Approximatesolutionsofasum-type fractionalintegro ......AkbariKojabadandRezapourAdvancesinDiﬀerenceEquations20172017:351 DOI10.1186/s13662-017-1404-y RESEARCH OpenAccess Approximatesolutionsofasum-type

Akbari Kojabad and Rezapour Advances in Difference Equations (2017) 2017:351 DOI 10.1186/s13662-017-1404-y

R E S E A R C H Open Access

Approximate solutions of a sum-typefractional integro-differential equation byusing Chebyshev and Legendre polynomialsEisa Akbari Kojabad and Shahram Rezapour*

*Correspondence:[email protected] of Mathematics,Azarbaijan Shahid MadaniUniversity, Tabriz, Iran

AbstractWe investigate the existence of solutions for a sum-type fractional integro-differentialproblem via the Caputo differentiation. By using the shifted Legendre and Chebyshevpolynomials, we provide a numerical method for finding solutions for the problem. Inthis way, we give some examples to illustrate our results.

MSC: 26A33; 34A08; 34K37

Keywords: approximate fixed point; Chebyshev polynomial; Legendre polynomial;numerical solution; sum-type fractional integro-differential equation

1 IntroductionIn , Reinermann investigated some problems by using approximate fixed point prop-erty ([]). In , Yamamoto and Ohtsubo published a paper on subspace iteration accel-erated by using Chebyshev polynomials for eigenvalue problems ([]). There has been pub-lished some work about different fractional integro-differential equations by using Cheby-shev polynomials ([, ] and []) or by using Legendre wavelets ([–] and []). Recently,different techniques for solving some fractional integro-differential equations have beenused (see [, –]). In this paper by using an approximate fixed point result and theshifted Legendre and Chebyshev polynomials, we investigate the existence of solutionsfor a sum-type fractional integro-differential problem.

As is well known, the Caputo fractional derivative of order β for a continuous functionf : (,∞) →R is defined by cDβ f (t) =

�(n–β)∫ t

f (n)(s)

(t–s)β–n+ ds, where n = [β] + ([, ]). Thefractional integral of order β for a function f : (,∞) → R is defined by Iβ f (t) =

�(β)∫ t

(t –s)β–f (s) ds ([, ]). Let (X, d) be a metric space, T a selfmap on X and α : X ×X → [,∞)a map. We say that T is α-admissible whenever α(x, y) ≥ implies α(Tx, Ty) ≥ . Also, T iscalled α-contraction whenever there exists λ ∈ (, ) such that α(x, y)d(Tx, Ty) ≤ λd(x, y)for all x, y ∈ X. We say that T has approximate fixed point property whenever there existsa sequence {xn}n≥ in X such that d(xn, Txn) → . We need the following results.

Lemma . ([]) Let q > , n = [q] + and v ∈ C([, ],R). Then the fractional differentialequation cDqx(t) = v(t) has a solution in the form

x(t) = Iqv(t) + c + ct + · · · + cn–tn–.© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

http://dx.doi.org/10.1186/s13662-017-1404-yhttp://crossmark.crossref.org/dialog/?doi=10.1186/s13662-017-1404-y&domain=pdfmailto:[email protected]

Akbari Kojabad and Rezapour Advances in Difference Equations (2017) 2017:351 Page 2 of 18

Lemma . ([]) Let (X, d) be a metric space and T an α-contractive and α-admissibleselfmap on X such that α(x, Tx) ≥ for some x ∈ X. Then T has the approximate fixedpoint property. If X is complete and T is continuous, then T has fixed point.

2 Main resultNow, we are ready to study the existence of solution of the sum-type fractional integro-differential equation

cDqx(t) = f(t, x(t), cDβ x(t), . . . , cDβn x(t)

)+ g

(t, x(t), Iβ x(t), . . . Iβn x(t)

)(.)

with boundary value conditions∑n

i=(aicDβi x()) = αx′() and∑n

i=(biIβi x()) = αx′(),where < q < , a, . . . , an, b, . . . , bn ∈R and f , g : [, ] ×Rn+ →R are two maps.

Lemma . Let < q < and v ∈ C(I,R). Then the unique solution for the fractional dif-ferential equation cDqx(t) = v(t) with boundary conditions

∑ni=(aicDβi x()) = αx′() and∑n

i=(biIβi x()) = αx′() is given by

x(t) = Iqv(t) –(∑n

i=ai

�(–βi)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

biIq+βi v()

–∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))αIq–v()

+∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

aiIq–βi v()

+αtIq–v() –

∑ni=(taiIq–βi v())∑n

i=ai

�(–βi)– α

,

where α, α, a, . . . , an, b, . . . , bn are some real numbers.

Proof By using Lemma ., general solution for the equation cDqx(t) = v(t) is given byx(t) =

�(q)∫ t

(t – s)q–v(s) ds + c + ct, where c, c ∈R. By applying the boundary condition

∑ni=(aicDβi x()) = αx′(), we get

n∑

i=

(ai

�(q – βi)

∫

( – s)q–βi–v(s) ds +

aic�( – βi)

+ )

=α

�(q – )

∫

( – s)q–v(s) ds + αc

and by using the boundary condition∑n

i=(biIβi x()) = αx′(), we get

n∑

i=

(bi

�(q + βi)

∫

( – s)q+βi–v(s) ds +

bic�(βi + )

+bic

�(βi + )

)

= αc.


