Fractional Riccati Equation Rational Expansion Method For Fractional Differential Equations

Transcription

1 Appl. Mth. Inf. Sci. 7 No (2013) 1575 Applied Mthemtics & Informtion Sciences An Interntionl Journl Frctionl Riccti Eqution Rtionl Expnsion Method For Frctionl Differentil Equtions Yongfeng Zhng nd Qinghu Feng School of Science Shndong University of Technology Zibo Shndong Chin Received: 28 Nov Revised: 6 Dec Accepted: 6 Jn Published online: 1 Jul Abstrct: In this pper new frctionl Riccti eqution rtionl expnsion method is proposed to estblish new exct solutions for frctionl differentil equtions. For illustrting the vlidity of this method we pply it to the nonliner frctionl Shrm-Tsso- Olever (STO) eqution the nonliner time frctionl biologicl popultion model nd the nonliner frctionl fom dringe eqution. Compred with the existing results in the literture more exct solutions re obtined by the proposed method. We lso illustrte the ppliction of the estblished exct solutions. Keywords: frctionl Riccti eqution method frctionl differentil equtions exct solutions nonliner frctionl Shrm-Tsso- Olever (STO) eqution nonliner time frctionl biologicl popultion model nonliner frctionl fom dringe eqution 1 Introduction Frctionl differentil equtions re generliztions of clssicl differentil equtions of integer order. In recent decdes frctionl differentil equtions hve gined much ttention s they re widely used to describe vrious complex phenomen in mny fields such s the fluid flow signl processing control theory systems identifiction biology nd other res. Mny rticles hve investigted some spects of frctionl differentil equtions such s the existence nd uniqueness of solutions to Cuchy type problems the methods for explicit nd numericl solutions nd the stbility of solutions [ ]. Among the investigtions for frctionl differentil equtions reserch for seeking exct solutions nd numericl solutions of frctionl differentil equtions is n importnt topic which cn lso provide vluble reference for other relted reserch. Mny powerful nd efficient methods hve been proposed to obtin numericl solutions nd exct solutions of frctionl differentil equtions so fr. For exmple these methods include the Adomin decomposition method [9 10] the vritionl itertive method [ ] the homotopy perturbtion method [14 15] the differentil trnsformtion method [16] the finite difference method [17] the finite element method [18] nd so on. Bsed on these methods vriety of frctionl differentil equtions hve been investigted nd solved. Recently Zhng et l. [19] first proposed new direct lgebric method nmed frctionl sub-eqution method bsed on the homogeneous blnce principle modified Riemnn-liouville derivtive by Jumrie [20] nd the frctionl Riccti eqution. The min ide of this method lies in tht the solutions of certin frctionl differentil equtions re supposed to hve the form u(ξ) = n i φ i where φ = φ(ξ) stisfies the frctionl Riccti eqution D α ξ φ = σ + φ 2. With the id of mthemticl softwre the uthors estblished successfully new exct solutions for some frctionl differentil equtions. Then in [21 22] the uthors improved this method to be suitble for more generl cses in which the solutions of certin frctionl differentil equtions re supposed to hve the forms u(ξ) = n i φ i u(ξ) = 0 + n i ( σb+dφ i= n D+Bφ )i respectively where φ = φ(ξ) stisfies the frctionl Riccti eqution D α ξ φ = σ+ φ 2. Motivted by the works bove in this pper by introducing new nstz with more generl form we propose new frctionl Riccti eqution rtionl expnsion method for solving frctionl differentil equtions in which the solutions u(ξ) of certin frctionl differentil equtions re supposed to hve the i=0 Corresponding uthor e-mil: fqhu@sin.com Nturl Sciences Publishing Cor.

