FUNDAMENTAL SOLUTIONS FOR FRACTIONAL ...FUNDAMENTAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS...
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FUNDAMENTAL SOLUTIONS FORFRACTIONAL DIFFERENTIAL EQUATIONS
INVOLVING FRACTIONAL POWERS OFFINITE DIFFERENCES OPERATORS
J. Gonzalez-Camus (USACH) and P.J. Miana (UZ)
Workshop on Banach spaces and Banach latticesICMAT, September 2019, 9th-13th
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1. Introduction
We present the solution of fractional differential equation{Dβt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n) n ∈ Z,(1)
Bf (n) = (K ∗ f )(n), with K ∈ l∞(Z), f ∈ lp(Z), p ∈ [1,∞] and
β ∈ (1, 2]. We recall that Dβt denotes the Caputo fractionalderivative given by
Dβt v(t) =1
Γ(2− β)
∫ t
0(t − s)1−βv ′′(s)ds = (g2−β ∗ v ′′)(t),
gα(t) :=tα−1
Γ(α), for α > 0.
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1. Introduction
We present the solution of fractional differential equation{Dβt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n) n ∈ Z,(1)
Bf (n) = (K ∗ f )(n), with K ∈ l∞(Z), f ∈ lp(Z), p ∈ [1,∞] and
β ∈ (1, 2]. We recall that Dβt denotes the Caputo fractionalderivative given by
Dβt v(t) =1
Γ(2− β)
∫ t
0(t − s)1−βv ′′(s)ds = (g2−β ∗ v ′′)(t),
gα(t) :=tα−1
Γ(α), for α > 0.
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1. Introduction
We present the solution of fractional differential equation{Dβt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n) n ∈ Z,(1)
Bf (n) = (K ∗ f )(n), with K ∈ l∞(Z), f ∈ lp(Z), p ∈ [1,∞] and
β ∈ (1, 2]. We recall that Dβt denotes the Caputo fractionalderivative given by
Dβt v(t) =1
Γ(2− β)
∫ t
0(t − s)1−βv ′′(s)ds = (g2−β ∗ v ′′)(t),
gα(t) :=tα−1
Γ(α), for α > 0.
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1. IntroductionFor 1 ≤ p ≤ ∞, the Banach space (`p(Z), ‖ ‖p) are formed byf = (f (n))n∈Z ⊂ C such that
‖f ‖p : =
( ∞∑n=−∞
|f (n)|p) 1
p
<∞, 1 ≤ p <∞;
‖f ‖∞ : = supn∈Z|f (n)| <∞.
`1(Z) ↪→ `p(Z) ↪→ `∞(Z), (`p(Z))′ = `p′(Z) with 1
p + 1p′ = 1 for
1 < p <∞ and p = 1 and p′ =∞.In the case that f ∈ `1(Z) and g ∈ `p(Z), then f ∗ g ∈ `p(Z) where
(f ∗ g)(n) :=∞∑
j=−∞f (n − j)g(j), n ∈ Z,
and ‖f ∗ g‖p ≤ ‖f ‖1 ‖g‖p for 1 ≤ p ≤ ∞. Note that (`1(Z), ∗) is acommutative Banach algebra with unit (we write δ0 = χ{0}).
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1. Introduction
We apply Guelfand theory to get
σ`1(Z)(f ) = F(f )(T), f ∈ `1(Z),
whereF(f )(θ) :=
∑n∈Z
f (n)e inθ, θ ∈ T.
Given a = (a(n))n∈Z ∈ `1(Z), define A ∈ B(`p(Z)) by convolution,
A(b)(n) := (a ∗ b)(n), n ∈ Z, b ∈ `p(Z),
for all 1 ≤ p ≤ ∞, ‖A‖ = ‖a‖1 and
σB(`p(Z))(A) = σ`1(Z)(a) = F(a)(T) (2)
for all 1 ≤ p ≤ ∞, (Wiener’s Lemma).
