The non-linear Dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization

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1 Available online at Math. Nat. Sci., 27, 7 Research Article Journal Homepage: The non-linear Dodson diffusion equation: Approimate solutions and beyond with formalistic fractionalization Jordan Hristov Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia 756, 8 Kliment Ohridsky, blvd. Bulgaria. Communicated by O. K. Matthew Abstract The Dodson mass diffusion equation with eponentially diffusivity is analyzed through approimate integral solutions. Integral-balance solutions were developed to integer-order versions as well as to formally fractionalized models. The formal fractionalization considers replacement of the time derivative with a fractional version with either singular Riemann-Liouville or Caputo or non-singular fading memory. The solutions developed allow seeing a new side of the Dodson equation and to separate the formal fractional model with Caputo-Fabrizio time derivative with an integral-balance allowing relating the fractional order to the physical relaation time as adequate to the phenomena behind. c 27 All rights reserved. Keywords: Non-linear diffusion, singular fading memory, non-singular fading memory, formal fractionalization, Caputo-Fabrizio derivative, integral balance approach. 2 MSC: 47H, 54H25.. Introduction.. Dodson diffusion equation The Dodson diffusion equation. is related to diffusion in minerals [, 2] related to cooling history in geology C, t = D ep t 2 C, t t 2, = /τ,. where the diffusion in solids is thermally promoted process [5, 4]. The same equation is relevant to other solid diffusion process in modern materials [3, 38]. Moreover, this is a common approach to model diffusion in solids [3, 38]. In solid diffusion, the dependence of the diffusion coefficient on the absolute temperature T can be simply epressed by the Arrhenius equation.2 DT = D ep E..2 RT address: jordan.hristov@mail.bg Jordan Hristov doi:.22436/mns... Received

2 J. Hristov, Math. Nat. Sci., 27, 7 2 The etreme value D of the diffusion coefficient is related to processes at very high temperatures where the eponent in Arrhenius law approached unity; R is the universal gas constant and E is the diffusion activation energy. According to original work of Dodson [] over a certain temperature range, the inverse temperature /T [, 2, 5, 33, 35, 4] varies linearly in time. With this empirically derived relationship the Arrhenius can be epressed as D = D ep E t = D e t d E τ, = RT τ dt RT τ, = τ, with initial conditions D = Dt =, T = Tt =. With this construction of the diffusion coefficient hence, the diffusion equation can be epressed as [, 2, 4, 5, 33, 35, 4] as.. Introducing the diffusional time of the systems as t D = a 2 /D and the ratio τd /a 2 = τ/t D the nondimensalization of. results in [] c θ = τd a 2 e θ 2 c c,.3 2 with c = C/C, θ = t/τ and = /a. In the original work [] C = C. is the initial concentration at t =. In the Dirichlet problem considered further at this article we will accept zero initial condition C = C. =. In.3 the parameter a is the length scale of the finite area at issue. It was demonstrated in [27] that the ratio τ/t D is the Deborah number of the diffusion process defined as De = τd /a 2. Hence, the alternative form of.3 c = De θ However, the eponential term remains and the equation is still non-linear. Further, with the nonlinear transformations [] q = e θ c, u = e θ Equation.4 can be transformed in a homogeneous form as [] t 2 c..4 Dz = M e θ, q u = 2 q 2, q s = M u λτ,.5 with initial condition q = at θ =. Moreover, for θ = we have u = and for θ we have u M. The parameter λ depends on geometry of the system at issue. For the sake of clarity of the above equations in the original work of Dodson [] the symbol M = τd /a 2 was used instead the Deborah number De introduced in [27]. The solution of.5 developed by Dodson [] is based on the results of Carslaw and Jaeger [9] especially the eample at page 4, Equation 3 where for -D rectangular system it is q = 2 i= i+ cosi /2π i /2π Γλτ + λτ i /2 2 π 2..6 M Certainly, the solution.6 is useful for computer calculations but quite inconvenient for engineering applications due to large number of terms of the series required. Moreover, the post-solution analyzes of the diffusion phenomena modelled by it are practically impossible since the terms of.6 have no physical meaning.