2 1576 Y. Zhng Q. Feng: Frctionl Riccti Eqution Rtionl... form u(ξ)= 0 + m φ i 1 ( i φ + b i D α ξ φ) ( + µ 1 φ + µ 2 D α ξ φ)i where φ = φ(ξ) stisfies the frctionl Riccti eqution D α ξ φ = σ + φ 2. We orgnize the rest of this pper s follows. In Section 2 we give some definitions nd properties of Jumrie s modified Riemnn-liouville derivtive nd the description of the frctionl Riccti eqution rtionl expnsion method. Then in Section 3 we pply the method to solve the nonliner frctionl Shrm-Tsso-Olever (STO) eqution the nonliner time frctionl biologicl popultion model nd the nonliner frctionl fom dringe eqution. Some conclusions re presented t the end of the pper. 2 Jumrie s modified Riemnn-liouville derivtive nd description of the frctionl Riccti eqution rtionl expnsion method The Jumrie s modified Riemnn-Liouville derivtive of order α is defined by the following expression [20]: 1 Dt α Γ(1 α) d t0 dt (t ξ) α ( f(ξ) f(0))dξ f(t)= 0<α < 1 ( f (n) (t)) (α n) n α < n+1 n 1. We list some importnt properties for the modified Riemnn-Liouville derivtive s follows (see [20 (3.10)-(3.13)]: D α t t r = Γ(1+r) Γ(1+r α) tr α (1) D α t ( f(t)g(t))=g(t)d α t f(t)+ f(t)d α t g(t) (2) D α t f[g(t)]= f g[g(t)]d α t g(t)=d α g f[g(t)](g (t)) α. (3) Suppose tht frctionl prtil differentil eqution sy in two or three independent vribles xyt is given by P(uu t u x u y D α t ud α x ud α y u...)=0 (4) where u = u(x y t) is n unknown function P is polynomil in u = u(x y t) nd its vrious prtil derivtives in which the highest order derivtive nd nonliner term re involved. Step 1. Suppose tht u(xyt)= U(ξ) ξ = ξ(xyt) (5) nd then Eq. (4) cn be turned into the following frctionl ordinry differentil eqution with respect to the vrible ξ : P(U U U D α ξ U...)=0. (6) Step 2. Suppose tht the solution of (6) cn be expressed in φ s follows: U(ξ)= 0 + m φ i 1 ( i φ + b i D α ξ φ) (7) ( + µ 1 φ + µ 2 Dξ α φ)i where φ = φ(ξ) stisfies the following frctionl Riccti eqution: D α ξ φ = σ+ φ 2 (8) nd 0 i b i c i i = 12...m µ 1 µ 2 re ll constnts to be determined lter. The positive integer m cn be determined by considering the homogeneous blnce between the highest order derivtive nd nonliner term ppering in (6). In [23] by using the generlized Exp-function method Zhng et l. first obtined the following solutions of Eq. (8): σ tnh α ( σξ) σ < 0 σ coth α ( σξ) σ < 0 φ(ξ)= σ tnα ( σξ) σ > 0 σ cot α ( (9) σξ) σ > 0 Γ(1+α) ξ α ω is constnt σ = 0 + ω where the generlized hyperbolic nd trigonometric functions re defined s sin α (ξ)= E α(iξ α ) E α ( iξ α ) 2i cos α (ξ)= E α(iξ α )+E α ( iξ α ) 2 sinh α (ξ)= E α(ξ α ) E α ( ξ α ) 2 cosh α (ξ)= E α(ξ α )+E α ( ξ α ) 2 tn α (ξ)= sin α(ξ) cos α (ξ) cot α(ξ)= cos α(ξ) sin α (ξ) tnh α (ξ)= sinh α(ξ) cosh α (ξ) coth α(ξ)= cosh α(ξ) sinh α (ξ) ξ k where E α (ξ) = α > 0 is the k=0 Γ(1+kα) Mittg-Leffler function. Step 3. Substituting (7) into (6) nd using (8) the left-hnd side of (6) is converted to nother polynomil in φ j ( σ+ φ 2 ) i fter eliminting the denomintor. Equting ech coefficient of this polynomil to zero Nturl Sciences Publishing Cor.

