On a method for constructing a solution of integro-differential equations of fractional order

On a method for constructing a solution of integro-differential equations of fractional order

Batirkhan Kh. Turmetov

Akhmet Yasawi International Kazakh-Turkish University, 29 B. Sattarkhanov Ave, Turkestan, 161200, Kazakhstan

Received 31 May 2017, appeared 11 May 2018 Communicated by Nickolai Kosmatov

Abstract. In this paper, we propose a new method for constructing a solution of the integro-differential equations of Volterra type. The particular solutions of the homo- geneous and of the inhomogeneous equation will be constructed and the Cauchy type problems will be investigated. Note that this method is based on construction of nor- malized systems functions with respect to the differential operator’s fractional order.

Keywords: integro-differential equation of Volterra type, Riemann–Liouville fractional integrals and derivatives, generalized Mittag-Leffler function, method normalized sys- tems of functions.

2010 Mathematics Subject Classification: 45J05, 26A33, 33E12.

1 Introduction

Letα>0,ν ≥0,β∈ R,λ6=0. On the domain 0<t ≤d<_∞consider an integro-differential equation of the following form:

D^αy(t) =λt^βI^νy(t) + f(t), 0<t ≤d<_∞, (1.1) where for any δ>_{0 :}

I^δy(t) = ¹ Γ(δ)

Z _t

0

(t−τ)^δ−1y(τ)dτ, andD^α is the derivative ofαorder in the Riemann–Liouville sense, i.e.

D^αy(t) = ^d

m

dt^mI^m−αy(t), m= [α] +1.

We denote

C_δ[0,d] =ⁿf(t):∃δ ∈[0, 1),t^δf(t)∈ C[0,d]^o.

By a solution of equation (1.1) we mean a function y(t) _{such that} ∃δ ∈ [_0,m−α]_,y(t) ∈ C_δ[0,d],D^αy(t)∈C[0,d]andy(t)satisfies equation (1.1) at all points t∈(0,d).

BEmail: turmetovbh@mail.ru

Questions related to theorems about existence and uniqueness of solutions of Cauchy type and Dirichlet type problems for linear and nonlinear fractional order differential equations have been developed in sufficient detail (see [15] for the main results, review of papers, and references). In [1], equation (1.1) was studied in the case β = 0 . In a more general case, an equation of the type (1.1) and a Cauchy-type problem for them were studied in [3,5,7–9, 16]. Theorems on existence and uniqueness of a solution of Cauchy-type problem have been proved. We note that explicit solutions have been constructed only for certain types of linear differential equations of fractional order. Solutions of certain elementary homogeneous and inhomogeneous equations, obtained by the selection method or by expansion of the desired solution into a quasi-power series, are known. Moreover, explicit solutions of a Cauchy-type problem for certain differential equations of fractional order were found in [12] by the method of reduction to an equivalent Volterra integral equation. Further, in [14], using the properties of Mittag-Leffler type functions:

E_α,m,l(z) =

∑

c_izⁱ, c₀ =1, c_i =

i−1

∏

Γ[α(km+l) +1]

Γ[α(km+l+1) +1]^{, i}≥1, (1.2) an algorithm for constructing a solution of the differential equation (1.1) in the caseν=0 was proposed. Moreover, the case, when f(t) =0 and f(t)is a quasi-polynomial, was considered.

Further, in [13] an analogous algorithm was used to construct a solution of the equation (1.1) in the caseα= β=0.

In this paper we propose a new method for constructing an explicit solution of integro- differential equations of fractional order. This method is based on construction of normalized systems with respect to a pair of operators D^α,λt^β

(see Section2). Moreover, in contrast to [14], we construct particular solutions of the inhomogeneous equation for a more general class of functions f(t). We also note that this method was used in [2,17] to construct solutions of certain linear differential equations of integer and fractional order with constant coefficients.

2 Normalized systems

In this section we give some information on normalized systems related to linear differential operators. LetL₁andL₂be linear operators, acting from the functional spaceXtoX,L_kX⊂X, k = 1, 2. Let functions from X be defined in a domain Ω ⊂ Rⁿ. Let us give the definition of normalized systems [10].