3 J. Hristov, Math. Nat. Sci., 27, 7 3 The Dodson equation is attractive for solutions due to its eponential non-linearity vanishing in time. In the book of Crank [] which is a basic reference source for many scientists, however, this problem is mentioned briefly as Eample 7. at page 4 Chapter 7 without details beyond citing the results.5 and.6. To complete this section it is noteworthy to mention that the Dodson equation. is a special integerorder form a time-fractional diffusion equation [27] epressed in terms of Caputo-Fabrizio fractional derivatives, when the product t obeys the condition t <<..2. Problem formulation The present study focuses on approimate solutions and relevant analyzes of a generalized form of the Dodson equation µ C, t t µ = D e t 2 C, t 2, = /τ..7 In the general form of epression.7 the time derivative µ C, t/ t µ could be a left-sided timefractional derivative of Riemann-Liouville, Caputo type with singular kernels [39] or Caputo-Fabrizio derivative with eponential non-singular kernel [7, 8] as well as the integer time-derivative when µ =. Before any further analyzes, we have to stress the attention that the form.7 is outcome of the socalled formalistic fractionalization where simply the time derivatives of well-known integer-order models are mechanistically replaced by time-fractional derivative, a common approach in the eisting literature [3, 6, 7, 3, 32, 4]. 2. Mathematical preliminaries 2.. Time-fractional integral and derivatives In accordance with the Riemann-Liouville approach the fractional integral of order µ > is a natural result of the Cauchy formula reducing calculations of the m-fold primitive of a function ft resulting in a single integral of convolution type [2] for arbitrary positive number µ >, namely I µ ft = Γµ t t z µ fzdz, t >, n R +. For sake of convenience, we will also use the notation D µ ft for I µ ft. Further, the law of eponents for fractional integrals means D µ D γ ft = D µ γ ft = D γ D µ ft. The Laplace transform of the fractional integral is defined by the convolution theorem as L [ [ ] D µ t µ t ft] = L L [ft; s] = s µ Fs, Γµ where Rs >, Rs > and Fs is the Laplace transform of ft Riemann-Liouville time-fractional derivative Therefore, we may define the fractional derivative D µ ft with by the relations [2] as: D µ I µ = I but I µ D µ I. Therefore, D µ is left-inverse, but not right-inverse, to the integral operator I µ. Hence, introducing a positive integer m such that m < µ < m, the natural definition of the Riemann-Liouville left-sided fractional derivative of order µ > is D µ ft = D m I m µ ft = d m Γm µ dt m t fz dz, m < µ < m, m N. t z µ+ m Consequently, it follows that D = I = I, that is, D µ I µ = I for µ.

4 J. Hristov, Math. Nat. Sci., 27, 7 4 In addition, the fractional derivative of power-law function and a constant, frequently used in this section are D µ t Γ + = Γ + + µ t µ, D µ C = C t µ, µ >, >, t >. Γ µ Similarly, the fractional integrals of the power-law function and a constant are D µ t Γ + = Γ + + µ tµ+, D µ C C = Γ + µ tµ, µ, 2,. The Laplace transform of the Riemann-Liouville fractional derivative for m N is [ d m ] m L dt m ft; s m = s m Fs s m k f m + = s m Fs s k D µ k t f+, k= where Rs >, Rs > and m < µ < m Caputo time-fractional derivative The Caputo derivative of a casual function ft, i.e., ft = for t <, is defined as [2] CD µ t ft = m µ dm I t m ft = D m µ t f m = where m < µ < m. The Laplace transform of Caputo derivative is Γm µ k= L [ m CD µ t ft; s] = s µ Fs f m s µ k. k= t f m z dz, t z µ+ m Caputo derivative of a constant is zero, i.e., C D µ t C = that matches the common knowledge we have from the integer order calculus and for this reason the Caputo fractional derivative is the preferred one among engineers [39]. If f = f = f = = f n =, then both Riemann-Liouville and Caputo derivatives coincide. In particular for µ, and f = one has C D µ t ft = RL D µ t ft. Further in this article we will use the notations C D µ t ft and RLD µ t ft to discriminate the effects on the solutions when both derivatives are used. In addition in some situations we will also use the notation µ C, t/ t µ meaning a time-fractional derivative without specification of the type Caputo-Fabrizio time-fractional derivative Caputo and Fabrizio [7] constituted a time-fractional derivative with a non-singular stretched- eponential kernel defined as CFD α t ft = Mα α t [ αt s ep α ] dft dt ds = α t [ ] αt s dft ep ds. 2. α dt In 2. Mα such that M = M = is normalizing function. From the definition 2. it follows that if ft = C = const., then CF D α t = as in the classical Caputo derivative [39]. An alternative definition of is given by [34] CFD α t CFD α α t [ t = α 2 [ft fa] ep α ] t s ds, t >. a α The Laplace transform of CF D α t with a = has the following Laplace transform given with variable [4] L [ CF D α t ft] = sl [ft f] s + α s. 2.2

