Exact solutions of a nonlinear diffusion equation with absorption and production
aa r X i v : . [ n li n . S I] A ug Exact solutions of a nonlinear diffusion equation with absorption and production
Robert Conte
Universit´e Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli,F-94235, Cachan, France.Department of mathematics, The University of Hong Kong, Pokfulam road, Hong Kong.ORCID https://orcid.org/0000-0002-1840-509525 August [email protected]
We provide closed form solutions for an equation which describes the transport of turbulent kinetic energy inthe framework of a turbulence model with a single equation.
Keywords : Nonlinear diffusion equation; turbulent kinetic energy; exact solutions2020 Mathematics Subject Classification: 34A05, 35C05, 76F60
1. Introduction
The partial differential equation (PDE) [10] x real , t > v t + ( v m ) xx + cv p − dv q = , m > , p > , c > , q > , d > , v ( x , ) = v ( x ) ≥ , (1.1)governs the transport of turbulent kinetic energy k in the framework of a turbulence model with asingle equation. This minimal model retains the essential physical ingredients: time evolution ( ∂ t ),space diffusion ( ∂ x ), one space dimension only, nonlinearity ( m = cv p ) and produc-tion ( dv q ). Our goal in the present paper is to find closed form solutions to serve as a validationtest for the numerical schemes. In order to achieve that, we will not design new methods but applyexisting methods to generate several new solutions of physical interest.The so-called “model equation” [10] corresponds to m = p = / , c = d = , q = /
2, and thevery special case c = d = v m − = w , w t − mww xx − mm − w x + ( m − ) cw p + m − m − − ( m − ) dw q + m − m − = , (1.2)with the choice p = m , q = − m , m arbitrary (since it contains the case of physical interest p = / q = / m = / t ) [10, Eq. (3.24)] λ v m − = w : E ( w ) ≡ w t − bww xx − w x + cw − d = , b > , c > , d > . (1.3)Since there could exist physical systems not requiring the positivity of b , c , d , we will also mentiona few solutions with b , c , d not all positive. onte / Nonlinear diffusion equation The paper is organized as follows. In section 2 we first recall the ingenious, however not gener-alizable, method which has allowed Maire to obtain a solution matching all the physical constraintsin (1.3).Section 3 is devoted, using the method of infinitesimal Lie point symmetries, to the constructionof all the reductions of the PDE (1.3) to an ordinary differential equation (ODE), then to theirintegration.In section 4, we study the local behaviour of the field w ( x , t ) near its movable singularities, aprerequisite to the search for closed form solutions.The next section 5 is devoted to a search for new exact solutions based on the singularity struc-ture.
2. Method of Galaktionov and Posashkov
The PDE (1.3) belongs to the class w t − P ( ∂ x , w ) = , P differential polynomial , (2.1)with the additional property of existence of a finite number of functions f k ( x ) whose linear combi-nations are stable under P ( ∂ x , w ) , ∀ ( α , . . . , α K ) ∃ ( β , . . . , β K ) : P ( ∂ x , K ∑ k = α k f k ( x )) = K ∑ k = β k f k ( x ) . (2.2)Then Galaktionov and Posashkov [7] observed that the (kind of “adiabatic”) assumption w ( x , t ) = K ∑ k = γ k ( t ) f k ( x ) , (2.3)amounts to solving a nonlinear system of ODEs (no more PDEs) for the functions γ k ( t ) . In thepresent case (1.3), w ( x , t ) = γ ( t ) + γ ( t ) cosh ( kx ) , one thus obtains four solutions leaving the physical parameters b , c , d unconstrained and positive.The most physically relevant solution [10, Eq. (3.37)], λ v m − = w = A cosh ( ω t ) − cosh ( kx ) sinh ( ω t ) , ω = Ac + b + b , k = c + b , A = dc , (2.4)does not depend on any movable (i.e. function of the initial conditions) constant, excluding of coursethe arbitrary origins x and t , and it provides a good description [10] of the transport of the turbulentkinetic energy. onte / Nonlinear diffusion equation The second solution is stationary, w = A cosh ( kx ) , k = c + b , A = ( + b ) dc , (2.5)and the third one homogeneous in space w = ω c tanh ω t , ω = cd . (2.6)Finally, the fourth solution is characterized by a second order nonlinear ODE, w = g ( t ) + g ( t ) cosh ( kx ) , k = c + b , g = g − dc = , − ( b + ) g ′′ − ( b + ) cg g ′ − ( b + ) cg ( cg − d ) = , (2.7)and, if one excludes the particular values g listed in the three previous solutions, its only physical( b positive) solutions are multivalued [3, Eq. (9.2)], but nevertheless expressible via quadratures [9].The same kind of ODE will again be encountered hereafter, see Eq. (5.17) for F .If one relaxes the constraint b >
0, there also exists a fifth solution, w = ω c tanh ω t + A cos ( kx ) , ω = cd + c A , k = c , b = − , A = arbitrary . (2.8)Since the above ingenious method only applies to the restricted class (2.1)–(2.2), let us in addi-tion apply the two main classes of methods able to find explicit solutions of algebraic PDEs : • Those based on the symmetries of the PDE, which generate reductions to ODEs; • Those based on the movable singularities of the PDE, which (after a double study, localthen global) generate closed form solutions w ( x , t ) .
