FUNDAMENTAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS - PowerPoint PPT Presentation

FUNDAMENTAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING FRACTIONAL POWERS OF FINITE DIFFERENCES OPERATORS J. G´ onzalez-Camus (USACH) and P.J. Miana (UZ) Workshop on Banach spaces and Banach lattices ICMAT, September 2019, 9th-13th pjmiana@unizar.es

1. Introduction We present the solution of fractional differential equation � D β t u ( n , t ) = Bu ( n , t ) + g ( n , t ) , n ∈ Z , t > 0 . (1) u ( n , 0) = ϕ ( n ) , u t ( n , 0) = φ ( n ) n ∈ Z ,

1. Introduction We present the solution of fractional differential equation � D β t u ( n , t ) = Bu ( n , t ) + g ( n , t ) , n ∈ Z , t > 0 . (1) u ( n , 0) = ϕ ( n ) , u t ( n , 0) = φ ( n ) n ∈ Z , Bf ( n ) = ( K ∗ f )( n ), with K ∈ l ∞ ( Z ), f ∈ l p ( Z ), p ∈ [1 , ∞ ] and β ∈ (1 , 2]. We recall that D β t denotes the Caputo fractional derivative given by � t 1 D β ( t − s ) 1 − β v ′′ ( s ) ds = ( g 2 − β ∗ v ′′ )( t ) , t v ( t ) = Γ(2 − β ) 0

1. Introduction We present the solution of fractional differential equation � D β t u ( n , t ) = Bu ( n , t ) + g ( n , t ) , n ∈ Z , t > 0 . (1) u ( n , 0) = ϕ ( n ) , u t ( n , 0) = φ ( n ) n ∈ Z , Bf ( n ) = ( K ∗ f )( n ), with K ∈ l ∞ ( Z ), f ∈ l p ( Z ), p ∈ [1 , ∞ ] and β ∈ (1 , 2]. We recall that D β t denotes the Caputo fractional derivative given by � t 1 D β ( t − s ) 1 − β v ′′ ( s ) ds = ( g 2 − β ∗ v ′′ )( t ) , t v ( t ) = Γ(2 − β ) 0 g α ( t ) := t α − 1 Γ( α ), for α > 0.

1. Introduction For 1 ≤ p ≤ ∞ , the Banach space ( ℓ p ( Z ) , � � p ) are formed by f = ( f ( n )) n ∈ Z ⊂ C such that � 1 ∞ � p � | f ( n ) | p � f � p : = < ∞ , 1 ≤ p < ∞ ; n = −∞ � f � ∞ : = sup | f ( n ) | < ∞ . n ∈ Z → ℓ ∞ ( Z ), ( ℓ p ( Z )) ′ = ℓ p ′ ( Z ) with 1 ℓ 1 ( Z ) ֒ → ℓ p ( Z ) ֒ p + 1 p ′ = 1 for 1 < p < ∞ and p = 1 and p ′ = ∞ . In the case that f ∈ ℓ 1 ( Z ) and g ∈ ℓ p ( Z ), then f ∗ g ∈ ℓ p ( Z ) where ∞ � ( f ∗ g )( n ) := f ( n − j ) g ( j ) , n ∈ Z , j = −∞ and � f ∗ g � p ≤ � f � 1 � g � p for 1 ≤ p ≤ ∞ . Note that ( ℓ 1 ( Z ) , ∗ ) is a commutative Banach algebra with unit (we write δ 0 = χ { 0 } ).

1. Introduction We apply G¨ uelfand theory to get f ∈ ℓ 1 ( Z ) , σ ℓ 1 ( Z ) ( f ) = F ( f )( T ) , where � f ( n ) e in θ , F ( f )( θ ) := θ ∈ T . n ∈ Z

1. Introduction We apply G¨ uelfand theory to get f ∈ ℓ 1 ( Z ) , σ ℓ 1 ( Z ) ( f ) = F ( f )( T ) , where � f ( n ) e in θ , F ( f )( θ ) := θ ∈ T . n ∈ Z Given a = ( a ( n )) n ∈ Z ∈ ℓ 1 ( Z ), define A ∈ B ( ℓ p ( Z )) by convolution, b ∈ ℓ p ( Z ) , A ( b )( n ) := ( a ∗ b )( n ) , n ∈ Z , for all 1 ≤ p ≤ ∞ , � A � = � a � 1 and σ B ( ℓ p ( Z )) ( A ) = σ ℓ 1 ( Z ) ( a ) = F ( a )( T ) (2) for all 1 ≤ p ≤ ∞ , (Wiener’s Lemma).

