du dθ We wish to approximate the initial value problem for the fractional ordinary differential equation of order β :

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1 Mixed Collocation for Fractional Differential Equations François Dubois and Stéphanie Mengué December 5, 3 This contribution is dedicaced to Claude Brézinski at the occasion of his sixtieth birthday. Abstract We present the mixed collocation method for numerical integration of fractional differential equations of the type D β u Φ(u, t). Given a regular mesh with constant discretization step, the unknown u(t) is considered as continuous and affine in each cell, and the dynamics Φ(u, t) as a constant. After a fractional integration, the equation is written strongly at the mesh vertices and the dynamics weakly in each cell. The Semidif software has been developed for the particular case of numerical integration of order. The validation for analytical results and published solutions is established and experimental convergence as the mesh size tends to zero is obtained. Good results are obtained for a nonlinear model with a strong singularity. Keywords : Mixed finite elements, numerical combustion. AMS(MOS) classification : 65N3. Applications Scientifiques du Calcul Intensif, bâtiment 56, B.P. 67, F-943 Orsay, Union Européenne. Mel : dubois@asci.fr. Conservatoire National des Arts et Métiers, Equipe de recherche associée n o 396, 5, rue Marat, F-78 Saint-Cyr-L Ecole, Union Européenne. Université de Marne-La-Vallée, Laboratoire Systèmes de Communication, bâtiment Copernic, F Marne-La-Vallée cedex, Union Européenne. Mel : smengue@univ-mlv.fr. Numerical Algorithms, volume 34, numéro, p. 33-3, dec 3, édition du mars 9.

2 François Dubois and Stéphanie Mengué ) Introduction Let β be a real number, < β <, Γ( ) the classical Euler function, and IR [, [ (u,t) Φ(u,t) IR a regular mapping. Following for example the now classical work of Caputo [Ca67], we define the fractional differential operator D β ( ) by () (D β t du dθ u)(t) Γ( β) dθ (t θ) β. We wish to approximate the initial value problem for the fractional ordinary differential equation of order β : { () D β (u u ) Φ(u (t),t), t > u u, t. We introduce a discretization step h, and denote by K h the interval of j+ the form K h j+ ]t j, t j+ [ ]jh, (j + )h[ for j IN. We use the discrete spaces P h and Q h defined as follows. The linear space P h is composed by functions that are continuous and affine in each mesh element K h. A j+ function v P h can be expanded on the basis ( ) ϕ h k of classical hat k IN functions : { ϕ h (θ) θ h, θ h, θ h ( θ t k ), t k θ t k+, ϕ h k (θ) h + ( θ t k ), t k θ t k, k, h, θ t k h and by using the nodal values v k v ( t k) at the grid vertices, we have : v k v k ϕ h k for v P h. The discrete space Q h is composed by functions that take a constant value in each element K h. The basis functions of the discrete space Q h j+ are null everywhere except inside the interval K h where it is equal to the j+ unity. Then, w Q h can be expanded on the χ h basis according to the k+ relation w k w k+ χh. k+ ) Mixed collocation scheme We integrate the differential equation () with the fractional integrator I β of order β defined by

3 Mixed Collocation for Fractional Differential Equations I β t (v ( ),t) Γ(β) (t θ)β v (θ) dθ and satisfying the functional equation I β D β Id when applied for a regular function u satisfying u(). The relation () can therefore be rewritten as (3) u (t) u I β (Φ(u ( ), t)), t. The P Q mixed collocation method consists in choosing a discrete state u h ( ) satisfying (4) u h u h j ϕ h j, u h P h, j and the so-called flux Φ(u ( ),t) by a discontinuous time function f h ( ) that is constant in each cell K h : j+ (5) f h, f h Q h. j f h j+ χ h j+ On one hand, the equation (3) is written strongly at the mesh vertices jh (j IN) and we can speak of a collocation method : (6) u h (jh) u I β ( f h ( ), jh ), j IN. On the other hand, it is a method inspired from mixed finite elements. We can not impose that for each time value t, we have f h (t) Φ(u h (t), t). Then the P Q mixed collocation method claims only that for the mean values, the approached flux f h ( ) and the exact flux Φ ( u ( ), ) h of the approached solution have the same mean value in each interval ]jh, (j +)h[. We impose : (j+)h jh f h (θ) dθ (j+)h jh Φ ( u h (θ),θ ) dθ, j IN. This projection step on the discrete space Q h can be written as : (7) f h j+ Φ ( u h j ( θ) + θu h j+, jh + θh ) dθ, j IN. Proposition. State-flux constraint for the P Q scheme With the choice (4) (5), the relation (6) can be written in the following way : (8) u h j+ h β Γ(β + ) fh j+ u + 3 h β Γ(β + ) j k α j k f h k+, j IN,