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1. Introduction
We apply Guelfand theory to get
σ`1(Z)(f ) = F(f )(T), f ∈ `1(Z),
whereF(f )(θ) :=
∑n∈Z
f (n)e inθ, θ ∈ T.
Given a = (a(n))n∈Z ∈ `1(Z), define A ∈ B(`p(Z)) by convolution,
A(b)(n) := (a ∗ b)(n), n ∈ Z, b ∈ `p(Z),
for all 1 ≤ p ≤ ∞, ‖A‖ = ‖a‖1 and
σB(`p(Z))(A) = σ`1(Z)(a) = F(a)(T) (2)
for all 1 ≤ p ≤ ∞, (Wiener’s Lemma).
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Aims of the talk
The main aim of this talk is to study the fractional differentialequations in `p(Z) for 1 ≤ p ≤ ∞. To do this.
(i) We apply Guelfand theory to describe convolution operators.
(ii) We calculate the kernel of the convolution fractional powers.
(iii) We solve some fractional evolution equation in `p(Z).
(iv) Finally we obtain explict solutions for fractional evolutionequation for some fractional powers of finite differenceoperators.
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Aims of the talk
The main aim of this talk is to study the fractional differentialequations in `p(Z) for 1 ≤ p ≤ ∞. To do this.
(i) We apply Guelfand theory to describe convolution operators.
(ii) We calculate the kernel of the convolution fractional powers.
(iii) We solve some fractional evolution equation in `p(Z).
(iv) Finally we obtain explict solutions for fractional evolutionequation for some fractional powers of finite differenceoperators.
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2. Finite difference operators on `1(Z)
Finite difference operators A ∈ B(`p(Z)) given by
Af (n) :=m∑
j=−ma(j)f (n − j), aj ∈ C,
for some m ∈ N, i.e. a = (a(n)n∈Z) ∈ cc(Z) are convolutionoperator and the discrete Fourier Transform of a is a trigonometricpolynomial
F(a)(θ) =m∑
j=−ma(j)e ijθ.
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2. Finite difference operators on `1(Z)
1. D+f (n) := f (n)− f (n + 1) = ((δ0 − δ−1) ∗ f )(n);
2. D−f (n) := f (n)− f (n − 1) = ((δ0 − δ1) ∗ f )(n);
3. ∆d f (n) := f (n+1)−2f (n)+f (n−1) = ((δ−1−2δ0+δ1)∗f )(n);
4. Df (n) := f (n + 1)− f (n − 1) = ((δ−1 − δ1) ∗ f )(n);
5. ∆++f (n) := f (n + 2)− 2f (n + 1) + f (n) =((δ−2 − 2δ−1 + δ0) ∗ f )(n);
6. ∆−−f (n) := f (n)−2f (n−1)+f (n−2) = ((δ0−2δ1+δ2)∗f )(n);
7. ∆dd f (n) := f (n + 2)− 2f (n) + f (n − 2) =((δ−2 − 2δ0 + δ2) ∗ f )(n);
for n ∈ Z, [Bateman, 1943].
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2. Finite difference operators on `1(Z)
Proposition
The following equalities hold:
(i)
∆d = −(D+ + D−) = −D+D−,
D = −(D+ − D−) = (−D+ + 2I )D− = (D− − 2I )D+,
∆dd = (∆++ − 2∆d + ∆−−) = D2f .
(ii) (D+)′ = D−; (D−)′ = D+; (∆d)′ = ∆d ; (D)′ = D;(∆dd)′ = ∆dd .
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2. Finite difference operators on `1(Z)
-4 -2 2 4Real axis
-2
-1
1
2
Imaginary axisSpectrum sets of finite difference operators
D+,D-
Δd ,Δdd
Δ++,Δ--
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2. Finite difference operators on `1(Z)
eza :=∞∑n=0
zna∗n
n!, z ∈ C.
cosh(za) :=∞∑n=0
z2na∗2n
(2n)!, z ∈ C.
