572 (11)
and the linear local fractional operators of order less than 2120572
denote
119877120572=
Γ (1 + 120572)
119910120572
1205972120572
119906 (119909 119910)
1205971199102120572 (12)
Define the 2120572-fold local fractional integral operator
119871(minus2120572)
120572119898(119904) =
0119868119909
(120572)
0119868119909
(120572)
119898(119904) (13)
so that we obtain the local fractional iterative formula asfollows
119871(minus2120572)
120572119871(2120572)
119909119909119906 + 119871(minus2120572)
120572119871(minus2120572)
120572119877120572119906 = 0 (14)
which leads to
119906 (119909) = 1199060(119909) + 119871
(minus2120572)
120572119871(minus2120572)
120572119877120572119906 (15)
Therefore for 119899 ge 0 we obtain the recurrence formula in theform
119906119899+1
(119909) = 119871(minus2)
120572119877120572119906119899(119909)
1199060(119909) = 119903 (119909)
(16)
Finally the solution of (9) reads
119906 (119909) = lim119899rarrinfin
120601119899(119909) = lim
119899rarrinfin
infin
sum
119899=0
119906119899(119909) (17)
4 The Nondifferentiable Numerical Solutions
In this section we discuss the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions
Example 1 We consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form [19]
119906 (0 119910) = 0 (18)
119906 (119897 119910) = 0 (19)
119906 (119909 0) =119909120572
Γ (1 + 120572) (20)
120597120572
119906 (119909 0)
120597119909120572=
119909120572
Γ (1 + 120572) (21)
Using (20)-(21) we structure the recurrence formula in theform
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+
119909120572
Γ (1 + 120572)
119910120572
Γ (1 + 120572)
(22)
Abstract and Applied Analysis 3
Hence for 119899 = 0 the first term of (22) reads
1199061(119909 119910) = 0 (23)
For 119899 = 1 the second term of (22) is given as
1199062(119909 119910) = 0 (24)
Hence we obtain
1199060(119909 119910) = 119906
1(119909 119910) = sdot sdot sdot = 119906
119899(119909 119910) = 0 (25)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (19)ndash(21) can be written as
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+
119909120572
Γ (1 + 120572)
119910120572
Γ (1 + 120572)
(26)
which is in accordance with the result from the local frac-tional variational iteration method [19]
Example 2 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) = 0
120597120572
119906 (119909 0)
120597119909120572= cos120572(119909120572
)
(27)
In view of (27) we set up the recurrence formula in the form
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) = cos
120572(119909120572
)119910120572
Γ (1 + 120572)
(28)
Hence from (28) we get the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(cos120572(119909120572
)119910120572
Γ (1 + 120572))]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(29)
Finally we obtain the solution of (9) with the local fractionalderivative boundary value conditions (27) namely
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
= cos120572(119909120572
)119910120572
Γ (1 + 120572)
(30)
whose graph is shown in Figure 1
Example 3 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) =119909120572
Γ (1 + 120572)
120597120572
119906 (119909 0)
120597119909120572= sin120572(119909120572
)
(31)
4 Abstract and Applied Analysis
0 0204
0608
1
0
05
1
0
05
1
15
x
y
u(xy)
Figure 1 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (27) when 120572 = ln 2 ln 3
Making use of (31) the recurrence formula can be written as
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(32)
Appling (32) gives the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572
times(119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)) ]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
0
02
0406
08
1
0
05
1
0
05
1
15
2
xy
u(xy)
Figure 2 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (31) when 120572 = ln 2 ln 3
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(33)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (31) reads
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(34)
and its graph is shown in Figure 2
5 Conclusions