3. Solutions obtained by symmetries
For generic values of b , c , d , the only Lie point symmetries of the PDE (1.3) are arbitrary translationsof both x and t . The resulting characteristic system,d x α = d t β = d w , α , β arbitrary constants , (3.1)admits the two invariants w and β x − α t , with α , β not both zero, thus defining the unique reductionto an ODE, w ( x , t ) = W ( ξ ) , ξ = β x − α t , − α W ′ − β ( bWW ′′ + W ′ ) + cW − d = . (3.2)One must distinguish β = β = β =
0, since the positive parameter b is never equal to − ( − / n ) , with n some signedinteger, the general solution W ( ξ ) is multivalued [11] and generically cannot be obtained in closed onte / Nonlinear diffusion equation form. A notable exception is β = α = ξ = x , w ( x , t ) = W ( x ) , − bWW ′′ − W ′ + cW − d = , (3.3)which admits the first integral [10] K = h ( + b )( W ′ − d ) + cW i W / b . (3.4)For K =
0, the general solution is physically acceptable and has already been obtained, see Eq. (2.5),and for K = w ( x , t ) = W ( x ) , x = x + Z (cid:18) + b ( + b ) d − cW − KW − / b (cid:19) / d W . (3.5)The only cases of invertibility of this quadrature are − / b = , , ,
4, yielding expressions W ( x ) trigonometric ( b = − , − ) or elliptic ( b = − / , − / b > β = x , Eq. (2.6).
4. Structure of singularities
Any closed form solution depends on arbitrary functions or constants, such as x , t in (2.4), whichmay define movable singularities. For instance, the solution (2.4) definies two families of movablesingularities : on one side the movable poles of w located at the points t = t + ni π / ω , n ∈ Z , on theother side movable poles of 1 / w (movable zeroes of w ) located on the singular manifold ϕ ( x , t ) = ϕ ( x , t ) ≡ cosh ω ( t − t ) − cosh k ( x − x ) = . (4.1)A prerequisite to the systematic search for solutions is therefore the determination of the struc-ture of the movable singularities of (1.3), w ∼ ϕ → w ϕ p , p / ∈ N . (4.2)Since the highest derivative term ww xx displays the singularity w =
0, one must also study themovable zeroes of w (movable poles of w − = f ), λ v m − = w = f − : E ( f ) ≡ − f f t + b f f xx − ( b + ) f x − d f + c f = . (4.3)This is a classical computation [6, § ϕ x = ww xx does not contribute to the leading order, and w (as well as f ) presents one family of movablesimple poles, ϕ x = w ∼ ϕ → c − ϕ t ϕ − , (4.4) ϕ x = w − ∼ ϕ → d − ϕ t ϕ − . (4.5) onte / Nonlinear diffusion equation (2) If ϕ x = w has no movable poles and w − = f presentstwo families of movable simple poles, f ∼ ϕ → f ϕ x ϕ − , d f − ϕ t ϕ x f + = , f = . (4.6)To finish this local analysis, one must then compute the Fuchs indices of the linearized equationof (1.3) in the neighborhood of ϕ =
0. Indeed, a necessary condition of singlevaluedness is thatall Fuchs indices be integers of any sign. For the families (4.4) or (4.5), the unique Fuchs index is i = −
1. For each of the two families (4.6), one finds i = − , d f − b , (4.7)and the noninteger value (even nonrational) of the second index reflects the high level of noninte-grability of the initial PDE.In order to build solutions of this kind of nonintegrable PDE, the various methods based onthe singularity structure are reviewed in summer school lecture notes [5], where the proper originalreferences can be found. Let us now investigate a few of them.