Proposition
Let A ∈ B(lp(Z)), with Af = a ∗ f , f ∈ `p(Z) and a ∈ `∞(Z).Then, the Fourier transform of the semigroup {eat}t≥0 generatedby a is given by
F(eat)(θ) = eF(a)(θ)t .
F(cosh(at))(θ) = cosh(F(a)(θ)t).
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2. Finite difference operators on `1(Z)
eza :=∞∑n=0
zna∗n
n!, z ∈ C.
cosh(za) :=∞∑n=0
z2na∗2n
(2n)!, z ∈ C.
Proposition
Let A ∈ B(lp(Z)), with Af = a ∗ f , f ∈ `p(Z) and a ∈ `∞(Z).Then, the Fourier transform of the semigroup {eat}t≥0 generatedby a is given by
F(eat)(θ) = eF(a)(θ)t .
F(cosh(at))(θ) = cosh(F(a)(θ)t).
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2. Finite difference operators on `1(Z)
Operator F(·)(z) Associated semigroup
−D+ z − 1 e−z zn
n!χN0(n) =: gz,+(n)
−D−1z − 1 e−z z−n
(−n)!χ−N0(n) =: gz,−(n)
∆d z + 1z − 2 e−2z In(2z)
D z − 1z Jn(2z)
−D 1z − z J−n(2z)
−D+ + 2 z + 1 ez zn
n!χN0(n)
−D− + 2 1z + 1 ez zn
n!χ−N0(n)
∆++ z2 − 2z + 1 ez (i√
2z)−nH−n(2iz)(−n)! χN0(n) =: hz,+(n)
∆−−1z2 − 2 1
z + 1 ez (i√
2z)nHn(2iz)n! χ−N0(n) =: hz,−(n)
∆dd z2 − 2 + 1z2 e−2z In(2z)χ2Z(n)
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2. Finite difference operators on `1(Z)
Theorem
(i) The Bessel function Jn has a factorization expression given by
Jn(2z) = (g−z,+ ∗ gz,−)(n), n ∈ Z, z ∈ C.
(ii) The Bessel function In admits factorization product given by
e−2z In(2z) = (gz,+ ∗ gz,−)(n),
In(2z) = (jz,+ ∗ jz,−)(n).
(iii) The Bessel function e−2z In(2z)χ2Z(n) admits a factorizationgiven by
In(2z)χ2Z(n) = hz,+(n) ∗ In(2z) ∗ e−2z In(2z) ∗ hz,−(n).
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3. Fractional powers of discrete operators
The Generalized Binomial Theorem is given by
(a + b)α =∞∑j=0
(α
j
)aα−jbj , α ∈ C.
For α > 0,
(α
j
)∼ 1
jα+1and ‖a‖ ≤ 1
(δ0 + a)α =∞∑j=0
(α
j
)aj , α > 0.
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3. Fractional powers of discrete operators
The Generalized Binomial Theorem is given by
(a + b)α =∞∑j=0
(α
j
)aα−jbj , α ∈ C.
For α > 0,
(α
j
)∼ 1
jα+1and ‖a‖ ≤ 1
(δ0 + a)α =∞∑j=0
(α
j
)aj , α > 0.
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3. Fractional powers of discrete operators
The Generalized Binomial Theorem is given by
(a + b)α =∞∑j=0
(α
j
)aα−jbj , α ∈ C.
For α > 0,
(α
j
)∼ 1
jα+1and ‖a‖ ≤ 1
(δ0 + a)α =∞∑j=0
(α
j
)aj , α > 0.
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3. Fractional powers of discrete operators
For 0 < α < 1, the Balakrishnan’s formula is expressed by
(−A)αx =1
Γ(−α)
∫ ∞0
(T (t)x − x)dt
t1+α, x ∈ D(A).
TheoremLet 0 < α < 1, and A ∈ B(`p(Z)), 1 ≤ p ≤ ∞ a generator of auniformly bounded semigroup, with Af = a ∗ f , f ∈ `p(Z) anda ∈ `1(Z). Then the fractional powers (−A)α is well-posedness andit is expressed by
(−A)αf = (−a)α ∗ f ,
where
(−a)α(n) :=1
2π
∫ 2π
0(−F(a)(θ))αe−inθdθ.
