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Green function method for a fractional–order delay differential equation

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UDC 517.91

M.G. Mazhgikhova

Institute of Applied Mathematics and Automation, Nalchik, Russia (E-mail: mazhgihova.madina@yandex.ru)

Green function method for a fractional–order delay differential equation

In this paper, we investigated a boundary value problem with the Sturm-Liouville type conditions for a linear ordinary differential equation of fractional order with delay. The condition for the unique solvability of the problem is obtained in the form 4 6= 0. The Green function of the problem, in terms of which the solution of the boundary value problem under study is written out, is constructed. The existence and uniqueness theorem for the solution of the problem is proved. It is also showed that in the case when the condition of unique solvability is violated, i.e4= 0, then the solution of the boundary value problem is not unique. Using the notation of the generalized Mittag-Leffler function via the generalized Wright function, we also studied the properties of the function4asλ→ ∞andλ→ −∞. Using asymptotic formulas for the generalized Wright function, a theorem on the finiteness of the number of eigenvalues of a boundary value problem with the Sturm-Liouville type conditions is proved.

Keywords:Fractional differential equation, delay differential equation, Green function, generalized Mittag- Leffler function, generalized Wright function.

Introduction Consider the equation

dα

dtαu(t)−λu(t)−µH(t−τ)u(t−τ) =f(t), 0< t <1, (1) where dtdαα is the Riemann-Liouville fractional derivative [1], 1 < α≤2, λ, µ are the arbitrary constants, τ is the fixed positive number,H(t)denotes the Heaviside function.

At present, the number of studies on fractional calculation has noticeably increased. This is due to the fact that fractional order differential equations are used in mathematical modeling of processes that occur in various fields of natural science, such as physics, chemistry, biology, sociology, etc.

The most general references to the theory of fractional calculus one can find in [2–5] (see also the references in these works). A linear ordinary differential equation of fractional order was considered by Barrett [6] in 1954.

Existence and uniqueness theorem for a fractional-order differential equation is proved in [7] by Dzhrbashyan and Nersesyan. Sturm-Liuville type boundary value problem for fractional differential operator was investigated by Dzhrbashyan in [8]. The initial value problem for a linear ordinary differential equation of fractional order was studied by Pskhu in [9].

Significant works were devoted to the delay differential equations (difference-differential equations) by Norkin in [10], Bellman and Cooke in [11], Elsgolts and Norkin in [12], Myshkis [13], Hale in 1977 [14].

The initial-value problem and the problem with general linear two-point boundary conditions, the Dirichlet and the Neumann problems for linear ordinary differential equation with Caputo derivative with delay in [15–17]

respectevely were solved.

The Cauchy problem for Eq.(1) was considered in [18], and the solutions to the Dirichlet and the Neumann problems were obtained in [19].

In this paper, we construct the Green function of the Sturm-Liouville type boundary value problem for Eq.(1) and prove the finiteness theorem for the number of real eigenvalues of the study problem.

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Auxiliary The Riemann-Liouville fractional operator is define by the formula

dα

dtαu(t) =Datαu(t) = 1 Γ(n−α)

dn dtn

t

Z

a

u(ξ)dξ

(t−ξ)α−n+1, n−1< α≤n, n∈N,

where Γ(z) =

R

0

e−ttz−1dtis the Euler gamma function.

The Mittag-Leffler function is given by the power series [20]

Eα,β(z) =

X

k=o

zk Γ(αk+β), and the generalized Mittag-Leffler function defines by the series [21]

Eα,βρ =

X

k=o

(ρ)kzk Γ(αk+β)k!,

where (ρ)k = Γ(ρ+k)Γ(ρ) is the Pokhhammer symbol. The generalized Mittag-Leffler function reduces to Eα,β(z) when we setρ= 1.

Consider the function

Wν(t) =Wν(t,τ;λ, µ) =

X

m=0

µm(t−mτ)αm+ν−1+ Eα,αm+νm+1 (λ(t−mτ)α+), ν ∈R, (2) where

(t−mτ)+ =

t−mτ, t−mτ >0, 0, t−mτ≤0.