This implies

c

( n∑

i=

ai�( – βi)

– α

)

=α

�(q – )

∫

( – s)q–v(s) ds –

n∑

i=

(ai

�(q – βi)

∫

( – s)q–βi–v(s) ds

)

and

n∑

i=

[

c(

bi�(βi + )

– α)

+(

bi�(βi + )

)

c]

= –n∑

i=

bi�(q + βi)

∫

( – s)q+βi–v(s) ds.

Hence,

c = –(∑n

i=bi

�(q+βi)∫

( – s)q+βi–v(s) ds)(

∑ni=

ai�(–βi)

– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

–(∑n

i=(bi

�(βi+)– α))( α�(q–)

∫ ( – s)

q–v(s) ds)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

+(∑n

i=(bi

�(βi+)– α))(

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–v(s) ds))

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

and

c =α

�(q–)∫

( – s)q–v(s) ds –

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–v(s) ds)∑n

i=ai

�(–βi)– α

.

Thus,

x(t) =

�(q)

∫ t

(t – s)q–v(s) ds

–(∑n

i=bi

�(q+βi)∫

( – s)q+βi–v(s) ds)(

∑ni=

ai�(–βi)

– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

–(∑n

i=(bi

�(βi+)– α))( α�(q–)

∫ ( – s)

q–v(s) ds)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

+(∑n

i=(bi

�(βi+)– α))(

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–v(s) ds))

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

+tα

�(q–)∫

( – s)q–v(s) ds – t

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–v(s) ds)∑n

i=ai

�(–βi)– α

= Iqv(t) –(∑n

i=ai

�(–βi)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

biIq+βi v()

–∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))αIq–v()


+∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

aiIq–βi v()

+αtIq–v() –

∑ni=(taiIq–βi v())∑n

i=ai

�(–βi)– α

.

One can check that the given x(t) is a solution for the problem cDqx(t) = v(t) with theboundary conditions. This completes our proof. �

Let X = {x : x, cDβ x, cDβ x, . . . , cDβn x ∈ C(I,R)} be endowed with the metric

d(x, y) = supt∈I

∣∣x(t) – y(t)

∣∣ + sup

t∈I

∣∣cDβ x(t) – cDβ y(t)

∣∣ + · · · + sup

t∈I

∣∣cDβn x(t) – cDβn y(t)

∣∣.

It is clear that (X , d) is a complete metric space (see []). By using Lemma ., a func-tion x ∈ X is a solution for the fractional differential equation (.) whenever it sat-isfies the boundary conditions and there exist functions v, v′ ∈ L[, ] such that v(t) =f (t, x(t), cDβ x(t), . . . , cDβn x(t)), v′(t) = g(t, x(t), Iβ x(t), . . . , Iβn x(t)) and

x(t) = Iq(v(t) + v′(t)

)

–(∑n

i=ai

�(–βi)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

biIq+βi(v() + v′()

)

–∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))αIq–

(v() + v′()

)

+∑n

i=(bi

�(βi+)– α)

(∑n

i=ai

�(–βi)– α)(

∑ni=(

bi�(βi+)

))

n∑

i=

aiIq–βi(v() + v′()

)

+αtIq–(v() + v′()) –

∑ni=(taiIq–βi (v() + v′()))∑n

i=ai

�(–βi)– α

for all t ∈ I .

Theorem . Let ξ : R(n+) → R be a map, λ ∈ (, ) and f , g : [, ] × Rn+ → R twofunctions such that

∣∣f (t, x, x, . . . , xn+) – f (t, y, y, . . . , yn+)

∣∣ +

∣∣g(t, x, x, . . . , xn+) – g(t, y, y, . . . , yn+)

∣∣

≤ λ� + n�

(|x – y| + · · · + |xn+ – yn+|)

for all t ∈ I = [, ] and x, . . . , xn, y, . . . , yn ∈R with

ξ (x, x, . . . , xn+, y, y, . . . , yn+) ≥ ,

where

� =[∣∣∣∣

�(q + )

∣∣∣∣ +

| α�(q) | +

∑ni= |i λ|

|∑ni= λi – α|

+|∑ni= λi – α||

∑ni=

i λ| + |

∑ni= λ

i – α|(| α�(q) | + |

∑ni=

i λ|)

|(∑ni= λi – α)∑n

i= λi |

]

,


� = max≤j≤n

(∣∣∣∣

�(q – βj + )

∣∣∣∣ +

∣∣∣∣

α�(q)

�( – βj)(∑n

i= λi – α)

∣∣∣∣

)

,

λi =ai

– �( – βi), λi =

bi�(βi + )

, λi =bi

�(q + βi), i λ =

bi�(q + βi + )

,

λi =bi

�(βi + ), λi =

ai�(q – βi)

and i λ =ai

�(q – βi + ).

