The Solution of the Variable Coefficients Fourth-Order Parabolic Partial Differential Equations by the Homotopy Perturbation Method

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1 The Soltion of the Variable Coefficients Forth-Order Parabolic Partial Differential Eqations by the Homotopy Pertrbation Method Mehdi Dehghan and Jalil Manafian Department of Applied Mathematics, Faclty of Mathematics and Compter Science, Amirkabir University of Technology, No 44, Hafez Avene, Tehran 594, Iran Reprint reqests to M D; mdehghan@atacir Z Natrforsch 64a, (009; received September 4, 008 / revised October 4, 008 In this work, the homotopy pertrbation method proposed by Ji-Han He [] is applied to solve both linear and nonlinear bondary vale problems for forth-order partial differential eqations The nmerical reslts obtained with minimm amont of comptation are compared with the eact soltion to show the efficiency of the method The reslts show that the homotopy pertrbation method is of high accracy and efficient for solving the forth-order parabolic partial differential eqation with variable coefficients The reslts show also that the introdced method is a powerfl tool for solving the forth-order parabolic partial differential eqations Key words: Homotopy Pertrbation Method; Forth-Order Parabolic Eqation Introdction We consider a forth-order parabolic partial differential eqation, with variable coefficients [,, 3, 4] µ(,y,z 4 4 λ (,y,z 4 y 4 η(,y,z 4 z 4 = g(,y,z,t, a <,y, z < b,t > 0, ( where µ(,y,z,λ (,y,z,η(,y,z are variable, sbject to the initial conditions [, 3] (,y,z,0= f 0 (,y,z, (,y,z,0=f (,y,z, ( and the bondary conditions (a,y,z,t=g 0 (y,z,t, (b,y,z,t=g (y,z,t, (3 (,a,z,t=k 0 (,z,t, (,b,z,t=k (,z,t, (4 (,y,a,t=h 0 (,y,t, (,y,b,t=h (,y,t, (5 (a,y,z,t=g 0(y,z,t, (b,y,z,t=g (y,z,t, y (,a,z,t=k 0(,z,t, y (,b,z,t=k (,z,t, (6 (7 z (,y,a,t=h 0(,y,t, z (,y,b,t=h (,y,t, (8 where the fnctions f i, g i, k i, h i, g i, k i, h i, i = 0, are continos The main focs of researchers was to obtain nmerical soltions by sing several techniqes sch as eplicit and implicit finite difference schemes sed in particlar in [3, 4] In [3] Andrade and Mc- Kee stdied alternating direction implicit (ADI methods for forth-order parabolic eqations with variable coefficients In [5] Byn and Wang investigated forth-order parabolic eqations with weak bonded mean oscillation (BMO coefficients in Reifenberg domains Also in [6] Caglar and Caglar investigated fifthdegree B-spline soltion for forth-order parabolic partial differential eqations Conte [4] investigated a stable implicit difference approimation to a forth-order parabolic eqation Also in [7] Danaee and Evans investigated the forth-order parabolic eqation by sing the Hopscotch method Evans [8] epressed ( in two space variables as a system of two second-order parabolic eqations where finite difference methods were employed Moreover, in [9] Evans and Yosef investigated the forth-order parabolic eqation with constant coefficients by sing the alternating grop eplicit (AGE method Gorman [0] stdied forthorder parabolic partial differential eqations in one / 09 / $ 0600 c 009 Verlag der Zeitschrift für Natrforschng, Tübingen

