Solving Nonlinear Stochastic Diffusion Models with Nonlinear Losses Using the Homotopy Analysis Method

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1 Applied Matheatics, 4, 5, 5-7 Published Olie Jauary 4 ( Solvig Noliear Stochastic Diffusio Models with Noliear Losses Usig the Hootopy Aalysis Method Aisha A. Fareed, Haafy H. El-Zoheiry, Magdy A. El-Tawil, Mohaed A. El-Beltagy 3, Hay N. Hassa Departet of Basic Scieces, Egieerig Faculty, Beha Uiversity, Beha, Egypt Departet of Egieerig Matheatics & Physics, Egieerig Faculty, Cairo Uiversity, Cairo, Egypt 3 Departet of Electrical & Coputer Egieerig, Egieerig Faculty, Effat Uiversity, Jeddah, KSA Eail: aisha.farid@yahoo.co, elbeltagy@effatuiversity.edu.sa Received October, 3; revised Noveber, 3; accepted Noveber 9, 3 Copyright 4 Aisha A. Fareed et al. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. I accordace of the Creative Coos Attributio Licese all Copyrights 4 are reserved for SCIRP ad the ower of the itellectual property Aisha A. Fareed et al. All Copyright 4 are guarded by law ad by SCIRP as a guardia. ABSTRACT This paper deals with the costructio of approiate series solutios of diffusio odels with stochastic ecitatio ad oliear losses usig the hootopy aalysis ethod (H). The ea, variace ad other statistical properties of the stochastic solutio are coputed. The solutio techique was applied successfully to the D ad D diffusio odels. The schee shows iportace of choice of covergece-cotrol paraeter ħ to guaratee the covergece of the solutios of oliear differetial Equatios. The results are copared with the Wieer-Herite epasio with perturbatio (WHEP) techique ad good agreeets are obtaied. KEYWORDS H Techique; WHEP Techique; Stochastic PDEs; Diffusio Models. Itroductio The deteriistic differetial equatios of the for t at t costitute the basic for of so-called diffusio or trasport probles which appear i relevat odels such as: the growth populatio geoetric (or Malthusia) odel i biology, at represets the per capita growth rate; the eutro ad gaa ray trasport odel i physics, coefficiet at ivolves the geoetry of the cross-sectios of the ediu; the cotiuous coposed iterest rate odels for studyig the evolutio of a ivestet uder tie-variable iterest rate r() t which ca be take as at rt, etc. Despite the usefuless of these basic odels, they do ot ofte cover all possible situatios observed fro a practical poit of view. I fact, as a siple but illustrative eaple, if at a, the Malthus odel predicts uliited growth of a species despite the fact that resources are always liited. The, the logistic (or Verhulst) odel itroduces a oliear ter i order to overcoe this drawback by cosiderig the differetial equatio t a t t b t, a, b, the oliearity itesity is give by paraeter b. I ay practical situatios it is appropriate to assue that the oliear ter affectig the pheoea uder study is sall eough; the its itesity is cotrolled by eas of a frak sall paraeter, say. Stochastic differetial equatios based o the white oise process provide a powerful tool for dyaically odelig these cople ad ucertai aspects. Over the last few years, ew ad relevat ethods for fidig the eact solutios of such Equatios have bee developed. They iclude the hootopy perturbatio (HPM) ethod [,], Wieer-Herite epasio with perturbatio ethod (WHEP Cortes []) [3] ad the ep-fuctio ethod [4,5]. H is a aalytical techique for solvig o liear differetial equatios. Proposed by Liao i 99, [6], the techique is superior to the traditioal perturbatio ethods, i which it leads to coverget series solutios of strogly oliear probles, idepedet of ay sall or large physical paraeter associated with the prob-

