Analytical Solutions of an Economic Model by the Homotopy Analysis Method

Transcription

1 Applied Mahemaical Sciences, Vol., 26, no. 5, HIKARI Ld, hp://dx.doi.org/.2988/ams Analyical Soluions of an Economic Model by he Homoopy Analysis Mehod Jorge Duare ISEL-Engineering Superior Insiue of Lisbon Deparmen of Mahemaics Rua Conselheiro Emídio Navarro, Lisboa, Porugal and Cener for Mahemaical Analysis, Geomery and Dynamical Sysems Mahemaics Deparmen, Insiuo Superior Técnico Universidade de Lisboa, Av. Rovisco Pais, 49- Lisboa, Porugal Crisina Januario ISEL-Engineering Superior Insiue of Lisbon Deparmen of Mahemaics Rua Conselheiro Emídio Navarro, Lisboa, Porugal Nuno Marins Cener for Mahemaical Analysis, Geomery and Dynamical Sysems Mahemaics Deparmen, Insiuo Superior Técnico Universidade de Lisboa, Av. Rovisco Pais, 49- Lisboa, Porugal Copyrigh c 26 Jorge Duare, Crisina Januario and Nuno Marins. This aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac The Bouali economic model is a chaoic sysem of hree ordinary differenial equaions gahering he variables of profis, reinvesmens and financial flow of borrowings. Using a echnique developed o find analyic soluions for srongly nonlinear problems, he sep homoopy analysis mehod (SHAM), we presen he explici series soluion of his dynamically rich model from he lieraure. Wih he homoopy soluion, we invesigae he dynamical effec of a meaningful bifurcaion

2 2484 Jorge Duare, Crisina Januario and Nuno Marins parameer, he deb rae s. Given he resuls presened in his aricle, showing ha he SHAM is a powerful and effecive mehod applicable o difficul nonlinear problems in science, his work is likely o appeal o a wide audience. Mahemaics Subjec Classificaion: Primary 34A5; Secondary 34A34 Keywords: Analyic soluions, Nonlinear differenial equaions, Economic model, Homoopy analysis mehod, Sep homoopy analysis mehod Preliminaries. Descripion of he model In order o analyse relevan aspecs of a firm dynamics, S. Bouali proposed recenly in [2] a sysem of ordinary differenial equaions involving hree dynamical variables: profis, reinvesmens and financial flow of borrowings. In his aricle, we give a brief descripion of he Bouali economic model and we presen he noaions and basic definiions of a general analyical echnique - he homoopy analysis mehod (HAM). We show he applicabiliy and effeciveness of HAM o he Bouali equaions when reaed has an algorihm in a sequence of inervals - he sep homoopy analysis mehod (SHAM) - for finding accuraly an analyical soluion. The Bouali firm model is given by he following auonomous sysem of ordinary differenial equaions [2] dx d = v (x 2 + x 3 ), () dx 2 d = mx + n ( ) x 2 x2, (2) dx 3 d = x 2x + sx 2. (3) This sysem consiss of hree dynamical variables: (i) he variable x ha corresponds o he profis, (ii) he variable x 2 ha measures he reinvesmens and (iii) he dynamical variable x 3 ha represens he financial flow of borrowings. The parameers of he economic model are: v, m, n, r, s and we will consider hroughou our sudy v = 4., m =.4, n =.2, r =.68 and ake he deb rae s as a conrol parameer ( s.5). In a real economic conex, he heurisic Bouali sysem can be used o analyze he asympoic behavior of he financial dynamics of a firm, where he unis of ime are chosen so ha ime could be anyhing from hours o days.