Definition 2.1. A sequence of functions {f_i(x)}^∞_i=0, f_i(x) ∈ X is called f-normalized with respect to (L1,L2) on Ω, having the base f0(x), if on this domain the following equality holds:

L₁f₀(x) = f(x), L₁f_i(x) =L₂f_i−1(x), i≥1.

If L2 = E is a unit operator, then f-normalized with respect to(L₁,I)system of functions is called f-normalized with respect toL₁, i.e.

L₁f₀(x) = f(x), L₁f_i(x) = f_i−1(x), i≥1.

If f(x) =0, then the system of functions{f_i(x)}is called just normalized.

The main properties of the f-normalized systems of functions with respect to the oper- ators (L₁,L₂) on Ω have been described in [11]. Let us consider the main property of the

f-normalized systems.

Proposition 2.2. If a system of functions {f_i(x)}^∞_i=0 is f -normalized with respect to(L₁,L₂)on Ω, then the functional series y(x) =_∑^∞_i=0 f_i(x),x∈Ω, is a formal solution of the equation:

(L₁−L₂)y(x) = f(x), x∈_Ω. (2.1) The next proposition allows to construct an f-normalized system with respect to a pair of operators(L₁,L₂).

Proposition 2.3. If for L₁ there exists a right inverse operator L⁻₁¹, i.e. L₁·L⁻₁¹ = E, where E is a unit operator and L₁f₀(x) = f(x), then a system of the functions

f_i(x) =L⁻₁¹·L₂i

f₀(x),i≥1, is f -normalized with respect to a pair of the operators(L₁,L₂)on Ω.

Proof. SinceL₁·L⁻₁¹= Eis the unit operator, then for alli=1, 2, . . . , we have L₁f_i(x) =L₁

L⁻₁¹·L2

i

f0(x) =L₁

L⁻₁¹·L2 L⁻₁¹·L2

i−1

f0(x)

= L₂

L⁻₁¹·L₂i−1

f₀(x) =L₂f_i−1(x).

Consequently, L₁f_i(x) = L₂f_i−1(x)and by assumption of the theorem L₁f₀(x) = f(x)i.e. the system f_i(x) = L⁻₁¹·L₂i

f₀(x), i ≥ 0, is f-normalized with respect to the pair of operators L₁,L₂

.

3 Properties of operators I

and D

Consider some properties of the operators I^α andD^α. The following statements are known [15].

Lemma 3.1. Letα>0. Then for all f(t)∈ C_δ[0,d]the equality

D^α[I^α[f]] (t) = f(t) (3.1) holds for all t ∈(0,d]. If f(t)∈C[0,d],then(3.2)holds for all t ∈[0,d].

Lemma 3.2. Letα>0,m= [α] +1and s∈ R. Then the following equalities hold:

I^αt^s= ^Γ(s+₁) Γ(s+1+α)^t

s+α, s >−1, (3.2)

D^αt^s= ^Γ(s+1) Γ(s+1−α)^t

s−α, s >α−1, (3.3)

D^αt^s=_{0, s} =α−j, j=1, 2, . . . ,m. (3.4) Corollary 3.3. Letα>0,m= [α] +1. Then the equality D^αy(t) =0holds if and only if

y(t) =

∑

c_jt^α−j, where c_j are arbitrary constants.

Lemma 3.4. Letα>0,m= [α] +1, 0≤δ <1and f(t)∈C_δ[a,b]. Then 1) ifα<δ, then I^αf(t)∈C_δ−α[a,b]and

kI^αfk_C

δ−α[a,b] ≤ Mkfk_C

δ[a,b], M= ^Γ(1−δ)

Γ(1+α−δ)^{; (3.5)} 2) ifα≥δ, then I^αf(t)∈C[a,b]and

kI^αfk_C[a,b] ≤ Mkfk_C

δ[a,b], M = (b−a)^α−δ_Γ(1−δ)

Γ(1+α−δ) ^{. (3.6)}

4 Construction of 0-normalized systems

In this section we construct 0 - normalized systems with respect to the pair of the operators D^α,λt^βI^ν

. To do it from Proposition2.3 it follows that it is necessary to find all solutions of the equation D^αy(t) = 0 and a right inverse for the operator D^α. According to statement of Lemma3.1the right inverse of the the operator D^α is the operator I^α, and due to (3.4) linear independent solutions of the equation D^αy(t) =0 are functions t^sj,s_j = α−j,j= 1, 2, . . . ,m.