Assume that

ξ(u(t), cDβ u(t), cDβ u(t), . . . , cDβn u(t), v(t), cDβ v(t), cDβ v(t), . . . , cDβn v(t)

) ≥

implies

ξ(Tu(t), cDβ Tu(t), . . . , cDβn Tu(t), Tv(t), cDβ Tv(t), . . . , cDβn Tv(t)

) ≥ ,

where the operator T : X →X is defined by

Tu(t) =

�(q)

∫ t

(t – s)q–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]

× α�(q – )

∫

( – s)q–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

)

+tα

�(q–)∫

( – s)q–f (s, u(s), cDβ u(s), . . . , cDβn u(s)) ds

∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–f (s, u(s), cDβ u(s), . . . , cDβn u(s)) ds)

∑ni= λ

i – α

+

�(q)

∫ t

(t – s)q–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]


× α�(q – )

∫

( – s)q–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

)

+tα

�(q–)∫

( – s)q–g(s, u(s), Iβ u(s), . . . , Iβn u(s)) ds

∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–g(s, u(s), Iβ u(s), . . . , Iβn u(s)) ds)

∑ni= λ

i – α

for all t ∈ I . If there exists u ∈X such that

ξ(u(t), cDβ u(t), . . . , cDβn u(t), Tu(t), cDβ Tu(t), . . . , cDβn Tu(t)

) ≥

for all t ∈ [, ], then the problem (.) has an approximate solution.

Proof We define α : X ×X → [,∞) by

α(u, v) =

⎧⎨

⎩

, ξ (u(t), cDβ u(t), . . . , cDβn u(t), v(t), cDβ v(t)), . . . , cDβn v(t)) ≥ ,∀t ∈ I,, else.

We show that T is an α-admissible and α-contractive selfmap on X . Let u, v ∈X be suchthat ξ (u(t), cDβ u(t), . . . , cDβn u(t), v(t), cDβ v(t), . . . , cDβn v(t)) ≥ for all t ∈ [, ]. Then wehave

∣∣Tu(t) – Tv(t)

∣∣

=

∣∣∣∣∣

{

�(q)

∫ t

(t – s)q–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]

× α�(q – )

∫

( – s)q–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

)


+tα

�(q–)∫


∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–f (s, u(s), cDβ u(s), . . . , cDβn u(s)) ds)

∑ni= λ

i – α

+

�(q)

∫ t

(t – s)q–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]α

�(q – )

∫

( – s)q–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

)

+tα

�(q–)∫


∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–g(s, u(s), Iβ u(s), . . . , Iβn u(s)) ds)

∑ni= λ

i – α

}

–

{

�(q)

∫ t

(t – s)q–f

(s, v(s), cDβ v(s), . . . , cDβn v(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–f

(s, v(s), cDβ v(s), . . . , cDβn v(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]

× α�(q – )

∫

( – s)q–f

(s, v(s), cDβ v(s), . . . , cDβn v(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–f

(s, v(s), cDβ v(s), . . . , cDβn v(s)

)ds

)

+tα

�(q–)∫

( – s)q–f (s, v(s), cDβ v(s), . . . , cDβn v(s)) ds

∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–f (s, v(s), cDβ v(s), . . . , cDβn v(s)) ds)

∑ni= λ

i – α


+

�(q)

∫ t

(t – s)q–g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q+βi–g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)ds

)

–[ ∑n

i= λi – α

(∑n

i= λi – α)(

∑ni= λ

i )

]α

�(q – )

∫

( – s)q–g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)ds

–[ ∑n

i= λi – α

(∑n

i= λi – α)

∑ni= λ

i

]

×n∑

i=

(

λi

∫

( – s)q–βi–g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)ds

)

+tα

�(q–)∫

( – s)q–g(s, v(s), Iβ v(s), . . . , Iβn v(s)) ds

∑ni= λ

i – α

–t∑n

i=(λi∫

( – s)q–βi–g(s, v(s), Iβ v(s), . . . , Iβn v(s)) ds)

∑ni= λ

i – α

}∣∣∣∣∣

≤∫ t

∣∣∣∣(t – s)q–

�(q)

∣∣∣∣

× ∣∣f (s, u(s), cDβ u(s), . . . , cDβn u(s)) – f (s, v(s), cDβ v(s), . . . , cDβn v(s))∣∣ds

+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)

∑ni= λ

i

∣∣∣∣

×n∑

i=

(∣∣λi

∣∣∫

( – s)q+βi–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)

– f(s, v(s), cDβ v(s), . . . , cDβn v(s)

)∣∣ds

)

+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)(

∑ni= λ

i )

∣∣∣∣

α

�(q – )

∫

( – s)q–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds

+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)

∑ni= λ

i

∣∣∣∣

n∑

i=

(

λi

∫

( – s)q–βi–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds)

+∣∣∣∣

tα�(q–)∑n

i= λi – α

∣∣∣∣

∫

( – s)q–βi–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds

+∣∣∣∣

t∑n

i= λi∑n

i= λi – α

∣∣∣∣

∫

( – s)q–βi–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds

+∫ t

∣∣∣∣(t – s)q–

�(q)

∣∣∣∣∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)– g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds


+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)

