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2 Chpter 5 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions Njeeb Alm Khn, Amber Shikh, Fqih Sultn nd Asmt Ar Additionl informtion is vilble t the end of the chpter Abstrct This chpter offers numericl simultion of frctionl differentil equtions by utilizing Chebyshev-simulted nneling neurl network (ChSANN) nd Legendre-simulted nneling neurl network (LSANN). The use of Chebyshev nd Legendre polynomils with simulted nneling reduces the men squre error nd leds to more ccurte numericl pproximtion. The comprison of proposed methods with previous methods confirms the ccurcy of ChSANN nd LSANN. Keywords: neurl network, frctionl Riccti, Legendre polynomil, Chebyshev polynomil, simulted nneling 1. Introduction During the lst few decdes, frctionl clculus hs gined mssive ttention of physicists nd mthemticins becuse of its numerous interdisciplinry pplictions. Mny recent reserches re ended up demonstrting the significnce of frctionl-order differentil equtions s vluble instruments to model severl physicl phenomen such s the nonliner oscilltion of erthquke nd the fluid dynmics trffic cn be elegntly modelled with frctionl derivtives [1, 2]. Vrious physicl processes show frctionl-order behviour tht might chnge with respect to time or spce. The doption of frctionl clculus concepts is well known in mny scientific res such s physics, diffusion nd wve propgtion, het 2016 The Author(s). Licensee InTech. This chpter is distributed under the terms of the Cretive Commons Attribution License ( which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited.

3 98 Numericl Simultion - From Brin Imging to Turbulent Flows trnsfer, viscoelsticity nd dmping, electronics, robotics, electromgnetism, signl processing, telecommunictions, control systems, trffic systems, system identifiction, chos nd frctls, biology, genetic lgorithms, filtrtion, modelling nd identifiction, chemistry, irreversibility, s well s economy nd finnce [3 5]. Modelling of different physicl phenomen gve rise to specil differentil eqution known s Riccti differentil eqution tht ws nmed fter n Itlin mthemticin Count Jcopo Frncesco Riccti. Due to mny pplictions of frctionl Riccti differentil equtions such s in stochstic controls nd pttern formtion, mny reserchers studied it to get the exct or pproximte solutions. Such s frctionl vritionl itertion method ws pplied in [6] to give n pproximte nlyticl solution of non-liner frctionl Riccti differentil eqution. His modified homotopy perturbtion method (MHPM) ws used on qudrtic Riccti differentil eqution of frctionl order [7]. The results of frctionl Riccti differentil eqution were lso obtined on the bsis of Tylor colloction method [8]. Frctionl Riccti differentil equtions were solved by mens of vritionl itertion method nd homotopy perturbtion Pde technique [9, 10], nd the numericl results were ttined by using Chebyshev finite difference method [11]. Adomin decomposition method ws presented for frctionl Riccti differentil eqution [12], the problem ws described by mens of Bernstein colloction method [13], nd enhnced homotopy perturbtion method (EHPM) ws used to study this problem [14]. Recently, rtificil neurl network nd sequentil qudrtic progrmming hve been utilized to obtin the solution of Riccti differentil eqution [15]. The problem ws lso explined by Legendre wvelet opertionl mtrix method [16] nd the results of frctionl Riccti differentil eqution by new homotopy perturbtion method (NHPM) [17] were obtined. In recent yers, rtificil neurl network (ANN) is one of the methods tht re ttining mssive ttention of reserchers in the re of mthemtics s well s in different physicl sciences. The concept of ANN strted to develop in 1943 when neurophysiologist nd young mthemticin [18] gve the ide on working of neuron with the help of n electric circuit. Lter, book [19] ws written to clrify the working of neurons then in Bernrd Widrow nd Mrcin Hoff developed model MEDALIN tht ws used to study the first rel-world problem of neurl network. Reserchers continued to study the singlelyered neurl network, but in 1975, the concept of multilyer perceptron (MLP) ws introduced, which ws computtionlly exhustive due to multilyer rchitecture. The excessive trining time nd high computtionl complexity of MLP gve rise to functionl neurl network by which the complexity of multilyers ws overcome by introducing vrible functions [20]. Functionl link neurl network hs been implemented to severl problems such s modified functionl link neurl network for denoising of imge [21], ctive control of non-liner noise processes through functionl link neurl network [22], nd the problem of chnnel equliztion in digitl communiction system ws solved by functionl link neurl network [23]. Due to less computtionl effort with esy to implement procedure, functionl link neurl network ws lso implemented to solve differentil equtions [24, 25].