Hereinafter, we denote L₁ = D^α and L2 = λt^βI^ν. Then the equation (1.1) is represented as (2.1). For real numbersα>0,ν≥0,δ >0,s∈ Rwe introduce the following coefficients:

C_α,ν(δ,s,i) =

i−1

∏

Γ(δk+s+1)

Γ(δk+s+1+ν)·^Γ[δ(k+1) +s+1−α]

Γ[δ(k+1) +s+1] ^{, i}≥1, C_α,ν(δ,s, 0) =1, s∈ R.

From the properties of the gamma function we haveCα,ν(δ,s,i)6=0. It’s obvious that Cα,0(δ,s,i) =

i−1 k

∏

Γ[δ(k+₁) +₁+s−α] Γ[δ(k+1) +1+s]) ^.

Lets_j = α−j, j=1, 2, . . . ,mand f_0,sj(t) = t^sj, then due to (3.4): L₁f_0,sj(t) =0, j=1, 2, . . . ,m.

We consider the system of functions:

f_i,j(t) =I^α·λt^βI^νi

f_0.sj(t)_, i≥1. (4.1)

Since (D^α)⁻¹ = I^α and D^αf_0,sj(t) = 0, then Proposition 2.3 implies that the system (4.1) is 0-normalized with respect to the pair of the operators D^α,λt^βI^ν

. We find explicit form of the system of functions f_i(t). Hereinafter, everywhere we will assume that α > 0, m = [α] +1, ν≥0, β>−{α} −ν. The following proposition is valid.

Lemma 4.1. Let s≥α−m,g_i(t) = I^αt^βI^νi

t^s,i≥1.Then 1) for the function g_i(t)the following equality holds:

g_i(t) =C_α,ν(α+β+ν,s,i)t^{(α+β+ν)i+s}, i≥1; (4.2) 2) the function g_i(t)at least belongs to the class Cm−α[0,d].

Proof. Note that due to (3.2) fors≥ α−mthe equality I^νt^s= ^Γ(s+1)

Γ(s+₁+ν)^t

s+ν

holds. Let i=1. Then due to the inequality s+β+ν ≥α−m+β+ν=−1+{α}+β+ν>

−1 and properties of the operator I^α, for the functiong₁(t)we have g₁(t) = I^α

t^βI^νt^s

= ^Γ(s+1) Γ(s+1+ν)^I

αt^s+ν+β = ^Γ(s+1) Γ(s+1+ν)

Γ(β+ν+s+1) Γ(α+ν+β+s+1)^t

s+ν+β+α

=C_α,ν(α+β+ν,s, 1)t^α+β+ν+s.

Due to the inequalityα+β+ν+s>α−mit follows that at leastg₁(t)∈C_m−α[0,d].

Further, in general case by the mathematical induction method it is possible to show valid- ity of the equality (4.2). Indeed, let for some positive integerr the equality (4.2) holds. Then forr+1 we get:

g_r+1(t) =I^α·t^βI^νr+₁

t^s =I^α·t^βI^ν I^α·t^βI^νr

t^s = I^αh

t^βI^νg_r(t)ⁱ

=C_α,ν(α+β+ν,s,r)I^αh

t^βI^νt^{r(α+β+ν)+s}i

=C_α,ν(α+β+_ν,s,r) ^Γ(r(α+β+ν) +s+1) Γ(r(α+β+ν) +s+1+ν)^I

αt^{r(α+β+ν)+β+ν+s}

=C_α,ν(α+β+ν,s,r) ^Γ(r(α+β+ν) +s+1) Γ(r(α+β+ν) +s+1+ν)

×^Γ(_r(α+β+ν) +α+β+ν+_s+₁−α) Γ(r(α+β+ν) +α+β+ν+s+1) ^t

r(α+β+ν)+α+β+ν+s

=C_α,ν(α+β+_ν,s,r+₁)t^{(r+1)(α+β+ν)+s}.