∑ni= λ

i

∣∣∣∣

n∑

i=

(∣∣λi

∣∣∫

( – s)q+βi–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds

)

+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)(

∑ni= λ

i )

∣∣∣∣

α

�(q – )

∫

( – s)q–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds

+∣∣∣∣

∑ni= λ

i – α

(∑n

i= λi – α)

∑ni= λ

i

∣∣∣∣

n∑

i=

(

λi

∫

( – s)q–βi–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds)

+∣∣∣∣

tα�(q–)∑n

i= λi – α

∣∣∣∣

∫

( – s)q–βi–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds

+∣∣∣∣

t∑n

i= λi∑n

i= λi – α

∣∣∣∣

∫

( – s)q–βi–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds

≤[∣∣∣∣

�(q + )

∣∣∣∣ +

| α�(q) | +

∑ni= |i λ|

|∑ni= λi – α|

+|∑ni= λi – α||

∑ni=

i λ| + |

∑ni= λ

i – α|(| α�(q) | + |

∑ni=

i λ|)

|(∑ni= λi – α)∑n

i= λi |

]

×(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣)

+[∣∣∣∣

�(q + )

∣∣∣∣ +

| α�(q) | +

∑ni= |i λ|

|∑ni= λi – α|

+|∑ni= λi – α||

∑ni=

i λ| + |

∑ni= λ

i – α|(| α�(q) | + |

∑ni=

i λ|)

|(∑ni= λi – α)∑n

i= λi |

]

×(

supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

= �(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

.

Let j ∈ {, , . . . , n} be given. Then we have∣∣cDβj Tu(t) – cDβj Tv(t)

∣∣

=∣∣∣∣

{

�(q – βj)

∫ t

(t – s)q–βj–f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)ds

+αt

–βj

�(q–)∫


�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))


–t–βj

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–f (s, u(s), cDβ u(s), . . . , cDβn u(s)) ds)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

}

+{

�(q – βj)

∫ t

(t – s)q–βj–g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)ds

+αt

–βj

�(q–)∫


�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

–t–βj

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–g(s, u(s), Iβ u(s), . . . , Iβn u(s)) ds)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

}

–{

�(q – βj)

∫ t

(t – s)q–βj–f

(s, v(s), cDβ v(s), . . . , cDβn v(s)

)ds

+αt

–βj

�(q–)∫

( – s)q–f (s, v(s), cDβ v(s), . . . , cDβn v(s)) ds

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

–t–βj

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–f (s, v(s), cDβ v(s), . . . , cDβn v(s)) ds)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

}

–{

�(q – βj)

∫ t

(t – s)q–βj–g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)ds

+αt

–βj

�(q–)∫

( – s)q–g(s, v(s), Iβ v(s), . . . , Iβn v(s)) ds

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

–t–βj

∑ni=(

ai�(q–βi)

∫ ( – s)

q–βi–g(s, v(s), Iβ v(s), . . . , Iβn v(s)) ds)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

}∣∣∣∣

≤∣∣∣∣

�(q – βj)

∣∣∣∣

∫ t

(t – s)q–βj–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds

+∣∣∣∣

αt–βj

�(q–)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

∣∣∣∣

×∫

( – s)q–

∣∣f

(s, u(s), cDβ u(s), . . . , cDβn u(s)

)


)∣∣ds

+∣∣∣∣

�(q – βj)

∣∣∣∣

∫ t

(t – s)q–βj–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)

– g(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds

+∣∣∣∣

αt–βj

�(q–)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

∣∣∣∣

×∫

( – s)q–

∣∣g

(s, u(s), Iβ u(s), . . . , Iβn u(s)

)– g

(s, v(s), Iβ v(s), . . . , Iβn v(s)

)∣∣ds


≤(∣∣

∣∣

�(q – βj + )

∣∣∣∣ +

∣∣∣∣

α�(q)

�( – βj)(∑n

i=ai

–�(–βi)– α)(

∑ni=(

bi�(βi+)

))

∣∣∣∣

)

×(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

= �(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

.

Thus, we get

d(Tu, Tv)

= supt∈I

∣∣Tu(t) – Tv(t)

∣∣ + sup

t∈I

∣∣cDβ Tu(t) – cDβ Tv(t)

∣∣ + · · ·

+ supt∈I

∣∣cDβn Tu(t) – cDβn Tv(t)

∣∣

≤ �(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

+ n�(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

= (� + n�)(

supt∈I

∣∣f

(t, u(t), cDβ u(t), . . . , cDβn u(t)

)– f

(t, v(t), cDβ v(t), . . . , cDβn v(t)

)∣∣

+ supt∈I

∣∣g

(t, u(t), Iβ u(t), . . . , Iβn u(t)

)– g

(t, v(t), Iβ v(t), . . . , Iβn v(t)

)∣∣)

≤ λ(

supt∈I

∣∣u(t) – v(t)

∣∣ + sup

t∈I

∣∣cDβ u(t) – cDβ v(t)

∣∣ + · · · + sup

t∈I

∣∣cDβn u(t) – cDβn v(t)

∣∣)