2 6 le, [7]. The H provides a ore viable alterative to o perturbatio techiques such as the Adoia decopositio ethod (ADM) [8] ad other techiques that caot guaratee the covergece of the solutio series ad ay be oly valid for weakly oliear probles, [7]. We ote here that He s HPM ethod, [9] is oly a special case of the H. I recet years, this ethod has bee successfully eployed to solve ay probles i sciece ad egieerig such as the viscous flows of o-newtoia fluids [,], the KdV-type equatios [], Glauert-jet proble [3], Burgers-Huley equatio [4], tie-depedet Ede-Fowler type equatios [5], differetial-differece equatio [6], two-poit oliear boudary value probles [7]. The H provides the solutio i the for of a rapidly coverget series with easily coputable copoets usig sybolic coputatio software such as Matheatica. This paper deals with the solutio of D stochastic differetial odels of the for t a t t t t;, t, the diffusio coefficiet a t ad iitial coditio are deteriistic, is a sall paraeter ad t ; is the white oise process, whose itesity is give by paraeter, which has the followig iportat properties: ; ; ; E t E t t t t E deotes the eseble average operator, is the Dirac delta fuctio. Ad is a rado outcoe for a triple probability space, A, p, is a saple space, A is a -algebra associated with ad P is a probability easure. The curret work also deals with the solutio of D stochastic quadratic oli- ; as o-hoogeeity. ear equatio with, ; geeity ter ; u t, ; t u u t,,, l, ut ut l u ;,,, ad,. ut is the diffusio process, is a deteriistic scale for the oliear ter. The o-hoo- is spatial white oise scaled by. The paper is orgaized as follows. Sectio suarizes the basic idea of the H ethod. I Sectio 3, the H is applied i order to obtai fourth order approiatio of the solutio of D diffusio odel. I Sectio 4, the H is applied up to the third order approiatio for the solutio of D diffusio odel. I additio, we copute approiatios for the ai statistical oets such as the ea ad variace. A copariso is doe with the results obtaied with the (WHEP Cortes [], WHEP El-Beltagy [3]) techique [4,5]. The results are show i Sectio 5 alog with coets o the results.. The Basic Idea of H A presetatio of the stadard H for deteriistic probles ca be foud i [9]. The followig subsectio is a brief descriptio of H. Cosider the followig differetial equatio: N is a oliear operator ad, N u t, (3) ut is the ukow fuctio. By eas of geeralizig the traditioal HPM ethod, Liao [6] costructs the so-called zero-order deforatio equatio, ql tqut qhtn tq, ;,,, ;, (4) q deotes the so-called ebeddig paraeter, is a auiliary paraeter ad L is a auiliary liear operator. ut, tq, ; tq, ; is a ukow The H is based o a kid of cotiuous appig, () ()

3 7, H t deotes a o-zero auiliary fuctio. It is obvious that whe the ebeddig paraeter q ad q, Equatio (3) becoes fuctio, u t is a iitial guess for ut,, ad, t ut t ut, ;,,, ;,, (5) respectively. Thus as q icreases fro to, the solutio tq, ; varies fro the iitial guess u, the solutio, otopy tq, ;. Havig the freedo to choose the auiliary paraeter, the auiliary fuctio, proiatio u, that the solutio tq, ; of the zero-order deforatio Equatio (4) eists for q. Epadig tq, ; i Taylor series with respect to q, oe has, t to ut. I topology, this kid of variatio is called deforatio; Equatio (3) costructs the ho- H t, the iitial apt, ad the auiliary liear operator L, we ca assue that all of the are properly chose so tq, ; u t, u tq,, (6) u t, tq, ;! q Assue that the auiliary paraeter, the auiliary fuctio H t,, the iitial approiatio u t ad the auiliary liear operator L are so properly chose that the series (6) coverges at q ad q t ut u t, ;,,, (8) which ust be oe of the solutios of the origial oliear Equatio, as proved by Liao [9]. As, H t,, Equatio (4) becoes ad ql tq ut qn tq, ;,, ;,, (9) This is ostly used i the HPM ethod. Accordig to defiitio (8), the goverig equatio ad the correspodig u t, ca be deduced fro the zero-order deforatio Equatio (4). Defie the vector iitial coditio of u tutututut,,,,,,,,,. Differetiatig Equatio (4) ties with respect to the ebeddig paraeter q ad the settig q ad fially dividig the by!, we have the so-called th -order deforatio equatio: ad The solutio is coputed as:,,, ; () Lu t u t H tru R u It should be ephasized that, N t, ; q! q q whe, () otherwise ut, ui t,. i u t for is govered by the liear Equatio () with liear boudary coditios that coe fro the deteriistic proble, which ca be solved by ay sybolic coputatio, (7)