3 Appl. Mth. Inf. Sci. 7 No (2013) / yields set of lgebric equtions for 0 i b i c i 2...m µ 1 µ 2. Step 4. Solving the equtions system in Step 3 nd by using the solutions of Eq. (8) we cn construct vriety of exct solutions of Eq. (4). 3 Applictions In this section we will pply the described method in Section 2 to some frctionl differentil equtions. 3.1 nonliner frctionl Shrm-Tsso-Olever (STO) eqution We consider the nonliner frctionl Shrm-Tsso-Olever (STO) eqution with spce- nd time-frctionl derivtives of the form Dt α u+3(dx α u) 2 + 3u 2 Dx α u+3udx 2α u+dx 3α u=0 0<α 1 (10) which is the vrition of the following nonliner frctionl Shrm-Tsso-Olever (STO) eqution [24 25] with timefrctionl derivtive of the form D α t u+3u 2 x+ 3u 2 u x + 3uu xx + u xxx = 0. To begin with we suppose u(xt)= U(ξ) where ξ = kx+ ct+ξ 0. Then by use of Eq. (3) Eq. (10) cn be turned into c α D α ξ U+ 3k2α (D α ξ U)2 + 3U 2 k α D α ξ U +3k 2α UD 2α ξ U+ k3α D 3α ξ U = 0. (11) Suppose tht the solution of Eq. (11) cn be expressed by polynomil in φ s follows: U(ξ)= 0 + m φ i 1 ( i φ + b i Dξ αφ) (12) ( + µ 1 φ + µ 2 Dξ α φ)i where φ = φ(ξ) stisfies Eq. (8). Blncing the order between the highest order derivtive term nd nonliner term in Eq. (11) we cn obtin m=1. So we hve U(ξ)= φ + b 1 D α ξ φ (13) + µ 1 φ + µ 2 D α φ ξ Substituting (13) into (11) nd collecting ll the terms with the sme power of φ j ( σ+ φ 2 ) i together equting ech coefficient to zero yields set of lgebric equtions. Solving these equtions yields: Cse 1: 0 = 0 1 = k α b 1 = 0 σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 2: 0 = 0 1 = 2 k α b 1 = 0 σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 3: 0 = 0 1 = k α b 1 = 0 σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 4: 0 = 0 1 = 2 k α b 1 = 0 σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 5: 0 = 0 1 = k α b 1 =± 2 kα σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 6: 0 = 0 1 = k α b 1 =± k α σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 7: 0 = 0 1 = 2 kα b 1 =± 2 kα σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 8: 0 = 0 1 = 2 kα b 1 =± k α σ = cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Nturl Sciences Publishing Cor.

4 1578 Y. Zhng Q. Feng: Frctionl Riccti Eqution Rtionl... Cse 9: 0 = 0 1 = k α b 1 =± 2 kα σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 10: 0 = 0 1 = k α b 1 =± k α σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 11: 0 = 0 1 = 2 kα b 1 =± 2 kα σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Cse 12: 0 = 0 1 = 2 kα b 1 =± k α σ = 4cα k 3α µ 1 = 0 µ 2 = 0 = where is n rbitrry constnt. Substituting the results bove into Eq. (13) nd combining with the solutions of Eq. (8) s denoted in (9) we cn obtin rich vriety of exct solutions to the nonliner frctionl Shrm-Tsso-Olever (STO) eqution with spce- nd time-frctionl derivtives. From Cses 1-4 we hve the following generlized exct solutions for Eq. (10) u 11 (xt)=m cα k α tnh α [n cα k 3α (kx+ct+ ξ 0 )] < 0 u 12 (xt)=m cα k α coth α [n cα k 3α (kx+ct+ ξ 0 )] < 0 u 13 (xt)= m tn α [n c α k 3α (kx+ct+ ξ 0 )] > 0 u 14 (xt)=m cot α [n c α k 3α (kx+ct+ ξ 0 )] > 0 (14) where m = 1 n = 1 or m = 4 n = 2 or m = 2 n = 2 or m=2 n=1. From Cses 5-12 we hve the following generlized exct solutions for Eq. (10) u 21 (xt)=m cα k α tnh α [p cα k 3α (kx+ct+ ξ 0 )] ±n {1 tnh 2 α[p cα k 3α (kx+ct+ ξ 0 )]} < 0 u 22 (xt)=m cα k α coth α [p cα k 3α (kx+ct+ ξ 0 )] ±n {1 coth 2 α[p u 23 (xt)= m tn α [p c α k 3α ±n {1+tn 2 α[p u 24 (xt)=m cot α [p ±n {1+cot 2 α[p cα k 3α (kx+ct+ ξ 0 )]} < 0 (kx+ct+ ξ 0 )] cα k 3α (kx+ct+ ξ 0 )]} > 0 c α k 3α (kx+ct+ ξ 0 )] cα k 3α (kx+ct+ ξ 0 )]} > 0 (15) where m=1 n= 1 2 or m=1 n=1 or m= 2 1 n= 1 2 or m= 2 1 n=1 nd p is determined by p=1 or p= nonliner time frctionl biologicl popultion model We consider the nonliner time frctionl biologicl popultion model [1922]: D α t u=(u 2 ) xx +(u 2 ) yy + h(u 2 r) 0<α 1 (16) where h r re constnts. Similr s in [1922] we suppose u(xyt) = U(ξ) where ξ = kx+iky+ct + ξ 0 k c ξ 0 re ll constnts with k c 0 nd i is the unit of imginry numbers. Then by use of Eq. (3) Eq. (16) cn be turned into c α D α ξ U = h(u 2 r). (17) Suppose tht the solution of Eq. (17) cn be expressed by polynomil in φ s follows: U(ξ)= 0 + m φ i 1 ( i φ + b i D α ξ φ) (18) ( + µ 1 φ + µ 2 D α φ)i ξ where φ = φ(ξ) stisfies Eq. (8). Blncing the order between the highest order derivtive term nd nonliner term in Eq. (17) we cn obtin m=1. So we hve U(ξ)= φ + b 1 D α ξ φ (19) + µ 1 φ + µ 2 D α φ ξ Nturl Sciences Publishing Cor.