= λd(u, v)

for all u, v ∈ X . This implies that T is α-contraction. Let u, v ∈ X be such that α(u, v) ≥ .Then ξ (u(t), cDβ u(t), . . . , cDβn u(t), v(t), cDβ v(t), . . . , cDβn ) ≥ Hence, ξ (Tu(t), cDβ Tu(t),. . . , cDβn Tu(t), Tv(t), cDβ Tv(t), . . . , cDβn Tv(t)) ≥ for all t ∈ [, ] and so α(Tu, Tv) ≥ . Itmeans that T is α-admissible. Finally, it is easy to check that α(u, Tu) ≥ . Now by us-ing Lemma ., T has approximate fixed point which is an approximate solution for theproblem (.). �

By using Lemma ., one can easily check that the sum-type fractional integro-differential equation (.) has at least one exact solution whenever the functions f , g arecontinuous.

3 Numerical methodIn this section, we use the Chebyshev and Legendre polynomials for finding approx-imate solutions of the problem (.). The shifted Chebyshev polynomials be defined


on [, ] by T∗n+(x) = (x – )T∗n (x) – T∗n–(x) for all n ≥ , where T∗ (x) = x – andT∗ (x) = ([]). The analytical form of the shifted Chebyshev polynomials T∗n (x) isgiven by T∗n (x) = n

∑ni=(–)n–i

i(n+i–)!(i)!(n–i)! x

i for all n ≥ ([]). We have the orthogonalitycondition

∫

T∗n (x)T∗m(x)√x–x

dx = whenever m �= n, ∫ T∗n (x)T∗m(x)√

x–xdx = π whenever m = n �=

and∫

T∗n (x)T∗m(x)√

x–xdx = π whenever m = n = ([]). Every function u ∈ L([, ]) can

be expressed by the shifted Chebyshev polynomials as u(x) =∑∞

i= ciT∗i (x), where c =π

∫

u(t)T∗ (t)√t–t

dt and ci = π∫

u(t)T∗i (t)√

t–tdt for all i ≥ ([]). Denote the first (m + )-terms of

the shifted Chebyshev polynomials by um(x) =∑m

i= ciT∗i (x) for all m ≥ ([]).

Theorem . Let α > be given. Then we have cDα(um(x)) =∑m

i=�α∑i

k=�α ciw(α)i,k xk–α and

Iα(um(x)) =∑m

i=∑i

k= ci(α)i,k x

k+α , where (α)i,k = (–)i–kk i(i+k–)!�(k+)(i–k)!(k)!�(k++α) ,

(α), =

�(α+) and

w(α)i,k = (–)i–kk i(i+k–)!�(k+)(i–k)!(k)!�(k+–α) .

Proof By using the linear properties of the Caputo fractional derivative, we get

cDα(um(x)

)= cDα

(cT∗ (x)

)+

m∑

i=

cicDα(T∗i

)(x)

= cDα(cT∗ (x)

)+

m∑

i=

i∑

k=

ci(–)i–kki(i + k – )!

(i – k)!(k)!cDα

(xk

).

Since cDα(xk) = whenever k = , , . . . , �α – and cDα(xk) = �(k+)�(k+–α) x

k–α whenever k ≥�α, we have

cDα(um(x)

)=

m∑

i=�α

i∑

k=�αci(–)i–k

ki(i + k – )!�(k + )(i – k)!(k)!�(k + + α)

xk–α =m∑

i=�α

i∑

k=�αciw(α)i,k x

k–α .

Also by using the linear properties of the Riemann-Liouville fractional integral, we get

Iα(um(x)

)= Iα

(cT∗ (x)

)+

m∑

i=

ciIα(T∗i

)(x)

= Iα(cT∗ (x)

)+

m∑

i=

i∑

k=

ci(–)i–kki(i + k – )!

(i – k)!(k)!Iα

(xk

).

Since Iαxk = �(k+)�(k++α) x

k+α , we obtain

Iα(um(x)

)=

cxk

�(α + )+

m∑

i=

i∑

k=

ci(–)i–kki(i + k – )!�(k + )

(i – k)!(k)!�(k + + α)xk+α

=m∑

i=

i∑

k=

ci(α)i,k xk+α .

This completes the proof. �

For solving the problem (.) by using the Chebyshev method, we approximate x(t) by

x(t) ∼=m∑

i=

ciT∗i (t). (.)


By substituting the estimates (.) in (.) and applying Theorem ., we obtain

m∑

i=�q

i∑

s=�qciw

(q)i,s t

s–q

= f

(

t,m∑

i=

ciT∗i (t),m∑

i=�β

i∑

s=�βciw(β)i,s t

s–β , . . . ,m∑

i=�βn

i∑

s=�βnciw(βn)i,s t

s–βn

)

+ g

(

t,m∑

i=

ciT∗i (t),m∑

i=

i∑

s=

ci(β)i,s ts+β ,

m∑

i=

i∑

s=

ci(β)i,s ts+β , . . . ,

m∑

i=

i∑

s=

ci(βn)i,s ts+βn

)

. (.)