4 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions Definitions nd preliminries The Riemnn-Liouville, Grünwld-Letnikov nd Cputo definitions of frctionl derivtives of order α > 0 re used more frequently mong severl definitions of frctionl derivtives nd integrls, but in this chpter, the Cputo definition will be used for working out the frctionl derivtive in subsequent procedure. The definitions of commonly used frctionl differentil opertors re discussed in the study of Sontkke nd Shikh [26]. Definition 1: The Riemnn-Liouville frctionl derivtive opertor cn be defined s follows: g ( b ) ( x ) x x 1 d D g( x) = db, x -1 < < x x 1 x G( x -) dx ò - b + - where here α >0, x >, α, ndx R. Definition 2: The definition of frctionl differentil opertor ws introduced by Cputo in lte 1960s tht cn be defined s follows [27]: ( x g ) ( b ) ( x ) 1 D g( x) = db, x -1 < < x x * G( x -) ò - b + 1 x where here α >0, x >, α, ndx R. The Cputo frctionl derivtive stisfies the importnt ttribute of being zero when pplied to constnt. In ddition, it is ttined by computing n ordinry derivtive followed by the frctionl integrl, while the Riemnn-Liouville is cquired in the contrry order. 3. ChSANN nd LSANN 3.1. Methodology The functionl mpping of (LSANN) nd (ChSANN) is shown in Figure 1 demonstrting the structure of both methods, but for the convenience of the reder, stepwise explntion of both the methods is lso presented.

5 100 Numericl Simultion - From Brin Imging to Turbulent Flows Figure 1. Model digrm of ChSANN nd LSANN. The combined steps for both the methods re explined becuse of the structurl similrity in them, except the polynomil bsis tht ffects the ccurcy of the results. Step 1: The summtion of the product of network dptive coefficients (NAC) nd Chebyshev or Legendre polynomils is clculted for the independent vrible of frctionl differentil eqution for n rbitrry vlue of m s shown in Figure 1. Step 2: The ctivtion of μ or η will be performed by the first three terms of the series of tngent hyperbolic function tnh ( ), these terms hve been mentioned in Figure 1. Step 3: The tril solution will be clculted by using initil conditions s in the study of Lgris nd Fotidis [28]. Step 4: Required derivtives of the tril solution will be clculted. Step 5: The optimiztion of men squre error (MSE) or lerning of NAC will be executed by the therml minimiztion methodology known s simulted nneling. The eqution used to clculte MSE would be discussed in next section. Before optimiztion, the independent vrible will be discretized by n rry of tril points.

6 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions Step 6: If the vlue of MSE is in n cceptble rnge, then the vlues of tril points nd NAC will be replced in tril solution to get the output. On the other hnd, the procedure will be repeted from step 1 with different vlue of m Implementtion on frctionl Riccti differentil eqution In this section, the ChSANN nd LSANN re employed for the frctionl Riccti differentil eqution of the type: d y( t) = f ( t, y), y(0) = A, 0 < 1 (1) dt For implementing both methodologies, Eq. (1) cn be written in the following form: [ ] Ñ y ( t, y ) - F( t, y ( t, y )) = 0, t Î 0,1 (2) tr tr y tr (t, ψ) cn be defined s tril solution, where ψ is defined s NAC, generlly known s weights, nd is defined s differentil opertor. Tril solution will be obtined by pplying Tylor series on the ctivtion of μ by using the initil conditions, while μ being the sum of the product of network dptive coefficients nd Chebyshev polynomils. For obtining the tril solution of LSANN, the bove procedure will be pursued, but η will be clculted in spite of μ, tht is the sum of the product of NAC nd Legendre polynomils. Here, tnh ( )is used s ctivtion function, but for frctionl derivtive bsed on Cputo sense, first three terms of the Tylor series of tnh ( ) re considered tht re given for ChSANN s follows: 3 5 m 2m N = m - + (3) 3 15 while for ChSANN, μ cn be defined s follows: m m = å y iti - 1 (4) i= 1 where here T i 1 re Chebyshev polynomils with the following recursive formul: ( ) ( ) ( ) T x = 2xT x - T x, m ³ 2 (5) m+ 1 m m-1 while hile T 0 (x)=1 nd T 1 (x)= x. For LSANN, the ctivtion function nd η cn be defined s follows:

7 102 Numericl Simultion - From Brin Imging to Turbulent Flows 3 5 h 2h N = h - + (6) 3 15 m h = å y L - (7) i= 1 i i 1 wheres heres L i 1 re the Legendre polynomils with the recursive formul: 1 1 L 1 = (2m + 1) x L ( x) - m L 1( x), m ³ 2 m+ ( m + 1) m ( m + 1) m- (8) where here L 0 (x)=1 nd L 1 (x)= x, nd vlue of m is djustble to rech the utmost ccurcy. For Eq. (1), the tril solution cn be written s defined in the study of Lgris nd Fotidis [28], but N will be used ccording to the method. y ( t, y ) = A + t N (9) tr Tril solution cn be written in expnded form for ChSANN t m =2 s follows: 1 2 ytr ( t, y ) = A + t æ y + ty - y + ty + y + ty è 3 15 ( ) ( ) 3 5 ç ö ø (10) Frctionl derivtive in Cputo sense of Eq. (10) is s follows: G2 æ y 2y ö 2 G6 Ñ = ç ç G - è ø G æ ö 5-4 ytr ( t, y ) t y 1 t y 1y 2 ( ) ( 2 ) 3 15 è 3 ø ( 7 ) æ 2 ö G7 6-5 G3 2- æ ö + ç t ( y 2 ) + t ç y 2 - y 1y 2 + y 1y 2 è15 ø G 7 - G 3 - è 3 ø ( ) 3 G4 3- æ ö G5 4- æ y ö 2 + t ç y 1y 2 - y 1y 2 + t ç y 1y 2 - G( 4 - ) è 3 ø G( 5 - ) è 3 3 ø ( ) (11) The men squre error (MSE) of the Eq. (1) will be clculted from the following: ( ( ) ( ( ))) 2 n 1 MSE( y ) = Ñ y t, y - F t, y t, y, t Îéë 0,1ùû å i tr j i j tr j i j = 1 n (12)

8 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions wheres here s n cn be defined s number of tril points. The lerning of NAC will be performed from Eq. (10) by minimizing the MSE to the lowest possible cceptble minimum vlue. The therml minimiztion methodology nd simulted nneling is pplied here for the lerning of NAC. The process of simulted nneling cn be described s physicl model of nneling, where metl object is first heted nd then slowly cooled down to minimize the system energy. Here, we hve implemented the procedure by Mthemtic 10, but the interested reders cn lern the detils of simulted nneling from the study of Ledesm et l. [29]. Exmple 1: Consider the following Riccti differentil eqution with initil condition s: ( ) d y t dt ( t) ( ) = = < y 1 0, y 0 0, 0 1 The exct solution for α =1 is given by the following: ( ) y t e = e 2t 2t The bove frctionl Riccti differentil eqution is solved by implementing the ChSANN nd LSANN lgorithms for vrious vlues of α nd the results re compred with severl methods to exhibit the strength of proposed neurl network lgorithms. The ChSANN nd LSANN methods re employed on the bove eqution for α =1 with 20 equidistnt trining points nd 6 NAC nd ttined the men squre error up to nd for ChSANN nd LSANN, respectively. Figure 2 shows the combined results of ChSANN for different Figure 2. ChSANN results t different vlues of α =1.