5 Appl. Mth. Inf. Sci. 7 No (2013) / Substituting (19) into (17) nd collecting ll the terms with the sme power of φ j ( σ+ φ 2 ) i together equting ech coefficient to zero yields set of lgebric equtions. Solving these equtions yields Cse 1: 0 = 2hrµ 1c α 1 = 4h2 rµ 2 1 c α µ 2 0 c α 2h b 1 = 0 µ 2 =± c α h 4r σ = 4h2 rc 2α = µ 1 = µ 1 where µ 1 re rbitrry constnts. Cse 2: 0 = 2hrµ 1c α 1 = 4h2 rµ 2 1 c α µ 2 0 c α 2h b 1 = 4h2 rµ 1 µ 2 c α ± 4h 2 rµ 0 2(µ2 2 µ2 1 )+ µ4 0 c2α 2h µ 2 = µ 2 σ = 4h 2 rc 2α = µ 1 = µ 1 where µ 1 µ 2 re rbitrry constnts. Cse 3: 0 = hrµ 1c α 1 = h2 rµ 2 1 c α µ 2 0 c α h b 1 = 0 µ 2 = 0 σ = h 2 rc 2α = µ 1 = µ 1 where µ 1 re rbitrry constnts. Substituting the results in the three cses bove into Eq. (19) nd combining with (9) we cn obtin the following exct solutions to Eq. (16). Fmily 1: u 11 (xyt)= 2hrµ 1c α + (4h 2 rµ 1 2 c 2α 2 ) r tnh α (2hc α rξ) p r>0 11 u 12 (xyt)= 2hrµ 1c α + (4h2 rµ 1 2 c 2α 2 ) r coth α (2hc α rξ) p r>0 12 u 13 (xyt)= 2hrµ 1c α µ + 0 (4h 2 rµ 1 2 c 2α 2 ) r tn α (2hc α rξ) p r<0 13 u 14 (xyt)= 2hrµ 1c α + (4h2 rµ 1 2 c 2α 2 ) r cot α (2hc α rξ) p r<0 14 (20) where ξ 0 k c µ 1 re ll rbitrry constnts with k c 0 ξ = kx+iky+ct+ ξ 0 nd p 11 = µ µ 1 hc α r tnh α (2hc α rξ) ±µ tnh 2 α(2hc α rξ) p 12 = 2 2 µ 1 hc α r coth α (2hc α rξ) 1 coth 2 α(2hc α rξ) ±µ 2 0 p 13 = µ 1 hc α r tn α (2hc α rξ) 1+tnα(2hc 2 α rξ) ±µ 2 0 p 14 = 2 2 µ 1 hc α r cot α (2hc α rξ) 1+cotα(2hc 2 α rξ) Fmily 2: ±µ 2 0 u 21 (xyt)= 2hrµ 1c α + (4h2 rµ 2 1 c 2α µ 2 0 ) r tnh α (2hc α rξ) p 21 + ϕ 21 q 21 r>0 u 22 (xyt)= 2hrµ 1c α + (4h2 rµ 2 1 c 2α µ 2 0 ) r coth α (2hc α rξ) p 22 + ϕ 22 q 22 r>0 u 23 (xyt)= 2hrµ 1c α + (4h2 rµ 1 2 c 2α 2 ) r tn α (2hc α rξ) p + ϕ q r<0 23 u 24 (xyt)= 2hrµ 1c α + (4h2 rµ 1 2 c 2α 2 ) r cot α (2hc α rξ) p + ϕ q r<0 24 u 25 (xyt)= c α Γ(1+α) h{ (ξ α + ω) µγ(1+α)} r=0 (21) where ξ 0 ω k c µ = µ 1 ± µ 2 re ll rbitrry constnts with k c 0 ξ = kx+iky+ct+ ξ 0 nd p 21 = 2 2 µ 1 hc α r tnh α (2hc α rξ) +2i µ 2 hc α r 1 tnh 2 α(2hc α rξ) p 22 = 2 2 µ 1 hc α r coth α (2hc α rξ) +2i µ 2 hc α r 1 coth 2 α(2hc α rξ) p 23 = µ 1 hc α r tn α (2hc α rξ) +2 µ 2 hc α r 1+tnα(2hc 2 α rξ) p 24 = 2 2 µ 1 hc α r cot α (2hc α rξ) +2 µ 2 hc α r 1+cotα(2hc 2 α rξ) q 21 = 2 2 µ 1 hc α r tnh α (2hc α rξ) +2i µ 2 hc α r 1 tnh 2 α(2hc α rξ) q 22 = 2 2 µ 1 hc α r coth α (2hc α rξ) +2i µ 2 hc α r 1 coth 2 α(2hc α rξ) q 23 = µ µ 1 hc α r tn α (2hc α rξ) Nturl Sciences Publishing Cor.