In this work we discussed the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions by using the local fractionaldecompositionmethod and their plots were also shown in theMatLab software
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgment
This work was supported by NSF110361048 of China andYunnan province NSF Grant no 2011FB090
References
[1] F Tricomi On Second-Order Linear Partial Differential Equa-tions of Mixed Type Leningrad Moscow Russia 1947
[2] A R Manwell ldquoThe Tricomi equation with applications to thetheory of plane transonic flowrdquo ReconTechnical ReportA 276171979
[3] D Lupo and K R Payne ldquoA dual variational approach toa class of nonlocal semilinear Tricomi problemsrdquo NonlinearDifferential Equations and Applications vol 6 no 3 pp 247ndash266 1999
[4] J U Kim ldquoAn 119871119901 a priori estimate for the Tricomi equation in
the upper half spacerdquo Transactions of the American Mathemati-cal Society vol 351 no 11 pp 4611ndash4628 1999
[5] J M Rassias ldquoUniqueness of quasi-regular solutions for abi-parabolic elliptic bi-hyperbolic Tricomi problemrdquo ComplexVariables Theory and Application vol 47 no 8 pp 707ndash7182002
[6] K Yagdjian ldquoGlobal existence for the 119899-dimensional semilinearTricomi-type equationsrdquoCommunications in PartialDifferentialEquations vol 31 no 4ndash6 pp 907ndash944 2006
[7] K Yagdjian ldquoA note on the fundamental solution for theTricomi-type equation in the hyperbolic domainrdquo Journal ofDifferential Equations vol 206 no 1 pp 227ndash252 2004
[8] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012
[9] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013
[10] W H Su D Baleanu X J Yang et al ldquoDamped wave equationand dissipative wave equation in fractal strings within the localfractional variational iteration methodrdquo Fixed PointTheory andApplications vol 2013 no 1 pp 1ndash11 2013
[11] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013
[12] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X Yang ldquoLocal fractional variational iteration and decom-position methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014
[13] X-J Yang D Baleanu and J T Machado ldquoSystems of Navier-Stokes equations on Cantor setsrdquo Mathematical Problems inEngineering vol 2013 Article ID 769724 8 pages 2013
[14] G Yi ldquoLocal fractional Z transform in fractal spacerdquo Advancesin Digital Multimedia vol 1 no 2 pp 96ndash102 2012
[15] X Yang H M Srivastava J H He et al ldquoCantor-typecylindrical-coordinate method for differential equations withlocal fractional derivativesrdquo Physics Letters A vol 377 no 28ndash30 pp 1696ndash1700 2013
[16] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012
[17] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractional
derivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014
[18] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on Cantor sets a local fractionalFourier series approachrdquoAdvances inMathematical Physics vol2014 Article ID 561434 7 pages 2014
[19] A M Yang Y Z Zhang and X L Zhang ldquoThe non-differentiable solution for local fractional Tricomi equationarising in fractal transonic flow by loca l fractional variationaliteration methodrdquo Advances in Mathematical Physics vol 2014Article ID 983254 6 pages 2014
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Definition 1 (see [8ndash19]) For |119909 minus 1199090| lt 120575 for 120576 120575 gt 0 and
120576 isin 119877 we give the function 119891(119909) isin 119862120572(119886 119887) when
1003816100381610038161003816119891 (119909) minus 119891 (1199090)1003816100381610038161003816 lt 120576120572
0 lt 120572 le 1 (4)
is valid
Definition 2 (see [8ndash19]) Let (119905119895 119905119895+1
) 119895 = 0 119873minus1 1199050= 119886
and 119905119873
= 119887withΔ119905119895= 119905119895+1
minus119905119895andΔ119905 = maxΔ119905
0 Δ1199051 be
a partition of the interval [119886 119887] The local fractional integralof 119891(119909) in the interval [119886 119887] is defined as
119886119868119887
(120572)
119891 (119909) =1
Γ (1 + 120572)int
119887
119886
119891 (119905) (119889119905)120572