4 8 software such as Matheatica, Maple, or Matlab. 3. Applicatio to the D Diffusio Model To deostrate the above preseted ethod it will be used to fid the ea ad variace of D stochastic diffusio proble as follows. The auiliary liear operator will be chose as Furtherore, we defie the oliear operator as d tq ; L t; q dt d tq ; N t; q att; qt; q t ; dt We costruct the zero-order deforatio equatio, qlx t X t q Ht R X. The th -order deforatio equatio for ad Ht is Subject to the iitial coditio L X t X t R, X () t X, dx R X a t X t X t X t t ; i i dt i Now the solutio of the th -order deforatio Equatio () for becoes d X, t LX t X t a t X t X i t Xi t t dt i The first order approiatio is obtaied by settig i () as follows The t LX t R X dx RX a t X t X t dt t dx LX t a t X t X t t, dt t dx t X t a t X t X t t d, t dt The eseble average of the first order approiatio is t dex t EX t a t EX t EX t d, t dt E X t.47 5t

5 9 The covariace of the first order solutio will be Cov X t, X t E X t EX t X t EX t The variace of the first order solutio will be Var t t t t E t dt t dt Et t dt dt t t t t tdtd t d t t t X t EX t EX t E t dt t ad I this aer, we ca have ore results of E X t Var X t The fial epressio of the ea of the 4 th order solutio will be M 4 E X t E X t.5.99 t Sice X t X t obtaied at,3, 4, i i t t t t t t t N i i The the fial epressio of the variace of the d order solutio will be N N N N Var Xi t Var Xi t Cov Xi t, X j t Var X t 4. Applicatio to the D Diffusio Model i i i ji v, Var Var X t Var X t Co X t X t X t t t ht 4. h t h t t H will be used to fid ea ad variace of stochastic quadratic oliear diffusio proble as follows. The auiliary liear operator is chose as L t, ; q tq, ; tq, ; t We have ay choices i guessig the iitial approiatio together with its iitial coditios which greatly affects the cosequet approiatio.the choice u is a desig proble which ca be take as follows: π, e t u t B si π B si d Oe ca otice that the selected value fuctio satisfies the iitial ad boudary coditios ad it depeds o the paraeter which is totally free. Oe ca also otice that selectio could cotrol the solutio covergece. Furtherore, we defie the oliear operator as tq, ; tq, ; N t, ; q t, ; q ; t (3)

6 We costruct the zero-order deforatio Equatio, The th -order deforatio Equatio for Ad subject to the boudary coditios Ad the iitial coditio qlu t u t qhtru,,,. H t, is ad Lu t, u t, R u, u t,, u t, l,, u,, u t u t Ru u i t, ui t, ;. t i Now the th -order deforatio equatio for becoes,, u t u t Lut, u t, u itu, it, ;. t i The first order approiatio is obtaied by substitutig to get,, u t u t Lut, u ;. (4) t The approiated first order solutio of (4) ca be obtaied usig Eige fuctio epasio as follows, π u t, I,tsi t π t, e, I t F d,, u t u t π F, t u ; si d, t the eseble average of the first order approiatio is π u t, EI tsi, t π t, e, E I t E F,, d L u t u t π EF, t u si du t, t π t t π t π e h3e ππ 8 8e siπ. 3 3 π π

7 The covariace of the first order solutio ca be coputed as Cov u t,, u t, E u t, Eu t, u t, Eu t, π π EI, t EI, t si I, t EI, t si. The covariace is obtaied fro the followig fial epressio Cov u t,, u t, t t π π 4 π π π π t t si si si si d e e dd u Cov u t,, t, π t h e siπ si π. 4 π The variace of the first order solutio will be coputed as To give π Var ut, Eut, Eu t, EI, tei,tsi. (5) t t π π 4 π π π π t t Var u t, si si si si d e e d d π t h π e si 4 π I this aer, we ca have ore results of E u t, ad Var u t, The fial epressio of ea of the 3 rd order solutio will be M Eut, E u t,.,,,,, Eu t u t Eu t Eu t Eu3 t Eut, (9e π t obtaied at,3, 4, 6e h3e ππ 8 8e π t t π t π t 3 9 π π e h 8 π 3 9π π 7π 3 t t π 6 3π 6π8 e 8e π 3 tπ π t 3 9e π π 7e π 6 3π 6π 8 π π π 3 si π. N, i,. Sice ut u t i The the fial epressio of the variace of the d order solutiowill be

