COMPLETELY MONOTONE FUNCTIONS IN THE STUDY OF A CLASS OF FRACTIONAL - PowerPoint PPT Presentation

COMPLETELY MONOTONE FUNCTIONS IN THE STUDY OF A CLASS OF FRACTIONAL EVOLUTION EQUATIONS Emilia Bazhlekova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Dedicated to the memory of Professor Mirjana Stojanovi´ c

Inspired by: Mainardi, F., Mura, A., Gorenflo, R., Stojanovi´ c, M. The two forms of fractional relaxation of distributed order, J. Vib. Control. 13 (2007) pp. 1249–1268. Gorenflo, R., Luchko, Yu., Stojanovi´ c, M. Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density, Fract. Calc. Appl. Anal. 16, No.2 (2013) pp. 297–316. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 2/27

Completely monotone functions ( CMF ) and Bernstein functions ( BF ) A function f : (0 , ∞ ) → R is called completely monotone if it is of class C ∞ and ( − 1) n f ( n ) ( t ) ≥ 0 , for all t > 0 , n = 0 , 1 , ... (The simplest example: e − λt , λ > 0 .) Bernstein’s theorem: f ( t ) ∈ CMF if and only if � ∞ e − tx dg ( x ) , f ( t ) = 0 where g ( x ) is nondecreasing and the integral converges for 0 < t < ∞ . A C ∞ function f : (0 , ∞ ) → R is called a Bernstein function if f ( t ) ≥ 0 and f ′ ( t ) ∈ CMF . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 3/27

Some useful properties Proposition: (a) The class CMF is closed under pointwise addition and multiplication; (b) The class BF is closed under pointwise addition, but, in general not under multiplication; (c) If f ∈ CMF and ϕ ∈ BF , then the composite function f ( ϕ ) ∈ CMF ; (d) If f ∈ BF , then f ( t ) /t ∈ CMF ; (e) Let f ∈ L 1 loc ( R + ) be a nonnegative and nonincreasing function, such that lim t → + ∞ f ( t ) = 0 . Then ϕ ( s ) = s � f ( s ) ∈ BF ; loc ( R + ) and f ∈ CMF , then � (f) If f ∈ L 1 f ( s ) admits analytic extension to the sector | arg s | < π and | arg � f ( s ) | ≤ | arg s | , | arg s | < π. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 4/27

The operators of fractional integration and differentiation J α t - the Riemann-Liouville fractional integral of order α > 0 : � t 1 J α ( t − τ ) α − 1 f ( τ ) dτ, t f ( t ) := α > 0 , Γ( α ) 0 where Γ( · ) is the Gamma function. D α t - the Riemann-Liouville fractional derivative C D α t - the Caputo fractional derivative D 1 t = C D 1 t = J 1 − α D 1 t = D 1 t J 1 − α C D α D α t = d/dt ; t , , α ∈ (0 , 1) . t t Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 5/27

Mittag-Leffler function Fractional relaxation equation ( λ > 0 , 0 < α ≤ 1 ): C D α t u ( t ) + λu ( t ) = f ( t ) , t > 0 , u (0) = c 0 . The solution is given by: � t τ α − 1 E α,α ( − λτ α ) f ( t − τ ) dτ. u ( t ) = c 0 E α ( − λt α ) + 0 Mittag-Leffler function ( α, β ∈ R , α > 0 ): � ∞ ( − t ) k E α,β ( − t ) = Γ( αk + β ) , E α ( − t ) = E α, 1 ( − t ) . k =0 E 1 ( − t ) = e − t ∈ CMF E α ( − t ) ∈ CMF , iff 0 < α < 1 (Pollard, 1948) E α,β ( − t ) ∈ CMF , iff 0 ≤ α ≤ 1 , α ≤ β (Schneider, 1996; Miller, 1999) Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 6/27

Plots of E α ( − t α ) for different values of α ∈ (0 , 1] . α = 1 - exponential decay, α ∈ (0 , 1) - algebraic decay ( t − α ). Completely monotone functions. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 7/27

Plots of E α ( − t α ) for different values of α ∈ (1 , 2] . No more complete monotonicity for α > 1 ! Damped oscillations. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 8/27

Plots of t α − 1 E α,α ( − t α ) for different values of α ∈ (0 , 1] . Completely monotone functions. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 9/27

Fractional evolution equation of distributed order Two alternative forms: � 1 µ ( β ) C D β t u ( t ) dβ = Au ( t ) , t > 0 , (1) 0 and � 1 µ ( β ) D β u ′ ( t ) = t Au ( t ) dβ, t > 0 , (2) 0 A - closed linear unbounded operator densely defined in a Banach space X Initial condition: u (0) = a ∈ X. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 10/27

Two cases for the weight function µ : • discrete distribution m � µ ( β ) = δ ( β − α ) + b j δ ( β − α j ) , (3) j =1 where 1 > α > α 1 ... > α m > 0 , b j > 0 , j = 1 , ..., m, m ≥ 0 , and δ is the Dirac delta function; • continuous distribution µ ∈ C [0 , 1] , µ ( β ) ≥ 0 , β ∈ [0 , 1] , (4) and µ ( β ) � = 0 on a set of a positive measure. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 11/27