=1
Γ (1 + 120572)limΔ119905rarr0
119895=119873minus1
sum
119895=0
119891 (119905119895) (Δ119905119895)120572
(5)
As the inverse operator of (6) local fractional derivative of119891(119909) of the order 120572 in the interval (119886 119887) is presented as [8ndash19]
119889120572
119891 (1199090)
119889119909120572= 119863119909
(120572)
119891 (1199090) =
Δ120572
(119891 (119909) minus 119891 (1199090))
(119909 minus 1199090)120572
(6)
where
Δ120572
(119891 (119909) minus 119891 (1199090)) cong Γ (1 + 120572) [119891 (119909) minus 119891 (119909
0)] (7)
The formulas of local fractional derivative and integral whichappear in the paper are valid [8]
119889120572
119889119909120572
119909119899120572
Γ (1 + 119899120572)=
119909(119899minus1)120572
Γ (1 + (119899 minus 1) 120572) 119899 isin 119873
119889120572
119889119909120572119864120572(119909120572
) = 119864120572(119909120572
)
119889120572
119889119909120572sin120572(119909120572
) = cos120572(119909120572
)
119889120572
119889119909120572cos120572(119909120572
) = minus sin120572(119909120572
)
0119868119909
(120572)119909(119899minus1)120572
Γ (1 + (119899 minus 1) 120572)=
119909119899120572
Γ (1 + 119899120572) 119899 isin 119873
0119868119909
(120572)cos120572(119909120572
) = sin120572(119909120572
)
(8)
3 Analysis of the Method
In this section we give the local fractional decompositionmethod [12] We consider the following local fractionaloperator equation in the form
119871(2)
120572119906 + 119877120572119906 = 0 (9)
where 119871(2)120572
is linear local fractional operators of the order 2120572with respect to119909 and119877
120572is the linear local fractional operators
of order less than 2120572 We write (9) as
119871(2120572)
119909119909119906 + 119877120572119906 = 0 (10)
where the 2120572-th local fractional differential operator denotes
119871(119899)
120572= 119871(2120572)
119909119909=
1205972120572
1205971199092120572 (11)
and the linear local fractional operators of order less than 2120572
denote
119877120572=
Γ (1 + 120572)
119910120572
1205972120572
119906 (119909 119910)
1205971199102120572 (12)
Define the 2120572-fold local fractional integral operator
119871(minus2120572)
120572119898(119904) =
0119868119909
(120572)
0119868119909
(120572)
119898(119904) (13)
so that we obtain the local fractional iterative formula asfollows
119871(minus2120572)
120572119871(2120572)
119909119909119906 + 119871(minus2120572)
120572119871(minus2120572)
120572119877120572119906 = 0 (14)
which leads to
119906 (119909) = 1199060(119909) + 119871
(minus2120572)
120572119871(minus2120572)
120572119877120572119906 (15)
Therefore for 119899 ge 0 we obtain the recurrence formula in theform
119906119899+1
(119909) = 119871(minus2)
120572119877120572119906119899(119909)
1199060(119909) = 119903 (119909)
(16)
Finally the solution of (9) reads
119906 (119909) = lim119899rarrinfin
120601119899(119909) = lim
119899rarrinfin
infin
sum
119899=0
119906119899(119909) (17)
4 The Nondifferentiable Numerical Solutions
In this section we discuss the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions
Example 1 We consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form [19]
119906 (0 119910) = 0 (18)
119906 (119897 119910) = 0 (19)
119906 (119909 0) =119909120572
Γ (1 + 120572) (20)
120597120572
119906 (119909 0)
120597119909120572=
119909120572
Γ (1 + 120572) (21)
Using (20)-(21) we structure the recurrence formula in theform
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+
119909120572
Γ (1 + 120572)
119910120572
Γ (1 + 120572)
(22)
Abstract and Applied Analysis 3
Hence for 119899 = 0 the first term of (22) reads
1199061(119909 119910) = 0 (23)
For 119899 = 1 the second term of (22) is given as
1199062(119909 119910) = 0 (24)
Hence we obtain
1199060(119909 119910) = 119906
1(119909 119910) = sdot sdot sdot = 119906
119899(119909 119910) = 0 (25)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (19)ndash(21) can be written as
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+
119909120572
Γ (1 + 120572)
119910120572
Γ (1 + 120572)
(26)