5 J. Hristov, Math. Nat. Sci., 27, 7 5 Applications of the Caputo-Fabrizio time-fractional derivative form a hot contemporary area of research which is very attractive for the scientists in the last 2 years after the seminal works [7, 8]. The basic idea was followed by numerous articles [2 4, 6, 6, 3, 32, 34, 4], initiated new computational techniques [7, 28], new definition of transient [26] and steady-state diffusion [25] equations, and boosted the creation of new fractional derivatives with non-singular kernels [, 5]. With 2.2 the Caputo-Fabrizio derivative of eponential function ep t is at issue in the present work. Here we take into account that by definition CF D α t ft is a convolution of ft and the eponential kernel as well the factor / α. Hence, the Laplace transform L [ CF D α t ept] for > is L [ CF D α t ept] = α s s + A, A = α α. 2.3 The inverse Laplace transform of 2.3 yields CFD α t ept = ept ep At α For α the second eponential term in the nominator of 2.4 goes to zero and lim α CF D α t ept = ept. For < the Laplace transform of CF D α t ep t is L [ CF D α t ep t] = Then, the inverse Laplace transform of 2.5 is CFD α t ep t = α s + For α the second term in the nominator of 2.6 goes to zero and 2.2. The integral-balance method s + A. 2.5 ep t + ep At. 2.6 A lim α CF D α t ep t= ep t. The method used the concept that the diffusant penetrates the medium at a final depth δ and therefore the speed of the flu, thus this concept corrects the unphysical infinite speed of the flu inherent of the parabolic models. The introduction of the finite penetration depth δ transforms the problem with boundary conditions at infinity C, t = and C, t/ = as a two-point problem with boundary conditions at the front δt Cδ, t Cδ, t =, =. 2.7 The conditions 2.7 define a sharp-front movement δt of the boundary between disturbed and undisturbed medium when an appropriate boundary condition at = is applied. The position of δt is unknown and should be determined trough the solution. When the diffusivity is concentration and time-independent, the integration of, the integration of. for =, over a finite penetration depth δ yields 2.7 Applying the Leibniz rule to left-side of 2.8 we get C, t δ 2 C, t d = t 2 d. 2.8