3 Analyical soluions of an economic model The sep homoopy analysis mehod analyic soluions The analyical approach of HAM will be used in a sequence of inervals, giving rise o he sep homoopy analysis mehod (SHAM). In he following paragraph, we ouline he descripion of SHAM applied o he Bouali equaions () - (3). 2. Explici series soluion Le us consider he Eqs. ()-(3) subjec o he iniial condiions x i () = IC i, which are aken in he form x i, () = IC i for i =, 2, 3, as our iniial approximaions of x (), x 2 () and x 3 (), respecively. In our analysis, we will consider IC = , IC 2 =.4872, IC 3 = As auxiliary linear operaors, we choose L [φ i (; q)] = φ i (; q) + φ i (; q), i =, 2, 3, wih he propery L [C i e ] =, where C i (i =, 2, 3) are inegral consans. The equaions of he Bouali model lead o he following nonlinear operaors N, N 2 and N 3 : N [φ (; q), φ 2 (; q), φ 3 (; q)] = φ (; q) v φ 2 (; q) v φ 2 (; q) v φ 3 (; q), N 2 [φ (; q), φ 2 (; q), φ 3 (; q)] = φ 2 (; q) mφ (; q) nφ 2 (; q)+nφ 2 (; q) φ 2 (; q), N 3 [φ (; q), φ 2 (; q), φ 3 (; q)] = φ 3 (; q) + rφ (; q) sφ 2 (; q). Considering q [, ] and h he non-zero auxiliary parameer, he zero h -order deformaion equaions are ( q) L [φ i (; q) x i, ()] = qhn i [φ (; q), φ 2 (; q), φ 3 (; q)], (4) wih i =, 2, 3 and subjec o he iniial condiions φ (; q) = , φ 2 (; q) =.4872, φ 3 (; q) =

4 2486 Jorge Duare, Crisina Januario and Nuno Marins Obviously, for q = and q =, he above zero h -order equaions (4) have he soluions and φ (; ) = x, (), φ 2 (; ) = x 2, (), φ 3 (; ) = x 3, () (5) φ (; ) = x (), φ 2 (; ) = x 2 (), φ 3 (; ) = x 3 (), respecively. (6) When q increases from o, he funcions φ (; q), φ 2 (; q) and φ 3 (; q) vary from x, (), x 2, () and x 3, () o x (), x 2 () and x 3 (). Following he heory in [3], he m h -order deformaion equaions are L [x i,m () χ m x i,m ()] = hr i,m [x,m (), x 2,m (), x 3,m ()], (7) wih he following iniial condiions x,m () =, x 2,m () =, x 3,m () =. (8) Defining he vecor u m = (x,m (), x 2,m (), x 3,m ()), we derive R,m [ u m ] = x,m () v x 2,m () v x 3,m () R 2,m [ u m ] = x 2,m () m x,m () n x 2,m () + +n m k= [( k ) ] x,k j () x,j () x 2,m k () j= and R 3,m [ u m ] = x 3,m () + r x,m () s x 2,m (). According o he noaions and definiions provided above, we solve he linear Eqs (7) a iniial condiions (8), for all m, and we obain x i,m () = χ m x i,m () + h e e τ R i,m [ u m ] dτ, (i =, 2, 3). (9) A M h -order approximae analyic soluion (which corresponds o a series soluion wih M + erms) is given by x i,m () = x i, () + M x i,m (). () m=

5 Analyical soluions of an economic model 2487 Wihin he purpose of having an effecive analyical approach of Eqs. () - (3) for higher values of, we use he sep homoopy analysis mehod, in a sequence of subinervals of ime sep and he 6-erm HAM series soluions (5 h -order approximaions) x i () = x i, () + 5 x i,m (), (i =, 2, 3) () m= a each subinerval. Accordingly o SHAM, he iniial values x,, x 2, and x 3, change a each subinerval, i.e., x i ( ) = ICi = x i, (i =, 2, 3), and he iniial condiions x,m ( ) = x 2,m ( ) = x 3,m ( ) = should be saisfied for all m. As a consequence, he analyical approximae soluion for each dynamical variable is given by x i () = x i ( ) + 5 x i,m ( ), (i =, 2, 3) (2) m= In general, we only have informaion abou he values of x (), x 2 () and x 3 () a =, bu we can obain he values of x (), x 2 () and x 3 () a = by assuming ha he new iniial condiions are given by he soluions in he previous inerval. Anoher illusraion of he use of SHAM can be seen in []. The homoopy erms depend on boh he physical variable and he convergence conrol parameer h. The arificial parameer h can be freely chosen o adjus and conrol he inerval of convergence, and even more, o increse he convergence a a reasonable rae, forunaely a he quickes rae. Wih he purpose of deermining he inerval of convergence and he opimum value of h, corresponding o each dynamical variable, we sae in wha follows a very recen convergence crierion addressed in [4]. 2.2 Inerval of convergence and opimum value from an appropriae raio Le us consider k + homoopy erms X (), X (), X 2 (),..., X k () of an homoopy series X() = X () + + m= Taking a ime inerval Ω, he raio [X k ()] 2 d Ω β = [X k ()] 2 d Ω X m (). (3)