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3. Fractional powers of discrete operators
For 0 < α < 1, the Balakrishnan’s formula is expressed by
(−A)αx =1
Γ(−α)
∫ ∞0
(T (t)x − x)dt
t1+α, x ∈ D(A).
TheoremLet 0 < α < 1, and A ∈ B(`p(Z)), 1 ≤ p ≤ ∞ a generator of auniformly bounded semigroup, with Af = a ∗ f , f ∈ `p(Z) anda ∈ `1(Z). Then the fractional powers (−A)α is well-posedness andit is expressed by
(−A)αf = (−a)α ∗ f ,
where
(−a)α(n) :=1
2π
∫ 2π
0(−F(a)(θ))αe−inθdθ.
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3. Fractional powers of discrete operators
Λα(m) :=(−α−1+m
m
)= (−1)m
(αm
), for m ∈ N0.
Fractional power Kernel Explicit expression
Dα+ Kα
+ Λα(n)χN0
Dα− Kα
− Λα(n)χ−N0
(−∆d)α Kαd
(−1)nΓ(2α+1)Γ(1+α+n)Γ(1+α−n)
Dα KαD+
in
2Γ(α+1)
Γ(α2
+ n2
+1)Γ(α2− n
2+1)
(−D)α KαD−
(−i)n2
Γ(α+1)Γ(α
2+ n
2+1)Γ(α
2− n
2+1)
(−D+ + 2I )α Kα(−D++2I ) (−1)mΛα(n)χN0
(−D− + 2I )α Kα(−D−+2I ) (−1)mΛα(n)χ−N0
∆α++ Kα
D++Λ2α(n)χN0
∆α−− Kα
D−−Λ2α(n)χ−N0
(−∆dd)α Kαdd
(−i)n2
Γ(2α+1)Γ(α+ n
2+1)Γ(α− n
2+1)
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3. Fractional powers of discrete operators
TheoremLet α ∈ (0, 1). These kernels admit following factorizationequalities.
1. Kαd = Kα
− ∗ Kα+,
2. KαD+
= Kα−D−+2 ∗ Kα
+,
3. KαD− = Kα
−D++2 ∗ Kα−,
4. Kα++ = Kα
D+∗ KαD+
,
5. Kα−− = Kα
D− ∗ KαD− ,
6. Kαdd = Kα
D− ∗ KαD+
.
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3. Fractional powers of discrete operators
TheoremLet α ∈ (0, 1). These kernels admit following factorizationequalities.
1. Kαd = Kα
− ∗ Kα+,
2. KαD+
= Kα−D−+2 ∗ Kα
+,
3. KαD− = Kα
−D++2 ∗ Kα−,
4. Kα++ = Kα
D+∗ KαD+
,
5. Kα−− = Kα
D− ∗ KαD− ,
6. Kαdd = Kα
D− ∗ KαD+
.
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4. Fundamental solutions of fractional evolution equations
We consider the operator Bf (n) = (b ∗ f )(n), with b ∈ `∞(Z),f ∈ `p(Z), p ∈ [1,∞]. We obtain an explicit representation ofsolutions of{
Dβt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n) n ∈ Z.
Here β ∈ (1, 2] is real number. We recall that Dβt denotes theCaputo fractional derivative given by
Dβt v(t) =1
Γ(2− β)
∫ t
0(t − s)1−βv ′′(s)ds = (g2−β ∗ v ′′)(t).
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4. Fundamental solutions of fractional evolution equationsFor β, γ > 0, and b ∈ `∞(Z) we define
Bβ,γ(n, t) :=∞∑j=0
b∗j(n)gjβ+γ(t) =∞∑j=0
b∗j(n)t jβ+γ−1
Γ(jβ + γ), n ∈ Z, t > 0.
