Diffusion equations for nonhomogeneous media. Existence of similarity solutions

Volume 11 lA, number 4 PHYSICS LETTERS 9 September 1985 DIFFUSION EQUATIONS FOR NONHOMOGENEOUS MEDIA. EXISTENCE OF SIMILARITY SOLUTIONS F. ROMERO. J.L. ROMERO and J.F.R. ARCHILLA Depurrrm~cv~to de Fkiw T&rcu, Fucultad de Fiwu, P. 0. Em 1065. Ser~,llr, Sprwt Received 23 May 1985: accepted for publication 28 June 1985 We study the invariance of the diffusion equation SP( x. t )/St = (6/&v )[ D( x)6P( x, t)/6x] under continuous groups of transformations. We show the conditions which D(x) must satisfy for the existence of similarity solutions. Recently considerable attention has been paid to the problem of diffusion in a medium whose diffusion coefficient varies in space. In one dimension x, it may be necessary to study the equation where P(x, t) represents the probability density for a diffusing particle to be at the point x at time f, and D(x) gives the value of the diffusion coefficient at each point x. There are many situations in physics where this problem must be studied. As an example we shall mention the diffusion of hot electrons in velocity space [ 1,2]. There are no general solutions for this kind of problem, and special difficulties arise when contour behaviours must be taken into account. In this paper we show that for a large class of diffusion coefficients, D(x), it is possible to obtain exact solutions for eq. (1). A possible way for obtaining exact solutions, called similarity solutions, for a given differential equation is by investigating its invariance under continuous groups of transformations [3,4]. Similarity’solutions for eq. (1) could be obtained only for certain forms of D(x). We think that it is important to establish the conditions which D(x) must satisfy in order to assure the existence of similarity solutions. Thus, if for a given problem one has a particular form ofD(x), it may be possible to check immediately if there exist similarity solutions. If that is the case, one may benefit from this solution method. Eq. (1) enters in the context of a more general equation, usually called the Fokker-PlanckSmoluchowski equation (FPS): afyx, t) a2 ~ = s M-4 P(x, t>l at -g P(x)fYx, r)l . This equation may be written in the form: H(x,P,P,,P,,,P,)-aP,,+od:+PP-P, =O, where (2) (3) ai=2a’-b, (3=a”-b’. (4) We introduce the group of transformations in the space (x, t, P) given infinitesimally by x*=x+C;(x,t,P)AE, (5) t*=t+T(X,r,P)&, (6) P*=P+Q(X,t,P)hE, (7) where E is a continuous parameter and ,$, 7,~ are called the infinitesimals of the group of transformations. If a(x) and b(x) are such that it is possible to find the infinitesimals for eq. (3) to be invariant under the group of transformations, then there exist similarity 0.375-9601/85/S 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 179 Volume 1 llA, number 4 PHYSICS LETTERS 9 September 1985 solutions, which may be obtained by solving the characteristic equation 6x/ ~(x, t. P) = 8t /r(x, t,P) = 8P/rl(x, t, P) . (8) It is not difficult to show that the infinitesimals which leave the FPS equation invariant have the following dependence: }= ~(x, t), r= r(t), 7? =f(x, t)P(x, t) + g(x, t) , (9) wilere ~, r, f and g are functions which must satisfy the following determining equations: agxx + °~gx + ~g - gt = O, (10) afx x +O~fx + ~' +~r'--ft =0, (ll) a(2fx ~xx) + a'~ + a(r' - ~x) + ~t = O, (12) a'~ + a(r' - 2~x) = 