which is in accordance with the result from the local frac-tional variational iteration method [19]
Example 2 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) = 0
120597120572
119906 (119909 0)
120597119909120572= cos120572(119909120572
)
(27)
In view of (27) we set up the recurrence formula in the form
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) = cos
120572(119909120572
)119910120572
Γ (1 + 120572)
(28)
Hence from (28) we get the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(cos120572(119909120572
)119910120572
Γ (1 + 120572))]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(29)
Finally we obtain the solution of (9) with the local fractionalderivative boundary value conditions (27) namely
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
= cos120572(119909120572
)119910120572
Γ (1 + 120572)
(30)
whose graph is shown in Figure 1
Example 3 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) =119909120572
Γ (1 + 120572)
120597120572
119906 (119909 0)
120597119909120572= sin120572(119909120572
)
(31)
4 Abstract and Applied Analysis
0 0204
0608
1
0
05
1
0
05
1
15
x
y
u(xy)
Figure 1 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (27) when 120572 = ln 2 ln 3
Making use of (31) the recurrence formula can be written as
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(32)
Appling (32) gives the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572
times(119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)) ]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
0
02
0406
08
1
0
05
1
0
05
1
15
2
xy
u(xy)
Figure 2 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (31) when 120572 = ln 2 ln 3
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(33)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (31) reads
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(34)
and its graph is shown in Figure 2
5 Conclusions
In this work we discussed the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions by using the local fractionaldecompositionmethod and their plots were also shown in theMatLab software
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgment
This work was supported by NSF110361048 of China andYunnan province NSF Grant no 2011FB090
References
[1] F Tricomi On Second-Order Linear Partial Differential Equa-tions of Mixed Type Leningrad Moscow Russia 1947
[2] A R Manwell ldquoThe Tricomi equation with applications to thetheory of plane transonic flowrdquo ReconTechnical ReportA 276171979
[3] D Lupo and K R Payne ldquoA dual variational approach toa class of nonlocal semilinear Tricomi problemsrdquo NonlinearDifferential Equations and Applications vol 6 no 3 pp 247ndash266 1999
[4] J U Kim ldquoAn 119871119901 a priori estimate for the Tricomi equation in
the upper half spacerdquo Transactions of the American Mathemati-cal Society vol 351 no 11 pp 4611ndash4628 1999
[5] J M Rassias ldquoUniqueness of quasi-regular solutions for abi-parabolic elliptic bi-hyperbolic Tricomi problemrdquo ComplexVariables Theory and Application vol 47 no 8 pp 707ndash7182002
[6] K Yagdjian ldquoGlobal existence for the 119899-dimensional semilinearTricomi-type equationsrdquoCommunications in PartialDifferentialEquations vol 31 no 4ndash6 pp 907ndash944 2006
[7] K Yagdjian ldquoA note on the fundamental solution for theTricomi-type equation in the hyperbolic domainrdquo Journal ofDifferential Equations vol 206 no 1 pp 227ndash252 2004
[8] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012
[9] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013
[10] W H Su D Baleanu X J Yang et al ldquoDamped wave equationand dissipative wave equation in fractal strings within the localfractional variational iteration methodrdquo Fixed PointTheory andApplications vol 2013 no 1 pp 1ndash11 2013
[11] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013
[12] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X Yang ldquoLocal fractional variational iteration and decom-position methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014
