A generalization of truncated M-fractional derivative and applications to fractional differential equations

Esin İlhan 1  and İ. Onur Kıymaz 2
  • 1 Mucur Vocational School, 40500, Kırşehir, Turkey
  • 2 Dept. of Mathematics, Faculty of Science, 40100, Kırşehir, Turkey
Esin İlhan and İ. Onur Kıymaz
  • Corresponding author
  • Dept. of Mathematics, Faculty of Science, Ahi Evran University, 40100, Kırşehir, Turkey
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Abstract

In this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

1 Introduction

Fractional analysis is a field that is frequently studied by scientists because of its many applications used to model real-world problems. In some recent studies, it is seen that mathematical models obtained by using various fractional derivatives have better overlapping with experimental data rather than the models with integer order derivatives. However, unlike integer order derivatives, different fractional derivative definitions may be used for different types of problems. This situation led scientists to identify more general fractional operators.

Especially in the last five years, several generalizations of some well-known fractional derivative operators have been addressed by many authors (see, for example [2, 3, 5, 6, 11, 18, 19, 33]). In addition to these studies, different fractional derivative operators having many features provided by the integer order derivative operator were also studied (see [16, 17, 27, 28, 29, 30, 31] and the references therein).

In 2014, Khalil et al. [17] introduced a new type of fractional derivative for f : [0,∞) → ℝ, t > 0 and α ∈ (0,1) as

Tαf(t)=limɛ0f(t+ɛt1α)f(t)ɛ.
They called it conformable fractional derivative.

In the same year, Katugampola [16] introduced the alternative and truncated alternative fractional derivatives for f : [0,∞) → ℝ as

Dα(f)(t)=limɛ0f(teɛtα)f(t)ɛ,t>0,α(0,1)
and
Diα(f)(t)=limɛ0f(teiɛtα)f(t)ɛ,t>0,α(0,1)
respectively. Here eix=k=0ixkk! is the truncated exponential function.

Recently, Sousa and de Oliveira [27, 29] introduced the M-fractional and truncated M-fractional derivatives for f : [0,∞) → ℝ as

DMα;βf(t)=limɛ0f(tEβ(ɛtα))f(t)ɛ,β,t>0,α(0,1)
and
iDMα;βf(t)=limɛ0f(tiEβ(ɛtα))f(t)ɛ,β,t>0,α(0,1)
respectively, by means of one parameter Mittag-Leffler function [12]
Eβ(z)=k=0zkΓ(βk+1),(β)>0,z,
and its truncated version.

All the derivatives given above satisfies some properties of classical calculus, e.g. linearity, product rule, quotient rule, function composition rule and chain rule. Besides, for all the operators given above the α-order derivative of a function is a multiple of t1αdfdt .

In 2009, generalized M-series defined by Sharma and Jain [25, 26]

pMqβ,γ(z):=pMqβ,γ[a1apc1cq;z]=k=0(a1)k(ap)k(c1)k(cq)kzkΓ(βk+γ)
where β, γ, z ∈ ℂ, p,q ∈ ℕ, ℜ(β) > 0, ci ≠ 0,−1,−2,…(i = 1,2,...,q). Here, (α)k is the Pochhammer symbol [1] which given by
(α)ν=Γ(α+ν)Γ(α),α,ν
with the assume (α)0 = 1. Note that if aj (j = 1,2,..., p) equals to zero or a negative integer, then the series reduces to a polynomial.

Generalized M-series is convergent for all z if pq; it is convergent for |z| < δ = αα if p = q + 1; and divergent if p > q + 1. When p = q + 1 and |z| = δ, the series can converge on conditions depending on the parameters. For more information about M-series we refer [25, 26] and the references therein.