5. Truncations
They consist in requiring the Laurent series of a single family to terminate. For the respective localbehaviours (4.4), (4.5) and (4.6), the corresponding possible solutions are defined by ( ϕ x = ) : w = c − χ − + w , (5.1) ( ϕ x = ) : f = d − χ − + f , (5.2) ( ϕ x = ) : f = f χ − + f , f = , (5.3)in which the expansion variable χ ( x , t ) is any homographic transform of ϕ ( x , t ) vanishing with ϕ .There exists an optimal choice of χ [4], characterized by its gradient ( ( ξ , η ) represents either ( x , t ) or ( t , x ) ) χ ξ = + S χ , χ η = − C + C ξ χ − ( CS + C ξξ ) χ , (5.4)and the constraint S η + C ξξξ + C ξ S + CS ξ = . (5.5)After substitution in equations (1.3) and (4.3), the LHS E of these equations become Laurentseries in χ which also terminate, E ≡ − q ∑ j = E j χ j + q , − q ∈ N , (5.6)and the coefficients E j only depend on w , w (or f , f ) and ( S , C ) . The solutions are then providedby solving the determining equations ∀ j = , ..., − q : E j ( w , w , S , C ) = . (5.7) onte / Nonlinear diffusion equation Characteristic one-family truncation of w The truncation (5.1) with χ x = E ≡ , E ≡ − bc − w , xx + w = , E ≡ − bw w , xx − w , x + w , t + cw − S c − d = , (5.8)whose general solution is w = A cos ( kx ) , S = − cd − c A , k = c , ( b + ) A = . (5.9)The value of χ results by integrating the Riccati equation (5.4), χ − = ω ω t , ω = cd + c A . (5.10)The bifurcation ( b + ) A = w , (2.6) and (2.8). Characteristic one-family truncation of f The truncation (5.2) with χ t = E ≡ , E ≡ f = , E ≡ cd + S − d f − d f , t = , E ≡ cd f + S f − d f − d f f , t + bd f , xx = , E ≡ E ( f ) + S d f = , (5.11)whose general solution is f = , S = − cd . (5.12)After integration of the Riccati equation (5.4), χ − = ω ω t , ω = cd , (5.13)one obtains f = d − ω ω t , ω = cd , (5.14)identical to (2.6).To conclude, these characteristic truncations yield nothing new. onte / Nonlinear diffusion equation Noncharacteristic one-family truncation of f Let us finally consider one of the two families (4.6), whose Fuchs indices are (4.7), and let us assume bd =
0. The truncation (5.3) defines the system E ≡ d f + C f + = , E ≡ ( − d f − C f + b ) f + f C x − f f , t + ( + b ) f , x = , E j ≡ E j ( f , f , S , C ) = , j = , , E ≡ b ( b + ) (cid:2) f S + f + f f , x − f f , x (cid:3) = , X ≡ S t + C xxx + C x S + CS x = , (5.15)and the factorization of E makes the resolution easy. It is even easier after the change of functions ( f , f ) → ( F , F ) : f = / F , f = f F − f , x . (5.16)In the case b = − /
2, the algebraic (i.e. nondifferential) elimination of F , t , F , t , S x yields themuch simpler equivalent system, b = −
12 : E ≡ F , t − ( b + ) F F , x + ( b + ) F F − dF = , E ≡ − bF , xx + ( b + ) F F , x + [( b + ) F , x − ( ( b + ) F + c )] F − F , t = , X ≡ bF , xx − ( b + ) F F , x + ( b + ) F − cF = , S = F , xx F − F , x F − F + F , x − F , x F F , C = − F − dF − . (5.17)This system (5.17) is solved in three steps:(i) integration of the ODE in F defined by X =
0, which introduces at most two arbitrary functionsof t ;(ii) determination of F by solving the overdetermined system ( E = , E = ) ;(iii) knowing the values of ( S , C ) , integration of the Riccati system (5.4) for χ .In the unphysical case b = − /
2, we could not find an equivalent system as simple as (5.17).Let us now perform the above mentioned three steps.5.3.1.