9 104 Numericl Simultion - From Brin Imging to Turbulent Flows vlues of α. Tble 1 depicts the comprison of results obtined from both the methods with exct solution nd the bsolute error vlues for both the methods. Absolute error (AE) vlues for ChSANN nd LSANN cn be viewed in Tble 1 but cn be better visulized in Figure 3. Implementtion of ChSANN nd LSANN for α = 0.75 with 10 equidistnt trining points nd 6 NAC on the bove eqution gve the men squre error up to for ChSANN nd for LSANN. Tble 2 shows the numericl comprison for α =0.75 with 10 equidistnt trining points of ChSANN nd LSANN with the methods in [13, 14], while Tbles 3 nd 4 demonstrte the numericl comprison of the proposed methods with the methods in [7, 13, 14] for α =0.5nd α =0.9 correspondingly. Numericl vlues of ChSANN for α =1 t t =1 re presented in Tble 5. x Exct ChSANN LSANN AE of ChSANN AE of LSANN Tble 1. Numericl comprisons of ChSANN nd LSANN vlues with exct vlues for frctionl Riccti differentil eqution.

10 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions Figure 3. Absolute error of ChSANN nd LSANN t α =1 for test exmple 1. x ChSANN LSANN IABMM [14] EHPM [14] MHPM [7] Bernstein [13] Tble 2. Numericl comprison for α =0.75. x ChSANN LSANN MHPM [7] Tble 3. Numericl comprison for α =0.5.

11 106 Numericl Simultion - From Brin Imging to Turbulent Flows x ChSANN LSANN IABMM [14] EHPM [14] MHPM [7] Bernstein [13] Tble 4. Numericl comprison for α =0.9. No of NAC No of trining points Men squre error y(t) Absolute error Tble 5. Numericl vlues of ChSANN t t =1 nd for α =1. Exmple 2: Consider the nonliner Riccti differentil eqution long with the following initil condition: ( ) d y t dt ( ) ( ) ( ) 2 + y t - 2y t - 1 = 0, y 0 = 0, 0 < 1 The exct solution when α =1 is given by [7]: ( ) y t æ 1 æ 2-1öö = 1+ 2 tnh 2 t + log ç 2 ç è è øø ChSANN nd LSANN lgorithms re executed on the bove test experiment with 6NAC, α =1 nd 20 equidistnt points tht gve the men squre error up to nd for ChSANN nd LSANN, respectively. Tble 6 shows the bsolute errors nd the numericl comprison with exct vlues for both the methods, while grphicl comprison cn be better envisioned through Figure 4. Tbles 7 nd 8 disply the numericl comprison of the proposed methods with the results obtined in [7] for α =0.75nd [13] for α =0.9, respectively, wheres the men squre error, number of trining points, nd NAC for different vlues of α re presented in Tble 9. The effects on ccurcy of results with vrible NAC nd trining points cn be understood through Tble 10.

12 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions x ChSANN LSANN Exct AE of ChSANN Tble 6. Numericl comprison of ChSANN nd LSANN vlues with exct vlues t α =1 for frctionl Riccti differentil eqution test exmple 2. Figure 4. Absolute error of ChSANN nd LSANN t α =1 for test exmple 2.

13 108 Numericl Simultion - From Brin Imging to Turbulent Flows x ChSANN LSANN MHPM [7] Tble 7. Numericl comprison for α =0.75. x ChSANN LSANN Reference [13] Tble 8. Numericl comprison for α =0.9. α ChSANN LSANN NAC Trining points MSE NAC Trining points MSE Tble 9. Vlue of men squre error t different vlues of α. No. of NAC No of trining points MSE y(t) AE Tble 10. Numericl vlues of ChSANN t t =1 nd for α =1.