[13] X-J Yang D Baleanu and J T Machado ldquoSystems of Navier-Stokes equations on Cantor setsrdquo Mathematical Problems inEngineering vol 2013 Article ID 769724 8 pages 2013
[14] G Yi ldquoLocal fractional Z transform in fractal spacerdquo Advancesin Digital Multimedia vol 1 no 2 pp 96ndash102 2012
[15] X Yang H M Srivastava J H He et al ldquoCantor-typecylindrical-coordinate method for differential equations withlocal fractional derivativesrdquo Physics Letters A vol 377 no 28ndash30 pp 1696ndash1700 2013
[16] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012
[17] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractional
derivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014
[18] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on Cantor sets a local fractionalFourier series approachrdquoAdvances inMathematical Physics vol2014 Article ID 561434 7 pages 2014
[19] A M Yang Y Z Zhang and X L Zhang ldquoThe non-differentiable solution for local fractional Tricomi equationarising in fractal transonic flow by loca l fractional variationaliteration methodrdquo Advances in Mathematical Physics vol 2014Article ID 983254 6 pages 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Hence for 119899 = 0 the first term of (22) reads
1199061(119909 119910) = 0 (23)
For 119899 = 1 the second term of (22) is given as
1199062(119909 119910) = 0 (24)
Hence we obtain
1199060(119909 119910) = 119906
1(119909 119910) = sdot sdot sdot = 119906
119899(119909 119910) = 0 (25)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (19)ndash(21) can be written as
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+
119909120572
Γ (1 + 120572)
119910120572
Γ (1 + 120572)
(26)
which is in accordance with the result from the local frac-tional variational iteration method [19]
Example 2 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) = 0
120597120572
119906 (119909 0)
120597119909120572= cos120572(119909120572
)
(27)
In view of (27) we set up the recurrence formula in the form
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) = cos
120572(119909120572
)119910120572
Γ (1 + 120572)
(28)
Hence from (28) we get the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(cos120572(119909120572
)119910120572
Γ (1 + 120572))]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(29)
Finally we obtain the solution of (9) with the local fractionalderivative boundary value conditions (27) namely
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
= cos120572(119909120572
)119910120572
Γ (1 + 120572)
(30)
whose graph is shown in Figure 1
Example 3 Let us consider the initial-boundary value condi-tions for the local fractional Tricomi equation in the form
119906 (0 119910) = 0
119906 (119897 119910) = 0
119906 (119909 0) =119909120572
Γ (1 + 120572)
120597120572
119906 (119909 0)
120597119909120572= sin120572(119909120572
)
(31)
4 Abstract and Applied Analysis
0 0204
0608
1
0
05
1
0
05
1
15
x
y
u(xy)
Figure 1 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (27) when 120572 = ln 2 ln 3
Making use of (31) the recurrence formula can be written as
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(32)
Appling (32) gives the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572
times(119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)) ]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
0
02
0406
08
1
0
05
1
0
05
1
15
2
xy
u(xy)
Figure 2 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (31) when 120572 = ln 2 ln 3
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(33)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (31) reads
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(34)
and its graph is shown in Figure 2
5 Conclusions
In this work we discussed the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions by using the local fractionaldecompositionmethod and their plots were also shown in theMatLab software
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgment
This work was supported by NSF110361048 of China andYunnan province NSF Grant no 2011FB090
References
[1] F Tricomi On Second-Order Linear Partial Differential Equa-tions of Mixed Type Leningrad Moscow Russia 1947