Values of F As already mentioned about the similar ODE (2.7), we will discard the multivalued solutions of theODE X = F ( t ) , refering the interested reader to Ref. [9].All singlevalued solutions of the ODE for F are obtained by two methods: for the generalsolution, by looking in the classical exhaustive tables [8, 11]; for particular solutions, by lookingfor Darboux polynomials. One thus finds exactly seven solutions, five of them with a negative b (unphysical for the diffusion problem, but possibly admissible for other systems) and two with an onte / Nonlinear diffusion equation arbitrary value of b , b = −
43 : F = − ∂ x log [ cosh ( kx − g ( t )) − cosh ( K ( t ))] , k = − c , (5.18) b = −
45 : F = − ∂ x log [ ℘ ( x − g ( t )) − e ] , e = − c , g = e , g ( t ) , (5.19) b = −
12 : F = ∂ x log [ ℘ ( x − g ( t )) − e ] , e = c , g = e , g ( t ) , (5.20) b = −
23 : F = p ℘ ( x − g ( t )) − e , e = − c , g = e − K ( t ) , g = − e + e K ( t ) , (5.21) b + = F = − k k ( x − g ( t )) , k = cb + , (5.22) b + = F = − b ( b + ) k k ( x − g ( t )) , k = ( b + ) cb , (5.23) b = − F = , (5.24)in which ℘ ( x , g , g ) is the elliptic function of Weierstrass, and g , K two arbitrary functions of t .The first four depend on two arbitrary functions of t , the next two on one arbitrary function of t .For b = − /
2, the obtained solution is a particular case of the solution, which we could notobtain, resulting from the system (5.15).For b = − c =
0, the ODE for F has no singlevalued solution.Let us next determine F . By the elimination of F between E = E =
0, the value of F is the root of a sixth degree polynomial whose coefficients are polynomial in F and its derivatives.However, this computation is only tractable for the two tanh solutions (which only depend on g ( t ) ),, and one finds constant values for F and g ′ ( t ) , b + = F = db + , g ′ ( t ) = +( b + ) k F , (5.25) b + = F = db + , g ′ ( t ) = − ( b + ) k F . (5.26)In the four other cases b = − / , − / , − / , − / E = F as follows.One first notices that, in its homogeneous part, the simple pole of F with residue r = − F , whose Fuchs indices i , the roots of bi − ( b + r + br ) i + ( b + ) r + ( b + ) r = , (5.27)are irrational for the four values of b . Since F is necessarily an algebraic function of F and itsderivatives, the only such algebraic solution of the homogeneous part of E = F =
0, see theexample Eq. (5.34) hereafter. One then computes F as a particular solution of the inhomogeneousequation E =
0. Once this value of F obtained, the nonlinear equation E = g ( t ) and K ( t ) . This method is exemplified in section 5.3.5.5.3.2. Solution, case of the first tanh value of F Given b + = F = a , a = b + d , g ′ ( t ) = +( b + ) k F , (5.28) onte / Nonlinear diffusion equation one finds successively S = − k , C = − b + a , (5.29)then, by integration of the Riccati system (5.4)–(5.4) χ − = k k ( x + b + a t ) , (5.30)the solution w − = f = a k (cid:20) tanh k ( x + b + a t ) − tanh k ( x − b + a t ) (cid:21) , k = cb + Solution, case of the second tanh value of F Solving ( E , E ) for ( F ( x , t ) , g ( t )) is again quite easy, F ( x , t ) = const = a , a = b + d , g ( t ) = − ( b + ) k a t , then one obtains S = − bb + k + b ( b + ) ( b + ) k k ( x + b + a t ) , C = − b + a · The ODE for χ − is a Lam´e equation in its Riccati form, whose solution is singlevalued for b = − + / n , n ∈ Z , multivalued otherwise. For the present diffusion problem, this new solution leaves b , c , d unconstrained and can indeed be used to test the validity of numerical schemes.5.3.4. Case b = −