In equation (.) for t = xp and p = , . . . , m + – �q, we obtain

m∑

i=�q

i∑

s=�qciw

(q)i,s x

s–qp

= f

(

xp,m∑

i=

ciT∗i (xp),m∑

i=�β

i∑

s=�βciw(β)i,s x

s–βp , . . . ,

m∑

i=�βn

i∑

s=�βnciw(βn)i,s x

s–βnp

)

+ g

(

xp,m∑

i=

ciT∗i (xp),m∑

i=

i∑

s=

ci(β)i,s xs+βp ,

m∑

i=

i∑

s=

ci(β)i,s xs+βp , . . . ,

m∑

i=

i∑

s=

ci(βn)i,s xs+βnp

)

. (.)

For calculating the unknowns c, . . . , cm, we consider the roots of T∗m+–�q(t) and use the∑nj=(ajcD

βj x()) = αx′() and∑n

j=(bjIβj x()) = αx′(). Then we get

n∑

j=

ajm∑

i=�βj

i∑

k=�βjciw

(βj)i,k = α

m∑

i=

i∑

k=

ciw()i,k (.)

and

n∑

j=

bjm∑

i=

i∑

s=

ci(βj)i,s = . (.)

Note that equations (.) and (.) and (.) generate m + nonlinear equations whichcan be solved by using the Newton iterative method. Thus, we can find the unknownsc, . . . , cm and so one can calculate x(t). Similarly, the shifted Legendre polynomials on[, ] defined by L∗n+(x) =

(n+)(x–)n+ L

∗n(x) –

nn+ L

∗n–(x) for all n ≥ , where L∗(x) = and

L∗ (x) = x – ([]). In fact, L∗n(x) =∑n

i=(–)n+i(n+i)!

(n–i)!(i!) xi for all n ≥ , ∫ L∗n(x)L∗m(x) dx =

whenever m �= n and ∫ L∗n(x)L∗m(x) dx = m+ whenever m = n ([]). Every function u ∈L([, ]) can be expressed by the shifted Legendre polynomials by u(x) =

∑∞i= ciL∗i (x),

where ci = (i + )∫

u(t)L∗i (t) dt for i ≥ ([]). Denote the first (m + )-terms shifted


Legendre polynomials by

um(x) =m∑

i=

ciL∗i (x). (.)

By applying a similar proof of Theorem ., one can prove next result.

Theorem . Let α > be given. Then we have cDα(um(x)) =∑m

i=�α∑i

k=�α ciA(α)i,k x

k–α andIα(um(x)) =

∑mi=

∑ik= ciB

(α)i,k xk+α , A

(α)i,k = (–)i+k

(i+k)!(i–k)!(k)!�(k+–α) and

B(α)i,k = (–)i–k(i + k)!

(i – k)!(k)!�(k + + α).

Now, we approximate x(t) by

x(t) ∼=m∑

i=

diL∗i (t). (.)

By using estimates (.) in the problem (.) and applying Theorem ., we obtain

m∑

i=�q

i∑

s=�qdiA(q)i,s ts–q

= f

(

t,m∑

i=

diL∗i (t),m∑

i=�β

i∑

s=�βdiA(β)i,s ts–β , . . . ,

m∑

i=�βn

i∑

s=�βndiA(βn)i,s ts–βn

)

+ g

(

t,m∑

i=

diL∗i (t),m∑

i=

i∑

s=

diB(β)i,s ts+β ,m∑

i=

i∑

s=

diB(β)i,s ts+β , . . . ,

m∑

i=

i∑

s=

diB(βn)i,s ts+βn)

. (.)

Now, we collocate (.) at m + – �q points xp (p = , . . . , m + – �q) as

m∑

i=�q

i∑

s=�qdiA(q)i,s xs–qp

= f

(

t,m∑

i=

diL∗i (xp),m∑

i=�β

i∑

s=�βdiA(β)i,s xs–βp , . . . ,

m∑

i=�βn

i∑

s=�βndiA(βn)i,s xs–βnp

)

+ g

(

xp,m∑

i=

diL∗i (xp),m∑

i=

i∑

s=

diB(β)i,s xs+βp ,m∑

i=

i∑

s=

diB(β)i,s xs+βp , . . . ,

m∑

i=

i∑

s=

diB(βn)i,s xs+βnp

)

, (.)

where xp (p = , . . . , m + – �q) are roots of the polynomial P∗m+–�q(t). Also by sub-stituting equation (.), Theorem . and the conditions

∑nj=(ajcD

βj x()) = αx′() and


∑nj=(bjI

βj x()) = αx′(), we get

n∑

j=

ajm∑

i=�βj

i∑

k=�βjdiA

(βj)i,k = α

m∑

i=

i∑

k=

diA()i,k (.)

and

n∑

j=

bjm∑

i=

i∑

s=

diB(βj)i,s = . (.)

Note that equations (.) and (.) and (.) generate m + nonlinear equations whichcan be solved by using the Newton iterative method to obtain the unknown d, . . . , dm.Thus, one can calculate the solution x(t) of the problem. Here, we provide two examplesto illustrate our numerical methods. There is much work which provides some methodsfor numerical solutions of some types fractional differential equations (see [, ] and[]). Our aim is not to introduce a method that can be answered with greater accuracyand speed. The following examples illustrate our main results and we show that numericalapproximations could be exact sometimes.