Therefore, (4.2) is true also for the case r+1. It is obvious that for any r ≥ 1 at least g_r+1(t)∈ C_m−α[0,d]andD^αg_r+1(t)∈ C(0,d).

From the lemma in the cases_j =α−j, j=1, 2, . . . ,m, we obtain C_α,ν(δ,α−j,i)

=

i−1

∏

Γ[kδ+1+α−j]

Γ[kδ+1+α−j+ν] · ^Γ[(k+1)δ+1−j]

Γ[(k+1)δ+1+α−j]^{, i}≥1, δ =α+β+ν. (4.3) Consider the function

u_j(z) =

∑

C_α,ν(α+β+ν,s_j,i)zⁱ, (4.4) wherez is a complex number. If in (4.3) β=0, then

C_α,ν(α+ν,α−j,i) = ^Γ(1+α−j)

Γ(i(α+ν) +1+α−j)^{, i}≥1, and

u_j(z) =

∑

Γ(1+α−j) Γ(i(α+ν) +1+α−j)^z

i =_Γ(1+α−j)E_α+ν,1+α−j(z), where E_ρ,δ(z)is a Mittag-Leffler type function [15].

It is easy to show that atν=0 the equality

C_α,0(α+_β,α−j,i) =c_i

holds, i.e. these coefficients coincide with coefficients of expansion of the function (1.2), with indexesm=1+ ^β

α, `=1+ ^β−j

α . In [6] it is shown that for the coefficients of the function (1.2) the following asymptotic estimate holds:

c_i c_i+1

= ^Γ[α(im+l+1) +1]

Γ[α(im+l) +1] ∼(αmi)^α (i→_∞).

Thus, the function (1.2) is entire. Denote δ = α+β+ν and rewrite the coefficients C_α,ν(δ,α−j,i)as follows:

C_α,ν(δ,α−j,i) =

i−1

∏

Γh ν

k^δ_ν+ ^α−j

ν

+1i Γh

ν

k^δ_ν+ ^α−j

ν +1

+1i· ^Γ h

α

k_α^δ +^δ−j

α

+1i Γh

α

k^δ_α +^δ−j

α +1 +1

i, i≥1, ν>0.

Further, the asymptotic estimate Cα,ν(δ,α−j,i)

C_α,ν(δ,α−j,i+1) ∼(δi)^ν+α →_∞ (i→_∞)

yields thatu_j(z)_,j =1, 2, . . . ,m, from (4.4) are also entire functions. Lemma4.1 and Proposi- tion2.3implies the following lemma.

Lemma 4.2. Let s_j =α−j, j=1, 2, . . . ,m. Then at all values j =1, 2, . . . ,m the system of functions f_i,j(t) =λⁱC_α,ν(α+β+ν,s_j,i)t^{(α+β+ν)i+sj}, i=0, 1, . . .

is 0-normalized with respect to the pair of operators D^α,λt^βI^ν

on the domain t>0.

Using the main property of normalized systems we get the following theorem.

Theorem 4.3. Let s_j = α−j, j=1, 2, . . . ,m. Then at all values j=1, 2, . . . ,m the functions y_j(t) =

∑

f_i,j(t) =t^sj

∑

λⁱC_α,ν(α+β+ν,s_j,i)t^(α+β+ν)i (4.5) are linearly independent solutions of the homogeneous equation(1.1).

Moreover, for all j=1, 2, . . . ,m−1, y_j(t)∈C[0,d]and y_m(t)∈C_m−α[0,d]. Proof. Consider the function

u_j(t) =

∑

λⁱC_α,ν(α+β+_ν,s_j,i)t^(α+β+ν)i.