[2] A R Manwell ldquoThe Tricomi equation with applications to thetheory of plane transonic flowrdquo ReconTechnical ReportA 276171979
[3] D Lupo and K R Payne ldquoA dual variational approach toa class of nonlocal semilinear Tricomi problemsrdquo NonlinearDifferential Equations and Applications vol 6 no 3 pp 247ndash266 1999
[4] J U Kim ldquoAn 119871119901 a priori estimate for the Tricomi equation in
the upper half spacerdquo Transactions of the American Mathemati-cal Society vol 351 no 11 pp 4611ndash4628 1999
[5] J M Rassias ldquoUniqueness of quasi-regular solutions for abi-parabolic elliptic bi-hyperbolic Tricomi problemrdquo ComplexVariables Theory and Application vol 47 no 8 pp 707ndash7182002
[6] K Yagdjian ldquoGlobal existence for the 119899-dimensional semilinearTricomi-type equationsrdquoCommunications in PartialDifferentialEquations vol 31 no 4ndash6 pp 907ndash944 2006
[7] K Yagdjian ldquoA note on the fundamental solution for theTricomi-type equation in the hyperbolic domainrdquo Journal ofDifferential Equations vol 206 no 1 pp 227ndash252 2004
[8] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012
[9] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013
[10] W H Su D Baleanu X J Yang et al ldquoDamped wave equationand dissipative wave equation in fractal strings within the localfractional variational iteration methodrdquo Fixed PointTheory andApplications vol 2013 no 1 pp 1ndash11 2013
[11] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013
[12] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X Yang ldquoLocal fractional variational iteration and decom-position methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014
[13] X-J Yang D Baleanu and J T Machado ldquoSystems of Navier-Stokes equations on Cantor setsrdquo Mathematical Problems inEngineering vol 2013 Article ID 769724 8 pages 2013
[14] G Yi ldquoLocal fractional Z transform in fractal spacerdquo Advancesin Digital Multimedia vol 1 no 2 pp 96ndash102 2012
[15] X Yang H M Srivastava J H He et al ldquoCantor-typecylindrical-coordinate method for differential equations withlocal fractional derivativesrdquo Physics Letters A vol 377 no 28ndash30 pp 1696ndash1700 2013
[16] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012
[17] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractional
derivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014
[18] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on Cantor sets a local fractionalFourier series approachrdquoAdvances inMathematical Physics vol2014 Article ID 561434 7 pages 2014
[19] A M Yang Y Z Zhang and X L Zhang ldquoThe non-differentiable solution for local fractional Tricomi equationarising in fractal transonic flow by loca l fractional variationaliteration methodrdquo Advances in Mathematical Physics vol 2014Article ID 983254 6 pages 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
0 0204
0608
1
0
05
1
0
05
1
15
x
y
u(xy)
Figure 1 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (27) when 120572 = ln 2 ln 3
Making use of (31) the recurrence formula can be written as
119906119899+1
(119909 119910) = 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
119906119899(119909 119910)
1205971199102120572]
1199060(119909 119910) =
119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(32)
Appling (32) gives the following equations
1199061(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199060(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572
times(119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)) ]
= 0
1199062(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199061(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
1199063(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199062(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
0
02
0406
08
1
0
05
1
0
05
1
15
2
xy
u(xy)
Figure 2 The plot of the solution of (9) with the local fractionalderivative boundary value conditions (31) when 120572 = ln 2 ln 3
1199064(119909 119910) = 119871
(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1199063(119909 119910)
1205971199102120572]
= 119871(minus2)