It follows from (2) that

Wk(i)(0) =

0, k6=i+ 1,

1, k=i+ 1. (3)

Remark 1. For somem the expression t−mτ < 0, therefore the series in (2) contains a finite number of termsN≤[τt] + 1.

Function (2) satisfies the following properties [16]

D0tαWν(t) =Wν−α(t), α∈R, ν >0, (4) Wν(t) =λWν+α(t) +µWν+α(t−τ) +tν−1

Γ(ν), α >0, ν∈R, (5) which are clear by the formula of differentiation [21]

dm dzm

zβ−1Eα,βρ (zα)

=zβ−m−1Eα,β−mρ (zα)) and by the autotransformation formula [22]:

Eα,βρ (t)−Eα,βρ−1(t) =tEα,α+βρ (t)

of the generalized Mittag-Leffler function.

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Main results

A functionu(t)is calleda regular solutionof Eq.(1) ifDα−20t u(t)∈C2(0,1),u(t)∈L(0,1)andu(t)satisfies Eq.

(1) for all0< t <1.

The problem we solve here is to find the regular solution to equation (1) satisfying the conditions alim

t→0D0tα−1u(t) +blim

t→0D0tα−2u(t) = 0, clim

t→1Dα−10t u(t) +dlim

t→1D0tα−2u(t) = 0,

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wherea2+b26= 0 andc2+d26= 0.

Green function Assume G(t, ξ)is given by

G(t, ξ) =H(t−ξ)Wα(t−ξ) + (cW1(1−ξ) +dW2(1−ξ))bWα(t)−aWα−1(t)

4 (7)

withλandµsatisfying the following condition

4=ac(λWα(1) +µWα(1−τ)) + (ad−bc)W1(1)−bdW2(1)6= 0. (8) Here the functionWν(t)is defined via (2).

We demonstrate the validity of the following properties for the functionG(t, ξ)(7).

1. The functionG(t, ξ)is continuous for all values of t andξfrom the closed interval [0,1].

This property implies from relation (7) and condition (8).

2. The functionG(t, ξ)satisfies the conditions

ε→0lim[Dα−20t Gξ(t, ξ)|ξ=t+ε−Dα−20t Gξ(t, ξ)|ξ=t−ε] = 1. (9)

Indeed,

Dα−0t2Gξ(t,ξ) =−H(t−ξ)W1(t−ξ)

cλWα(1−ξ)+cµWα(1−ξ−τ) +dW1(1−ξ)bW2(t)−aW1(t)

4 . (10)

Insert (10) into (9) asξ=t+εandξ=t−ε. Passing to the limit asε→0we get the property (9).

3. The functionG(t, ξ)is the solution to the equation

αG(t, ξ)−λG(t, ξ)−µH(1−τ−ξ)G(t, ξ+τ) = 0. (11) Here∂0tα is the Caputo derivative [23; 11] defines as

0tαu(t) =D0tα−2u00(t) = 1 Γ(2−α)

t

Z

0

u00(ξ)dξ (t−ξ)α−1. This property implies the presentation of the function (7) and the relations (4), (5).

4. The functionG(t, ξ)satisfies the boundary conditions

 alim

ξ→0D0tα−2Gξ(t, ξ) +blim

ξ→0Dα−20t G(t, ξ) = 0 clim

ξ→1Dα−20t Gξ(t, ξ) +dlim

ξ→1D0tα−2G(t, ξ) = 0 . (12) This property obviously implies the relations (4), (5).

The function G(t, ξ)that possesses properties 1-4 is called Green function for problem (1), (6).

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Existence and uniqueness theorem

Theorem 1. Assume the function f(t)∈L(0,1)∩C(0,1) and the condition (8)is satisfied. Then 1) there exists a regular solution to problem (1),(6)in the form of

u(t) =

1

Z

0

f(ξ)G(t, ξ)dξ; (13)

2) the solution to problem (1),(6)is unique if and only if condition (8) is satisfied.