Since the function (4.4) is entire, then it is obvious that y_j(t) = t^α−ju_j(t) ∈ C[0,d] at j=1, 2, . . . ,m−1 andym(t) =t^α−mum(t)∈Cm−α[0,d]. Moreover, for allj=1, 2, . . . ,m:

D^αf_0,j(t) =_0,D^αf_i,j(t) =λI^νf_i−1,j(t) =λⁱC_α,ν(α+β+_ν,s_j,i−1)I^νt^{(α+β+ν)i+sj}

=λⁱC_α,ν(α+β+ν,s_j,i−1) ^Γ

(α+β+ν)i+s_j+1 Γ

(α+β+ν)i+s_j+₁+νt^{(α+β+ν)i+ν+sj}, i≥1.

Consequently, the series∑^∞i=0D^αf_i,j(t),∑^∞i=0I^νf_i,j(t)uniformly converge on any closed domain [_ε,d]_{, 0} < ε < d and, therefore, termwise use of the operators D^α and I^ν to the series (4.5) is rightful. Then functions y_j(t) from (4.5) are solutions of the homogeneous equation (1.1).

Proof of linearly independence of the solutions (4.5) we will show below in Theorem 6.2 of Section6.

Remark 4.4. In the case ν=0 the functionsy_j(t)are represented in the form:

y_j(t) =t^α−jE

α,1+^β_α,1+^β−_α^j

λt^α+β

, j=1, 2, . . . ,m.

This representation of solution of the equation (1.1) coincides with the result of [14] (see Theorem 1, formulas (19) and (21)).

5 Construction of f -normalized systems

Now we turn to construction of a solution of inhomogeneous equation. Let f(t) ∈ C[0,d]. Then by the statement of Lemma 3.1 for the function f₀(t) = I^αf(t)the following equality is true:

L₁f₀(t) =D^αI^αf(t) = f(t). Consider the system

f_i(t) =I^αλt^βI^νi

f₀(t)≡λⁱ

I^αt^βI^νi

f₀(t), i=1, 2, . . . (5.1) Lemma 5.1. Let f(t)∈ C[0,d], d < ∞. Then the system of functions (5.1)is f(t)-normalized with respect to the pair of operators D^α,λt^βI^ν

on the domain t>₀.

Proof. Since f(t) ∈ C[0,d], then due to statement of Lemma 3.4 f0(t) = I^αf(t) ∈ C[0,d]. Moreover,

|f₀(t)|=|I^αf(t)| ≤ ¹ Γ(α)

Zt

0

(t−τ)^α−1|f(τ)|dτ≤ kfk_C[0,d] ^t

α

Γ(α+1)^. Thus

|f₀(t)| ≤ kfk_C[0,d] Γ(α+1)^t

α, kf₀k_C[0,d] ≤ ^d

α

Γ(α+₁)kfk_C[0,d]. Hereinafter, we denoteM = ^k_Γ^f₍^kC[0,d]

α+1). Then|f₀(t)| ≤ Mt^α. Since|f₀(t)| ≤ Mt^α, then

λ

I^αt^βI^ν f₀(t)

≤ M|λ|I^αt^βI^ν t^α. Therefore, for anyi≥1 the following estimate holds:

|f_i(t)|=

λⁱ

I^αt^βI^νi

f0(t)

≤ M|λ|ⁱI^αt^βI^νi

t^α. Further, due to (4.2) the function I^αt^βI^νi

t^α is represented as follows:

I^αt^βI^νi

t^α = C_α,ν(α+β+ν,α,i)t^{(α+β+ν)i+α}, thus,

|f_i(t)| ≤M|λ|ⁱC_α,ν(α+β+_ν,α,i)t^{i(α+β+ν)+α}. (5.2)

Consequently, at anyi≥1 we have f_i(t)∈ C[0,d]as well as (5.2).

