GEOMETRIC PROPERTIES OF FRACTIONAL DIFFUSION EQUATION OF THE PROBABILITY DENSITY FUNCTION IN A COMPLEX DOMAIN

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1 Pak. J. Statist Vol. 31(5), GEOMETRIC PROPERTIES OF FRACTIONAL DIFFUSION EQUATION OF THE PROBABILITY DENSITY FUNCTION IN A COMPLEX DOMAIN Rabha W. Ibrahim 1, Hiba F. Al-Jaaby 2 ad M.Z. Ahmad 2 1 Istitute of Mathematical Scieces, Uiversity Malaya, 50603, Malaysia 2 Istitute of Egieerig Mathematics, Uiversiti Malaysia Perlis, Arau, Perlis, Malaysia aljaaby.hiba@yahoo.com, mzaii@uimap.edu.my Correspodig author rabhaibrahim@yahoo.com ABSTRACT I this study, we itroduce a ew class of fractioal aalytic fuctio i the ope uit disk. We deal with the subordiatio ad superordiatio of a class, which ivolves the fractioal differetial operator i the sese of Srivastava-Owa operators. The fractioal probability desity fuctio of oliear stochastic system is studied. By applyig the ew class to fid approximate solutios of the time fractioal master equatio i the complex domai. KEYWORDS Fractioal calculus; fractioal diffusio equatio; subordiatio ad superordiatio; uit disk; aalytic fuctio. 1. INTRODUCTION I the last few decades, the type of fractioal differetial equatios has bee cerebrated to be felicitous models of real life pheomeo. Oe of the fudametal implemetatios of the fractioal calculus is formulated by the itermediate physical process. A very wohy class is the wave equatios ad fractioal diffusio. It has bee established that the acoustic, diversity of the uiversal electromagetic, mechaical ad resposes ca be formed accurately employig fractioal diffusio-wave equatios equatios (Podluby, 1999), (Hilfer, 2000), (Kilbas et al., 2006), (Sabatier et al., 2007), (Lakshmikatham et al., 2009), (Jumarie, 2013). Differet seekig of the fractioal diffusio equatios have bee itroduced i various fractioal operators such as the Riema-Liouville, Caputo ad Rize fractioal differetial operators (He et al., 2012), (Guo, et al., 2012). I additio, the authors i (Ibrahim, 2012) ad (Ibrahim ad Jalab, 2013) studied a maximal solutio of the timespace fractioal heat equatio i a complex domai. The fractioal time is cosidered i the sese of the Riema-Liouville operator, while the fractioal space is itroduced i the sese of Srivastava-Owa operator for complex variables. The fractioal order plays a sigificat role i of the dyamical critical expoet. It was show that there is a relatio betwee the fractioal equatio ad cotiuous time. Therefore, the characteristic waitig time desity is foud by utilizig the Mettage- Leffler fuctio (Hilfer,1994) Pakista Joural of Statistics 601

2 602 Geometric propeies of fractioal diffusio equatio The cocepts of the subordiatio ad superordiatio are useful tools to describe the upper ad lower solutios of fractioal differetial equatios i a complex domai (Ibrahim ad Darus, 2008a) ad (Ibrahim ad Darus, 2008b). These cocepts are defied i the geometric fuctio theory, uivalet fuctio theory ad aalytic fuctio theory. Moreover, they provided a geometric explaatio for these solutios. I this ivestigatio, we suggest a ew class of fractioal aalytic fuctio i the ope uit disk. We study the subordiatio ad superordiatio of a class that ivolves the fractioal differetial operator i the sese of Srivastava-Owa operators. The fractioal probability desity fuctio of oliear stochastic system is studied. By employig this class to fid approximate solutios of the fractioal Tim- Space master equatio i the complex domai. 2. MATERIALS AND METHODS Let H be the class of fuctios aalytic i the uit disk U = { : <1} ad for C ad N. Let H[, ] be the subclass of H cosistig of fuctios of the form 1 1 ( ) =. Let A be the class of fuctios, aalytic i U ad ormalized by the coditios (0) = (0) 1 = 0. The a fuctio A is called covex or starlike if it maps U ito a covex or starlike regio, respectively. Correspodig classes are symbolized by K * ad S : It is well kow that K S * ; that both are subclasses of the class of uivalet fuctios ad have the followig aalytical represetatios ad * () S > 0, U () () K 1 > 0, U. () Let be aalytic i U, aalytic ad uivalet i U ad (0) = (0). The, by the symbol ( ) ( ) ( subordiate to ) i U, we shall mea ( U) ( U). 2 Let : C C ad let be uivalet i U. If is aalytic i U ad satisfies the differetial subordiatio ( ( )), ( )) ( ) the is called a solutio of the differetial subordiatio. The uivalet fuctio q is called a domiat of the solutios of the differetial subordiatio, q. If ad ( ), ( ) ad satisfy the differetial superordiatio ( ) ( ), ( ) are uivalet i U the is called a solutio of the differetial superordiatio. A aalytic fuctio q is called subordiate of the solutio of the differetial superordiatio if q.