Aims of the talk

Aims of the talk The main aim of this talk is to study the fractional differential equations in ℓ p ( Z ) for 1 ≤ p ≤ ∞ . To do this. (i) We apply G¨ uelfand theory to describe convolution operators. (ii) We calculate the kernel of the convolution fractional powers. (iii) We solve some fractional evolution equation in ℓ p ( Z ). (iv) Finally we obtain explict solutions for fractional evolution equation for some fractional powers of finite difference operators.

2. Finite difference operators on ℓ 1 ( Z ) Finite difference operators A ∈ B ( ℓ p ( Z )) given by m � Af ( n ) := a ( j ) f ( n − j ) , a j ∈ C , j = − m for some m ∈ N , i.e. a = ( a ( n ) n ∈ Z ) ∈ c c ( Z ) are convolution operator and the discrete Fourier Transform of a is a trigonometric polynomial m � a ( j ) e ij θ . F ( a )( θ ) = j = − m

2. Finite difference operators on ℓ 1 ( Z ) 1. D + f ( n ) := f ( n ) − f ( n + 1) = (( δ 0 − δ − 1 ) ∗ f )( n ); 2. D − f ( n ) := f ( n ) − f ( n − 1) = (( δ 0 − δ 1 ) ∗ f )( n ); 3. ∆ d f ( n ) := f ( n +1) − 2 f ( n )+ f ( n − 1) = (( δ − 1 − 2 δ 0 + δ 1 ) ∗ f )( n ); 4. D f ( n ) := f ( n + 1) − f ( n − 1) = (( δ − 1 − δ 1 ) ∗ f )( n ); 5. ∆ ++ f ( n ) := f ( n + 2) − 2 f ( n + 1) + f ( n ) = (( δ − 2 − 2 δ − 1 + δ 0 ) ∗ f )( n ); 6. ∆ −− f ( n ) := f ( n ) − 2 f ( n − 1)+ f ( n − 2) = (( δ 0 − 2 δ 1 + δ 2 ) ∗ f )( n ); 7. ∆ dd f ( n ) := f ( n + 2) − 2 f ( n ) + f ( n − 2) = (( δ − 2 − 2 δ 0 + δ 2 ) ∗ f )( n ); for n ∈ Z , [Bateman, 1943].

2. Finite difference operators on ℓ 1 ( Z ) Proposition The following equalities hold: (i) ∆ d = − ( D + + D − ) = − D + D − , D = − ( D + − D − ) = ( − D + + 2 I ) D − = ( D − − 2 I ) D + , (∆ ++ − 2∆ d + ∆ −− ) = D 2 f . ∆ dd = (ii) ( D + ) ′ = D − ; ( D − ) ′ = D + ; (∆ d ) ′ = ∆ d ; ( D ) ′ = D ; (∆ dd ) ′ = ∆ dd .

2. Finite difference operators on ℓ 1 ( Z ) Spectrum sets of finite difference operators Imaginary axis 2 D + , D - 1 Δ d , Δ dd  Real axis - 4 - 2 2 4 Δ ++ , Δ -- - 1 - 2

2. Finite difference operators on ℓ 1 ( Z ) ∞ z n a ∗ n e za := � z ∈ C . , n ! n =0 ∞ z 2 n a ∗ 2 n � cosh( za ) := (2 n )! , z ∈ C . n =0

2. Finite difference operators on ℓ 1 ( Z ) ∞ z n a ∗ n e za := � z ∈ C . , n ! n =0 ∞ z 2 n a ∗ 2 n � cosh( za ) := (2 n )! , z ∈ C . n =0 Proposition Let A ∈ B ( l p ( Z )) , with Af = a ∗ f , f ∈ ℓ p ( Z ) and a ∈ ℓ ∞ ( Z ) . Then, the Fourier transform of the semigroup { e at } t ≥ 0 generated by a is given by F ( e at )( θ ) = e F ( a )( θ ) t . F ( cosh ( at ))( θ ) = cosh ( F ( a )( θ ) t ) .