2A computation similar to the above one yields F = , F = A cosh kx , S = − c (cid:20) −
32 tanh kx (cid:21) , C = − A + dd cosh kx , k = − c , (5.32)then χ − = ∂ x log (cid:20) cosh ( kx ) − / (cid:18) − Ω Ak cosh ( Ω t ) + sinh ( Ω t ) cosh ( kx ) (cid:19)(cid:21) , Ω = ( A + d ) c , (5.33)i.e. a solution identical to (2.8) already found in section 2.5.3.5. Solutions, case b = − / b = − / , − / , − / , − / b = − / onte / Nonlinear diffusion equation For this value b = − /
3, the ODE E = F = g + ( t ) [ cosh K ( t ) cosh ( kx − g ( t )) + sinh K ( t ) sinh ( kx − g ( t )) − ] √ ( cosh ( kx − g ( t )) − cosh K ( t )) √ − + g − ( t ) [ cosh K ( t ) cosh ( kx − g ( t )) + sinh K ( t ) sinh ( kx − g ( t )) − ] −√ ( cosh ( kx − g ( t )) − cosh K ( t )) −√ − + [ K ′ sinh K ( t ) sinh ( kx − g ( t )) + g ′ cosh K ( t ) cosh ( kx − g ( t )) − g ′ ] k sinh K ( t ) , (5.34)in which g ± ( t ) are two other arbitrary functions of t .As already argued in section 5.3.1, since the relation between F and e kx − g ( t ) is necessarilyalgebraic, the two functions g + and g − must vanish. Equation E = g ′′ = , K ′′ = , g ′ K ′ = , g ′ + K ′ = cd , (5.35)solved as g ( t ) = ω t , K ( t ) = Ω t + k , ω Ω = , ω + Ω = cd , (5.36)in which ω , Ω , k are constant.The system (5.17) therefore has for solution, in the first case ω = , Ω = F = − ∂ x log [ cosh ( kx − ω t ) − cosh k ] , k = − c , ω = cd , F = ω k sinh k [ cosh k cosh ( kx − ω t ) − ] , S = − k (cid:20) − k ( cosh k cosh ( kx − ω t ) − ) (cid:21) , C = − ω k ( cosh k cosh ( kx − ω t ) − ) − sinh k sinh k ( cosh k cosh ( kx − ω t ) − ) , (5.37)and in the second case ω = , Ω = F = − ∂ x log [ cosh kx − cosh Ω t ] , F = Ω sinh kx k sinh Ω t , k = − c , Ω = cd , S = − k (cid:18) + kx (cid:19) , C = Ω k (cid:20) sinh Ω t sinh kx − kx sinh Ω t (cid:21) . (5.38)The integration of the Riccati system (5.4)–(5.4) introduces another arbitrary constant t , thenthe solutions w are defined by (5.3) and (5.16). One thus obtains two solutions w , respectively χ − = k a sinh ξ ( cosh k sinh ξ + cosh k − ) + a ( cosh k sinh ξ − ξ + cosh k )[ a ( sinh ξ − cosh k ) + a sinh ξ ][ cosh k sinh ξ − ] , a a = ω tanh ( ω ( t − t )) ω , ξ = kx − ω t , k = − c , ω = cd , ω = (cid:18) − k (cid:19) cd , w = ω ( cosh k − cosh ξ ) k sinh k h ω − ω ( ω ( t − t ))( cosh k − cosh ξ ) i , (5.39) onte / Nonlinear diffusion equation and χ − = k a ( cosh kx − ) − a cosh kx [ a cosh kx + a ] sinh kx , k = − c , a a = ( Ω t ) − Ω ( t − t ) Ω t − ( Ω t ) − Ω ( t − t ) cosh Ω t ) , Ω = cd , w = Ω k [ cosh Ω t − cosh kx ][ a cosh kx + a ] sinh Ω t [ a cosh Ω t + a ] . (5.40)Each of these two new solutions is outside the class (2.4) and depends on a single arbitrary constant( k in the first one, t in the second one).5.3.6. Case b = − / F , the function F , algebraic in the derivatives of F , is necessarily an affinefunction of ζ , ℘ ′ and x − g [1, § F = R + R ℘ ′ + R ζ + ( x − g ( t )) R , (5.41)whose coefficients R j are rational in ℘ ( x − g ( t )) . One then proves that, since d is nonzero, thediscriminant g − g must vanish, thus reducing F and F to simply periodic functions, F = − ∂ x ψ ( x , t ) , ψ = ( k tanh ( k ( x − g ( t )))) − ( / ) k − e , k = ε e , ε = , F = − g ′ (cid:2) + ( l − / ) e ψ (cid:3) , ( ε − ) l = . (5.42)Equation E = ε = , g ′ = d , (5.43)thus restricting this solution to the particular case b = − / Case b = − / F of E =