2 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations 4 space variable arises in the transverse vibrations of a niform fleible beam Forth-order parabolic eqations of variable coefficients were also stdied by Khaliq and Twizell [] where method of lines (MOL approach was sed to obtain a nmerical approimation In [] Biazar and Ghazvini investigated the forth-order parabolic eqation with variable coefficients by sing the variational iteration method (VIM Also forth-order parabolic partial differential eqations of constant coefficients were stdied by Wazwaz [] where Adomian s decomposition method (ADM was sed and the noise terms phenomenon were investigated Wazwaz [3] investigated forthorder parabolic partial differential eqations in higherdimensional spaces with variable coefficients where Adomian s decomposition method was sed to solve them See also [4] for another research work of this athor on forth-order parabolic partial differential eqations The approach in this paper is different, as we employ a semi-analytic techniqe which is based on the homotopy pertrbation method The homotopy pertrbation method [, 5 0] is developed to search the accrate asymptotic soltions of nonlinear problems Also, homotopy pertrbation method (HPM will be effectively sed to solve ( It is well known in the literatre that the homotopy pertrbation method provides the soltion in a rapidly convergent series This series may provide the soltion in a closed form This techniqe has been sccessflly applied to many problems sch as fnctional integral eqations [], Laplace transform [], qadratic Riccati differential eqation [3], hyperbolic partial differential eqations [4], integro-differential eqations arising in oscillating magnetics fields [5] and parabolic partial differential eqations sbject to temperatre overspecification [6], the second kind of nonlinear integral eqations [7], nonlinear eqations arising in heat transfer [8], soltions of generalized Hirota-Satsma copled KdV eqation [9], nmerical soltions of the nonlinear Volterra-Fredholm integral eqations [30], eact soltions for nonlinear integral eqations [3], Fredholm integral eqations [3], nmeric-analytic soltion of system of ODEs [33], nonlinear biochemical reaction model [5], non-linear Fredholm integral eqations [34], periodic soltions of nonlinear Jerk eqations [35], non-linear system of second-order bondary vale problems [36], inverse problem of diffsion eqation [37], delay differential eqations [38], heat transfer flow of a third grade flid between parallel plates [39] Song and Zhang [40] stdied application of the etended HPM to a kind of nonlinear evoltion eqation Also Yildirim [4] investigated soltions of bondary vale problems (BVP for forth-order integro-differential eqations by HPM The athors of [4] applied the homotopy pertrbation method to solve the Bossinesq partial differential eqation arising in modeling of flow in poros media The homotopy pertrbation method is sed in [44] to solve the differential algebraic eqations HPM is investigated by athors of [45] to solve the second Painleve eqation This paper is organized as follows: In Section, we describe the homotopy pertrbation method briefly and apply this techniqe to forth-order parabolic partial differential eqations Section 3 contains several test problems to show the efficiency of the new method Also a conclsion is given in Section 4 Finally some references are given at the end of this report Homotopy Pertrbation Method The homotopy pertrbation method is a powerfl tool for solving varios nonlinear eqations, especially nonlinear partial differential eqations Recently this method has attracted a wide class of adience in all fields of science and engineering This method was proposed by the Chinese mathematician J H He [] In this work, He s homotopy pertrbation method is adopted to stdy the forth-order parabolic partial differential eqations To illstrate the basic idea of the homotopy pertrbation method, consider the following nonlinear eqation A(v= f (r, r Ω, (9 sbject to the bondary condition ( B, = 0, r Γ, (0 n where A is a general differential operator, B is a bondary operator, f (r is a known analytic fnction, Γ is the bondary of the domain Ω and denotes differentiation along the normal vector drawn otwards Ω The n operator A can generally be divided into two parts L, N Therefore (9 can be rewritten as follows: A(v=L(vN(v, r Ω ( He [6] constrcted a homotopy v(r, p : Ω [0,] R

3 4 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations which satisfies H(v, p=( p[l(v L( 0 ] p[a(v f (r] ( = 0, and is eqivalent to H(v, p=l(v L( 0 pl( 0 p[n(v f (r] = 0, (3 where p [0,] is an embedding operator, and 0 is an initial approimation of (9 Obviosly, we have H(v,0=L(v L( 0, H(v,=A(v f (r (4 The change process of p from zero to nity is jst that of H(v, p from L(v L( 0 to A(v f (r In topology, this is called deformation and L(v L( 0 and A(v f (r are called homotopic According to the homotopy pertrbation method, the parameter p is sed as a small parameter, and the soltion of (3, can be epressed as in p in the form v = v 0 v p v p (5 When p, (3 corresponds to the original one, (9 Ths (5 becomes the approimate soltion of (9, i e = lim v = v k (6 p For solving ( by the homotopy pertrbation method, we have L :=, N := µ(,y,z 4 4 λ (,y,z 4 y 4 η(,y,z 4 (7 z 4, where g(,y,z,t is a known fnction Beginning with 0 (,y,z,t = f 0 (,y,z f (,y,zt, the approimate soltion of ( can be determined 3 Test Problems To illstrate the soltion procedre and show the ability of the method some eamples are provided Eample Consider the following one dimensional variable coefficients forth-order parabolic partial differential eqation [, 3, 0,, 4] ( (,t 4 0 < <, t > 0, 4 4 (,t=0, (8 sbject to the initial conditions 5 (, 0=0, (,0= 0, (9 and the bondary conditions ( ( (05 5,t = sin(t, 0 (,t= (0 0 sin(t, (,t = 6 ( Now by (3, we have: 3 sin(t, v (,t 0 0 (,tp (,t ( p 4 4 v 0 4 (,t=0 From the initial conditions we have (,t= 6 sin(t ( (,0= 0 (,0 (,0 (,0 ( k (,0= 0 Using 0 (,t=(,0t t (,0, thenweget 0 (,t= ( 5 t, 0 0 (,0=0, (3 and so, we have (,t= (,t= t=0 = 0, (,t = 5 t=0 0 = 0 (,0 (,0 k (,0 Ths we can write (4 0 5 (,0= 0, (,0= (,0== (5 k (,0== 0 Sbstitting (5 into (, and eqating coefficients of like powers of p, we obtain p 0 : v 0 (,t 0 (,t=0,