6 J. Hristov, Math. Nat. Sci., 27, 7 6 d dt C, t C, td = D. 2.9 Equation 2.9 is the principle relationship of the simplest version of the single-integration method [2] known as Heat-balance Integral Method HBIM [8, 9]. After this first step, replacing C, t by an assumed profile C a epressed as a function of the relative space co-ordinate /δ the integration in 2.9 results in an ordinary differential equation about δ [8, 9, 2, 36]. The principle problem emerging in application of 2.9 is that its right-side depends on the type of the assumed profile.the principle problem emerging in application of 2.9 is that its right-side depends on the type of the assumed profile [8, 9, 2, 36, 37]. The principle drawback of HBIM can be avoided by application of the double integration approach DIM [3, 22, 23, 36, 37]. In accordance with the technique of DIM the first is integration of the diffusion equations is from to and then the resulting equation is integrated again from to δ. for the sake of simplicity, in eplanation of the method, we assume that the diffusion coefficient is constant D, the principle relationship of DIM is [24, 36, 37] d dt C, td = D C, t. An alternative epression of the DIM principle relationship applicable to fractional in time diffusion equations can be easily derived, too [22, 23], namely δ δ C, t d d = D C, t, 2. t d dt δ C, td d = D C, t. 2. When time-dependent term is epressed though a time-fractional derivative of order µ, then we have [22, 23] δ δ µ C, t t µ d d = D C, t. 2.2 Since the Leibniz rule is not applicable yet to left-hand side of the integral relations. The integral relations presented by 2. and 2.2 will be used further in this work in the development of the problems at issue. 3. Integer-order Dodson diffusion equation 3.. Approimate solutions 3... HBIM solution If the concept of finite penetration depth of the diffusant δt, which evolves in time, is assumed then the infinite speed of the flu propagation, which is inherent for the parabolic model., could be ad hoc avoided. The finite penetration depth is the basic concept of the integral balance method of Goodman [3, 8, 9, 2, 22] which is a simple mass balance over the depth of the diffusion layer with a thickness δ, namely C, t d = t Dt 2 C, t 2 d d dt C, td = Dt C, t, 3.

7 J. Hristov, Math. Nat. Sci., 27, 7 7 Dt = D e t, = /τ, with conditions 2.7 at the front δt. The second version of 3. is a result of application of the Leibniz rule for integration under the integral sign as was demonstrated by 2.9. The thickness of the diffusion layer δt can be determined as a step of the solution by using assumed profile epressed as a function of the dimensionless space variable /δ. With the transform u = e t leading to C/ t = C/ u u/ te t we may epress. as C, u u Then the integral-balance equation 2.9 of HBIM is d dt = D C, utd = D 2 C, u C, ut. Suggesting the assumed profile as a parabolic profile with unspecified eponent C a, ut = C s /δ n [3, 2, 24, 26, 36, 37] and for the sake of simplify considering the Dirichlet problem with C s =. Then, the replacement of C, t in 3.2 by C a leads to [27] dδ n + du = D n δ δ2 = D u[2nn + ]. In terms of the original variables the penetration depth δt is D δ = e t [2nn + ]. 3.3 The ratio D / has dimension of [m 2 ], while the non-linear time function e t, growing in time, saturates rapidly to as /e t for large times, i.e., t. In this etreme case δ D / = const., that is the penetration depth stops to evolve in time and the diffusion process ceases DIM solution Applying the integral relationship 2. to the transformed equation 3.2 we get d δ δ C, u d = D C, u. 3.4 dt With assumed profile C a, ut = C s /δ n and C, t = C, u = C s = the integral relation 3.4 results in d n + n + 2 dt δ2 = D δ = D e t n + n + 2. The ratio DIM = δ/ D / is dimensionless and actually defines the numerical factor depending on the eponent n and, the more important issue, the relaation function e t Approimate integral-balance solutions From the HBIM and DIM solutions we get two approimate solutions C, t HBIM = D e t [2nn + ] n, 3.5

8 J. Hristov, Math. Nat. Sci., 27, 7 8 C, t DIM = D e t [n + n + 2] n. 3.6 For large times, i.e., t, when /e t the steady-state profiles defined by the integral solutions are n C, HBIM = D, 3.7 2nn + 2 n C, DIM = D. 3.8 n + n + 2 The approimate profiles 3.5 and 3.6 define a non-conventional similarity variable defined as η u = D u = D e t. 3.9 In the space, u 3.5 and 3.6 have known optimal eponents [36, 37], namely: for HBIM solution n opthbim and n optdim 2.28 for DIM solution. This allows us to complete the integralbalance solutions 3.2. Eact similarity solution With the nonlinear change of variables u = ep t the original equation. can be epressed in the, u space as 3.2. Using the well-known technology of the similarity solution through the Boltzmann variable we define inspired by the integral-balance solutions and the homogenous form of 3.2 in the, u space, a similarity variable η u = D u, D = D. 3. Even though the approimate solutions are at the focus of this study, we stress the attention on a possibility to develop a solution with a technique well-known from the tetbooks. However, now we will use the unconventional similarity variable η u defined by 3.. Looking for a solution in the form C, t = u k gη u, where eponent k and the function gη u have to be determined through the solution. With the new similarity variable transforming 3.2 we have Hence, the equation that should be solved is u kg k 2 η dg u du d2 g du 2 =. dg d 2 u η dg u = kg. 3. du Now, let us consider the special case with k =. From the definition C, t = u k gη u, we immediately get the Boltzmann similarity variable. For k =, equation 3. reduces to dg d 2 u η dg u du = dg du = B ep ηu 4 η2 u gη u = B ep 4 η2 u dη u, where B should be defined through the solution. Therefore, we get the well-known solution epressed through the Gauss error-function erfη u /2.