6 1580 Y. Zhng Q. Feng: Frctionl Riccti Eqution Rtionl µ 2 hc α r 1+cotα(2hc 2 α rξ) q 24 = 2 2 µ 1 hc α r cot α (2hc α rξ) +2 µ 2 hc α r 1+cotα(2hc 2 α rξ) ϕ 21 = ic α r( 4h 2 rµ 1 µ 2 c α ± 4h 2 rµ 0 2(µ2 2 µ2 1 )+ µ4 0 c2α ) 1 tnhα(2hc 2 α rξ) ϕ 22 = ic α r( 4h 2 rµ 1 µ 2 c α ± 4h 2 rµ 0 2(µ2 2 µ2 1 )+ µ4 0 c2α ) 1 cothα(2hc 2 α rξ) ϕ 23 = c α r( 4h 2 rµ 1 µ 2 c α ± 4h 2 rµ 0 2(µ2 2 µ2 1 )+ µ4 0 c2α ) 1+tnα(2hc 2 α rξ) ϕ 24 = c α r( 4h 2 rµ 1 µ 2 c α ± 4h 2 rµ 0 2(µ2 2 µ2 1 )+ µ4 0 c2α ) 1+cotα(2hc 2 α rξ). Fmily 3: u 31 (xyt)= hrµ 1c α + (h2 rµ 2 1 c 2α µ 2 0 ) r tnh α (hc α rξ) µ 2 0 µ 1 hc α r tnh α (hc α rξ) r>0 u 32 (xyt)= hrµ 1c α + (h2 rµ 1 2 c 2α 2 ) r coth α (hc α rξ) 2 µ 1 hc α r coth α (hc α rξ) r>0 u 33 (xyt)= hrµ 1c α (h2 rµ 1 2 c 2α 2 ) r tn α (hc α rξ) 2 + µ 1 hc α r tn α (hc α rξ) r<0 u 34 (xyt)= hrµ 1c α + (h2 rµ 1 2 c 2α 2 ) r cot α (hc α rξ) 2 µ 1 hc α r cot α (hc α rξ) r<0 u 35 (xyt)= c α Γ(1+α) h{ (ξ α + ω) µ 1 Γ(1+α)} (22) where ξ 0 k c µ 1 re ll rbitrry constnts with k c 0 nd ξ = kx+iky+ct+ ξ 0. Remrk 1. If we set = D µ 1 = B k = 1 c=λ then the solutions in (22) reduce to the solutions estblished in [22 Eqs. (36)-(40)]. If we set µ 1 = 0 then the solutions in (22) reduce to the solutions estblished in [19 Eqs. (16)-(20)]. 3.3 nonliner frctionl fom dringe eqution We consider the nonliner fom dringe eqution with time nd spce-frctionl derivtives [222627]: D α t u= u 2 Dα x D α x u 2u 2 D α x u+(d α x u) 2 0<α 1. (23) The fom dringe eqution is model of the flow of liquid through chnnels nd nodes (intersection of four chnnels) between the bubbles driven by grvity nd cpillrity [28]. First we suppose u(xyt)= U(ξ) where ξ = kx+ct+ ξ 0. Then by use of Eq. (3) Eq. (23) cn be turned into c α D α ξ U = U 2 k2α D α ξ Dα ξ U 2U 2 k α D α ξ U+ k2α (D α ξ U)2. (24) Suppose tht the solution of Eq. (24) cn be expressed by polynomil in φ s follows: U(ξ)= 0 + r=0 where µ 1 re rbitrry constnts. Cse 2: 0 = 0 1 = k α 2 m φ i 1 ( i φ + b i D α ξ φ) (25) ( + µ 1 φ + µ 2 Dξ α φ)i where φ = φ(ξ) stisfies Eq. (8). Blncing the order between the highest order derivtive term nd nonliner term in Eq. (24) we cn obtin m=1. So we hve U(ξ)= φ + b 1 Dξ αφ (26) + µ 1 φ + µ 2 Dξ α φ Substituting (26) into (24) nd collecting ll the terms with the sme power of φ j ( σ+ φ 2 ) i together equting