LemmaIf b ∈ `1(Z) then Bβ,γ(·, t) ∈ `1(Z), for t > 0 and β, γ > 0.
B1,1(n, t) =∞∑j=0
b∗j(n)t j
j!= etb(n);
B2,1(n, t) =∞∑j=0
b∗j(n)t2j
(2j)!= cosh(tb)(n);
(Bβ,γ(n, ·) ∗ gα)(t) = Bβ,γ+α(n, t), α, t > 0.
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4. Fundamental solutions of fractional evolution equationsFor β, γ > 0, and b ∈ `∞(Z) we define
Bβ,γ(n, t) :=∞∑j=0
b∗j(n)gjβ+γ(t) =∞∑j=0
b∗j(n)t jβ+γ−1
Γ(jβ + γ), n ∈ Z, t > 0.
LemmaIf b ∈ `1(Z) then Bβ,γ(·, t) ∈ `1(Z), for t > 0 and β, γ > 0.
B1,1(n, t) =∞∑j=0
b∗j(n)t j
j!= etb(n);
B2,1(n, t) =∞∑j=0
b∗j(n)t2j
(2j)!= cosh(tb)(n);
(Bβ,γ(n, ·) ∗ gα)(t) = Bβ,γ+α(n, t), α, t > 0.
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4. Fundamental solutions of fractional evolution equationsFor β, γ > 0, and b ∈ `∞(Z) we define
Bβ,γ(n, t) :=∞∑j=0
b∗j(n)gjβ+γ(t) =∞∑j=0
b∗j(n)t jβ+γ−1
Γ(jβ + γ), n ∈ Z, t > 0.
LemmaIf b ∈ `1(Z) then Bβ,γ(·, t) ∈ `1(Z), for t > 0 and β, γ > 0.
B1,1(n, t) =∞∑j=0
b∗j(n)t j
j!= etb(n);
B2,1(n, t) =∞∑j=0
b∗j(n)t2j
(2j)!= cosh(tb)(n);
(Bβ,γ(n, ·) ∗ gα)(t) = Bβ,γ+α(n, t), α, t > 0.
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4. Fundamental solutions of fractional evolution equations
{Dβt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n) n ∈ Z.
TheoremLet b ∈ `1(Z), ϕ, φ, g(·, t) ∈ `p(Z) for t > 0 andsup
s∈[0,t]||g(·, s)||p <∞ . Then the function
u(n, t) =∑m∈Z
Bβ,1(n −m, t)ϕ(m) +∑m∈Z
Bβ,2(n −m, t)φ(m)
+∑m∈Z
∫ t
0Bβ,β(n −m, t − s)g(m, s)ds,
is the unique solution of the initial value problem on `p(Z) for1 ≤ p ≤ ∞.
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5. Explicit solutions for fractional powers
Given ϕ, φ, g(·, t) ∈ `p(Z), a ∈ `1(Z) and
Af = a ∗ f , f ∈ `p(Z).
Now we consider the following evolution problem{Dβt u(n, t) = ±(±A)αu(n, t) + g(n, t), n ∈ Z, t > 0.
u(n, 0) = ϕ(n), ut(n, 0) = φ(n), n ∈ Z.
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5. Explicit solutions for fractional powers
1. D+f (n) := f (n)− f (n + 1) = ((δ0 − δ−1) ∗ f )(n);
2. D−f (n) := f (n)− f (n − 1) = ((δ0 − δ1) ∗ f )(n);
3. ∆d f (n) := f (n+1)−2f (n)+f (n−1) = ((δ−1−2δ0+δ1)∗f )(n);
4. Df (n) := f (n + 1)− f (n − 1) = ((δ−1 − δ1) ∗ f )(n);
5. ∆++f (n) := f (n + 2)− 2f (n + 1) + f (n) =((δ−2 − 2δ−1 + δ0) ∗ f )(n);
6. ∆−−f (n) := f (n)−2f (n−1)+f (n−2) = ((δ0−2δ1+δ2)∗f )(n);
7. ∆dd f (n) := f (n + 2)− 2f (n) + f (n − 2) =((δ−2 − 2δ0 + δ2) ∗ f )(n);
for n ∈ Z.