Moreover,

L₁f_i(t) =D^α

I^α·λt^βi

f_0,sj(t) =D^αI^α·λt^β

I^α·λt^βi−1

f_0,sj(t)

=λt^βf_i−1(t) =L₂f_i−1(t), i≥1.

Thus, in the class of functions X=_C[_0,d]_{we get:}

L₁f₀(t) = f(t), L₁f_i(t) =L₂f_i−1(t), i≥1,

i.e. the system (5.1) is f-normalized with respect to the pair of operators D^α,λt^β .

Theorem 5.2. Let f(t)∈C[0,d]and functions f_i(t), i≥0be defined by(5.1). Then the function y_f(x) =

∑

f_i(t) (5.3)

is a particular solution of the equation(1.1)from the class C[0,d]. Proof. Estimate the series (5.3). Due to (5.2), we have

|y_f| ≤

∑

i=0

|f_i(t)| ≤ kfk_C[0,d]t^α Γ(α+1)

"

1+

∑

|λ|ⁱC_α,ν(α+β+ν,α,i)t^i(α+β)

# .

Since the last series is uniformly convergent on the domain 0 ≤ t ≤ d, then sum of the series, and hence the functiony_f(t)belong to the classC[0,d].

Now we study representation of the functions (5.1) for certain particular cases of func- tions f(t).

Lemma 5.3. Let f(t) =t^µ, µ>−1. Then a particular solution of the equation(1.1)has the following form:

y_f(t) = ^Γ(µ+1)t^α+µ Γ(µ+₁+α)

∑

λ^kC_α,ν(α+β+ν,µ+α,k)t^k(α+β+ν). Proof of the lemma follows from (4.2).

Theorem 5.4. Let f(t) =

∑p j=1

λ_jt^µj,µ_j >−1. Then a particular solution of the equation(1.1)has the following form:

y_f(t) =

∑

λ_jΓ(µ_j+1)t^α+µj Γ(µ_j+1+α)

∑

λ^kC_α,ν(α+β+ν,µ_j+α,k)t^k(α+β+ν). (5.4) In the caseν=0 the representation (5.4) of a particular solution of (1.1) coincides with the result of [14] (see Theorem 2, formula (27)).

Now we give an algorithm for constructing particular solutions of the inhomogeneous equation (1.1) in the case when f(t)is an analytic function.

Theorem 5.5. Let f(t)be an analytic function. Then a particular solution of the equation(1.1) has the form

y_f(t) =

∑

f^(k)(0)t^α+k

Γ(α+k+1)^yk+α(t), (5.5) where y_k+α(_t)is defined by the equality:

y_k+α(t) =

∑

λⁱC_α,ν(α+β,k+α,i)t^i(α+β).

Proof. Let f(t)be an analytical function. Then it can be represented in the form f(t) =

∑

f^(k)(0) k! t^k, and assuming that f₀(t) =I^αf(t), we have

f_i(t) =I^α·λt^βI^νi

f0(t) =I^α·λt^βI^νi ∞ k

∑

f^(k)(0) k! I^αt^k

!

=

∑

f^(k)(0) k!

Γ(k+1) Γ(k+1+α)

I^α·λt^βI^νi

t^k+α. Due to (4.2):

I^α·λt^βI^νi

t^k+α =λⁱCα,ν(α+β+ν,k+α,i)t^{(α+β+ν)i+k+α}. Then

f_i(t) =

∑

f^(k)(0) k!

Γ(k+1) Γ(k+1+α)^λ

iC_α,ν(α+β+ν,k+α,i)t^{(α+β+ν)i+k+α}. Hence for the functiony_f(t)we get (5.5):

y_f(t) =

∑

f_i(t) =

∑

f^(k)(0)t^k+α Γ(α+k+1)

∑

λⁱC_α,ν(α+β,k+α,i)t^i(α+β)=

∑

f^(k)(0)t^k+α

Γ(α+k+1)^yk+α(t).