8 5. Result Aalysis N N N N Var uit, Var uit, Cov uit,, uj t, i i i ji, t, Var ut, Var u t, Var u t, Cov u t, u Var u, t ht 4. h4..749th..735t.t Var u t, π 5.. D Diffusio Model Results 8 4 h si t π 4 π 3 e π π t 3 π t π t π e π e π e t tπ tπ π 3 e e π e π t π t π t π t e π 4 e h π e π t π t π 6e h e t 4 t π t e e e t π t si π π Figures ad show the plots of the -curves for the fourth order variace ad ea approiatios respec- tively for differet values of tie t at at,, ad.5 o the tie iterval [,]. Accordig to these -curves, it is easy to discover that the valid regio of is a horizotal lie segets, thus.9 Figures 3 ad 4 show the copariso of the epectatio ad variace as a fuctio of tie usig H ad WHEP which uses the Wieer Herite epasio ad perturbatio techique to solve a class of oliear partial differetial Equatios with a perturbed oliearity techiques ad good agreeet is obtaied. The ea ad variace results of the WHEP techique are obtaied fro [5] as: E t e at at e e a at a. 3 at at at at at Var t e e e e e 3 a a a The effect of o the variace is show i Figure 5. The variace is plotted with tie for differet values of. The peak variace decreases i agitude with the icrease of. Also, the tie of the peak variace decreases with the icrease of. 5.. D Diffusio Model Results I the followig figures, results of the solutio of D stochastic quadratic oliear diffusio odel usig H π techique are show at,,,,, si. Figure 6 shows the Plot of -curve of third order approiatio of ea for differet values of tie t ad space variable at,,,,, si π. Figure 7 shows the plot of -curve of

9 3 Variace t=. t=.5 t= t=.5 h Figure. The chage of variace of the solutio 5 t= X t with paraeter at differet t values. Mea.4.3. h t=. t=.5 t= t=.5 t=.5.5 Figure. The chage of ea of the solutio.5 X t with paraeter at differet t values... Mea Hootopy Aalysis Wieer Herite tie.5.5 Figure 3. Copariso of the epectatio obtaied by usig H at.9 for the D proble ad WHEP [8].

10 Variace Hootopy Aalysis Wieer Herite tie.5.5 Figure 4. Copariso of ad the variace obtaied by usig H ethod at.9 for the D ad WHEP [8] Variace Ɛ=. Ɛ=. 3 Ɛ=.35 Ɛ=.5 Ɛ=.9 tie.5.5 Figure 5. The effect of o Var[(t)]. Mea h.5 t=.,= t=.3,=. t=.4,=.5 t=.5,=.4 t=.,=.3 Figure 6. The chage of the ea u with paraeter at differet t, values..5

11 5 third order approiatio of ea for differet values. Accordig to these -curves, it is easy to discover that the valid regio of is a horizotal lie segets, thus.96. Figures 8 ad 9 show the plot of ea ad variace with tie for differet values. Figure shows the copariso betwee the ea of the first, the secod ad the third order approiatios. Figure shows the copariso betwee the variace of the first ad secod order approiatios. 6. Coclusio This paper shows that the H techique costitutes a powerful tool for costructig approiate solutios for the stochastic process for rado diffusio odels with oliear perturbatios ucertaity is cosidered by eas of a additive ter defied by white oise. The H ethod is eployed to give a statistical aalytic solutio for stochastic D ad D diffusio odels. Differet fro all other aalytic ethods, the H provides us with a siple way to adjust ad cotrol the covergece regio of the series solutio by eas of the auiliary paraeter ħ. Thus the auiliary paraeter ħ plays a iportat role withi the frae of the H which ca be deteried by the so called ħ-curves. The solutio obtaied by eas of the H is a ifiite power series for appropriate iitial approiatio, which ca be, i tur, epressed i a closed for. The accuracy for the ethod is verified o D diffusio odel by coparisos with WHEP techique ad good agreeets are obtaied. As show i Figures ad, we ca see that the valid ħ regio i the D eaple is.9 < <.4 ad i D eaple the iterval is.9 < <., as show i Figure 6. The results deostrate reliability ad efficiecy of the H ethod. Sice H was used to solve oly deteriistic probles, we.3 Mea. h Figure 7. The chage of the ea u with paraeter at differet β= 3 β= β= β= β=3 β values,, t = = Mea Ɛ=. Ɛ=. Ɛ=.4 Ɛ=.7 Ɛ= tie Figure 8. The chage of the ea u with tie t at differet values, =., β,.96.