120572[Γ (1 + 120572)
119910120572
1205972120572
1205971199102120572(0)]
= 0
119906119899(119909 119910) = 0
(33)
Finally the solution of (9) with the local fractional derivativeboundary value conditions (31) reads
119906 (119909 119910) = lim119899rarrinfin
120601119899(119909 119910)
= lim119899rarrinfin
infin
sum
119899=0
119906119899(119909 119910)
=119909120572
Γ (1 + 120572)+ sin120572(119909120572
)119910120572
Γ (1 + 120572)
(34)
and its graph is shown in Figure 2
5 Conclusions
In this work we discussed the nondifferentiable numericalsolutions for the local fractional Tricomi equation arisingin fractal transonic flow with the local fractional derivativeboundary value conditions by using the local fractionaldecompositionmethod and their plots were also shown in theMatLab software
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 5
Acknowledgment
This work was supported by NSF110361048 of China andYunnan province NSF Grant no 2011FB090
References
[1] F Tricomi On Second-Order Linear Partial Differential Equa-tions of Mixed Type Leningrad Moscow Russia 1947
[2] A R Manwell ldquoThe Tricomi equation with applications to thetheory of plane transonic flowrdquo ReconTechnical ReportA 276171979
[3] D Lupo and K R Payne ldquoA dual variational approach toa class of nonlocal semilinear Tricomi problemsrdquo NonlinearDifferential Equations and Applications vol 6 no 3 pp 247ndash266 1999
[4] J U Kim ldquoAn 119871119901 a priori estimate for the Tricomi equation in
the upper half spacerdquo Transactions of the American Mathemati-cal Society vol 351 no 11 pp 4611ndash4628 1999
[5] J M Rassias ldquoUniqueness of quasi-regular solutions for abi-parabolic elliptic bi-hyperbolic Tricomi problemrdquo ComplexVariables Theory and Application vol 47 no 8 pp 707ndash7182002
[6] K Yagdjian ldquoGlobal existence for the 119899-dimensional semilinearTricomi-type equationsrdquoCommunications in PartialDifferentialEquations vol 31 no 4ndash6 pp 907ndash944 2006
[7] K Yagdjian ldquoA note on the fundamental solution for theTricomi-type equation in the hyperbolic domainrdquo Journal ofDifferential Equations vol 206 no 1 pp 227ndash252 2004
[8] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012
[9] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013
[10] W H Su D Baleanu X J Yang et al ldquoDamped wave equationand dissipative wave equation in fractal strings within the localfractional variational iteration methodrdquo Fixed PointTheory andApplications vol 2013 no 1 pp 1ndash11 2013
[11] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013
[12] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X Yang ldquoLocal fractional variational iteration and decom-position methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014
[13] X-J Yang D Baleanu and J T Machado ldquoSystems of Navier-Stokes equations on Cantor setsrdquo Mathematical Problems inEngineering vol 2013 Article ID 769724 8 pages 2013
[14] G Yi ldquoLocal fractional Z transform in fractal spacerdquo Advancesin Digital Multimedia vol 1 no 2 pp 96ndash102 2012
[15] X Yang H M Srivastava J H He et al ldquoCantor-typecylindrical-coordinate method for differential equations withlocal fractional derivativesrdquo Physics Letters A vol 377 no 28ndash30 pp 1696ndash1700 2013
[16] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012
[17] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractional
derivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014
[18] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on Cantor sets a local fractionalFourier series approachrdquoAdvances inMathematical Physics vol2014 Article ID 561434 7 pages 2014
[19] A M Yang Y Z Zhang and X L Zhang ldquoThe non-differentiable solution for local fractional Tricomi equationarising in fractal transonic flow by loca l fractional variationaliteration methodrdquo Advances in Mathematical Physics vol 2014Article ID 983254 6 pages 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Acknowledgment
This work was supported by NSF110361048 of China andYunnan province NSF Grant no 2011FB090
References
