aa r X i v : . [ m a t h . G M ] M a r On the Basics of the Nonlinear Diffusion
Henrik Stenlund ∗ March 12th, 2020
Abstract
This study handles spatial three-dimensional solution of the nonlineardiffusion equation without particular initial conditions. The functionalbehavior of the equation and the concentration have been studied in newways. An auxiliary function for diffusion is given having an interestingrelationship with the concentration. A set of new integro-differential equa-tions is given for diffusion. diffusion, nonlinear diffusion, concentration dependence Mathematics Subject Classification: 34A34,34B15
Diffusion is a basic natural phenomenon occurring everywhere with practicallyall chemical compositions thinkable, in gaseous, liquid and solid form. It hap-pens as a self-diffusion and in mixtures. The diffusion equation is well knownand has a wealth of solutions with special initial conditions, especially in one-dimensional models [5]. Diffusion is closely connected to thermal conductionproblems as the equations are the same. Usually one is talking of initial valueproblems, not boundary value problems, as is done in this paper too. However,no specific initial value problems are treated here. In three dimensions the diffu-sion equation becomes awkward to solve and often numerical work is required. Ifa non-linearity is added things become really ugly. There are many cases wherean analytical solution in closed form would be welcome for further analysis. Inthis paper it is attempted to give new formulas for tackling nonlinear diffusionin three dimensions. Also other observations are made of the diffusion equation.An introductory discussion is in Shewmon [4]. The available work made by ∗ The author is grateful to Visilab Signal Technologies for supporting this work. Visilab Report
The analysis is started by the three-dimensional non-linear diffusion equationwith a dependence on concentration. D = D ( c ) (1)Here c is the concentration and D is the diffusion coefficient and its functionalform is supposed to be known. The diffusion equation in three dimensions willbe ∂c∂t = ∇ · ( D ( c ) ∇ c ) (2)where t, ¯ r are the time and spatial coordinate vector. No sources nor sinks arepresent as they will significantly complicate the analysis. One attempts to offera solution to the initial value problem presented in the following.The analysisgoes along the line of first transforming this equation to a nonlinear integralequation. Then the solution is expressed in terms of c (¯ r, t = 0) and its spatialderivatives ∂ n c (¯ r, t ) ∂t n (3)which are assumed to be known. The time derivatives are formally needed butas will become clear in the following, they are actually not required, but solvedinstead. At t = 0 the concentration may have either a piecewise continuousbehavior or it may even be discontinuous. However, at times t >
0, it behavescontinuously and so does its derivatives. By introducing the simple transforma-tion F ( c ) = Z cc D ( s ) ds (4)with c a constant, one will have ∇ · ( D ( c ) ∇ c ) = ∇ F ( c ) (5) F ( c ) is a scalar function of concentration and is implicitly four dimensional.Thus Eq.(2) becomes ∂c∂t = ∇ F ( c ) (6)One can continue from (6) integrating it once in terms of time c (¯ r, t ) = c (¯ r, t = 0) + ∇ Z t F ( c (¯ r, t ′ ) dt ′ (7)2his is likely the simplest integro-differential equation derivable for (2). AMacLaurin power series in t can be used for solving this equation. F ( c ) = X n =0 t n n ! ( ∂ n F ( c (¯ r, t )) ∂t n ) t =0 (8)The integral can be transformed to c (¯ r, t ) = c (¯ r, t = 0) + X n =0 t n +1 ( n + 1)! ∇ ( ∂ n F ( c (¯ r, t )) ∂t n ) t =0 (9)It is important to note that this is an initial value problem. To solve generaltime-varying problems other methods are needed. The functions( ∂ n F ( c (¯ r, t )) ∂t n ) t =0 (10)can be solved from the initial differential equation (6) at t = 0. The firstderivative becomes ∂F ( c (¯ r, t )) ∂t = F ′ ( c ) ∂c∂t = F ′ ( c ) ∇ F ( c ) (11)The second derivative becomes ∂ F ( c (¯ r, t )) ∂t = F ′′ ( c )( ∇ F ( c )) + F ′ ( c ) ∇ ( F ′ ( c ) ∇ F ( c )) (12)Continuing in the same way for higher derivatives one can replace ∂c∂t (13)by (2). All derivatives are taken at t = 0 and since one knows c (¯ r, t = 0) andall derivatives with ∇ of it and F ( c ), one knows all the terms( ∂ n F ( c (¯ r, t )) ∂t n ) t =0 (14)Thus (9) is the general solution to (2) and (7). This result serves both as astarting point for further analysis and for numerical work in practical initialvalue problems. The ease of solution is dependent on the complexity of thefunction F ( c ). In complicated cases one may approximate by breaking the series at index N . Toestimate the resulting error, one can use Lagrange’s expression for the remainderterm of the series above broken at N . R N = t N +1 ( N + 1)! ∇ ( ∂ N +1 F ( c (¯ r, t )) ∂t N +1 ) t =0 (15)3 Method for Solving the Poisson-type Differ-ential Equation