ech coefficient to zero yields set of lgebric equtions. Solving these equtions yields Cse 1: 0 = 2µ 1c α k 2α 1 = 4k 2α µ 2 1 c α + µ 2 0 k α 2 b 1 = 0 µ 2 =± 2 k 3 2 α c α 2 σ = 4c α k 3α = µ 1 = µ 1 b 1 =± k α 2 µ 2 = 0 σ = 4c α k 3α = µ 1 = 0 where is n rbitrry constnt. Cse 3: 0 = µ 1c α k 2α 1 = k 2α µ 2 1 c α + µ 2 0 k α b 1 = 0 µ 2 = 0 σ = c α k 3α = µ 1 = µ 1 where µ 1 re rbitrry constnts. Substituting the results in the three cses bove into Eq. (26) nd combining with (9) we cn obtin the following exct solutions to Eq. (23). Nturl Sciences Publishing Cor.

7 Appl. Mth. Inf. Sci. 7 No (2013) / Fmily 1: u 11 (xt)= 2µ 1c α k 2α (4k 3α µ 1 2 cα + 2 ) tnh α (2 c α k 3α ξ) l 11 < 0 u 12 (xt)= 2µ 1c α k 2α (4k 3α µ 1 2 cα + 2 ) coth α (2 c α k 3α ξ) l 12 < 0 u 13 (xt)= 2µ 1c α k 2α + (4k 3α µ 1 2 cα + 2 ) tn α (2 c α k 3α ξ) l 13 c α k α > 0 u 14 (xt)= 2µ 1c α k 2α (4k 3α µ 1 2 cα + 2 ) cot α (2 c α k 3α ξ) l 14 c α k α > 0 (27) where ξ 0 k c µ 1 re ll rbitrry constnts with k c 0 ξ = kx+ct+ ξ 0 nd l 11 = 2 2 µ 1 c α k 3α tnh α (2 c α k 3α ξ) ± 2 c α k 1 tnh 3α 2 α(2 c α k 3α ξ) l 12 = 2 2 µ 1 c α k 3α coth α (2 c α k 3α ξ) ± 2 c α k 1 coth 3α 2 α(2 c α k 3α ξ) l 13 = µ µ 1 c α k 3α tn α (2 c α k 3α ξ) ±µ 2 0 c α k 3α 1+tn 2 α(2 c α k 3α ξ) l 14 = µ µ 1 c α k 3α cot α (2 c α k 3α ξ) ± 2 c α k 1+cot 3α α(2 2 c α k 3α ξ) Fmily 2: u 21 (xt)= tnh α (2 c α k 3α ξ) ±i {1 tnh 2 α(2 c α k 3α ξ)} < 0 u 22 (xt)= coth α (2 c α k 3α ξ) ±i {1 coth 2 α(2 c α k 3α ξ)} < 0 u 23 (xt)= tn α (2 c α k 3α ξ) ±i {1+tnα(2 2 c α k 3α ξ)} > 0 u 24 (xt)= cot α (2 c α k 3α ξ) ±i {1+cotα(2 2 c α k 3α ξ)} > 0 (28) where ξ 0 k c re ll rbitrry constnts with k c 0 nd ξ = kx+ct+ ξ 0. Fmily 3: u 31 (xt)= µ 1c α k 2α (k 3α µ 1 2 cα + 2 ) tnh α ( c α k 3α ξ) 2 µ 0µ 1 c α k 3α tnh α ( c α k 3α ξ) < 0 u 32 (xt)= µ 1c α k 2α (k 3α µ 1 2 cα + 2 ) coth α ( c α k 3α ξ) µ 0 2 µ 0µ 1 c α k 3α coth α ( c α k 3α ξ) < 0 u 33 (xt)= µ 1c α k 2α + (k 3α µ 1 2 cα + 2 ) tn α ( c α k 3α ξ) 2 + µ 0µ 1 c α k 3α tn α ( c α k 3α ξ) > 0 u 34 (xt)= µ 1c α k 2α (k 3α µ 1 2 cα + 2 ) cot α ( c α k 3α ξ) 2 µ 0µ 1 c α k 3α cot α ( c α k 3α ξ) > 0 (29) where ξ 0 k c µ 1 re ll rbitrry constnts with k c 0 nd ξ = kx+ct+ ξ 0. Remrk 2. If we set = D µ 1 = B k = 1 c=λ then the solutions u 33 (xt) u 34 (xt) reduce to the solutions estblished in [22 Eq. (31)]. 