4 François Dubois and Stéphanie Mengué with (9) α k (k + ) β k β, k IN. Proof of proposition. We evaluate the right hand side of relation (6) for the discrete time t j+. We have : I β ( f h ( ),t j+) Γ(β) Γ(β) Γ(β) Γ(β) t j + j k j k j k Γ(β + ) h β Γ(β + ) ( t j+ θ ) β f h (θ)dθ (k+)h kh f h k+ f h k+ j k j k (t j+ θ) β f h (θ) dθ [(j + ) h (kh + hξ)] β h dξ h β ( β f h k+ and the relation (8) is a direct consequence of (6). ) [ (j + k ξ) β] ξ ξ h β [ (j + k) β (j k) β] f h k+ α j k taking into account (9) 3) First numerical tests We have compared in [DMk] several numerical schemes i.e. two and three point Grünwald-Letnikov scheme, Msallam scheme and P finite element scheme and we have also compared in [DM] thoses schemes with the mixed collocation method. These tests are done for a semi-derivation, i.e. β.5. The dynamics (u,t) Φ(u,t) is parameterized by the functions u f (u) and t g (t) and by the real numbers θ f and θ g : Φ(u,t) θ f f (u) + θ g g (t). The parameters θ f and θ g are chosen in order to obtain as solution the following particular functions : ( ) j () u j (t) t, j,,5. In a first approach, we consider θ f and θ g ; a simple semi-quadrature has to be done. The equation () corresponds to the following t g (t) choice : g (t) π, g (t) π t,, g3 (t) 3 4 π t, g4 (t) 8 3 π t t and g 5 (t) 5 6 π t. 4

5 Mixed Collocation for Fractional Differential Equations In a second approach, we take θ f and θ g. This dynamics is associated to the following functions : f (u) π, f (u) u, f3 (u) 3 π u 3, π 4 () f 4 (u) 8 3 π u 3 4, f5 (t) 5 4 π u 5. 6 In a third approach, whave also considered the following numerical test cases previously proposed by K. Diethelm [Di97], Diethelm and Ford [DF], Diethelm and Luchko [DL3], Blank [Bl96] and Lubich [Lu86] : () u, t, D / u u π t 3, t > : u (t) t ( (3) u, t, D / u u, t > : u (t) e t erf ) t + ( (4) u, t, D / u u, t > : u (t) e t erf ) t (5) u, t, D / u sin t, t > : u (t) ( ) tj t. π We remark that for the nonlinear cases proposed in (), the sufficient Lipschitz continuity condition for the function Φ(, ), revisited by Nagumo and presented e.g. the book [GV9] of Gorenflo and Vessela, that assumes that the nonlinear Abel-Volterra equation (3) has a unique continuous solution, is a priori not satisfied. Nevertheless, the functions presented at the relations () are clearly solution of problem (3) and our numerical results have captured this fact. The numerical simulation is done with our Fortran software Semidif. We have computed systematically the orders of convergence for mesh step sizes h of the type for 3 n 3. We have measured the L and L errors denotes respectively n by e n and e n and defined according to : e n max{ u(jh) u j, j,, n } and e n h u () u + n j ( ) j u u j + u () u n. The orders of convergence are summarized in the following table : n g (t) g (t) g 3 (t) g 4 (t) g 5 (t) () (3) (4) L L

6 François Dubois and Stéphanie Mengué Our results show a good agreement with the studies done by these authors. We refer to our report [DM] for the details. 4) Nonlinear model with a singularity We explain in this section how to consider the P Q method when we study the Joulin model [Jo85] for spherical flammes : { (6) D / (u) Φ(u (t),t), t > u, t (7) Φ(u (t),t) log u + E tγ ( t) H( t), u where E and γ.3 are positive constants and θ H(θ) the Heaviside fonction equal to if θ < end to if θ >. Proposition. Computation of the first point for Joulin equation For the equation (6), and the representation (8) of the unknown u h, the first unknown point u h is solution of the following equation whose unknown is denoted by x : h (8) x {log x + E ( hγ π x γ h )}. γ + Proof of proposition. We integration the equation (6) one half time on the interval [, h] : (9) u h h π (h θ) Φ ( u h (θ),θ ) dθ. Then we make the hypothesis that the flux Φ(, ) is equal to a constant f h on the interval [,h] : f h h h Φ( u h (θ),θ ) dθ. We inject this hypothesis at the right hand side of equation (9) : () u h h Φ ( u h (θ),θ ) dθ π h and we compute the term on the right of () by using the function Φ(, ) of the relation (7). We note the singularity in the vincinity of zero. The hypothesis of affine representation of u between and h, show the following expression : Φ ( u h (θ),θ ) ( ) log u h θ + E θ γ ( θ) h u h θ if h h log ( u h ) θ + log h + E h θγ ( θ) u h. 6