14 Numericl Simultion Using Artificil Neurl Network on Frctionl Differentil Equtions Results nd discussion In this chpter, two new lgorithms hve been developed nd verified for the Riccti differentil eqution with frctionl order, bsed on the functionl neurl network, Chebyshev nd Legendre polynomils nd simulted nneling for frctionl differentil equtions. Substntition of the methods is crried out by exmining two benchmrk exmples tht were lredy solved by some previously renowned methods. The numericl evlution with previously obtined results for frctionl-order derivtive exhibited the chievement of proposed methods. For test exmple 1, better results with less vlue of men squre error were obtined for ech method. Comprison of the men squre errors nd for ChSANN nd LSANN, respectively, showed tht the men squre error is less for LSANN when α =1. However, it cn be observed from Tble 1 nd Figure 2 tht ChSANN gve the better results with slightly more men squre error thn LSANN. It cn be noted from Tble 5 tht better results cn be ttined with vrible number of weights nd trining points, while the trend witnessed from Tble 5 indicted tht for ChSANN, decresing vlue of men squre error is directly proportionl to the bsolute error for α=1. The test exmple 2 showed quite similr trends s of exmple 1. Tbles 6 nd 9 exhibited tht for α =1, less men squre error for ChSANN thn LSANN ws noted due to which, more ccurte results were chieved by ChSANN t α =0.9 s compred to LSANN tht cn be viewed in Figure 4. The results obtined for frctionl vlues of derivtives re compred with MHPM for α =0.75 nd colloction-bsed method of Bernstein polynomils for α =0.9 s presented in Tbles 7 nd 8. The comprison showed tht the results chieved by ChSANN nd LSANN re quite similr to the results obtined by MHPM nd colloction-bsed method of Bernstein polynomils. However, ccording to the observtions from the cse of α =1, it cn be ssumed tht the results obtined for α =0.75 will be ccurte up to 2 3 deciml plces becuse the MSE ws detected up to for ChSANN nd for LSANN. While the results chieved for α =0.9 will be ccurte up to 3 4 deciml plces s the MSE ws noticed up to for ChSANN nd for LSANN. The methods proposed in this study re cpble of hndling highly non-liner systems. Both the proposed neurl rchitectures re less computtionl nd exhustive thn MLP. With ese of computtion, the suggested ctivtion function hs mde frctionl differentil equtions possible to solve. Trining of NAC by simulted nneling with Chebyshev nd Legendre neurl rchitecture minimized the MSE up to tolerble level tht leds to more ccurte numericl pproximtion. Simulted nneling is probbilistic procedure tht is mostly free of initil vlues nd cn esily escpe from locl optimum to globl optimum unlike other methods. As well s it cn successfully optimize the functions with crests nd plteus. The methods cn be enhnced by introducing more dvnced optimiztion techniques. The motivtion behind the work is the successful implementtion of neurl lgorithms in the field of clculus tht gve the solution of frctionl differentil equtions new direction with ese of implementtion.

15 110 Numericl Simultion - From Brin Imging to Turbulent Flows 5. Conclusion In this chpter, ChSANN nd LSANN hve been developed for frctionl differentil equtions nd successfully employed on two benchmrk exmples of Riccti differentil equtions. The proposed methods gve the excellent numericl pproximtion of the Riccti differentil eqution of frctionl order. The most remrkble dvntge of the proposed methods is the ccurte prediction of the result for the frctionl vlues of derivtive. These procedures re esy to implement nd cn be used to find the exct solution in the frctionl vlues of derivtive. ChSANN displyed more ccurte results thn LSANN for the similr pplied conditions. Both the proposed lgorithms re non-itertive nd cn be implemented through mthemticl softwre nd Mthemtic 10 ws used in this study to obtin ll the results displyed in Tbles 1 10 nd Figures 2 4. Author detils Njeeb Alm Khn 1*, Amber Shikh 1, Fqih Sultn 2 nd Asmt Ar 3 *Address ll correspondence to: njblm@yhoo.com 1 Deprtment of Mthemtics, University of Krchi, Krchi, Pkistn 2 Deprtment of Sciences nd Humnities, Ntionl University of Computer nd Emerging Sciences, Krchi, Pkistn 3 Deprtment of Computer Science, Mohmmd Ali Jinnh University, Krchi, Pkistn References [1] He JH. Nonliner oscilltion with frctionl derivtive nd its pplictions. Interntionl Conference on Vibrting Engineering. 1998;98: [2] He JH. Some pplictions of nonliner frctionl differentil equtions nd their pproximtions. Bulletin of Science, Technology & Society. 1999;15(2): [3] Grigorenko I, Grigorenko E. Chotic dynmics of the frctionl Lorenz system. Physicl Review Letters. 2003;91(3): [4] Podlubny I. Frctionl differentil equtions: n introduction to frctionl derivtives, frctionl differentil equtions, to methods of their solution nd some of their pplictions. Acdemic Press; ISBN [5] Podlubny I. Geometric nd physicl interprettion of frctionl integrtion nd frctionl differentition. Frctionl Clculus nd Applied Anlysis. 2002;5(4):

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