3 Ibrahim, Al-Jaaby ad Ahmad 603 I (Srivastava ad Owa, 1989) defied fractioal operators (derivative ad itegral) i the complex z-plae C as follows: Defiitio 2.1 The fractioal derivative of order is defied, for a fuctio () by 1 d ( ) ( ) := d; 0 < 1, 0 (1 ) d ( ) where the fuctio () is aalytic i simply-coected regio of the complex z-plae C ivolvig the origi ad the multiplicity of ( is extracted by demadig log( ) to be real whe ( ) > 0. Defiitio 2.2 The fractioal itegral of order is defied, for a fuctio ( ), by 1 1 I ( ) := ( )( ) d ; > 0, 0 ( ) where the fuctio () is aalytic i simply-coected regio of the complex z-plae ( C ) icludig the origi ad the multiplicity of log( ) to be real whe ( ) > 0. Remark 2.1 ) ( ) 1 is removed by demadig ad ( 1) =, > 1;0 < 1 ( 1) D ( 1) =, > 1; > 0. ( 1) I Note that the above fractioal operators are the exteded operators for the Riema- Liouville fractioal operators (Miller ad Mocau, 2003) 1 t ( t ) I f ( t) = f ( ) d. ( ) a t a by The fractioal (arbitrary) order differetial of the fuctio f of order > 0 is give d t ( t) D f ( t) = f ( ) d. dt (1 ) a t a

4 604 Geometric propeies of fractioal diffusio equatio Whe a = 0, we shall deote 0 Dt f ( t) := Dt f ( t) follow-up. t t ad 0 I f ( t) := I f ( t) i the Defiitio 2.3 (Miller ad Mocau, 2003) The set of all fuctios () that are aalytic ad uivalet o U ( ) where ( ) := U : lim ( ) = ad are such that ( ) 0 for U ( ). Lemma 2.1 (Shamugam et al., 2006) () Let () K ad ad C \{0} with 1 > 0. If () H ad () ( ) ( ) ( ) ( ), the ( ) ( ) ad is the best domiat. Lemma 2.2 (Miller ad Mocau, 2003) Let () K ad C with { } > 0. If H ( ) (0),1, with ( ) ( ) S the ( ) ( ) ( z) ( ) leads to ( ) ( ) ad () is the best subordiat. series Let A be a class of aalytic fuctios, which approximate by the ifiite power 1 1 h( ) =, (2 ) =2 1 0 < 1, 1 :=, U. (2 ) We have the followig result: Theorem 2.1 ' Let A, D ( ) D ( ) S ad C \{0}. If the subordiatios ' ( ) ( ) ( ) ( ) ( ) ( ), where 1 ad 2 satisfyig the coditios of Lemma 2.2 ad Lemma 2.1 respectively the ( ) D ( ) ( ), U. (1) 1 2 Proof. Puttig ( ) := ( ), A. By usig Remark 2.1, yields that D () A ad (0) = 1(0) = 2(0) = 0. By lettig =1 i Lemma 2.1 ad utilizig Lemma 2.2, the asseio (1) is followed directly.