Theorem 5.6. Let β = n, n = 0, 1, . . . ,f(t) ∈ C[0,d]. Then a particular solution of the equation (1.1)has the form:

y_f(t) =

Z _t

0 Gn,α+ν(t−τ,τ,λ)f(τ)dτ, (5.6) where Gn,α+ν(u,w,λ)is defined by the equality:

G_n,α+ν(u,w,λ) =

∑

G_n,α+ν,i(u,w,λ), G_n,α,ν,i(u,w,λ) = ^λ

i

Γ(α)

∑

. . .

∑

C_n^j1. . .C_n^j1C_α,ν(α+ν,j₁+j₂+· · ·+j_i+α−1,i)

×u^{i(α+ν)+j1+···+ji+α−1}w^{in−j1−···−ji},C_n^ji = ^n!

j_i!(n−j_i)!.

C_α,ν(α+ν,j₁+j₂+· · ·+j_i+α−1,i)

=

i−1 p

∏

Γ(p(α+ν) +j₀+· · ·+j_p+α)

Γ(p(α+ν) +j0+· · ·+jp+α+ν)· ^Γ

(p+1)(α+ν) +j₁+· · ·+j_p+1

Γ

(p+₁)(α+ν) +j₁+· · ·+j_p+1+α, (5.7) where j0 =0.

Proof. Leti=1, β=n, n=0, 1, . . . , f₀(t) = I^αf(t). Then f₁(t) = (I^α·λtⁿI^ν)f0(t) = (I^α·λtⁿ)I^ν+αf(t) = ¹

Γ(α)

Z _t

0

(t−τ)^α−1λτⁿI^ν+αf(τ)dτ

= ^λ

Γ(α)_Γ(ν+α)

Z _t

0 f(z)

Z _t

z

(t−τ)^α−1(τ−z)^α+ν−1τⁿdτdz.

Investigate the inner integral:

I_n,1 =

Z _t

z

(t−τ)^α−1(τ−z)^α+ν−1τⁿdτ.

After the change of variablesτ=z+ (t−z)ξ we have:

I_n,1 =

Z _t

z

(t−τ)^α−1(τ−z)^α+ν−1τⁿdτ= (t−z)^2α+ν−1

Z ₁

0

(1−ξ)^α−1ξ^α+ν−1((t−z)ξ+z)ⁿdξ.

Consequently, f₁(t) = ^λ

Γ(α+ν)

∑

C_n^j1 Γ(α+ν+j₁) Γ(α+ν+j₁+α)

Z _t

0

(t−z)^{j1+2α+ν−1}z^n−j1f(z)dz.

Further, for the function I^νf₁(t)we obtain:

I^νf1(_t) = _Γ( ^λ

ν)_Γ(α+ν)

∑

Cn^j1 Γ(α+ν+j₁) Γ(α+ν+j1+α)

Z _t

0

(_t−τ)^ν−1

Z _τ

0

(τ−z)^{j1+2α+ν−1}_z^n−j1_f(_z)_dzdτ

= _Γ( ^λ

ν)_Γ(α+ν)

∑

C_n^j1_Γ(^Γ_α⁽₊^α+_ν+^ν+_j^j1)

1+α) Z _t

0 z^n−j1f(z)

t

Z

z

(t−τ)^ν−1(τ−z)^{j1+2α+ν−1}dτdz

= _Γ( ^λ

ν)_Γ(α+ν)

∑

Cn^j1Γ(α+ν+j1) Γ(α+ν+j₁+α)

Z _t

0

(t−τ)^{j1+2α+2ν−1}z^n−j1f(z)

Z ₁

0

(1−ξ)^{j1+2α+ν−1}ξ^ν−1dξdz

= _Γ( ^λ

ν)_Γ(α+ν)

∑

Cn^j1Γ(α+ν+j₁) Γ(α+ν+j1+α)

Γ(ν)_Γ(j₁+2α+ν) Γ(j1+2α+2ν)

Z _t

0

(t−τ)^{j1+2α+2ν−1}z^n−j1f(z)dz

= _Γ(^λ

α+ν)

∑

C_n^j1_Γ(^Γ_α⁽₊^α+_ν+^ν+_j^j1)

1+α)

Γ(2(α+ν)+j1−ν) Γ(2(α+ν)+j₁)

Z _t

0

(t−τ)^{j1+2α+2ν−1}z^n−j1f(z)dz.

