A GENERALIZED COLE-HOPF TRANSFORMATION FOR A TWO-DIMENSIONAL BURGERS EQUATION WITH A VARIABLE COEFFICIENT

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1 Indian J. Pure Appl. Mah., 43(6: , December 2012 c Indian Naional Science Academy A GENERALIZED COLE-HOPF TRANSFORMATION FOR A TWO-DIMENSIONAL BURGERS EQUATION WITH A VARIABLE COEFFICIENT B. Mayil Vaganan, M. Senhilkumaran and T. Shanmuga Priya Deparmen of Applied Mahemaics and Saisics, Madurai Kamaraj Universiy, Madurai , India Deparmen of Mahemaics, Thiagarajar College, Madurai , India (Received 11 April 2012; afer final revision 12 June 2012; acceped 12 June 2012 The direc mehod is applied o he wo dimensional Burgers equaion wih a variable coefficien (u + uu x u xx x + s(u yy = 0 is ransformed ino he Riccai equaion H 1 2 H H = 0 via he ansaz u(x, y, = 1 H(ρ + y 2 ρ(x, y, = x y, provided ha s( = 3/2. Furher, a generalized Cole-Hopf ransformaions u(x, y, = y 2 2 U ρ (ρ, r U(ρ, r, ρ(x, y, = x y, r( = log is derived o linearize (u + uu x u xx x + 3/2 u yy = 0 o he parabolic equaion U r = U ρρ U ρ. Key words : Generalized Cole-Hopf ransformaions; wo-dimensional Burgers equaion wih variable coefficien; Riccai equaion.

2 592 B. MAYIL VAGANAN e al. 1. INTRODUCTION The Burgers equaion is linearised o he hea equaion u + uu x = δ 2 u xx, (1.1 φ = δ 2 φ xx, (1.2 hrough he Cole-Hopf ransformaion (Hopf [5] and Cole [3] u = δ φ x φ. (1.3 Nimmo and Crighon [11] have derived Bäcklund ransformaions for nonlinear parabolic equaions of he form u u xx + H(, x, u, u x = 0, (1.4 and proved ha besides he Burgers equaion (1.1 is inhomogeneous version, viz., u + uu x + a(x, = δ 2 u xx, (1.5 is also linearisable. Bu Lighhill [7] showed ha, unlike in (1.1 and (1.5, he viscosiy is funcion of. Taking his fac ino consideraion, Mayil Vaganan and Senhilkumaran [8,9] sudied he generalized Burgers equaions (GBEs wih ime dependen viscosiy, viz., u + u n u x = (a + bm u xx, (1.6 2 u + uu x + αu = e α 2 u xx, α > 0. (1.7 They repored exac linearizaions of (1.6 and (1.7 o Bernoulli s and he Kummer s equaions, respecively. I should be menioned ha he reduced linear equaions are no parial differenial equaions (PDEs bu ordinary differenial equaions (ODEs.

3 A GENERALIZED COLE - HOPF TRANSFORMATION 593 Subsequenly, Mayil Vaganan and Jeyalakshmi [10] have linearized he GBE (1.7 o he PDE via he GCHT u(x, = 2 G G + 2z z2 z + a G α τ + α2 z 2 G = 0, (1.8 [ ] ( 1 G(z, τ [G(z, τ] αe α z e 2α 2 1/2 z α eα + 1 (1.9, z(x, = x eα e α + 1, (1.10 τ( = a 4 0 ds 1 α cosh(αs. (1.11 For, hey embedded he concep of Cole-Hopf ransformaion ino he mehod inroduced by Kawamoa [6] o map nonlinaer evoluion equaions using Lie s classical mehod and laer used by Sachdev and Mayil Vaganan [13] in connecion wih he direc mehod of Clarkson and Kruskal [2]. In his paper we show ha he wo-dimensional variable coefficien Burgers equaion (u + uu x u xx x + 3/2 u yy = 0, (1.12 is linearizable. However he linearized PDE does no depend on hree independen variables as u does, bu depends only on wo independen variables. We furher presen, wihou proof, a ransformaion which linearizes he wodimensional Burgers equaions wih algebraic viscosiy u + uu x + m u xx + n u yy = 0. ( GENERALIZED COLE-HOPF TRANSFORMATIONS FOR ( Linearizaion of (1.12 o an ODE