5 Ibrahim, Al-Jaaby ad Ahmad RESULTS AND DISCUSSION The fractioal diffusio equatio of the probability desity fuctio P( t, r ) ca be expressed by 1 2 t 1 r 2 r D P( t, r) = ( D m( r) D ( r)) P( t, r), (2) where 1, 2 are real costats, r is the amplitude of the system, m( r) = a r ad 2 ( r) = b r. I viue of Theorem 2.1, the operator D r upper ad lower boud by covex fuctios, therefore, we may obtai the followig fractioal equatio: t D P( r, t) = ( r) P( r, t), 0 < 1, (3) =1 ( r, t ) = ( r,0) = ( r ), where P( t, r ) is the probability desity of computig diffusig etity at the positio r at time t ad () r is the fractioal trasitio rate. Our aim is to approximate the solutio of (3) i the ope uit disk i term of time t J := [0, T]. Eq. (3) has various applicatios ot oly i mathematics but i differet fields. It represeted to fractioal Browia motio, electrochemical respose ad the river flow system (Ibrahim et al., 2015). We eed the followig results i the sequel. The proof is similar to Theorems 7,8 i (Ibrahim ad Jalab, 2013). Lemma 3.1 Let Pt (, ) be uivalet fuctio i the uit disk for all t J. The 1 r P( t, ) rf (2),(1) ;(1 ) ;, 0 1 (1 ) (4) d ( :=, r = ; U \{0}), d where the equality holds true for the Koebe fuctio. Lemma 3.2 Let Pt (, ) be covex fuctio i the uit disk for all t J. The 1 r P( t, ) F (2),(1) ;(1 ) ;, 0 1 (1 ) r = ; U \{0}. (5) Lemma 3.3 Let Pt (, ) be uivalet fuctio i the uit disk for all t J ad let :=1,0 < <1. The

6 606 Geometric propeies of fractioal diffusio equatio r P( t, ) rf (2),(1) ;(1 ) ; (1 ) (6) d ( :=, r = ; U \{0}), d where the equality holds true for the Koebe fuctio. Lemma 3.4 Let Pt (, ) be covex fuctio i the uit disk for all t J ad let =1 defied. The r u( t, ) F (2),(1) ;(1 ) ;, (1 ) r = ; U \{0}. We have the followig result: Theorem 3.1 Cosider the iitial differetial equatio (2). If P(, t) is uivalet the Eq. (2) has a approximate solutio to the hypergeometric fuctio ( 1r 2) P( t, ) t rf (2),(1),(1) ;(1 ),(1 ) ;. r (1 ) (1 ) (7) Proof. By employig the upper boud of the operator D P( t, ) ad 3.1 ad Lemma 3.3), we have D 1 P( t, ) (Lemma 1 1r D P( t, ) rf (2),(1) ;(1 ) ; (1 ) 2r (1 ) rf (2),(1) ;(1 ) ; r (2) (1) 1 = ( ) r 1 2 (1 ) (8) =0 (1 )! 0 < = r <1;0 < <1. Operatig (8) by I ad utilizig Remark 2.1 yield

7 Ibrahim, Al-Jaaby ad Ahmad 607 ( 1r 2) (2) (1) 1 ( 1) P( t, ) = r t r (1 ) =0 (1 )! ( 1 ) ( 1r 2) (2) (1) (1) 1 = t ( ) (9) r (1 ) (1 ) (1 ) (1 )! =0 1r 2 t rf ( ) = (2),(1),(1) ;(1 ),(1 ) ; r (1 ) (1 ) Hece the proof. Theorem 3.2 Cosider the iitial differetial equatio (2). If P(, t) is covex the Eq. (2) has a approximate solutio to the hypergeometric fuctio ( 1r 2) P( t, ) t rf (2),(1),(1) ;(1 ),(1 ) ;. r (1 ) (1 ) Proof. By applyig the upper boud of the operator D P( t, ) ad ad Lemma 3.4), we attai to 1 P( t, ) (Lemma r D P( t, ) F (2),(1) ;(1 ) ; (1 ) 2r F (1 ) ( 1r 2) = F (2),(1) ;(1 ) ; r (1 ) (2),(1) ;(1 ) ; (10) (0 < = r <1;0 < <1). Operatig (10) by I ad usig agai Remark 2.1 imply the desired asseio. 4. CONCLUSION We coclude that the probability desity fuctios ca be studied i view of the geometric fuctio theory i a complex domai. It has bee show that the fuctio is bouded by a geeralized hypergeometric fuctio. The method based o fractioal differetial equatio type time-space. ACKNOWLEDGMENT This research is suppoed by Project No. RG312-14AFR from the Uiversity of Malaya.

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