9 J. Hristov, Math. Nat. Sci., 27, 7 9 However, this solution would face problems in calculations since the backward transform to the original variables in the, t space is non-linear and hard to be handled. Because of that, denoting Q = g [ epη 2 u/4 ] we may transform 3. as d du 2 Q 2 η d u du Q = k + Q, 2 The trivial solution for k = /2 is Q = b = const. From this point of view we have [ gη u = b ep ] 4 η2 u. Therefore, in the, u space the solution has a simple form as C, u = b ep 2. u 4D u Further, in terms of the original variables, t we have 2 C, t = b ep ep t 4D [ ep t ]. With the boundary condition C, t = C s we may denote b = C s ep t. Therefore the solution assuming C s = for the Dirichlet problem is C, t = ep 2 4D [ ep t] At large time when t from 3.2 we get a steady-state profile. 3.2 C, = ep D The steady-state solution 3.3 automatically defines the similarity variable / D, which we will see as naturally appearing dimensionless group in the approimate solutions when the Caputo-Fabrizio time derivative is involved in the fractional version of.. From 3.3 it is clear that it is impossible to determine the depth the diffusion layer the boundary between the disturbed C, t and the undisturbed area C, t = of the medium. If the result 3.3 is used, we may define, for convenience, and mainly from practical point of view, that the front is defined by point where C, = C final = ɛc = ɛ, where ɛ <<. Then final = D / 4 ln ɛ. This result differs from 3.7 and 3.8 where the parabolic profile defines finalhbim = D / 2nn + and finaldim = D / n + n + 2. As final comments, it is noteworthy that we used the classical diffusion equation in the space, u. However, the initial governing equation. in the, t space is non-linear. The change of variables t u which is linearizing the governing equation is a result of the non-linear transform u = t e z dz = e t see [, Chapter 7]. 4. Formal fractional models 4.. Integral-balance solution by DIM From now we will continue with DIM technology in the solution of the formal fractional versions of the Dodson equation, presented by.7. Applying the integral relation 2.2 we have in the left side the double integral of the fractional derivative of the assumed profile C a = /δ n, namely

10 J. Hristov, Math. Nat. Sci., 27, 7 µ t µ C a, td d = J 2 = µ t µ n d d. 4. δ As it was demonstrated in previous works [22, 23] the result of the integration in 4., irrespective of the type of fractional derivative used Riemann-Liouville or Caputo is [ J 2 = µ t µ δ 2 n + n + 2 ], 4.2 because we have a physically defined condition δt = =, that there is no diffusion at t =. However, for the fractional derivatives in 4.2 this means that we have zero initial condition that makes µ t δ 2 t = µ RL D µ t δ2 = C D µ t δ2 [39]. Therefore, the fractional ordinary equation with respect to δt relevant to the problem at issue can be epressed as µ [ δ 2 ] t µ = D e t C, t. n + n + 2 With C, t = corresponding to the assumed profile C a = /δ n we get µ t µ δ2 = D e t N, N = n + n For the special case with = we get [22, 23] from 4.3 and δ D t µ. However, now, we have to solve 4.3 with. δ 2 = D N Γ + µ tµ, Fractional models with Riemann-Liouville and Caputo derivatives With Riemann-Liouville derivative applying the equation 4.3 simply yields Equation 4.5 can be re-arranged as RLD µ t δ2 = D e t N. 4.5 d dt δ2 = RL D µ t D e t N. 4.6 The eponential function e t can be epressed as an infinite series through the Mittag-Leffler function e t = E, t = i= t i Γi + = i t i Γi +. i= Hence, we may present e t as Now, with 4.7 we have e t = i= i i t i Γi RLD µ t e t = RL D µ t With the simple rule RL D µ t t i = Γ i+ Γ i+µ ti+µ we may epress 4.8 as i= i i t i Γi