0, namely, in the elliptic subcase g − g = y = ℘ ( x − g ( t )) − e , F = y + g + e y (cid:20) g ′ − g ′ ∆ (cid:0) ( x − g ) g − e ζ (cid:1)(cid:21) − g ′ ∆ ( e y − ( g + e )( y − e )) ℘ ′ , ∆ = ( e − g ) , (5.44)and in the trigonometric subcase g − g = F = ∂ x ψ ( x , t ) , ψ = ( k tanh ( k ( x − g ( t )))) − ( / ) k − e , k = ± e , F = g ′ (cid:20) + ( − ε ) e ψ (cid:21) . (5.45)Equation E = g ′ = ε = − g ′ = d / b = − / onte / Nonlinear diffusion equation Case b = − /
6. Conclusion
By a systematic investigation, we have obtained several new solutions of this diffusion problem.Two of them (section 5.3.3, Eq. (2.7)) match all the physical constraints b > , c > , d > b , couldbe useful for other diffusion problems governed by (1.3) with b < w which take into accountone of the two movable poles. Taking account of both poles via the two-singular manifold method(see [6, § Remark . As suggested by the two new solutions Eqs. (5.39) and (5.40), the class w equal to asecond degree polynomial in cosh ( kx ) with time-dependent coefficients could also generate physi-cally interesting solutions. Acknowledgments
The author gratefully acknowledges the support of LRC M´eso and is happy to thank B.-J. Gr´ea,A. Llor and R. Motte for suggesting this interesting and challenging problem.
References [1] M. Abramowitz, I. Stegun,
Handbook of mathematical functions , Tenth printing (Dover, New York,1972).[2] D.G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math. LXIV (1964) 229–364.[4] R. Conte, Invariant Painlev´e analysis of partial differential equations, Phys. Lett. A (1989) 383–390.https://doi.org/10.1016/0375-9601(89)90072-8.[5] R. Conte, Exact solutions of nonlinear partial differential equations by singularity analysis,
Directand inverse methods in nonlinear evolution equations , 1–83, ed. A. Greco, Lecture notes in physics (Springer Verlag, Berlin, 2003). http://arXiv.org/abs/nlin.SI/0009024 CIME school, Cetraro, 5–12September 1999.[6] R. Conte and M. Musette,
The Painlev´e handbook (1994) 313–321.[8] B. Gambier, Sur les ´equations diff´erentielles du second ordre et du premier degr´e dont l’int´egraleg´en´erale est `a points critiques fixes, Acta Math. (1910) 1–55.[9] R.L. Lemmer and P.G.L. Leach, The Painlev´e test, hidden symmetries and the equation y ′′ + yy ′ + ky =
0, J. Phys. A (1993) 5017–5024.[10] Pierre-Henri Maire, ´Etude d’une ´equation de diffusion non-lin´eaire. Application `a la discr´etisation del’´equation d’´energie cin´etique turbulente pour un mod`ele de turbulence `a une ´equation, 81 pages, Rap-port CEA D01 03661 (2001).[11] P. Painlev´e, M´emoire sur les ´equations diff´erentielles dont l’int´egrale g´en´erale est uniforme,Bull. Soc. Math. France (1900) 201–261. onte / Nonlinear diffusion equation [12] Y. B. Zel’dovich and Y.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenom-ena, Vol. 2 (Academic press, New York, 1967).[12] Y. B. Zel’dovich and Y.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenom-ena, Vol. 2 (Academic press, New York, 1967).