4 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations 43 p : v 0 (,t (,t ( 4 4 v (,t=0, ( p : v (,t 4 4 v 0 4 (,t=0, p k : v k Therefore, we obtain ( (,t v 0 (,t= 0 (,t= 4 v 0 4 (,t=t, v (,t v k ( 5 t, 0 ( 5 t = 0, 0 then we have v ( (,t= 5 t 0! g(, (,t = v t=0 (,t = g( t=0 (,t=0 (6 4 (7 (8 By sing (5, (8, we obtain g(=0 v ( (,t= 5 t 0!, (9 where (9 gives v (,t= ( 5 t 3 0 3!, (30 with repeating this procedre we obtain v (,t= ( 5 t 5 0 5!, ( v n (,t=( n 5 t n 0 (n! Ths we can write (,t=lim p p k v k (,t= ( (t 5 t3 0 3! t5 5! t n ( n (n! = ( 5 sin(t, 0 which is the eact soltion of the test eample (3 (3 Eample Consider the following parabolic eqation [0,, 4] ( 4 (,t sin( 4 (,t=0, (33 0 < <, t > 0, with initial conditions (,0=( sin(, (,0= ( sin(, (34 and the bondary conditions (0,t=0, (,t=ep( t( sin(, (0,t=0, (35 (,t=ep( tsin( From the initial conditions we have (,t= k (,t= 0 (,t (,t (,t (36 sin(=(,0= 0 (,0 (,0 (,0 Using 0 (,t=(,0t t (,0, we obtain 0 (,t=( sin(( t, (,0= (,0== 0, (,0= 0 (,0 (,0 (,0 sin(= (,0= 0 (,0 (,0 (,0 (37 Ths we obtain 0 (,0= ( sin(, (38 (,0= (,0== 0 Commencing with 0 (,t =( sin(( t, and with eqating coefficients of like powers of p, we obtain p 0 : v 0 (,t 0 (,t=0, p : v 0 (,t (,t ( 4 sin( v 0 4 (,t=0,

5 44 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations ( p : v 4 (,t sin( v 4 (,t=0, p : v (,t=( 3 4 ( t! cos(t ( 6 ( 7 ( cos(t, p k : v k 4 7! (,t sin( v k (,t=0 (39 4 p : v (,t=4( ( t! t4 4! cos(t This gives ( 3 4 ( t v 0 (,t= 0 (,t=( sin(( t,! cos(t, ( t v (,t=( sin(! t3, p 3 : v 3 (,t= 4( ( t 3!! t4 4! cos(t, ( t 4 (40 v (,t=( sin( 4! t5 (45, 5! v i (,t=0, i = 4,5,6,, (46 v(,t= v k (,t, Using (40 yields where (,t=lim p p k v k (,t= v k (,t, (,t= k (,t=lim p p k v k (,t (,t=( sin( ( t t! t3 3! t4 4! t5 5! = 6 (47 7! 7 cos(t, = ep( t( sin(, (4 which is the eact soltion of the test eample Eample 3 Now we solve the following one dimensional non-homogeneos forth-order parabolic eqation [0,, 4] (,t( 4 4 (,t= ( ( cos(t, 7! 0 < <, t > 0, (4 sbject to the initial conditions ( 6 (,0= 7, (,0=0, (43 7! and the bondary conditions (0,t=0, (0,t=0, (,t= ( 6 cos(t, 7! (,t= 0 cos(t (44 Starting with 0 (,t =( 7! 6 7, and eqating coefficients of like powers of p, we obtain ( 6 p 0 : v 0 (,t= 7, 7! which is the eact soltion of the test eample 3 Eample 4 Consider the forth-order parabolic eqation in two space variables [, 3,, 3] ( (,y,t (,y,t ( y y4 4 y 4 (,y,t=0, (48 <,y <, t > 0, with initial conditions 6 (,y,0=0, (,y,0= y6, (49 and the bondary conditions [, 3] ( (,y,t = (, y,t= ( ( y (05 y6 sin(t, sin(t, y6 =,y,t (054 4 sin(t, (,y,t= 4 sin(t, (, =,t (054 4 sin(t, y (,,t= 4 sin(t (50