2. Finite difference operators on ℓ 1 ( Z ) Operator F ( · )( z ) Associated semigroup e − z z n − D + z − 1 n ! χ N 0 ( n ) =: g z , + ( n ) e − z z − n 1 − D − z − 1 ( − n )! χ − N 0 ( n ) =: g z , − ( n ) z + 1 e − 2 z I n (2 z ) ∆ d z − 2 z − 1 D J n (2 z ) z 1 −D z − z J − n (2 z ) e z z n − D + + 2 z + 1 n ! χ N 0 ( n ) e z z n 1 − D − + 2 z + 1 n ! χ − N 0 ( n ) √ e z ( i 2 z ) − n H − n (2 iz ) z 2 − 2 z + 1 ∆ ++ χ N 0 ( n ) =: h z , + ( n ) ( − n )! √ e z ( i 2 z ) n H n (2 iz ) z 2 − 2 1 1 ∆ −− z + 1 χ − N 0 ( n ) =: h z , − ( n ) n ! z 2 − 2 + 1 e − 2 z I n (2 z ) χ 2 Z ( n ) ∆ dd z 2

2. Finite difference operators on ℓ 1 ( Z ) Theorem (i) The Bessel function J n has a factorization expression given by J n (2 z ) = ( g − z , + ∗ g z , − )( n ) , n ∈ Z , z ∈ C . (ii) The Bessel function I n admits factorization product given by e − 2 z I n (2 z ) = ( g z , + ∗ g z , − )( n ) , I n (2 z ) = ( j z , + ∗ j z , − )( n ) . (iii) The Bessel function e − 2 z I n (2 z ) χ 2 Z ( n ) admits a factorization given by I n (2 z ) χ 2 Z ( n ) = h z , + ( n ) ∗ I n (2 z ) ∗ e − 2 z I n (2 z ) ∗ h z , − ( n ) .

3. Fractional powers of discrete operators

3. Fractional powers of discrete operators The Generalized Binomial Theorem is given by ∞ � α � ( a + b ) α = � a α − j b j , α ∈ C . j j =0

3. Fractional powers of discrete operators The Generalized Binomial Theorem is given by ∞ � α � ( a + b ) α = � a α − j b j , α ∈ C . j j =0 � α � 1 For α > 0, ∼ j α +1 and � a � ≤ 1 j ∞ � α � ( δ 0 + a ) α = � a j , α > 0 . j j =0

3. Fractional powers of discrete operators For 0 < α < 1, the Balakrishnan’s formula is expressed by � ∞ 1 ( T ( t ) x − x ) dt ( − A ) α x = t 1+ α , x ∈ D ( A ) . Γ( − α ) 0

3. Fractional powers of discrete operators For 0 < α < 1, the Balakrishnan’s formula is expressed by � ∞ 1 ( T ( t ) x − x ) dt ( − A ) α x = t 1+ α , x ∈ D ( A ) . Γ( − α ) 0 Theorem Let 0 < α < 1 , and A ∈ B ( ℓ p ( Z )) , 1 ≤ p ≤ ∞ a generator of a uniformly bounded semigroup, with Af = a ∗ f , f ∈ ℓ p ( Z ) and a ∈ ℓ 1 ( Z ) . Then the fractional powers ( − A ) α is well-posedness and it is expressed by ( − A ) α f = ( − a ) α ∗ f , where � 2 π ( − a ) α ( n ) := 1 ( −F ( a )( θ )) α e − in θ d θ. 2 π 0

3. Fractional powers of discrete operators � − α − 1+ m = ( − 1) m � α Λ α ( m ) := � � , for m ∈ N 0 . m m Fractional power Kernel Explicit expression D α K α Λ α ( n ) χ N 0 + + D α K α Λ α ( n ) χ − N 0 − − ( − 1) n Γ(2 α +1) ( − ∆ d ) α K α d Γ(1+ α + n )Γ(1+ α − n ) i n Γ( α +1) D α K α D + 2 + n 2 − n 2 Γ( α 2 +1)Γ( α 2 +1) ( − i ) n Γ( α +1) ( −D ) α K α 2 + n 2 − n D − 2 Γ( α 2 +1)Γ( α 2 +1) ( − D + + 2 I ) α K α ( − 1) m Λ α ( n ) χ N 0 ( − D + +2 I ) ( − D − + 2 I ) α K α ( − 1) m Λ α ( n ) χ − N 0 ( − D − +2 I ) Λ 2 α ( n ) χ N 0 ∆ α K α ++ D ++ ∆ α K α Λ 2 α ( n ) χ − N 0 −− D −− ( − i ) n Γ(2 α +1) ( − ∆ dd ) α K α Γ( α + n 2 +1)Γ( α − n dd 2 2 +1)

3. Fractional powers of discrete operators