Then for f₂(t)we get:

f₂(t) = (I^α·λtⁿI^ν)f₁(t) = ¹ Γ(α)

Z _t

0

(t−τ)^α−1λτⁿI^νf₁(τ)dτ

= ^λ

2

Γ(α)_Γ(α+ν)

∑

C_n^j1 Γ(α+ν+j₁) Γ(α+ν+j₁+α)

Γ(2(α+ν) +j₁−ν) Γ(2(α+ν) +j₁)

×

Z _t

0

z^n−j1f(z)

Z _t

z

(t−τ)^α−1(τ−z)^{j1+2α+2ν−1}τⁿdτdz.

Consequently, f2(t) = ^λ

2

Γ(α)

∑

∑

Cn^j1Cn^j2 Γ(α) Γ(α+ν)

Γ(α+ν+j₁+α) Γ(α+ν+j₁+α+ν)

Γ(α+ν+j₁) Γ(α+ν+j₁+α)

× ^Γ(2(α+ν) +j₁+j₂) Γ(2(α+ν) +j₁+j₂+α)

Z _t

0

(t−z)^{2(α+ν)+α+j1+j2−1}z^2n−j1−j2f(z)dz.

Put j₀ =0. Then for f₂(t)we obtain the following representation:

f₂(t) = ^λ

2

Γ(α)

∑

∑

C_n^j1C_n^j2

∏

Γ(p(α+ν) +j₀+· · ·+j_p+α) Γ(p(α+ν) +j0+· · ·+jp+α+ν)

× ^Γ

(p+₁)(α+ν) +j₀+· · ·+j_p+1

Γ

(p+1)(α+ν) +j₀+· · ·+j_p+1+α

t

Z

0

(t−z)^{2(α+ν)+α+j1+j2−1}z^2n−j1−j2f(z)dz.

In general case, using the representation of coefficientsC_α,ν(δ,s,i)for f_i(t), we get:

f_i(t) = ^λ

i

Γ(α)

∑

· · ·

∑

C_n^j1. . .C_n^jiCα,ν(α+ν,j₁+j2+· · ·+j_i+α−1,i)

×

Zt

0

(t−z)^{k(α+ν)+α+j1+j2+···+ji−1}z^{kn−j1−j2−···−ji}f(z)dz,

where coefficientsC_α,ν(α+_ν,j₁+j₂+· · ·+j_i+α−_1,i)are defined by the equality (5.7). Then a particular solution of the equation (1.1) is represented in the form (5.6).

Example 5.7. Letβ≡n=_{0. Then} j₁ =j₂= · · ·= j_i =_{0 ,} C_α,ν(α+ν, 0,i) =

i−1 p

∏

Γ(p(α+ν) +α)

Γ(p(α+ν) +α+ν)· ^Γ[(p+1)(α+ν)]

Γ[(p+1)(α+ν) +α] = ^Γ(α) Γ[i(α+ν) +α]^, G_0,α,i(u,w,λ) = ^λ

i

Γ(i(α+ν) +α)^u

i(α+ν)+α−1, G_0,α(u,v,λ) =

∑

λⁱu^{i(α+ν)+α−1}

Γ(i(α+ν) +α) =u^α−1Eα+ν,α(λu^α+ν). In this case

y_f(t) =

Z _t

0 G_0,α(t−τ,τ,λ)f(τ)dτ=

Z _t

0

(t−τ)^α−1Eα+_ν,α(λ(t−τ)^α+ν)f(τ)dτ.

This formula has been obtained in [1] (see formula (15)).

6 Solution of Cauchy type problem

Consider the following Cauchy type equation:

D^αy(t) =λt^βI^νy(t) +

∑

λ_jt^µj, 0<t ≤d<_∞, (6.1) D^α−ky(t)

t=₀=b_k, k=1, 2, . . . ,m−1, (6.2)