4 594 B. MAYIL VAGANAN e al. The wo-dimensional variable coefficien Burgers equaion is (u + uu x u xx x + s(u yy = 0, (2.1 We seek soluions of (2.1 in he form (Clarkson and Kruskal [2] u(x, y, = A(H(ρ + B(y,, ρ = ρ(x, y,. (2.2 Inserion of (2.2 ino (1.12 yields A H Aρ 2 x AH 2 ρ x + sb yy Aρ 3 x ρ H ρ 2 x BH ρ x AHH + sρ2 yh ρ x ρ 3 H ρ x x ρ 3 BH ρ xx x ρ 3 AHH ρ xx x ρ 3 x + 3H ρ xx ρ 2 x + sh ρ yy ρ 3 x + H + H ρ xxx ρ 3 x = 0. (2.3 We require in (2.3 ha he coefficiens are funcions of ρ so ha (2.3 becomes an ODE for H(ρ. Therefore, i follows from he coefficien of HH ha A + ρ x Γ(ρ = 0. This is saisfied if Γ = 1 and A = ρ x. Now he coefficien of HH, in view of Γ = 1 and A = ρ x, gives If Γ = Λ (ρ/λ(ρ, hen (2.4 becomes ρ xx ρ x + ρ x Γ(ρ = 0, (2.4 ρ xx ρ x + ρ x Λ (ρ Λ(ρ = 0. (2.5 Inegraing (2.5 wih respec o x and aking α(y, o be he funcion of inegraion, we find ha ρ x Λ(ρ = α(y,. If we wrie Ω (ρ for Λ in ρ x Λ(ρ = α(y, and inegrae he resuling equaion wih respec o x, we ge Ω(ρ = x α(y, +β(y,, where β(y, is anoher funcion of inegraion. We se Ω(ρ = ρ o obain Γ = 0 and ρ = x α(y, + β(y,. (2.6

5 A GENERALIZED COLE - HOPF TRANSFORMATION 595 In view of (2.6, A = ρ x becomes A = α(, where we have dispensed wih he dependence of α on y as A is assumed o be a funcion of only. I is easily checked ha he coefficien of H 2 becomes 1. Now from he coefficien of H we obain ha ρ /ρ 2 x B/ρ x + sρ 2 y/ρ 3 x = Γ(ρ, by using (2.6, i gives xα + β α 2 B α + sβ2 y = Γ(ρ. (2.7 α3 Equaion (2.7 demands ha Γ(ρ = aρ + b, and splis ino wo equaions α = aα 3, (2.8 β α 2 B α + sβ2 y α 3 = aβ + b. (2.9 A soluion of (2.8 is provided ha a = 1/2,(2.9 reduces o α( = 1, (2.10 β B + sβ2 y 3/2 = 1 β + b. ( I is also verified ha he coefficien of H becomes 1. Equaion (2.11 requires ha β = p(y. Now we rewrie (2.3 by insering he expressions obained for Γ(ρ o ge H H 2 HH b H + sb yy Aρ 3 + H = 0. (2.12 x Equaion (2.12 is meaningful only if B yy = 0 and herefore, we wrie B = T (y. Then (2.11 splis ino wo equaions 2 T ( + p( = 0 and s( 3/2 = b. (2.13

6 596 B. MAYIL VAGANAN e al. For breviy, we se b = 1, p = 1 o obain T ( = 1 2, s( = 3/2, B = y 2, (2.14 where we have used he fac ha equaion (1.12 is invarian under y y. The similariy ransformaion (2.2 hus akes he form u(x, y, = 1 H(ρ + y 2, (2.15 ρ(x, y, = x y, (2.16 where H(ρ saisfies he hird order nonlinear ODE H HH H H 2 + H = 0. (2.17 The ransformaion (2.15-(2.16 has also been derived by Güngör [4] by applying Lie s classical mehod (see Bluman and Kumei [1] and Olver [12] o (1.12. Equaion (2.17 may be inegraed wice o yield he Riccai equaion H 1 2 H H = 0, (2.18 where we have se he consans of inegraion equal o zero. We noe ha he Riccai equaion (2.18 may be linearized o he second order ODE U U = 0, (2.19 hrough he CHT H(ρ = 2 U (ρ U(ρ. (2.20 Now we derive (2.19 direcly from (2.15 afer replacing for H from he CHT (2.20. Equaions (2.15 and (2.20 give he generalized Cole-Hopf ransformaion (GCHT u(x, y, = y 2 2 U (ρ U(ρ. (2.21