6 2488 Jorge Duare, Crisina Januario and Nuno Marins Β x h Β x h Β x h Figure : The curves of raios β x, β x2 and β x3 versus h. These curves correspond o a 5 h -order approximaion of soluions x (), x 2 () and x 3 (). The solid lines correspond o a periodic regime and he dashed lines correspond o a chaoic behavior. In each regime, he opimum value of h, h, gives rise o he minimum value of β. The inervals of convergence of h and he respecive opimum values h, corresponding o he dynamical regimes are for he periodic behavior: < h x < wih h x =.5539, < h x2 <.9458 wih h x 2 =.65225,.4337 < h x3 < wih h x 3 = and for he chaoic behavior:.8498 < h x < wih h x =.52998, < h x2 <.787 wih h x 2 = and.3824 < h x3 < wih h x 3 = represens a more convinien way of evaluaing he convergence conrol parameer h. In fac, given order of approximaion, he curves of raio β versus h indicae no only he effecive region for he convergence conrol parameer h, bu also he opimal value of h ha corresponds o he minimum of β. Now, ploing β versus h, as well as by solving [X k ()] 2 d Ω [X k ()] 2 d Ω < and dβ dh =, he inerval of convergence and he opimum value for he parameer h can be simulaneously achieved. As an illusraion, a he order of approximaion M = 5, he curves of raio β versus h, corresponding o x (), x 2 () and x 3 (), are displayed in Fig.. In each represenaion, he solid line corresponds o a periodic behavior and he dashed line corresponds o a chaoic regime. In Fig. 2, we show he comparison beween he SHAM analyical soluions, for x, x 2 and x 3, and he numerical resuls, considering he wo dynamical regimes (periodic and chaoic behaviors) and using he opimum values presened in he capion of Fig. 2. This analysis provides an illusraion of how our undersanding of an economic dynamical variable arising in he conex of economics can be direcly

7 2489 Analyical soluions of an economic model P P R R F F Figure 2: Comparison of he SHAM analyical soluions (2) of P x (Profis), R x2 (Reinvesmens) and F x3 (Flow of borrowings) - solide lines - wih he respecive numerical soluions - doed lines - of he Bouali model. (Lef column) Time series corresponding o he periodic regime (s =.25). (Righ column) Time series corresponding o he chaoic regime (s =.366).

8 249 Jorge Duare, Crisina Januario and Nuno Marins enhanced by he use of he respecive analyical expression, for differen combinaions of a conrol parameer and ime. 3 Final consideraions In his aricle, we applied an analyical mehod for nonlinear differenial equaions - he sep homoopy analysis mehod (SHAM) - o solve he Bouali economic model. A new crierium of convergence presened in he lieraure for he conrol parameer h, associaed o he explici series soluions, represens a convenien way of conrolling he convergence of approximae series, which is a criical qualiaive difference in he analysis beween HAM/SHAM and oher mehods. As a consequence, our resuls are found o be remarkably accurae, in agreemen wih he numerical soluions. This work is likely o inspire applicaions of he presened analyical procedure for solving highly nonlinear problems in differen areas of knowledge. Acknowledgemens. This work was parially suppored by FCT/Porugal hrough UID/MAT/4459/23. References [] A. K. Alomari, M. S. M Noorani, R. Nazar, C. P. Li, Homoopy analysis mehod for solving fracional Lorenz sysem, Commun. Nonlinear Sci. Numer. Simula., 5 (2), hp://dx.doi.org/.6/j.cnsns [2] S. Bouali, The srange aracor of he firm, Chaper in Complexiy in Economics: Cuing Edge Research, Springer In. Publishing Swizerland, 24. hp://dx.doi.org/.7/ [3] S. J. Liao, Beyond Perurbaion: Inroducion o he Homoopy Analysis Mehod, CRC Press, Chapman and Hall, Boca Raon, 23. hp://dx.doi.org/.2/ [4] S. J. Liao, Advances in he Homoopy Analysis Mehod, World Scienific Publishing Connecing Grea Minds, 24. hp://dx.doi.org/.42/8939 Received: June 26, 26; Published: Augus 4, 26