11 J. Hristov, Math. Nat. Sci., 27, 7 RLD µ t e t = Now, from 4.6 and 4.8 we have Then, the integration in 4.9 yields Therefore, i= i Γi + µ i t i+µ = t µ d dt δ2 = D N δ 2 = D N RLδ µ = D N Etracting the i = term in 4. we have i= i= i= i Γi + µ i t i. i Γi + µ i t i+µ. 4.9 i ti+µ i Γi + µ i + µ δ2. i= i 2 Γ + i + µ i t i+µ. 4. RLδ µ = D t µ N Γ + µ i 2 Γ + i + µ ti. 4. Neglecting the sum in 4. we get δ µ D t µ N/Γ + µ. The reason for this approimation is the possibility to epress e t as a series e t t + t 2 /2 t 3 /6 + Ot 4 and neglecting all terms containing, or actually accepting =, which means D e t D corresponding to the case with constant diffusion coefficient [2, 24]. Therefore, the result 4. reduces to 4.4 for =. For we may epress approimately the penetration depth as Epressing δ µ = D t µ N Γ + µ t Γ2 + µ + 2 t 2 Γ3 + µ 3 t 3 Γ4 + µ + O4 t 4. RLδ µ = D tµ we try to etract the pre-factor have δ µ = i= N Γ + µ 2 t Γ2 + µ + 2 t 2 Γ3 + µ 4 t 3 Γ4 + µ + O5 t 4, 4.2 D tµ as in the integer order solution. Then, from 4. and 4.2 we D i 2 Γ + i + µ i+ t i+µ n + n i= The ratio µ = δ µ / D / is the scaled penetration depth. It is dimensionless and actually defines the numerical factor depending on the eponent n, and the more important issue, the relaation function i= i Γ +i+µ i+ t i+µ 2 which for µ approaches e t. The result 4.3, but more clearly 4., shows how δ µ t evolves in time and allows to see physically how the additional relaation effect introduced by the fractional derivative µ C, t/ t µ, formally replacing C, t/ t in the diffusion equation, alters the solution.

12 J. Hristov, Math. Nat. Sci., 27, Fractional models with Caputo derivative Representing the decaying eponential term of the diffusion coefficient as see 4.7 It is easy to see that e t = i= i i t i Γi + = + i= i i t i Γi +. CD µ t e t = C D µ t [ + i= ] i Γi + µ i t i+µ = i= i Γi + µ i t i+µ = t µ i= i Γi + µ i t i. Then, the solution is practically the same as that performed with the Riemann-Liouville derivative. The result about the penetration depth is Cδ µ = 4.2. Fractional models with Caputo-Fabrizio derivative D i 2 Γ + i + µ i+ t i+µ n + n i= Now, with the Caputo-Fabrizio derivative the equation about δt is we omit the prefi and use as symbol of the fractional order to distinguish the solution from the cases with singular fractional derivatives D α t δ 2 = D Ne t. 4.5 Here we will apply a different strategy for the solution. The Laplace transform of 4.5, taking into account that δ =, is sδ 2 s s + α s = D N δ 2 s = s + The inverse Laplace transform of 4.6 is α + α. 4.6 s + s s + δ 2 = D N [ α e t + αe t]. 4.7 Now, epressing 4.7 in a form similar to the solution of the integer-order problem, we get D CFδ α = N e t α + α e t. 4.8 For α =, 4.8 reduces to the integer order solution see 3.2 D CFδ α = N e t n + n + 2. The scaled penetration depth α = δ α / D / actually defines the same relation function e t as in the case of α =. From this point of view, more realistic formal fractionalization could be obtained by replacement of the time derivative in the Dodson equation by the Caputo-Fabrizio time derivative. The ratio δ α /δ α= is a measure of the deviation of the penetration depth from the case with α =. Hence we have CFδ α = α α. 4.9 δ α e t