Most of the famous special functions can be described as the special cases of the generalized M-series:

1M11,1(1;1;z)=k=0zkk!=ez,1M1β,1(1;1;z)=k=0zkΓ(βk+1)=Eβ(z),1M1β,γ(1;1;z)=k=0zkΓ(βk+γ)=Eβ,γ(z),1M1β,γ(σ;1;z)=k=0(σ)kzkΓ(βk+γ)=Eβ,γσ(z),1M11,1(a;c;z)=k=0(a)k(c)kzkk!=Φ(a;c;z),2M11,1(a,b;c;z)=k=0(a)k(b)k(c)kzkk!=2F1(a,b;c;z),pMq1,1(z)=k=0(a1)k(ap)k(c1)k(cq)kzkk!=pFq[a1apc1cq;z].
Here, Eβ, Eβ,γ, Eβ,γσ are the one [23], two [32] and three parameters [24] Mittag-Leffler functions; and also Φ, 2F1, pFq are the confluent, Gauss and generalized hypergeometric functions [1], respectively.

Motivated by the above studies and the frequent use of M-series in fractional operator theory (see [8, 9, 10, 14, 21]), with the help of M-series, we first define a more general fractional derivative (truncated ℳ-series fractional derivative) and investigate its properties like linearity, product rule, the chain rule, etc. Then we extend some of the classical results in calculus like Rolle’s theorem, mean value theorem etc. We also introduce the ℳ-series fractional integral and finally, we obtain the analytical solutions of ordinary and partial ℳ-series fractional linear differential equations.

2 Truncated ℳ-series Fractional Derivative

We first present the definitions of the truncated M-series and truncated ℳ-series fractional derivative operator.

Definition 1

The truncated M-Series is defined for β > 0 as

ip,qβ,γ(t)=ip,qβ,γ[a1apc1cq;t]:=k=0i(a1)k(ap)k(c1)k(cq)ktkΓ(βk+γ)
where β, γ,t ∈ ℝ, p,q ∈ ℕ, an,cm∈ ℝ, cm≠ 0,−1,−2,...(n = 1,2,..., p; m = 1,2,...,q).

Definition 2

Let f : [0,∞) → ℝ. For β > 0, t > 0 and α ∈ (0,1), the truncated ℳ-series fractional derivative of order α of a function f is

i𝒟αf(t)=i𝒟α[a1apc1cq;β,γ]f(t):=limɛ0f(Γ(γ)tip,qβ,γ(ɛtα))f(t)ɛ,
where α, β, γ ∈ ℝ, p,q ∈ ℕ, an,cm∈ ℝ, cm≠ 0,−1,−2,...(n = 1,2,..., p; m = 1,2,...,q) and ip,qβ,γ is the truncated M-series given with (6). If a truncated ℳ-series fractional derivative of a function f exists then we called the function f is ℳ-differentiable.

Note that, if f is ℳ-differentiable in some interval (0,a), a > 0 and

limt0+i𝒟αf(t)
exists, then we define
i𝒟αf(0)=limt0+i𝒟αf(t).

Because Sousa and de Oliveira showed in [29] that, truncated M-fractional derivative (5) is the generalization of the fractional derivative operators (1)(4), it is enough to choose γ = p = q = 1 and a1 = c1 in (7) for proving that the all the fractional derivative operators (1)(5) given above are the special cases of our definition.

For the sake of shortness, throughout the paper we assume that α,β, γ ∈ ℝ, p,q ∈ ℕ, β > 0, p > 0, q > 0, an,cm∈ ℝ and cm≠ 0,−1,−2,...(n = 1,2,..., p; m = 1,2,...,q). Also, we use the notation 𝒦 instead of the constant a1apc1cqΓ(γ)Γ(β+γ) .

Now we begin our investigation with an important theorem.

Theorem 1

If a function f : [0,∞) → ℝ is ℳ-differentiable at t0 > 0 for α ∈ (0,1], then f is continuous at t0.

Proof

Consider the identity

f(Γ(γ)t0ip,qβ,γ(ɛtα))f(t0)=f(Γ(γ)t0ip,qβ,γ(ɛtα))f(t0)ɛɛ.
Applying the limit for ɛ → 0 on both sides, we get
limɛ0f(Γ(γ)t0ip,qβ,γ(ɛtα))f(t0)=limɛ0(f(Γ(γ)t0ip,qβ,γ(ɛtα))f(t0)ɛ)limɛ0ɛ=i𝒟αf(t)limɛ0ɛ=0.
Then, f is continuous at t0.