Example Consider the fractional differential equation

cD x(t) =

[t + sin(t)

]+ ln

(∣∣sinh(t)

∣∣ +

)+

(x(t) + cD

x(t)

)

+[cD

x(t) + .

](.)

with the boundary conditions cD x() + cD x() = x′() and I x() + I x() = x′(). Con-sider the function f (t, x, x, x) = [t + sin(t)] + ln(| sinh(t)| + ) + x + [x + .] + x ,g(t, x, x, x) = and ξ ((x, x, x), (y, y, y)) = whenever x = and y = almost ev-erywhere and ξ ((x, x, x), (y, y, y)) = – otherwise. Put n = , λ = ., α = α = a =a = b = b = , β = , β =

and q =

. One can check that the problem (.) satisfy the

conditions of Theorem (.), where, thus, the problem (.) has an approximate solution.Check Tables and and Figure .

One can find the coefficients ci and di by using the explained Chebyshev and Legendremethods as in Table . Also, one can find difference of the numerical approximate solutionsin Figure and Table . Here, we denote the numerical solutions of the Chebyshev andLegendre methods by x̃ and x̂, respectively.

Table 1 Coefficients

i Coefficient value of Chebyshev method ci Coefficient value of Legendre method di1 –13.64912481 –13.895858472 32.68664752 32.610685853 0.739732907 0.9839148144 0.123903993 0.1881708985 0.004270165 0.0122996916 0.011334926 0.023029697 –0.004503085 –0.009980864


Table 2 Differences

t |x̃(t) – x̂(t)|0.0 4.76063632959267e–130.1 3.90798504668055e–130.2 3.26849658449646e–130.3 2.62900812231237e–130.4 1.98951966012828e–130.5 1.36779476633819e–130.6 7.19424519957101e–140.7 8.88178419700125e–150.8 5.59552404411079e–140.9 1.26121335597418e–131 1.84741111297626e–13

Figure 1 Chebyshev and Legendre method.

Example Consider the fractional integro-differential equation

cD√

x(t) = ecos(t) + ln(t + ) + ln

(t +

)+

((t + t

)x(t) + tcD

x(t) +

√t

t + cD

x(t)

+(

sin x(t)sin x(t) +

)cD

√

x(t) –I

√

x(t)

|I√

x(t)| +

I x(t) +

√t

I x(t)

–

I√

x(t)

)

(.)

with boundary conditions –cD x()+ cD = x′() and I x()–I√

x() = x′(). Consider

the continuous functions

f (t, x, x, x, x) = ecos(t) + ln(t + ) +

(

tx + tx +√

tt +

x +sin x(t)

sin x(t) + x

)

and g(t, x, x, x, x) = ln(t + ) + (tx – xx|x|+ +

√tx –

x ). Define the map ξ ((x, x,

x, x), (y, y, y, y)) = for all x, . . . , x, y, . . . , y ∈ R. Put n = , λ = ., a = –, a = ,a = , b = , b = , b = –, α = , α = , β = , β =

, β =

√

and q =√

. Thus by

using Theorem ., the problem (.) has an exact solution. Check Tables and andFigure . We present the coefficients c, c, . . . , c and d, d, . . . , d (for m = ) by usingthe Chebyshev and Legendre methods in Table . As one easily sees, the difference ofthe numerical approximate solutions by the Chebyshev and Legendre methods (which


Table 3 Coefficients

i Coefficient value of Chebyshev method ci Coefficient value of Legendre method di0 25.0832696245301 25.09237643435461 –100.27385020664 –100.2450263433282 –0.023785620114085 –0.01640383300464053 –0.045007244673989 –0.06069015950743664 –0.0234838615675559 –0.05646293603953885 –0.0127366109683591 –0.02587755879300646 0.0135562287330535 0.030046706107713

Table 4 Differences

t |x̃(t) – x̂(t)|0 8.17294676380698E–100.1 6.89240664542012E–100.2 5.59495560992218E–100.3 4.26595647695649E–100.4 2.9262992029544E–100.5 1.60159885354005E–100.6 3.02238234439756E–110.7 9.81437153768638E–110.8 2.27224461468722E–100.9 3.58717500148487E–101 4.90501861349912E–10

Figure 2 Chebyshev and Legendre method.

has been provided in Figure ) is inconsiderable. Denote the numerical solutions of theChebyshev and Legendre methods by x̃ and x̂, respectively. In Table , we show that thedifference of the approximate solutions obtained by Chebyshev and Legendre methods isnegligible.

4 ConclusionsWe first prove the existence of approximate solutions for a sum-type fractional integro-differential problem via Caputo differentiation. By using the shifted Legendre and Cheby-shev polynomials, we provide a numerical method for finding solutions for the problem.Also, we give two examples to illustrate our results from a numerical point of view. Ouraim is not to introduce a method that can be answered with greater accuracy and speed.