7 A GENERALIZED COLE - HOPF TRANSFORMATION 597 Insering (2.21, wih ρ given by (2.16, equaion (1.12 becomes U 2 U (iv 2UU U U 2 U UU 2 + 2U 2 U UU U + U 2 U + (ρ 2U 3 UU 2 = 0.(2.22 Equaion (2.22 may be pu in he form U 4 d ( L [U] ( dρ U 2 UU 2U 2 L[U] = 0, (2.23 where L[U] = U U, and L [U] = d L[U]. (2.24 dρ Equaion (2.23 is saisfied if Case 1 : UU 2U 2 = 0 and L [U] = 0, (2.25 or Case 2 : L[U] = 0. (2.26 In Case : 1, U has o saisfy wo equaions UU 2U 2 = 0, (2.27 L [U] U U U = 0. (2.28 I is easy o check ha here is no nonrivial soluion U saisfying boh he ODEs (2.27-(2.28. We herefore have L[U] = 0 which is nohing bu ( Linearizaion of (1.12 o a PDE We now shall prove ye anoher resul: Equaion (1.12 may be linearized o he PDE U r = U ρρ U ρ, (2.29

8 598 B. MAYIL VAGANAN e al. hrough he GCHT u(x, y, = y 2 2 U ρ (ρ, r U(ρ, r, (2.30 ρ(x, y, = x y, (2.31 r( = log. (2.32 For, we subsiue (2.30-(2.32 o ge U 2 U ρρρρ 2UU ρ U ρρρ U 2 U ρρρ UUρρ 2 + 2Uρ 2 U ρρ UU ρ U ρρ + U 2 U ρρ + (ρ 2Uρ 3 UUρ 2 U 2 U ρρr + 2UU ρ U ρr + UU r U ρρ 2U r U 2 ρ = 0. (2.33 Equaion (2.33 may be pu in he form ( ( 1 L1 [U] + U ( ρ L1 [U] ρ U ρ U U 2 = 0, (2.34 ρ U where L 1 [U] U ρρ U ρ U r. (2.35 (2.29. Equaion (2.34 is saisfied only if L 1 [U] = 0 which is nohing bu he PDE 3. FURTHER RESULTS I is easy o verify ha he ransformaion u = x + m 1/2 G(z, τ, z = y, τ = log, (2.36 (n+1/2

9 A GENERALIZED COLE - HOPF TRANSFORMATION 599 linearizes (1.13 o he PDE G τ + G zz n + 1 zg z + 2 ( m + 1 G = 0. ( ACKNOWLEDGEMENT The auhors are hankful o he anonymous referee for his useful suggesions. REFERENCES 1. G. W. Bluman and S. Kumei, Symmeries and Differenial Equaions, Springer- Verlag, New York, ( P. A. Clarkson and M. D. Kruskal, New similariy reducions of he Boussinesq equaions, J. Mah. Phys., 30 (1989, J. D. Cole, On a quasi-linear parabolic equaion occurring in aerodynamics, Quar. Appl. Mah., 9 (1951, F. Güngör, Symmeries and invarian soluions of he wo-dimensional variable coefficien Burgers equaion, J. Phys. A: Mah. Gen., 34 (1951, E. Hopf, The parial differenial equaion u + uu x = u xx, Commun. Pure Appl. Mah., 3, (1950, S. Kawamoo, Derivaion of nonlinear parial differenial equaions reducible o he Painlevé equaions, J. Phys. Sco. Jpn., 52 (1983, M. J. Lighhill, Viscosiy effecs in sound waves of finie ampliude. In Surveys in Mechanics (Eds: G.K. Bachelor and R.M. Davies, (1956, B. Mayil Vaganan and M. Senhil Kumaran, Exac Linearizaion and Invarian soluions of a Generalized Burgers equaion wih Linear Damping and Variable Viscosiy, Sud. Appl. Mah., 117, (2006, B. Mayil Vaganan and M. Senhil Kumaran, Kummer Funcion Soluions of Linearly Damped Burgers Equaions wih Time-dependen Viscosiy by Exac Linearizaion, Nonlinear Analysis: Real World Applicaions, 9 (2008, , doi: /j.nonrwa

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