12 6.9 Variace Ɛ=. Ɛ=. Ɛ=.35 Ɛ=.5 Ɛ=. tie Figure 9. The chage of the variace u with tie t at differet values, =., β,.96. Mea ea u ea u ea u3 tie Figure. Mea copariso betwee first u, secod u ad third order u 3 approiatios with tie t at, =., β, var u Var u Figure. Variace copariso betwee first ad secod approiatios u, u with tie t at, =., β,.96. Variacetie

13 7 ca say that this is the first tie to apply H ethod o stochastic probles ad we foud that it s easier tha WHEP ad ore geeral tha HPM sice HPM is a special case of H obtaied at ad its results is accurate. REFERENCES [] M. A. El-Tawil ad A. S. Al-Jihay, O the Solutio of Stochastic Oscillatory Quadratic Noliear Equatios Usig Differet Techiques, a Copariso Study, Topological Methods i Noliear Aalysis, Vol. 3, No., 8, pp [] M. A. El-Tawil ad N. A. Al-Mulla, Usig Hootopy WHEP Techique for Solvig A Stochastic Noliear Diffusio Equatio, Matheatical ad Coputer Modellig, Vol. 5, No. 9,, pp [3] J. C. Cortes, J. V. Roero, M. D. Rosello ad C. Sataaria, Solvig Rado Diffusio Models with Noliear Perturbatios by the Wieer-Herite Epasio 67 Method, Coputers & Matheatics with Applicatios, Vol. 6, No. 8,, pp [4] C. Q. Dai ad J. F. Zhag, Applicatio of He s Ep-Fuctio Method to the Stochastic KdV Equatio, Iteratioal Joural of Noliear Scieces ad Nuerical Siulatio, Vol., No. 5, 9, pp [5] M. El-Beltagy ad M. El-Tawil, Toward a Solutio of a Class of No-Liear Stochastic Perturbed PDEs Usig Autoated WHEP Algorith, Applied Matheatical Modelig, Vol. 37, No. -3, 3, pp [6] S. J. Liao, The Proposed Hootopy Aalysis Techique for the Solutio of Noliear Probles, Ph.D. Thesis, Shaghai Jiao Tog Uiversity, Shaghai, 99. [7] S. J. Liao, Notes o the Hootopy Aalysis Method: Soe Defiitios ad Theories, Couicatios i Noliear Sciece Nuerical Siulatio, Vol. 4, No. 4, 9, pp [8] G. Adoia, A Review of the Decopositio Method ad Soe Recet Results for Noliear Equatios, Coputers ad Matheatics with Applicatios, Vol., No. 5, 99, pp. -7. [9] J. H. He, Hootopy Perturbatio Method: A New Noliear Aalytical Techique, Applied Matheatics ad Coputatio, Vol. 35, No., 3, pp [] T. Hayat ad M. Sajid, Aalytic Solutio for Aisyetric Flow ad Heat Trasfer of a Secod Grade Fluid Past a Stretchig Sheet, Iteratioal Joural of Heat ad Mass Trasfer, Vol. 5, No. -, 7, pp [] S. Abbasbady, Solito Solutios for the 5th-Order KdV Equatio with the Hootopy Aalysis Method, Noliear Dyaics, Vol. 5, No. -, 8, pp [] Y. P. Liu ad Z. B. Li, The Hootopy Aalysis Method for Approiatig the Solutio of the Modified Korteweg-de Vries Equatio, Chaos, Solitos ad Fractals, Vol. 39, No., 9, pp. -8. [3] Y. Boureel, Eplicit Series Solutio for the Glauert-Jet Proble by Meas of the Hootopy Aalysis Method, Couicatio i Noliear Sciece Nuerical Siulatio, Vol., No. 5, 7, pp [4] A. Molabahrai ad F. Khai, The Hootopy Aalysis Method to Solve the Burgers-Huley Equatio, Noliear Aalysis Real World Applicatios, Vol., No., 9, pp [5] S. Abbasbady, E. Magyari ad E. Shivaia, The Hootopy Aalysis Method for Multiple Solutios of Noliear Boudary Value Probles, Couicatios i Noliear Sciece ad Nuerical Siulatio, Vol. 4, No. 9-, 9, pp [6] H. N. Hassa ad M. A. El-Tawil, Solvig Cubic ad Coupled Noliear Schrödiger Equatios Usig the Hootopy Aalysis Method, Iteratioal Joural of Applied Matheatics ad Mechaics, Vol. 7, No. 8,, pp [7] H. N. Hassa ad M. A. El-Tawil, A Efficiet Aalytic Approach for Solvig Two-Poit Noliear Boudary Value Probles by Hootopy Aalysis Method, Matheatical Methods i the Applied Scieces, Vol. 34, No. 8,, pp