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5. Explicit solutions for fractional powers
Fractional power Kernel Explicit expression
Dα+ Kα
+ Λα(n)χN0
Dα− Kα
− Λα(n)χ−N0
(−∆d)α Kαd
(−1)nΓ(2α+1)Γ(1+α+n)Γ(1+α−n)
Dα KαD+
in
2Γ(α+1)
Γ(α2
+ n2
+1)Γ(α2− n
2+1)
(−D)α KαD−
(−i)n2
Γ(α+1)Γ(α
2+ n
2+1)Γ(α
2− n
2+1)
(−D+ + 2I )α Kα(−D++2I ) (−1)mΛα(n)χN0
(−D− + 2I )α Kα(−D−+2I ) (−1)mΛα(n)χ−N0
∆α++ Kα
D++Λ2α(n)χN0
∆α−− Kα
D−−Λ2α(n)χ−N0
(−∆dd)α Kαdd
(−i)n2
Γ(2α+1)Γ(α+ n
2+1)Γ(α− n
2+1)
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5. Explicit solutions for fractional powers
Aα Bαβ,γ
Dα+ (−1)n
∞∑j=0
(αj
n
)gjβ+γ(t)χN0(n)
Dα− (−1)n
∞∑j=0
(αj
−n
)gjβ+γ(t)χ−N0(n)
∆α++ (−1)n
∞∑j=0
(2αj
n
)gjβ+γ(t)χN0(n)
∆α−− (−1)n
∞∑j=0
(2αj
−n
)gjβ+γ(t)χ−N0(n)
Dα in
2
∞∑j=0
Γ(αj + 1)
Γ(αj2 + n2 + 1)Γ(αj2 −
n2 + 1)
gjβ+γ(t)
(−D)α(−i)n
2
∞∑j=0
Γ(αj + 1)
Γ(αj2 + n2 + 1)Γ(αj2 −
n2 + 1)
gjβ+γ(t)
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5. Explicit solutions for fractional powers
−(−A)α Bαβ,γ
−(−∆d)α (−1)n∞∑j=0
(−1)jΓ(2αj + 1)
Γ(αj + n + 1)Γ(αj − n + 1)gβj+2(t)
−(−∆dd)α(−i)n
2
∞∑j=0
(−1)jΓ(2αj + 1)
Γ(αj + n2 + 1)Γ(αj − n
2 + 1)gβj+β(t)
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Bibliography
[AMT] L. Abadias, M. de Leon-Contreras and J.L. Torrea.Non-local fractional derivatives. Discrete and continuous. J. Math.Anal. Appl., (2017).[B] H. Bateman, Some simple differential difference equations andthe related functions. Bull. Amer. Math. Soc. (1943).[Bo] S. Bochner. Diffusion equation and stochastic processes.Proc. Nat. Acad. Sci. U. S. A. (1949).[CGRTV] O. Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea and J.L.Varona, Harmonic analysis associated with a discrete Laplacian J.Anal. Math. (2017).[GKLW] J. Gonzalez-Camus, V. Keyantuo, C. Lizama and M.Warma. Fundamental solutions for discrete dynamical systemsinvolving the fractional Laplacian. Mathematical Methods in theApplied Sciences, (2019).[LR] C. Lizama and L. Roncal HAulder-Lebesgue regularity andalmost periodicity for semidiscrete equations with a fractionalLaplacian. Discrete and Continuous Dynamical Systems, (2018).
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MUCHAS GRACIAS Pedro J. Miana, IUMA-UZ
GRAFFITIS MATEMATICOS: Antonio Ledesma López
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Index
1. Introduction
2. Finite difference operators on `1(Z)
3. Fractional powers of discrete operators
4. Fundamental solutions of fractional evolution equations
5. Explicit solutions for fractional powers
Bibliography