4 Illustrtion of the presented results In this section we will illustrte the ppliction of the results estblished bove. The nonliner frctionl Shrm-Tsso-Olever (STO) eqution (10) is KdV-like eqution nd ply n importnt role in describing the nonliner wve phenomen. Exct solutions for it with different forms cn describe different nonliner wves. For the estblished exct solutions u 11 (xt) u 12 (xt) u 21 (xt) u 22 (xt) with hyperbolic functions forms in (14)-(15) solitry wve phenomenon cn be demonstrted by them while for the estblished exct solutions u 13 (xt) u 14 (xt) u 23 (xt) u 24 (xt) with periodic functions forms in (14)-(15) periodic wve phenomenon cn be demonstrted. For the better understnding the solitry wve phenomenon nd periodic wve phenomenon we tke u 11 (xt) u 13 (xt) for exmple nd show them in Figs. 1-4 with some given prmeters in which the vlue of the vrible c represents wve velocity. From Figs. 1-2 one cn see with the incresing of the order α of the frctionl derivtive the time nd spce coordintes t which the solitry wve ppers get smller which implies the existing period of the solitry wve gets shorter. From Figs. 3-4 one cn see with the Nturl Sciences Publishing Cor.

8 1582 Y. Zhng Q. Feng: Frctionl Riccti Eqution Rtionl... incresing of the order α of the frctionl derivtive it gets more frequent tht the periodic wve reches its locl climx. In the nonliner time frctionl biologicl popultion model (16) the function u denotes the popultion density nd h(u 2 r) represents the popultion supply due to births nd deths. The reson of using frctionl differentil equtions to modeling biologicl popultion is tht frctionl differentil equtions re nturlly relted to systems with memory which exists in most biologicl systems. Also they re closely relted to frctls which re bundnt in biologicl systems. The resulting solutions spred fster thn the clssicl solutions nd my exhibit symmetry. For the ske of illustrting the vrition trend of the popultion density we tke the solution u 33 (xyt) in (22) for exmple nd show it in Fig. 5. The nonliner frctionl fom dringe eqution (23) nd the estblished solutions (27)-(29) for it ply fundmentl role in describing the fom dringe process where the vribles x nd t represent scled position nd time coordintes respectively. Foming occurs in mny distilltion nd bsorption processes. Recent reserch in foms hs centered on three topics which re often treted Nturl Sciences Publishing Cor.

