Analytical and numerical methods for the time-fractional nonlinear diusion

Transcription

1 Analytical and numerical methods for the time-fractional nonlinear diusion Šukasz Pªociniczak Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology BCAM, Bilbao, Spain 1 / 26

2 Introduction This talk is about a 1-D diusion of moisture in building materials (such as siliceous brick). By u(x, t) we denote the moisture concentration at x at time t. We consider the following initial-boundary conditions: u(0, t) = C, u(x, 0) = 0. Self-similarity - a characteristic feature of diusion in our experiment. Moisture concentration u(x, t) can be drawn on a single curve [1]: u(x, t) = U(η), η = x/ t, for U(0) = C i U( ) = 0. 2 / 26

3 Nature is tricky (and thus very interesting)! As it turns out, the diusion not always behaves as we are used to. In a number of experiments (ex. [2-4]) the so-called Boltzmann scaling η = x/t 1/2 is not observed. A more appropriate and accurate is the anomalous diusion scaling (Figure from [2]) u(x, t) = U(η), η = x/t α/2, 0 < α < 2. 3 / 26

4 How to describe this mathematically? The classical diusion equation does not work: for our initial-boundary conditions it possesses the x/ t scaling. In [2,4] the following modication of the constitutive equation has been proposed ( ) 1 u α 1 q = D(u). x Result: very complicated equations (doubly degenerate PME) and average tting accuracy. As it turned out, a more appropriate is to model this phenomenon by an equation with fractional derivative (see [5-7]) α u t α = ( D(u) u ). x x We obtain the sought scaling x/t α/2 with very small tting errors. 4 / 26

5 Model We consider a 1D time-fractional porous medium equation α u t α = ( u m u ), 0 < α < 1, m 1, x x with initial-boundary conditions u(0, t) = 1, u(x, 0) = 0. We seek for a self-similar slution u(x, t) = U(η), where η = x/t α/2. We obtain an ordinary integro-dierential equation [ (U m U ) = (1 α) α 2 η d ] F α U, F α := I 0,1 α, m 1, dη 2 α with U(0) = 1 and U( ) = 0, where the integral operator is of the Erdelyi-Kober type 5 / 26 I a,b c U(η) := 1 Γ(b) 1 0 (1 z) b 1 z a U(ηz 1 c )dz.

6 Physical derivation Time-fractional porous medium equation is not an ad-hoc choice. We assume Water can be trapped in some regions of porous medium for prolonged periods of time non-local conservation law. Consitutive relation Darcy's Law. Waiting times have a power-type weight emergence of Caputo fractional derivative. Zero initial condition equivalence of Riemann-Liouville and Caputo derivatives. The relevant paper with full derivation: Š.Pªociniczak, Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Communications in Nonlinear Science and Numerical Simulation 24 (13) (2015), / 26

7 Existence and uniqueness First, we will show that the equation [ (U m U ) = (1 α) α 2 η d dη ] F α U, m 1, with U(0) = 1 and U( ) = 0 possesses a unique compactly-supported solution. This is a free-boundary problem. The relevant paper: Š.Pªociniczak, M. witaªa, Existence and uniqueness results for a time-fractional nonlinear diusion equation, under review. 7 / 26

8 Existence and uniqueness We assume the compact support, i.e. there exists some η such that U(η) = 0 for η η (physically motivated). The key-idea is to use the transformation (borrowed from the works of Okrasi«ski [9]) U(η) = ( m(η ) 2) 1 m y(z), z = 1 η η, which turns our free-boundary problem into an initial-value one [ m(y m y ) = (1 α) + α 2 (1 z) d ] G α y, 0 < α < 1, 0 z 1, dz which y(0) = 0 (G α is related to F α ). What about the second condition? Theorem The solution y = y(z) of the above problem has to satisfy ( α ) 2 α Γ ( ) 2 α m y(z) 2 Γ ( z 2 α m ) 1 α + 2 α m (2 α) ( ) for z 0 +. m 1 8 / 26

9 Existence and uniqueness Next, transform the problem into an equivalent integral equation ( z y(z) = 0 ) 1 m+1 G(z, t)f α y(t)dt =: T (y)(z), for some kernel G. Nonlinear operator T is monotone and homogeneous of degree 1 m+1. Theorem ([10], P.J. Bushell, 1976) Suppose that T : K K is a monotone increasing mapping which is homogeneous of degree p with 0 < p < 1. If there exists ϕ K 0 such that γ 1 ϕ T (ϕ) γ 2 ϕ, for some positive constants γ 1,2 then T has a unique xed-point y K 0. 9 / 26