AcknowledgementsThe authors would like to thank the esteemed referees for their important comments, which improved the final version ofthis manuscript. The research of the authors was supported by Azarbaijan Shahid Madani University.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors have made equal contributions. The whole work was carried out, read and approved by the authors.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 19 June 2017 Accepted: 17 October 2017

References1. Reinermann, J: Uber Toeplitzsche Iterationsverfahren und einige ihrer Anwendungen in der konstruktiven

Fixpunkttheorie. Stud. Math. 32, 209-227 (1969) (in German)2. Yamamoto, Y, Ohtsubo, H: Subspace iteration accelerated by using Chebyshev polynomials for eigenvalue problems

with symmetric matrices. Int. J. Numer. Methods Eng. 10(4), 935-944 (1976)3. Khader, MM: Introducing an efficient modification of the variational iteration method by using Chebyshev

polynomials. Appl. Appl. Math. 7(1), 283-299 (2012)4. Khader, MM: Introducing an efficient modification of the homotopy perturbation method by using Chebyshev

polynomials. Arab J. Math. Sci. 18(1), 61-71 (2012)5. Maleknejad, K, Sohrabi, S, Rostami, Y: Numerical solution of nonlinear Volterra integral equations of the second kind

by using Chebyshev polynomials. Appl. Math. Comput. 188, 123-128 (2007)6. Chen, YM, Liu, LL, Sun, L, Sun, XH: Numerical solution of a nonlinear fractional integro-differential equation by using

Legendre wavelets. J. Hefei Univ. Technol. Nat. Sci. 36(8), 1019-1024 (2013) (in Chinese)7. Mirzaee, F, Fathi, S: Numerical solution of some class of integro-differential equations by using Legendre-Bernstein

basis. J. Hyperstruct. 3(2), 139-154 (2014)8. Yousefi, SA: Numerical solution of Abel’s integral equation by using Legendre wavelets. Appl. Math. Comput. 175,

574-580 (2006)9. Zhou, FY, Xu, XY: Numerical approximation of definite integrals by using Legendre wavelets and operator matrices.

Acta Math. Appl. Sin. 38(5), 862-873 (2015)10. Aydogan, SM, Baleanu, D, Mousalou, A, Rezapour, Sh: On approximate solutions for two higher-order Caputo-Fabrizio

fractional integro-differential equations. Adv. Differ. Equ. 2017, 221 (2017)11. Bhrawy, AH, Zaky, MA: Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional

differential equations. Appl. Math. Model. 40(2), 832-845 (2016)12. Jafarian, A, Rostami, F, Golmankhaneh, AK, Baleanu, D: Using ANNs approach for solving fractional order Volterra

integro-differential equations. Int. J. Comput. Intell. Syst. 10(1), 470-480 (2017)13. Kumar, D, Singh, J, Baleanu, D: Modified Kawahara equation within a fractional derivative with non-singular kernel.

Therm. Sci. (2017). doi:10.2298/TSCI160826008K14. Mashayekhi, S, Razzaghi, M: Numerical solution of distributed order fractional differential equations by hybrid

functions. J. Comput. Phys. 315, 169-181 (2016)15. Singh, J, Kumar, D, Nieto, JJ: A reliable algorithm for a local fractional Tricomi equation arising in fractal transonic flow.

Entropy 2016, 206 (2016)16. Singh, J, Kumar, D, Al Qurashi, M, Baleanu, D: A new fractional model for giving up smoking dynamics. Adv. Differ.

Equ. 2017, 88 (2017)17. Singh, J, Kumar, D, Al Qurashi, M, Baleanu, D: A novel numerical approach for a nonlinear fractional dynamical model

of interpersonal and romantic relationships. Entropy 2017, 375 (2017)18. Singh, H, Srivastava, HM, Kumar, D: A reliable numerical algorithm for the fractional vibration equation. Chaos

Solitons Fractals 103, 131-138 (2017)19. Suganya, S, Baleanu, D, Kalamani, P, Arjunan, MM: A note on non-instantaneous impulsive fractional neutral

integro-differential systems with state-dependent delay in Banach spaces. J. Comput. Anal. Appl. 20, 1302-1317(2016)

20. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier,Amsterdam (2006)

21. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)22. Miandaragh, MA, Postolache, M, Rezapour, Sh: Some approximate fixed point results for generalized α-contractive

mappings. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 75(2), 3-10 (2013)23. Su, X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett.

22, 64-69 (2009)24. Snyder, MA: Chebyshev Methods in Numerical Approximation. Prentice Hall, Englewood Cliffs (1966)25. Khader, MM: Numerical solution of nonlinear multi-order fractional differential equations by implementation of the

operational matrix of fractional derivative. Stud. Nonlinear Sci. 2(1), 5-12 (2011)

http://dx.doi.org/10.2298/TSCI160826008K

Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomialsAbstractMSCKeywords

IntroductionMain resultNumerical methodConclusionsAcknowledgementsCompeting interestsAuthors' contributionsPublisher's NoteReferences

Documents

Approximatesolutionsofasum-type fractionalintegro ......AkbariKojabadandRezapourAdvancesinDiﬀerenceEquations20172017:351 DOI10.1186/s13662-017-1404-y RESEARCH OpenAccess Approximatesolutionsofasum-type