The following property of the three-dimensional Dirac delta function is wellknown ∇ ( 1 | ¯ r − ¯ r | ) = − πδ | ¯ r − ¯ r | (16)Solving the Poisson equation ∇ V (¯ r ) = − k (¯ r ) (17)as V (¯ r ) = 14 π Z k ( ¯ r ′ ) | ¯ r − ¯ r ′ | d ¯ r ′ (18)can be done with the aid of the Dirac delta function. This can be verifiedby applying the Laplacian to it. One might argue that there is an additionalfunction φ (¯ r ) involved as follows. V (¯ r ) = 14 π Z k ( ¯ r ′ ) | ¯ r − ¯ r ′ | d ¯ r ′ + φ (¯ r ) (19)with the property ∇ φ (¯ r ) = 0 (20)However, it needs to comply with the original equation simultaneously ∇ φ (¯ r ) = − k (¯ r ) (21)Since k () is arbitrary the only possibility left is φ (¯ r ) = 0 (22)As an example of application one can transform equation (6) to an integral F ( c ) = − π Z ∂c (¯ r ,t ) ∂t | ¯ r − ¯ r | d ¯ r (23)This method is used in many instances here. The result from (2) and opening the diffusion equation to ∂c∂t = D ′ ( c )( ∇ c ) + D ( c ) ∇ c (24)and rearranging it to get ∂c∂t D ( c ) − D ′ ( c ) D ( c ) ( ∇ c ) = ∇ c (25)4y using the method above one will get c (¯ r, t ) = − π Z d ¯ r ′ | ¯ r − ¯ r ′ | (cid:2) D ( c ( ¯ r ′ , t )) ∂c ( ¯ r ′ , t ) ∂t − ∇ ′ ( ln ( D ( c ( ¯ r ′ , t ))) · ∇ ′ c ( ¯ r ′ , t ) (cid:3) (26)This is the general integro-differential equation for nonlinear diffusion. ∇ ′ isaffecting on the ¯ r ′ variable only. In the following the diffusion coefficient D is dependent on c . By differentiationone has ∂F ( c ) ∂t = D ( c ) ∂c∂t (27)and by substituting the diffusion equation to it the result will be ∂F ( c ) ∂t = D ( c ) ∇ F ( c ) (28)This is a nonlinear diffusion equation for the F ( c ). It is actually an amaz-ing equation since it is a differential equation for the integral of the diffusioncoefficient in terms of the concentration variable. The F ( c ) can be expressed as F ( c ) = − π Z d ¯ r ′ D ( c ( ¯ r ′ , t )) | ¯ r − ¯ r ′ | ∂F ( c ( ¯ r ′ , t )) ∂t (29)This is an integro-differential equation for the F ( c ). One can make an assump-tion for the existence of an auxiliary function φφ = ∇ F ( c ) (30)and one can see from the above that φ = 1 D ( c ) ∂F ( c (¯ r, t )) ∂t (31) F = − π Z d ¯ r ′ φ ( ¯ r ′ , t ) | ¯ r − ¯ r ′ | (32)Substitution to equation (27) will give D ( c ) φ = − π ∂∂t Z d ¯ r ′ φ ( ¯ r ′ , t ) | ¯ r − ¯ r ′ | (33)and application of a Laplacian to this produces finally ∂φ∂t = ∇ ( D ( c ) φ ) (34)This is a non-linear diffusion equation for the auxiliary function.5 Differential Equations for the Diffusion Coef-ficient
It is tempting to see if the same procedures can be applied as in the precedingsection to the diffusion coefficient itself. ∂D ( c ) ∂t = D ′ ( c ) ∂c (¯ r, t ) ∂t (35) ∇ D ( c ) = D ′ ( c ) ∇ c (36)One can solve for the gradient ∇ c = ∇ D ( c ) D ′ ( c ) (37)and use the original diffusion equation to get ∂D ( c ) ∂t = D ′ ( c ) ∇ · ( D ( c ) ∇ D ( c ) D ′ ( c ) ) (38)One has obtained a differential equation for the diffusion coefficient itself, aslong as the diffusion coefficient is non-linear. It is equivalent to the non-lineardiffusion equation as is very easy to see by starting to execute the differentialoperators. One is able to extend this thinking without a proof further to anyderivative of the coefficient to ∂D ( N ) ( c ) ∂t = D ( N +1) ( c ) ∇ · ( D ( c ) ∇ D ( N ) ( c ) D ( N +1) ( c ) ) (39) The author has not seen the general three-dimensional solution (9) given here inexisting publications. Therefore, this expression is believed to be new as mostof the other expressions presented. It will serve as a starting point both foranalytical investigations and numerical work with various initial conditions.The last part shows results for F ( c ), the diffusion coefficient integrated interms of the concentration having its own diffusion equation and an auxiliaryfunction. The diffusion coefficient itself seems also to have its own differentialequation. References [1]
Boltzmann, Ludwig : Zur Integration der Diffusionsgleichung bei vari-abeln Diffusionscoefficienten , Annalen der Physik,
53, 960 (1894) [2]
Matano, Chujiro : On the Relation between the Diffusion-Coefficientsand Concentrations of Solid Metals (The Nickel-Copper System) , JapaneseJournal of Physics,
8, 109 (1933)
Crank, John : Mathematics of Diffusion , Clarendon Press (1979) [4]
Shewmon, Paul G. : Diffusion in Solids , McGraw-Hill New York (1963) [5]
Churchill, Ruel : Operational Mathematics , McGraw-Hill Kogakusha,3rd edition (1972) , Tokyo[6]