10 Existence and uniqueness Theorem Let ϕ(z) := z 2 α m, then γ 1 ϕ T (ϕ) γ 2 ϕ, with γ 1 = ( ( α 2 ( ( α 2 ) 1 α Γ( 2 α m ) Γ(2 α+ 2 α ) 2 α Γ(1+ 2 α m ) Γ(2 α+ 2 α m ) m ) 2 α+m(3 α) 1 2 α ) 1 m+1 1, 0 < α 1 1 m+1 ; ) 1 m+1, 1 1 m+1 < α 1, γ 2 = Γ(3 α) 1 m+1. Existence and uniqueness follows from Bushell's Theorem. 10 / 26

11 Approximate solutions Now, we will nd some approximate solutions of our problem. The strategy is to substitute the E-K operator (nonlocal) by some local approximation. This can be accurate for α 1. The relevant papers: Š.Pªociniczak, Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Communications in Nonlinear Science and Numerical Simulation 24 (13) (2015), Š.Pªociniczak, Approximation of the Erdelyi-Kober fractional operator with application to the time-fractional porous medium equation, SIAM Journal of Applied Mathematics 74(4) (2014), Š.Pªociniczak, H.Okrasi«ska, Approximate self-similar solutions to a nonlinear diusion equation with time-fractional derivative, Physica D 261 (2013), / 26

12 Approximate solutions Theorem For analytic U and a > 1, b > 0, c > 0 we have the following representation I a,b c U(η) = k=0 λ k U (k) (η) ηk k!, where λ k = k ( k ) j=0 j ( 1) k j Γ(a+ j +1) c Γ(a+b+ j +1). c Moreover, we have an asymptotic expansion when k λ k ( 1) k c ( ) c(a+1) Γ (c(a + 1)) 1 Γ (c(a + 1)). Γ(b) k The series converges very fast, especially for η close to 0. We can use it and obtain a series of useful approximations U i. 12 / 26

13 Numerical results Wetting front position Cumulative moisture t t Figure: On the left: front position η (t) (solid line) and its approximation η 3 (t) = η 3 t α/2 (dashed line). On the right: cumulative moisture (solid line) and approximations I 1 (dashed line) i I 2 (dot-dashed line). Here, α = 0.95 and m = / 26

14 Numerical results 3 m 3 D UHhL h Figure: Fitting U3 with experimental data from [2]. On the left: a self-similar pro le; an the right: time evolution. Here α = 0.855, C = 0.71 m3 /m3, m = 6.98, D0 = 5.36 mm/s /

15 Inverse problems In applications it is possible to measure U, but how to infer about the material properties of the medium? The relevant quantity is the diusivity D(U). Our (inverse) problem is to nd D(U) provided we know U. The relevant paper: Š.Pªociniczak, Diusivity identication in a nonlinear time-fractional diusion equation, Fractional Calculus and Applied Analysis 19(4) (2016), / 26

16 Inverse Problems Denition A mathematical problem (for example a dierential or algebraic equation) is called a well-posed problem if it has all of the following properties it has a solution, it has only one solution, it is continuous with respect to the perturbations in the initial data (stability). If a problem is not well-posed it is called ill-posed. Inverse problems usually belong to this class. 16 / 26

17 Diusivity identication Existence and uniqueness (up to a constant D s ) - easy integration D(U(η)) = 1 [ ( U D s U (0) + 1 α ) η F α U(z)dz α ] (η) 2 2 ηf αu(η). What about stability and computational cost? For the latter we have an approximation (based on our previous results) 0 D(U(η)) = 1 U (η) [D su (0) ( ) 1 + Γ(1 α) + α η ] α U(z)dz 2Γ(2 α) 2Γ(2 α) ηu(η). As we can see, this formula does not require double integration (much cheaper in applications) / 26