13 J. Hristov, Math. Nat. Sci., 27, 7 3 For α = we get immediately δ α /δ α= =. Since from the analysis done in [27] the Dodson equation corresponds to the case when t <<, the denominator in the second term, i.e., e t approimates as e t t. Then, 4.9 can be approimated as CFδ α = α α δ α t<< t, 4.2 or alternatively as CFδ α α δ α t<< Now, recall that, from the analysis done in [27] the Dodson equation see also the Introduction corresponds to the case when << then when the product t <<, the time-dependent denominator in the second term of 4.8, i.e., can be approimated as e t t + O 2 t 2 t. Then, for large times 4.2 approimates as CFδ α α, δ α t<< and demonstrates eplicitly what the effect of fractional order α is the same results can be obtained from 4.2 and 4.2 if is neglected with respect α. Hence, physically the decrease in α reduces the penetration depth and the diffusion ceases rapidly in contrast to cases with large values of α. That is, the distance from the surface = at which the diffusant stops to penetrate in depth depends on the fractional order α, in fact decreases with decrease in α and vice versa. Moreover, as it was developed in [27] the rate coefficient is represented as = α/α and then for α =.5, for eample we get =, while for α =.25 we have 3. Therefore, the range of application of the Dodson equation, for < corresponds to the range.5 < α < and more realistically when α, since physically is defined as inverse of the relaation time, i.e., = /τ. Finally, the approimate solution in case of Caputo-Fabrizio time derivative is C α, t DIM = D N e t and for large times the steady-state profile can be approimated as C α, DIM = α + α D N α + When t << from 4.22 as well as from 4.23 neglecting we get C α, DIM = D N α n e t n n, The final concentration profiles C α,, α DIM are presented in Figure for stipulated eponent n = 3 see the comments in the net point. The two-dimensional profiles Figure a clearly demonstrate the retardation effect of the fractional order α. More generalized presentation by the 3-D profile in Figure b shows the evolution of the penetration depth αα in the front plane. It is clear that the profile for α =. goes rapidly to zero while at the same time the same value of the similarity variable η u the profile with α = no retardation effects reaches about.2 time of the surface concentration at =.

14 J. Hristov, Math. Nat. Sci., 27, 7 4 Figure : Steady-state concentration profiles: a Two-dimensional; b Three-dimensional plot. In both cases n = 3. To this end, we have to mention that similar retardation effect through the fractional order µ can be observed when the time-fractional derivative has singular fading memory. To be clear, look at 4. where if the sum is neglected due to the small value of as in the case presented by 4.24, then the penetration depth scales as RL δ µ = D t µ N/Γ + µ which is the result obtained in [23] and [22]. In this case the sum in is not neglected this causes a problem in definition of the similarity variable as well as difficulties in definition how many terms of the sum should be taking in account in the calculations. From this point of view, the use of the Caputo-Fabrizio time-derivative leads to a formal fractional models which is mimicking the epected real-world behavior, that is the retardation effect. A serious advantage of the model with the Caputo-Fabrizio time-derivative is the possibility to relate the fractional order α and the rate constant, which is task without success when the Riemann-Liouville and Caputo derivative with power-law memory are used The eponent of the approimate profile This study demonstrates the principle solution of the formally fractionalized versions of the Dodson equation. The eponent of the profile could be either stipulated or determined by optimization procedure minimizing the residual function, precisely the mean-squared error of approimation over the penetration length δ. In this case, the principle problem emerging in evaluation of the residual function is the timefractional derivative of composite function since the assumed profile is [ /δt] n. In the simple case when = and the derivatives were with singular kernels [23], the problem was resolved by epressing the profile as a power-law series of the similarity variable η u = / D t µ. However, in the case of the Dodson equation we have an additional nonlinearity due to the eponential term in the diffusion coefficient. If the fractional order is µ =, then we obtain the integer order model and the solution is with known optimal eponents if the similarity variable is defined by 3.9 see the comments about 3.9. Moreover, if the Caputo-Fabrizio fractional derivative is taken as time derivative, then the same problem emerge and should be resolved. The development of approimate fractional derivative of the approimate profiles is beyond the scope of the present work but we will mark some restrictions that should be obeyed by the eponent n. If the residual function is presented generally as [ t µ R = t µ n t nn D e δ δ 2 ] n 2. δ Then when at the vicinity of the front, the condition for a positive solution is as in the case of integerorder models [3, 22 24, 36, 37] the general condition is n > 2. Hence, the eponent n could be stipulated