[1] F Tricomi On Second-Order Linear Partial Differential Equa-tions of Mixed Type Leningrad Moscow Russia 1947
[2] A R Manwell ldquoThe Tricomi equation with applications to thetheory of plane transonic flowrdquo ReconTechnical ReportA 276171979
[3] D Lupo and K R Payne ldquoA dual variational approach toa class of nonlocal semilinear Tricomi problemsrdquo NonlinearDifferential Equations and Applications vol 6 no 3 pp 247ndash266 1999
[4] J U Kim ldquoAn 119871119901 a priori estimate for the Tricomi equation in
the upper half spacerdquo Transactions of the American Mathemati-cal Society vol 351 no 11 pp 4611ndash4628 1999
[5] J M Rassias ldquoUniqueness of quasi-regular solutions for abi-parabolic elliptic bi-hyperbolic Tricomi problemrdquo ComplexVariables Theory and Application vol 47 no 8 pp 707ndash7182002
[6] K Yagdjian ldquoGlobal existence for the 119899-dimensional semilinearTricomi-type equationsrdquoCommunications in PartialDifferentialEquations vol 31 no 4ndash6 pp 907ndash944 2006
[7] K Yagdjian ldquoA note on the fundamental solution for theTricomi-type equation in the hyperbolic domainrdquo Journal ofDifferential Equations vol 206 no 1 pp 227ndash252 2004
[8] X-J Yang Advanced Local Fractional Calculus and Its Applica-tions World Science New York NY USA 2012
[9] A M Yang Y Z Zhang and Y Long ldquoThe Yang-Fouriertransforms to heat-conduction in a semi-infinite fractal barrdquoThermal Science vol 17 no 3 pp 707ndash713 2013
[10] W H Su D Baleanu X J Yang et al ldquoDamped wave equationand dissipative wave equation in fractal strings within the localfractional variational iteration methodrdquo Fixed PointTheory andApplications vol 2013 no 1 pp 1ndash11 2013
[11] X J Yang D Baleanu and J A T Machado ldquoMathematicalaspects of Heisenberg uncertainty principle within local frac-tional Fourier analysisrdquoBoundary Value Problems no 1 pp 131ndash146 2013
[12] D Baleanu J A Tenreiro Machado C Cattani M C Baleanuand X Yang ldquoLocal fractional variational iteration and decom-position methods for wave equation on Cantor sets within localfractional operatorsrdquo Abstract and Applied Analysis vol 2014Article ID 535048 6 pages 2014
[13] X-J Yang D Baleanu and J T Machado ldquoSystems of Navier-Stokes equations on Cantor setsrdquo Mathematical Problems inEngineering vol 2013 Article ID 769724 8 pages 2013
[14] G Yi ldquoLocal fractional Z transform in fractal spacerdquo Advancesin Digital Multimedia vol 1 no 2 pp 96ndash102 2012
[15] X Yang H M Srivastava J H He et al ldquoCantor-typecylindrical-coordinate method for differential equations withlocal fractional derivativesrdquo Physics Letters A vol 377 no 28ndash30 pp 1696ndash1700 2013
[16] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012
[17] C-G Zhao A-M Yang H Jafari and A Haghbin ldquoThe Yang-Laplace transform for solving the IVPs with local fractional
derivativerdquo Abstract and Applied Analysis vol 2014 Article ID386459 5 pages 2014
[18] Z Y Chen C Cattani and W P Zhong ldquoSignal processing fornondifferentiable data defined on Cantor sets a local fractionalFourier series approachrdquoAdvances inMathematical Physics vol2014 Article ID 561434 7 pages 2014
[19] A M Yang Y Z Zhang and X L Zhang ldquoThe non-differentiable solution for local fractional Tricomi equationarising in fractal transonic flow by loca l fractional variationaliteration methodrdquo Advances in Mathematical Physics vol 2014Article ID 983254 6 pages 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Invariant Shape Features and Relevance Feedback for Weld ... · Sym [0 1] < 0.5 > 0.5 > 0.5 < 0.5 Sig [0 1] < 0.5 < 0.5 → 1 > 0.5 2.2 Generic Fourier descriptor](https://img.dokumen.tips/doc/110x75/5fb60fbe46489e03c70e3474/invariant-shape-features-and-relevance-feedback-for-weld-sym-0-1-05.jpg)
![[XLS] · Web view0 0.2 0.5 1 0 0.02 0.04 0.1 0.2 0.35 0.5 0.7 0.75 1 1.5 2.5 3.7 12.5 0 0.2 0.5 1 0 0.02 0.04 0.1 0.2 0.35 0.5 0.7 0.75 1 1.5 2.5 3.7 12.5 0 0.2 0.5 1 0 0.02 0.04](https://img.dokumen.tips/doc/110x75/5af0fdb97f8b9ac2468eca80/xls-view0-02-05-1-0-002-004-01-02-035-05-07-075-1-15-25-37-125-0.jpg)