Proof. First we illustrate that the solution to problem (1), (6) has the form (13). To clear this, multiply both sides of Eq. (1) (given in terms of variableξ) byDα−20t G(t, ξ)and integrate it with respect to variable ξ ranging fromεto 1−ε(ε→0):

1−ε

Z

ε

Dα−20t G(t, ξ)Dαu(ξ)dξ−λ

1−ε

Z

ε

u(ξ)Dα−20t G(t, ξ)dξ

−µ

1−ε

Z

ε

H(t−τ)u(ξ−τ)Dα−20t G(t, ξ)dξ=

1−ε

Z

ε

f(ξ)D0tα−2G(t, ξ)dξ, 0< t <1. (14) Integrate by parts the first term of equality (14):

1−ε

Z

ε

D0tα−2G(t,ξ)Dαu(ξ)dξ=Dα−20t G(t,ξ)Dα−1u(ξ)

1ε ε

t

Z

ε

D0tα−2Gξ(t,ξ)Dα1u(ξ)dξ

1−ε

Z

t

Dα−0t2Gξ(t,ξ)Dα−1u(ξ)dξ=Dα−0t2G(t,ξ)Dα−1u(ξ)

ξ=1−D0tα−2G(t,ξ)Dα−1u(ξ) ξ=0

+Dα−20t u(ξ)

Dα−20t Gξ(t,ξ)

ξ=t+0−Dα−20t Gξ(t,ξ) ξ=t−0

+Dα−20t Gξ(t,ξ)Dα−2u(ξ) ξ=0

−D0tα−2Gξ(t, ξ)Dα−2 u(ξ) ξ=1+

1

Z

0

D0tα−2Gξξ(t, ξ)Dα−2 u(ξ)dξ. (15) Applying to (15) the properties (9), (12) of function (7) and conditions (6) of the problem we get the following formula

Dα−20t u(ξ) +

1

Z

0

Dα−2u(ξ)D0tα−2Gξξ(t, ξ)dξ. (16) Replaceξwithξ−τ in the third integral on the left-hand side of the expression (14) to reduce it to

1

Z

0

H(ξ−τ)u(ξ−τ)G(t, ξ)dξ=

1

Z

0

H(1−τ−ξ)u(ξ)G(t, ξ+τ)dξ. (17) Put (16) and (17) into Eq. (14) and using the formula for fractional integration by parts [20, p. 15]

b

Z

a

g(s)Dasαh(s)ds=

b

Z

a

h(s)Dαbsg(s)ds,

arrive at identity

Dα−20t u(ξ) +D0tα−2

1

Z

0

u(ξ)h

Dα−2 Gξξ(t, ξ)−λG(t, ξ)−µH(1−t−ξ)G(t, ξ+τ)i dξ =

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=Dα−20t

1

Z

0

f(ξ)G(t, ξ)dξ.

Taking advantage of the third property of Green functionG(t, ξ)(11) and finding the derivative of orderD2−α0t we arrive at representation (13).

Next, we show that the function (13) is the solution to problem (1), (6). Formula (13) can be written out in terms of functionWν(t)in the form of bellow:

u(t) =

t

Z

0

f(ξ)Wα(t−ξ)dξ+bWα(t)−aWα−1(t) 4

1

Z

0

f(ξ) (cW1(1−ξ) +dW2(1−ξ))dξ.

Next, using formulas (4), (5) obtain by the previous relation that

Dα0tu(t) =f(t) +λ

1

Z

0

f(ξ)G(t, ξ)dξ+µ

1

Z

0

f(ξ)G(t, ξ−τ)dξ.