Besides, using the definition of the truncated M-series, we can write

f(Γ(γ)tip,qβ,γ(ɛtα))=f(Γ(γ)tn=0i(a1)k(ap)k(c1)k(cq)k(ɛtα)kΓ(βk+γ)).
If we apply the limit for ɛ → 0 on both sides and since f is continuous, we get
limɛ0f(Γ(γ)tip,qβ,γ(ɛtα))=f(Γ(γ)tlimɛ0k=0i(a1)k(ap)k(c1)k(cq)k(ɛtα)kΓ(βk+γ)).
Because
limɛ0k=0i(a1)k(ap)k(c1)k(cq)k(ɛtα)kΓ(βk+γ)=1Γ(γ),
we can write
limɛ0f(Γ(γ)tip,qβ,γ(ɛtα))=f(t).

The following theorem is about the basic properties of ℳ-series fractional derivative:

Theorem 2

Let α ∈ (0,1], a,b ∈ ℝ and f, g ℳ-differentiable functions at a point t > 0. Then

  1. (a)i𝒟α(af+bg)(t)=ai𝒟αf(t)+bi𝒟αg(t)
  2. (b)i𝒟α(fg)(t)=f(t)i𝒟αg(t)+g(t)i𝒟αf(t)
  3. (c)i𝒟α(fg)(t)=g(t)i𝒟αf(t)f(t)i𝒟αg(t)[g(t)]2
  4. (d)If f is differentiable, then
    i𝒟α(f)=𝒦t1αdf(t)dt,
  5. (e)If f′(g(t)) exists, then
    i𝒟α(fg)(t)=f(g(t))i𝒟αg(t).

Proof

The proof of the first three cases are quite simple and easily obtainable by following the same way with the corresponding proofs of classical calculus. For (d): from the definition of truncated M-series we can write

i𝒟αf(t)=limɛ0f(Γ(γ)tip,qβ,γ(ɛtα))f(t)ɛ=limɛ0f(Γ(γ)t(1Γ(γ)+a1apc1cqɛtαΓ(β+γ)+𝒪(ɛ2))f(t)ɛ=limɛ0f(t+ɛt1α(𝒦+𝒪(ɛ)))f(t)ɛ

Choosing h = ɛt1−α (𝒦 + 𝒪(ɛ)) we get the result

i𝒟αf(t)=t1αlimɛ0f(t+h)f(t)h𝒦+𝒪(ɛ)=𝒦t1αdf(t)dt.

For (e): If g is a constant function in a neighborhood of a. Then clearly i𝒟αf(g(a))=0 . Now, assume that g is not a constant function, that is, we can find an ɛ > 0 for any t1,t2 ∈ (a − ɛ,a + ɛ) such that g(t1) ≠ g(t2). Since g is continuous at a and for small enough ɛ, we have

i𝒟α(fg)(a)=limɛ0f(g(Γ(γ)aip,qβ,γ(ɛaα)))f(g(a))ɛ=limɛ0f(g(Γ(γ)aip,qβ,γ(ɛaα)))f(g(a))g(Γ(γ)aip,qβ,γ(ɛaα))g(a)g(Γ(γ)aip,qβ,γ(ɛaα))g(a)ɛ=limɛ10f(g(Γ(γ)aip,qβ,γ(ɛaα))f(g(a))ɛ1.limɛ0g(Γ(γ)aip,qβ,γ(ɛaα))g(a)ɛ=f(g(a))i𝒟αg(a),
with a > 0.