6 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations 45 Commencing with 0 (,y,t =( 6 y6 t, and eqating coefficients of like powers of p, we obtain p 0 : v 0 (,y,t 0 (,y,t=0, p : v 0 (,y,t (,y,t ( 4 4 v 0 4 (,y,t ( y y4 4 v 0 y 4 (,y,t=0, ( p : v (,y,t 4 ( y y4 4 v y 4 (,y,t=0, Ths we obtain v 0 (,y,t= ( 6 y6 t, v (,y,t= ( 6 y6 t 3 3!, v (,y,t= ( 6 y6 t 5 5!, It can be seen that 4 v 4 (,y,t (, y,t=lim p k v k (,y,t p = ( (t 6 y6 t3 3! t5 5! t7 7!, (, y,t= ( 6 y6 sin(t, which is the eact soltion of the test eample 4 (5 (5 (53 Eample 5 Sppose we solve the following partial differential eqation in three space variables [,3] ( y z 4 (,y,z,t cos( 4 (,y,z,t ( z 4 cos(y y 4 (,y,z,t ( y cos(z 4 z 4 (,y,z,t=0, 0 <,y,z < π, 3 t > 0, (54 sbject to the initial conditions (, y, z, 0=yz (cos(cos(ycos(z, (55 (,y,z,0=(cos(cos(ycos(z ( y z, and the bondary conditions [3] (0,y,z,t=ep( t( y z cos(y cos(z, ( π 3,y,z,t= ( π 3 ep( t 6 (,0,z,t= y z cos(y cos(z, ep( t( z cos( cos(z, (, π 3,z,t = ( π 3 ep( t z cos( cos(z, 6 (,y,0,t= (56 (57 ep( t( y cos( cos(y, (,y, π 3,t = ( π 3 ep( t y cos( cos(y, 6 (0,y,z,t= y (,0,z,t= z (,y,0,t=ep( t, ( π 3,y,z,t = (, π y 3,z,t = (,y, π ( 3 z 3,t = ep( t From (3 the homotopy pertrbation method will be obtained as v (,y,z,t 0 (,y,z,tp 0 (,y,z,t

7 46 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations [ ( y z 4 p cos( v 4 (,y,z,t ( z 4 cos(y v y 4 (,y,z,t (58 ( ] y 4 cos(z v z 4 (,y,z,t = 0 Starting with 0 (,y,z,t =( y z cos( cos(y cos(z( t, by sing (5 for (58 and eqating coefficients of like powers of p yield p 0 : v 0 (,y,z,t 0 (,y,z,t=0, p : v (,y,z,t 0 (,y,z,t ( y z 4 cos( v 0 4 (,t ( z 4 cos(y v 0 y 4 (,y,z,t ( y 4 cos(z v 0 z 4 (,y,z,t=0, Therefore, we have v 0 (,y,z,t= ( y z cos( cos(y cos(z( t, v (,y,z,t= ( t ( y z cos( cos(y cos(z v (,y,z,t= ( t 4 ( y z cos( cos(y cos(z! t3 3! 4! t5 5!,, (59 (60 Ths we can write (, y, z,t=lim p k v k (,,y,z,t= p ( y z cos( cos(y cos(z ( t t! t3 3! t4 4! =( y z cos( cos(y ( t cos(z n =( y z cos( cos(y n! cos(zep( t, (6 which is the eact soltion of the test eample 5 Eample 6 As the last eample, we consider the following three dimensional non-homogeneos forthorder parabolic eqation [, 3] (,y,z,t [ 4 4! z 4 (,y,z,t 4 y 4 (,y,z,t 4 ] y z 4 (,y,z,t = ( y y z z 5 y 5 (6 z 5 cos(t, <,y,z <, t > 0, with initial conditions (,y,z,0= y y z z, (,y,z,0=0, (63 and the bondary conditions [3] ( (,y,z,t = y y z z cos(t, ( (,y,z,t= y y z z cos(t, (, (,z,t = z z cos(t, ( (,,z,t= z z cos(t, (,y, (,t = y y cos(t, ( (,y,,t= y y cos(t, ( (,y,z,t = y 4z cos(t, ( (,y,z,t= y z cos(t, (, ( y,z,t = 4 cos(t, z ( y (,,z,t= cos(t, z (,y, ( z,t = 4y cos(t, ( z (,y,,t= y cos(t (64 Applying the He s homotopy pertrbation method and