Prove that the solutionu(t)satisfies the boundary conditions (6) (in view of relation (3)):

alim

t→0Dα−10t u(t) +blim

t→0D0tα−2u(t) = 1 4

1

Z

0

f(ξ) [cW1(1−ξ) +dW2(1−ξ)]

×

abW1(0)−a2λWα(0)−a2µWα(−τ) +b2W2(0)−abW1(0) dξ = 0;

clim

t→1Dα−10t u(t) +dlim

t→1D0tα−2u(t) =

1

Z

0

f(ξ)[cW1(1−ξ) +dW2(1−ξ)]×

×

1 +−ac(λWα(1) +µWα(1−τ))−(ad−cb)W1(1) +bdW2(1) 4

dξ=

=

1

Z

0

f(ξ)(cW1(1−ξ) +dW2(1−ξ))

1−4 4

dξ= 0.

The task is now to show that if the condition (8) is not satisfied 4= 0,

then the solution of the problem is not unique. Consider the function

¯

u(t) =C1Wα(t) +C2Wα−1(t), which is the solution to the problem

Dα0tu(t)¯ −λ¯u(t)−µH(t−τ)¯u(t−τ) = 0, alim

t→0Dα−10t u(t) +blim

t→0Dα−20t u(t) = 0, clim

t→1D0tα−1u(t) +dlim

t→1Dα−20t u(t) = 0.

(18) The conditions (18) can be written out in the form

aC1+bC2= 0,

C1[W1(1) +dW2(1)] +C2[cλWα(1) +cµWα(1−τ) +dW1(1)] = 0. (19) Then the determinant of the system (19) is equal to

4=

a b

cW1(1) +dW2(1) cλWα(1) +cµWα(1−τ) +dW1(1)

= 0.

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Thus, solution to problem (1), (6) is unique if and only if condition (8) is satisfied.

Remark.For all

λ, µ >0, (a−b)(c+d)>0 condition (8) is always satisfied.

On the finiteness of the number of real eigenvalues

Definition. The eigenvalues of problem (1), (6) are the values λ, such that problem (1), (6) has a regular solution that is not the identically zero.

The set of real eigenvalues for problem (1), (6) coincides with the set of real zeros for the function

Φ(λ) =ac(λWα(1) +µWα(1−τ)) + (ad−bc)W1(1)−bdW2(1). (20) Theorem 2. Problem (1),(6) has only a finite number of real eigenvalues.

The function Wν(λ)can be written out as [2, p. 45]

Wν(1, τ;λ, µ) =

X

m=0

µm

m!(1−mτ)αm+ν−1+ 1Ψ1

(m+ 1,1) (αm+ν, α)

λ(1−mτ)α+

,

where

pΨq

(al, αl)1,p

(bl, βl)1,q

z

=

X

k=0

Qp

l=1Γ(allk) Qq

l=1Γ(bllk) zk k!

is the generalized Wright function [24].

Function (20) is an integer function of parameter λ. Let us investigate the properties of the function (20) asλ→+∞andλ→ −∞.

As λ→+∞the following asymptotic formula holds true for the generalized Wright function [24], [25]:

1Ψ1

(m+ 1,1) (αm+ν, α)

λ(1−mτ)α+

−mλm(1−α)−ν+1α (1−mτ)m(1−α)−ν+1+ eλ1/α(1−mτ)+

1 +O 1

λα1

. LetN be the maximum value ofmthat satisfies the inequality(1−mτ)>0. Then the asymptotic formula for function (20) is in the form

Φ(λ) =

X

m=0

µmα−m

m λm−αmα+1h

(1−mτ)m+eλ1/α(1mτ)+

ac+(ad−bc)λ−1/α−bdλ−2/α

+acµ

λ(1−(m+ 1)τ)m+eλ1/α(1−(m+1)τ)i

×h

1 +O(λ−1/α)i . Hence, asλ→ ∞, the series above increases without limit.

The asymptotic formula for the generalized Wright function as λ→ −∞has form [24], [25]

1Ψ1

(m+ 1,1) (αm+ν, α)

λ(1−mτ)α+

=

n

X

l=0

(−1)m+l+1(l+m)!(1−mτ)−α(m+l+1)+

|λ|m+l+1Γ(ν−α−αl)(m+l+ 1)!