18 Main results Denition For η 0 > 0 the supremum norm of a function U : [0, ] R is dened by the formula U,η0 := sup U(η). η [0,η 0] Denition A function U : [0, ] [0, ] is called admissible if it is bounded along with its rst derivative, nonincreasing and vanishing at innity. Theorem Let U be admissible and x η 0 > 0 such that E η0 := sup η [0,η0] η/u (η) is nite. We then have D(U) D(U) E η0 (2A(α) U + B(α)η 0 U ),,η0 where A(α) and B(α) are O(1 α) as α 1 and explicitly known. 18 / 26

19 Main results. Regularization Dierentiation is not a stable operation, thus we have to regularize our formula for D. Let U δ be the moisture with a measurement noise, i.e. U U δ δ. Introduce a regularization strategy for U (ex. nite dierence) U (U δ ) h R(h, δ), for which there exists h 0 = h 0 (δ) such as R(h 0 (δ), δ) 0 as δ 0. Moreover, ( U δ) is a stable operator for each h > 0. h 19 / 26

20 Main results. Regularization Theorem Let D h be a family of regularization operators for the diusivity and x η 0 > 0. Then for U X such that there exists an ɛ > 0 with ɛ inf η [0,η0] U (η) < we have D(U) Dh (U δ ),η0 1 ( [D s 1 + R(h, δ) + U ɛ ɛ + η ( 0 Γ(2 α) δ + δ + U R(h, δ) ɛ ) R(h, δ) )]. Theorem Let the assumptions of the above Theorem be fullled. If the regularization strategy is such that R(h 0 (δ), δ) = O(δ µ ) for δ 0 then D(U) D h (U δ ),η0 = O(δ µ ) as δ / 26

21 Numerical methods This is a work in progress but we have some preliminary results. Discretization of the E-K operator n 1 Ic a,b U(η) = w n,i U(η i ) + R n (η). i=0 We have several schemes for choosing w n,i along with exact asymptotic behaviour of R n as n. Our results can be applied to any integro-dierential equations with E-K operator. The relevant work: Š.Pªociniczak, S. Sobieszek, Numerical schemes for integro-dierential equations with Erdélyi-Kober fractional operator, Numerical Algorithms 76(1) (2016), pp / 26

22 Numerical methods The second problem is a numerical solution of ( z ) 1 m+1 y(z) = G(z, t)f α y(t)dt, 0 which is not easy since the nonlinearity is not Lipschitzian. The above equation appears in the above integral formulation of time-fractional porous medium equation. Results: Convergence of a family of nite-dierence schemes. An explicit construction of a convergent method. Main assumption: kernel has to be positively bounded from below (now working how to remove this: for anomalous diusion we have G(z, u) C(z u) 1 α ). The relevant work: H. Okrasi«ska-Pªociniczak, Š.Pªociniczak, Finite dierence method for Volterra equation with a power-type nonlinearity, arxiv: [math.na] 22 / 26

23 Numerical methods The crucial step of the convergence proof is done with aid of the following Gronwall-Bellman's Lemma. Lemma Let {e n }, n = 1, 2,... be a sequence of positive numbers satisfying ( ) e n 1 n 1 A e i + B, n 2, (1) n i=1 for A > 0. If we dene M := max{e 1, B} 1 Aζ(1 A), 0 < A < 1 1, A 1, (2) then 23 / 26 e n M. (3) n1 A

24 Numerical methods: a sketch of the proof The method's error can be estimated as follows (m + 1)ξ m n e n (m + 1)ξ m n y(nh) y n = y(nh) m+1 yn m+1 hwd n 1 e i + δ(h) Some work is required to estimate ξ n in terms of y(nh). Then, a comparison theorem for integral equations helps us to obtain a recurrence inequality only for e i. Gronwall-Bellman's Lemma gives the assertion. In the above way we can prove the following estimate i=1 e n const. h 1 C m max{h p, δ(h)h 1 } as h 0 + with nh x n, where e 1 = O(h p ) as h 0 + with p > 0 and C is a (known) constant. 24 / 26

25 Numerical example Consider the following equation y(x) m+1 = which can be readily solved with x y(x) = 0 y(t)dt, x [0, 1], ( ) 1 m m + 1 x m. Our theorem gives us the order of convergence equal to at least 1 1 m (which is not large, though). To calculate the numerical order of convergence we use Aitken's method based on Richardson's extrapolation. As a quadrature we use the midpoint's method. 25 / 26 m Order Est. order

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