15 J. Hristov, Math. Nat. Sci., 27, 7 5 and the approimate solutions could be straightforwardly developed as it was demonstrated in Figure. In this case the error of approimation is predetermined. The alternative way with optimal eponent is still undeveloped. Despite the incomplete solutions in this direction we may comment and compare the results of the basis of the functional relationships of the front δt and this the problem discussed in the net point Comments on the integral-balance of the formal fractional models The approimate integral-balance solutions either HBI or DIM 3.3 or 3.7 of the integer-order model. provide that front propagates as δt D / e t and when the diffusion process ceases we have δ final D /. The same result follows from the eact solution 3.3 where final D /. The formal fractionalization implements ad-hoc a relaation through the time-fractional derivative. When the time fractional derivative is with a singular kernel either Riemann-Liouville or Caputo we have that δ µ t D t µ and eplicit effect the rate constant = /τ disappears. Moreover, these solutions see 4. and 4.4 involve infinite series that makes them hard to be implemented in practical calculations. A principle difficulty emerging in this type of fractionalization is that the fractional order µ and the rate constant are interrelated, that is, there are no physical reasons to create a functional relationship between them. However, when the Caputo-Fabrizio time derivative is ad-hoc used the front propagate as in the case of the integer-order model, i.e., CF δt D / e t and the retardation factor is approimately proportional to α. Then, the final depth is CF δ D /α. This behaviour is more physically adequate in contrast to the other approaches to fractionalize the model since the retardation effect is obvious. Moreover, as commented earlier and established in [27] we have = α/ α and < α = / +. Then, the solution can be epressed through the fractional order α or through the rate constant. To recapitulate, the formal fractionalization with the Caputo-Fabrizio derivative results in an approimate solution where the formal fractionalization does not change the law of the time evolution of the front, in contrast to the case when time-derivatives with singular kernels are used. Moreover, the solution about the penetration front eplicitly demonstrates the retardation effect epressed through the fractional order. As a support of the model with the Caputo-Fabrizio derivative we may refer to the results in [25, 26] and [27] where the correct fractionalization through definition of an eponential damping function of the diffusion flu does not affect the time derivative of the starting parabolic model.. 5. Conclusions This work developed approimate integral-balance solutions of the non-linear Dodson equation in its integer-order form and formally fractionalized versions. As a good eample, it was demonstrated that an eact similarity solution is also available after an initial non-linear transformation. In fact, these are the first developed solutions where the time evolution of the front of the solution can be eplicitly presented. As intermediate results relevant to solution developed it was presented how the time-fractional Caputo-Fabrizio derivative works with eponential function ep±t. This was missing information in articles published on problems involving Caputo-Fabrizio derivative. The comparative analysis indicates that the use of the Caputo-Fabrizio time derivative allows creating formally fractionalized models with an integral-balance solution which can be easily analyzed. Moreover, this solution directly generates a non-boltzmann similarity variable incorporating the diffusion coefficient and the rate constant inverse of the physically defined relaation time τ. Since the Dodson equation was originally developed to model the final penetration depth of the diffusion and recovering from that the age of the geological formation, the results developed for the steadystate regime when the Caputo-Fabrizio derivative is used are eplicit and physically adequate. Moreover, this solution allows directly relating the fractional order of the fractionalized equation to the physically

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