+O 1

|λ|m

. Therefore

Φ(λ) =

N

X

m=0

(−1)m+1µm (m+ 1)!|λ|m

ac Γ(−α)

(1−mτ)−1+ +µ(1−(m+ 1)+τ)−1

|λ|

+(ad−bc)(1−mτ)−α

|λ|Γ(1−α) −bd(1−mτ)1−α

|λ|Γ(2−α) +O 1

|λ|N+1

. (21)

Consider the limit relation in the case when µ6= 0

λ→−∞lim λNΦ(λ) =ac(−1)N+1µN(1−N τ)−1+

Γ(−α)(N+ 1)! 6= 0. (22)

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As µ= 0we have

λ→−∞lim λΦ(λ) =− ac

Γ(−α). (23)

SinceΦ(λ)is an entire function of the variableλ, it follows from relations (21), (22), and (23) that the series (20) may have only a finite number of real zeros. This establishes the theorem.

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М.Г. Мажгихова

Бөлшек реттi кешiкпелi аргументтi дифференциалдық теңдеу үшiн Грин функциясы әдiсi

Мақалада кәдiмгi сызықтық тұрақты коэффициенттi кешiкпелi аргументтi бөлшек реттi дифферен- циалдық теңдеу үшiн Штурм-Лиувилль типтi шеттiк есеп зерттелген. Қойылған есептiң бiрмәндi шешiлуi4 6= 0түрiнде алынды. Зерттелiп отырған есептi шешу үшiн Грин әдiсi қолданылды. Грин функциялары Миттаг-Леффлер жалпыланған функциялары терминiнде жазылды. Зерттелiп отыр- ған есептiң шешуiнiң бар болуы және жалғыздығы жайлы теорема дәлелдендi. Бiрмәндi шешiлу шарты бұзылған жағдайда, яғни4= 0болғанда, шеттiк есептiң шешуi жалғыз еместiгi нақтылан- ды. Сонымен қоса Миттаг-Леффлер жалпыланған функцияларын Райт жалпыланған функциялары арқылы жазуды қолданып,λ үлкен мәндерiнде 4функцияларының қасиеттерi, яғни λ → ∞және λ→ −∞ болғанда, оқылды. Райттың жалпыланған функциялары үшiн асимптотикалық формула- ларын қолданып, Штурм-Лиувилль типтi шарттарымен берiлген шеттiк есептiң меншiктi мәндерiнiң сандарының ақырлылығы жайлы теорема анықталды.

Кiлт сөздер: бөлшек реттi дифференциалдық теңдеулер, кешiкпелi аргументтi дифференциалдық теңдеулер, Грин функциясы, Миттаг-Леффлер жалпыланған функциясы, Райт жалпыланған функ- циясы.

М.Г. Мажгихова

Метод функции Грина для дифференциального уравнения дробного порядка с запаздывающим аргументом

В статье исследована краевая задача с условиями типа Штурма-Лиувилля для линейного обыкновен- ного дифференциального уравнения дробного порядка с запаздывающим аргументом с постоянными коэффициентами. Условие однозначной разрешимости поставленной задачи получено в виде4 6= 0.

Для решения исследуемой задачи авторами применен метод функции Грина, в терминах которой и выписано решение краевой задачи. Функции Грина, в свою очередь, записаны в терминах обоб- щенной функции Миттаг-Леффлера. Доказана теорема существования и единственности решения исследуемой задачи. Отмечено, что в случае, когда условие однозначной разрешимости нарушается, то есть при 4 = 0, решение краевой задачи не единственно. Используя запись обобщенной функ- ции Миттаг-Леффлера через обобщенную функцию Райта, изучены также свойства функции4при больших значениях λ, то есть при λ → ∞ и λ → −∞. Применяя асимптотические формулы для обобщенной функции Райта, определена теорема о конечности числа собственных значений краевой задачи с условиями типа Штурма-Лиувилля.

Ключевые слова: дифференциальное уравнение дробного порядка, дифференциальное уравнение с запаздывающим аргументом, функция Грина, обобщенная функция Миттаг-Леффлера, обобщенная функция Райта.

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