7 Mixed Collocation for Fractional Differential Equations Then after integration, we obtain h Φ ( u h (θ),θ ) dθ h log ( u h ) + log tdt + E u h h γ t γ ( ht) dt log ( u h ) + [t log t t] + E [ u h γ h ] γ + log ( ( ) u h E + u h γ h ). γ + and the proposition is established. The equation (8) is solved with the Newton algorithm. For the other grid points, some algebra and a numerical integration (see the details in [DM]) show that we have to solve the system composed by (8) and the following representation : f h j+ u h j+ uh j + E hγ [ u h j+ log u h j+ u h ] j+ ( j γ ( jh) u h j ) + (j + )γ ( (j + )h) u h j points 4 points 394 points 393 points 393 points 39 points 38 points 3 points 5 points Change of comportment between np393 and np393 mesh points 6 Simulations with E7.6 and t max Figure. Qualitative change of the solution according to various discretizations. We have followed in [DM] the working plan proposed by Audounet and Roquejoffre [AR98] for their simulations of the semidifferential equation (6) 7

8 François Dubois and Stéphanie Mengué (7) : γ.3 and 7.6 E 7.7. According to these authors, a bifurcation in the asymptotic comportment occurs for the solution of model (6)(7) with γ.3. For E 7.7, we have extinction of the flamme in a finite time whereas for E 7.8, its radius tends to infinity as time tends to infinity. We first observe that we have to control very precisely the discretization step h, as proposed in the following experiment. We consider the parameter value E 7.6 and t max 4. When we use less than N 393 mesh points, the comportment of the discrete solution is to tends to infinity and for more than N 393 mesh points, we observe an extinction in a finite time (see the figure )! points 5 points 5 points points 5 points 5 points points E Figure. Numerical solution of the model (6) (7), E In what follows, we systematically refine the mesh in order to make in evidence the independence of the numerical solution with the number of grid points. When we use sufficiently refined meshes, we observe (see [DM]) an extinction in finite time for E 7.6 and a diverging process for E 7.7. We have experimentally determined the value of bifurcation parameter E. We use a number of mesh points N according to the rule N, k,.5 k, 5 k, k+,. The same conclusions hold for the extinction parameter E 7.66 and we have a diverging flame for E 7.67 (Figure ), but the numerical proof needs meshes that use up to 5 points. The case E (Figure 3) shows an extinction of the flamme in a finite time, but the high cost of computer ressources (more than 8

9 Mixed Collocation for Fractional Differential Equations two millions mesh points to observe asymptotically numerical convergence with the time discretization!) makes in evidence the actual limitations of the method. Therefore, according to our numerical results, the bifurcation parameter E for the model (6) (7) satisfies the inequalities : < E < 7.67 with γ points 5 points 5 points points 5 points 5 points points 5 points 5 points points 5 points E Figure 3. Numerical solution of the model (6) (7), E ) Conclusion and aknowledgments In our Semidif software, the system of equations (7) (8) that characterizes the P Q mixed collocation method is solved with a Newton method. We have validated our approach with exact solutions and recent published numerical approximations about fractional differential equations. For a difficult test case issued from combustion modelling, we show that our mixed collocation method is relevant and conducts to significative results. We have noticed an accuracy of order for nonlinear test problems whereas the interpolation process is of order. The natural question is therefore to obtain second order accuracy in the future. Last but not least, we thank the unknown referee for helpful scientific comments on the first version of this article. 6) References. [AR98] J. Audounet and J.M. Roquejoffre. An asymptotic fractional differential model of spherical flame, ESAIM Proceedings, vol. 5, p. 5-7,

10 [Bl96] [Ca67] [DF] [Di97] [DL3] François Dubois and Stéphanie Mengué L. Blank. Numerical treatment of differential equations of fractional order, Numerical analysis report n o 87, Manchester University, March 996. M. Caputo. Linear models of dissipation whose Q is almost frequency independent, Part, Geophys. J.R. Astr. Soc., vol. 3, p , 967. K. Diethelm and N. Ford. Analysis of fractional differential equations, J. Math. Anal. Appl., vol. 65, p. 9-48,. K. Diethelm. An algorithm for the numerical solution of differential equations of fractional order, Electronic Transactions on Numerical Analysis, vol. 5, p. -6, 997. K. Diethelm and Y. Luchko. Numerical solution of linear multiterm differential equations of fractional order, J. Comput. Anal. Appl., 3, to appear. [DMk] F. Dubois and S. Mengué. Schémas numériques implicites pour les équations semi-différentielles, IAT-CNAM internal report n o 334, see also June. [DM] F. Dubois and S. Mengué. Collocation mixte pour les équations différentielles non-linéaires d ordre fractionnaire, IAT-CNAM internal report n o 348 and Semidif software, July. [GV9] R. Gorenflo and S. Vessella. Abel Integral Equations : Analysis and Applications, Lecture Notes in Mathematics, vol. 46, Springer, New York, 99. [Jo85] [Lu86] G. Joulin. Point source initiation of lean spherical flames of light reactants: an asymptotic theory, Combustion Science and Technology, vol. 43, p. 99-3, 985. C. Lubich. Discretized fractional calculus, SIAM Journal on Mathematical Analalysis, vol. 7, n o 3, p , 986.