8 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations 47 Fig The srface of the eact and the approimate soltions of (8 (a The eact soltion (b The approimate soltion obtained in this work Fig The srface of the eact and the approimate soltions of (33 (a The eact soltion (b The approimate soltion obtained in this work eqating coefficients of like power of p, we obtain p 0 : v 0 (,y,z,t= y y z z, ( p : v (,y,z,t= y y z z (cos(t ( 5 y 5 (cos(t z 5 t!, ( p : v (,y,z,t= 5 y 5 (cos(t t z 5! ( 70 z 9 y 9 ( t yz 9! t4 4! cos(t, (65 Ths we can write (,y,z,t= ( y y z z cos(t, (66 which is the eact soltion of the test eample 6 We illstrate the accracy and efficiency of homotopy pertrbation method (HPM by applying the method to forth-order parabolic eqations and comparing the approimate soltions with the eact soltions For this prpose, we calclate the nmerical reslts of the eact soltions and the mlti-terms approimate soltions of HPM At the same time, the srface graphics of the eact and mlti-terms approimate soltions are plotted in Figs,, 3, 4, 5 and 6 One can

9 48 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations Fig 3 The srface of the eact and the approimate soltions of (4 (a The eact soltion (b The approimate soltion obtained in this work Fig 4 The srface of the eact and the approimate soltions of (48 (a The eact soltion (b The approimate soltion obtained in this work see that the approimate soltions obtained by HPM are qite close to their eact soltions 4 Conclsion The main idea of this work was to propose a simple method for solving forth-order parabolic partial differential eqations We have achieved an analytical soltion by applying the He s homotopy pertrbation method (HPM The main advantage of the method is the fact that it gives an analytical approimation soltion The reslts are compared with those in the literatre, revealing that the obtained soltions are eactly the same with those obtained by the Adomian s decomposition method [ 4] Also soltions obtained by the homotopy pertrbation method are the same with He s variation iteration method [] In eamples we observed that the HPM with the initial approimations obtained from (6 yield eact soltions in few iterations only In all eamples we observed that the HPM soltions are more efficient than the modified Adomian s decomposition method HPM avoids the difficlties arising in finding the Adomian s polynomials [46 50] In addition, the calclations involved in HPM are very simple and straightforward It can be shown that the HPM is a promising tool for solv-

10 M Dehghan and J Manafian Variable Coefficients Forth-Order Parabolic Partial Differential Eqations 49 Fig 5 The srface of the eact and the approimate soltions of (54 (a The eact soltion (b The approimate soltion obtained in this work Fig 6 The srface of the eact and the approimate soltions of (63 (a The eact soltion (b The approimate soltion obtained in this work ing some linear and nonlinear partial differential eqations It is worth to point ot that this techniqe nlike the mesh points methods [43] does not provide any linear or nonlinear system of eqations Acknowledgements The athors are very gratefl to both referees for their comments and sggestions [] J H He, Compt Methods Appl Mech Eng 67, 69 (998 [] A Q M Khaliq and E H Twizell, Int J Compt Math 3, 63 (987 [3] C Andrade and S McKee, Int J Compt Math 3, (977 [4] S D Conte, J Assoc Compt Mach 4, 0 (957 [5] S S Byn and L Wang, J Differential Eqations 45, 37 (008 [6] H Caglar and N Caglar, Appl Math Compt 0, 597 (008 [7] A Danaee and D J Evans, Math Compt Simlation 4, 36 (98 [8] D J Evans, Compt J 8, 80 (965 [9] D J Evans and W S Yosef, Int J Compt Math 40, 93 (99 [0] D J Gorman, Free Vibrations Analysis of Beams and Shafts, Wiley, New York 975

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