0. (13) Eq. (1) corresponds to eq. (2) for a=D(x), a=D'(x), /3=0. (14) With these identifications, eqs. (10)-(13) become: D'gx + Dgxx - gt = O, (15) Dfxx +D'fx -/i =0, (16) D(2fx - }xx) +D'(r'- ~x) + ~f)" + ~t = 0, (17) D(r' - 2 }x) + ~D' = 0. (18) This system of equations is more easily treated if we introduce the variable: /, = D~ dz. (19) x0 In this variable, eq. (18) is a linear differential equation. Integrating it we obtain: = vC6(c + ' ,- ,i r x), (20) where C is an arbitrary constant. Here C represents the invariance of translation in x. We can take C = 0 because any condition relative to the solutions in determined values of 2 (and then of x) avoids the invariance of translation. If we substitute (20) into (17), and introducing the auxiliary function H = In D, we obtain after integrating: f(x,O = ~r x ~ --~-' (2 .... ' (H£2) +q(t) 1) where q(t) is some function of t. hathe new coordinates (x, t) eq. 16) becomes +~f~ .6=°. f2~ 1 (22) Substituting (2 l) into (22), we obtain 1 -, "~ ~r x- + [-~-r q'(t)] {r'r(2) = O, (23) where r(x) (H~)2~ 1 = +-~H~(H~2)7~. (24) The existence of similarity solutions requires that D(x), r(t) and q(t) have to be related in the way indicated by eq. (23). We distinguish the following two cases in which that equation is satisfied: Case L H = In D arbitrary, r' = 0 and q' = 0. The infinitesimals are = O, (25) r = r 0 , (26) r? = qo P. (27) This case yields a shnilarity solution of the form P(2, t) = P(2) exp(qot/ro), (28) where P(2) satisfies the ordinary differential equation: 1 P5,£ + ~HYcP2 - qo/ro = 0. (29) Of course it is well stablished that eq. (1) admits solutions in the form of separate variables. For this case our method does not give new results. Case II. ,,, , ,= ~ +gt¢17 , r =/xr , q - r" l , (30) and G~c +~ 1G~ G /aN2 + tq = 0, (31) where/~, K 1 are arbitrary constants and G = (D£/D)2. ILA. If/~ = 0, the infinitesimals are (for the sub180 Volume 111A, number 4 PHYSICS LETTERS 9 September 1985 group g = 0): = (rlt + r2)x , (32) r = rlt2 + 2r2t + r O, (33) ~ =p(_~_rlX 2 1 - a(rlt + r2) G + q), (34) with 1 2 1 (35) q=~lrl t +a(Klr2-2rl)t+q0, where r 1 , r2, r 0 and q0 are arbitrary constants. -' -2v ) -' ql-grl(K1 , q2-gr2(Kl+2V/-~). (39) Eq. (31) gives the condition that the diffusion coefficient D(x) must satisfy so that eq. (t) admits similarity solutions. These could be obtained by solving the characteristic equation (8) with the infinitesimals associated. We are now working in order to find the solutions corresponding to some physical interesting cases which can not be resolved by other more elementary methods. ll.B. Ifgt v ~ 0, the infinitesimals are (forg = 0) r = r 1 exp(v~t) + r 2 exp(-x/~t) + r 3 , (36) -- 1 = ~X/~ [r 1 exp(x/-~t) - r 2 exp(-x/-~)] 2, (37) ,~ 1 "-2 1 , rT=r~--~r x ~r G +q(t)), with q = ql exp(x/-~t) + q2 exp(-x/--/'tt) + q3, (38) where rl, r 2, r 3 and q3 are arbitrary constants, and References [ 1] P.I. Price, in: Fluctuation phenomena in solids, ed. R.E. Burgess(Academic Press, New York, 1965). [2] S.V. Gantsevich, V.L. Gurevich and R. Katilins, Riv. Nuovo Cimento 2 (1979) 1. [3] L.V. Ovsjannikov, Gruppovye Svoystva Differentsialny Uravneni (Novosibirsk, 1962) [Group properties of differential equations, translated by G. Bluman (1967)]. [4] G.W. Bluman, Construction of solutions to partial differential equations by the use of transformation groups, Ph.D. Thesis, California Institute of Technology (1967). 18l