Discrete distribution: Multi-term time-fractional equations in the Caputo sense m � α j C D α C D t u ( t ) + b j t u ( t ) = Au ( t ) , t > 0 , (5) j =1 and in the Riemann-Liouville sense m � α j u ′ ( t ) = D α t Au ( t ) + b j D t Au ( t ) , t > 0 (6) j =1 If m = 0 (single-term equations): problem (5) is equivalent to (6) with α replaced by 1 − α . All problems are generalizations of the classical abstract Cauchy problem u ′ ( t ) = Au ( t ) , t > 0; u (0) = a ∈ X. (7) Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 12/27

Solution u ( t ) of (5) with A = − 1 for: m = 1 , α = 0 . 75 , α 1 = 0 . 25 , m = 0 , α = 0 . 25 m = 0 , α = 0 . 75 . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 13/27

Solution u ( t ) of (5) with A = − 1 for: m = 2 , α = 0 . 75 , α 1 = 0 . 5 , α 2 = 0 . 25 m = 1 , α = 0 . 75 , α 1 = 0 . 25 , m = 0 , α = 0 . 25 m = 0 , α = 0 . 75 . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 14/27

Unified approach to the four problems Rewrite problems (1) and (2) as an abstract Volterra integral equation � t u ( t ) = a + k ( t − τ ) Au ( τ ) dτ, t ≥ 0; a ∈ X, 0 where k 1 ( s ) = ( h ( s )) − 1 , � � k 2 ( s ) = h ( s ) /s, In the continuous distribution case: � 1 µ ( β ) s β dβ. h ( s ) = 0 In the discrete distribution case: m � h ( s ) = s α + b j s α j . j =1 Define g i ( s ) = 1 / � k i ( s ) , i = 1 , 2 . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 15/27

Particular cases In the single-term case: k 1 ( t ) = t α − 1 t − α g 1 ( s ) = s α , g 2 ( s ) = s 1 − α , Γ( α ) , k 2 ( t ) = Γ(1 − α ) , In the double-term case: t − α t − α 1 k 1 ( t ) = t α − 1 E α − α 1 ,α ( − b 1 t α − α 1 ) , k 2 ( t ) = Γ(1 − α ) + b 1 Γ(1 − α 1 ) , s g 1 ( s ) = s α + b 1 s α 1 , g 2 ( s ) = s α + b 1 s α 1 = s � k 1 ( s )!!! In the case of continuous distribution in its simplest form: µ ( β ) ≡ 1 . g 1 ( s ) = s − 1 g 2 ( s ) = s log s log s , s − 1 . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 16/27

Properties of the kernels Theorem. Let µ ( β ) be either of the form (3) or of the form (4) with the additional assumptions µ ∈ C 3 [0 , 1] , µ (1) � = 0 , and µ (0) � = 0 or µ ( β ) = aβ ν as β → 0 , where a, ν > 0 . Then for i = 1 , 2 , : (a) k i ∈ L 1 loc ( R + ) and lim t → + ∞ k i ( t ) = 0 ; (b) k i ( t ) ∈ CMF for t > 0 ; (c) k 1 ∗ k 2 ≡ 1 ; (d) g i ( s ) ∈ BF for s > 0 ; (e) g i ( s ) /s ∈ CMF for s > 0 ; (f) g i ( s ) admits analytic extension to the sector | arg s | < π and | arg g i ( s ) | ≤ | arg s | , | arg s | < π. In the discrete distribution case a stronger inequality holds: | arg g i ( s ) | ≤ α | arg s | , | arg s | < π. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 17/27

The classical abstract Cauchy problem: u ′ ( t ) = Au ( t ) , t > 0; u (0) = a ∈ X. Main result: Assume that the classical Cauchy problem is well-posed with solution u ( t ) satisfying � u ( t ) � ≤ M � a � , t ≥ 0 . Then any of the problems � 1 µ ( β ) C D β t u ( t ) dβ = Au ( t ) , t > 0 , u (0) = a ∈ X, 0 � 1 µ ( β ) D β u ′ ( t ) = t Au ( t ) dβ, t > 0 , u (0) = a ∈ X 0 is well-posed with solution satisfying the same estimate. Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 18/27

The classical abstract Cauchy problem: u ′ ( t ) = Au ( t ) , t > 0; u (0) = a ∈ X. T ( t ) - solution operator (defined by T ( t ) a = u ( t ) , t ≥ 0 ); R ( s, A ) - resolvent operator: � ∞ R ( s, A ) = ( s − A ) − 1 = e − st T ( t ) dt, s > 0 , 0 The Hille-Yosida theorem states that the classical Cauchy problem is well-posed with solution operator T ( t ) such that � T ( t ) � ≤ M, t ≥ 0 , iff R ( s, A ) is well defined for s ∈ (0 , ∞ ) and � R ( s, A ) n � ≤ M s n , s > 0 , n ∈ N . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 19/27

Abstract Volterra integral equation � t u ( t ) = a + k ( t − τ ) Au ( τ ) dτ, t ≥ 0; a ∈ X, 0 The Laplace transform of the solution operator S ( t ) � ∞ e − st S ( t ) dt, H ( s ) = s > 0 0 is given by H ( s ) = g ( s ) g ( s ) = 1 / � s R ( g ( s ) , A ) , k ( s ) . The Generation Theorem (Pr¨ uss, 1993) states that the integral equation is well- posed with solution operator S ( t ) satisfying � S ( t ) � ≤ M, t ≥ 0 , iff � H ( n ) ( s ) � ≤ M n ! s n +1 , for all s > 0 , n ∈ N 0 . Pannonian Mathematical Modelling, Novi Sad, April 25-26, 2015. p. 20/27