9 Appl. Mth. Inf. Sci. 7 No (2013) / describing the fom dringe process we tke the solitry wve solution u 34 (xt) in (29) for exmple nd show it in Fig. 6 in which the vlue of the vrible c represents wve velocity. 5 Conclusions We hve proposed new frctionl Riccti eqution rtionl expnsion method for solving frctionl differentil equtions nd pplied it to find exct solutions of the nonliner frctionl Shrm-Tsso-Olever (STO) eqution the nonliner time frctionl biologicl popultion model nd the nonliner frctionl fom dringe eqution. As result some generlized nd new exct solutions for them hve been successfully found. Being concise nd powerful this method cn lso be pplied to solve other frctionl differentil equtions s long s the homogeneous blnce principle is stisfied. Acknowledgement The uthors re grteful to the nonymous referee for creful checking of the detils nd for helpful comments tht improved this pper. References seprtely but re in fct interdependent: dringe corsening nd rheology. Dringe plys n importnt role in fom stbility: indeed when fom dries its structure becomes more frgile; the liquid films between djcent bubbles being thinner then cn brek leding to fom collpse. The dringe of liquid foms involves the interply of grvity surfce tension nd viscous forces. Forced fom dringe my well be the best prototype for certin generl phenomen described by nonliner differentil equtions prticulrly the type of solitry wve which is most fmilir in tidl bores. For better understnding the function of the solutions (27)-(29) in [1] A. Sdtmndi nd M. Dehghn Comput. Mth. Appl (2010). [2] Y. Zhou F. Jio nd J. Li Nonliner Anl (2009). [3] L. Gleone nd R. Grrpp J. Comput. Appl. Mth (2009). [4] J.C. Trigessou N. Mmri J. Sbtier nd A. Oustloup Signl Process (2011). [5] W. Deng Nonliner Anl (2010). [6] F. Ghoreishi nd S. Yzdni Comput. Mth. Appl (2011). [7] J.T. Edwrds N.J. Ford nd A.C. Simpson J. Comput. Appl. Mth (2002). [8] M. Muslim Mth. Comput. Model (2009). [9] A.M.A. El-Syed nd M. Gber Phys. Lett. A (2006). [10] A.M.A. El-Syed S.H. Behiry nd W.E. Rsln Comput. Mth. Appl (2010). [11] J.H. He Commun. Nonliner Sci. Numer. Simul (1997). [12] G. Wu nd E.W.M. Lee Phys. Lett. A (2010). [13] S. Guo nd L. Mei Phys. Lett. A (2011) 309. [14] J.H. He Comput. Methods Appl. Mech. Engrg (1999). [15] J.H. He Inter. J. Non-Liner Mech (2000). [16] Z. Odibt nd S. Momni Appl. Mth. Lett (2008). [17] M. Cui J. Comput. Phys (2009). Nturl Sciences Publishing Cor.

10 1584 Y. Zhng Q. Feng: Frctionl Riccti Eqution Rtionl... [18] Q. Hung G. Hung nd H. Zhn Adv. Wter Resour (2008). [19] S. Zhng nd H.Q. Zhng Phys. Lett. A (2011). [20] G. Jumrie Comput. Mth. Appl (2006). [21] S.M. Guo L.Q. Mei Y. Li nd Y.F. Sun Phys. Lett. A (2012). [22] B. Lu Phys. Lett. A (2012). [23] S. Zhng Q.A. Zong D. Liu nd Q. Go Commun. Frct. Clc (2010). [24] L.N. Song Q. Wng nd H.Q. Zhng J. Comput. Appl. Mth (2009). [25] B. Lu J. Mth. Anl. Appl (2012). [26] H.H. Fdrvi H.S. Nik nd R. Buzhbdi Internt. J. Differ. Equtions (2011). [27] Z. Dhmni M.M. Mesmoudi nd R. Bebbouchi Electron. J. Qul. Theor. Differ. Equtions (2008). [28] D. Weire S. Hutzler S. Cox M.D. Alonso nd D. Drenckhn J. Phys. Condens. Mtter 15 S65-S73 (2003). Yongfeng Zhng is resercher in the domin of differentil equtions nd pplictions. She obtined her M.D. from Northest Norml University (Chin) in 2006 nd now is ssocite professor in Shndong University of Technology. She mjors in exct solutions qulittive nlysis nd numericl method of differentil equtions. Qinghu Feng is resercher in the domin of differentil equtions nd pplictions. He obtined his M.D. from Shndong University(Chin) nd is now PhD cndidte. He hs published more thn 30 reserch rticles in reputed interntionl journls of mthemticl nd engineering sciences. Nturl Sciences Publishing Cor.