Example 3

Now we give the truncated ℳ-series fractional derivatives of some well-known functions by using the result(8). Let n ∈ ℝ and α ∈ (0,1]. Then we have the following results

  1. (a)i𝒟α(const.)=0
  2. (b)i𝒟α(ent)=𝒦nt1αent
  3. (c)i𝒟α(sinnt)=𝒦nt1αcosnt
  4. (d)i𝒟α(cosnt)=𝒦nt1αsinnt
  5. (e)i𝒟α(tn)=𝒦ntnα
  6. (f)i𝒟α(tαα)=𝒦

Theorem 4 (Rolle’s theorem)

Let a > 0 and f : [a,b] → ℝ be a function such that:

  1. (a)f is continuous on [a,b],
  2. (b)f is ℳ-differentiable on (a,b) for some α ∈ (0,1),
  3. (c)f (a) = f (b).

Then, there exists c ∈ (a,b), such thati𝒟αf(c)=0 .

Proof

Let f is a continuous function on [a,b] and f (a) = f (b), then there exists a point c ∈ (a,b) at which the function f has a local extreme. Then,

i𝒟αf(c)=limɛ0f(Γ(γ)cip,qβ,γ(ɛtα))f(c)ɛ=limɛ0+f(Γ(γ)cip,qβ,γ(ɛtα))f(c)ɛ,

Since

limɛ0±ip,qβ,γ(ɛtα)=1Γ(γ),
the two limits have opposite sings. So i𝒟αf(c)=0 .

Theorem 5 (Mean value theorem)

Let a > 0 and f : [a,b] → ℝ be a function such that:

  1. (a)f is continuous on [a,b];
  2. (b)f is ℳ-differentiable on (a,b) for some α ∈ (0,1).

Then, there exists c ∈ (a,b), such that

i𝒟αf(c)=𝒦f(b)f(a)bααaαα.

Proof

Consider the following function:

g(t)=f(t)f(a)(f(b)f(a)bααaαα)(tααaαα).

The function g provides the conditions of the Rolle’s theorem. Then, there exists a point c ∈ (a,b), such that i𝒟αg(c)=0 . Applying the new truncated ℳ-series fractional derivative on both sides of the equality (9) and using the properties (a) and (f) of Example 1, we have the result.

Theorem 6 (Extended mean value theorem)

Let f,g : [a,b] → ℝ, a > 0 be two functions such that:

  1. (a)f,g are continuous on [a,b];
  2. (b)f,g are ℳ-differentiable on (a,b) for some α ∈ (0,1).

Then, there exists c ∈ (a,b), such that:

i𝒟αf(c)i𝒟αg(c)=f(b)f(a)g(b)g(a).

Proof

Consider the following function:

F(x)=f(t)f(a)(f(b)f(a)g(b)g(a))(g(t)g(a)).

The function F provides the conditions of the Rolle’s theorem. Then, there exists a point c ∈ (a,b), such that i𝒟αF(c)=0 . Applying the truncated ℳ-series fractional derivative on both sides of the equality (10) and using the property (a) of Example 1, we have the result.

Theorem 7

Let a > 0 and f : [a,b] → ℝ be a function such that:

  1. (a)f is continuous on [a,b];
  2. (b)f is ℳ-differentiable on (a,b) for some α ∈ (0,1).

If for all t ∈ (a,b) t(a,b)i𝒟αf(t)=0, then f is a constant function on [a,b].

Proof

Assume that, for all t ∈ (a,b), t(a,b),i𝒟αf(t)=0 , and let, t1,t2 ∈ [a,b], with t1< t2. Since f is also continuous in [t1,t2] and ℳ-differentiable in (t1,t2), from Rolle’s theorem, there exist a point c ∈ (t1,t2) with

i𝒟αf(c)=𝒦f(t2)f(t1)t2ααt1αα=0.
So, f (t1) = f (t2). Since t1< t2 are arbitrary chosen from [a,b], f has to be a constant function.

Corollary 8

Let a > 0 and f, g: [a,b] → ℝ be functions such that for all α ∈ (0,1) and t ∈ (a,b),

i𝒟αf(t)=i𝒟αg(t).

Then, there exists a constant c such that f (t) = g(t) + c

Proof

Apply Theorem 7 with choosing h(t) = f (t) − g(t).

Theorem 9

Let 𝒦 > 0 and f : [a,b] → ℝ be a function which continuous on [a,b] and ℳ-differentiable on (a,b) for some α ∈ (0,1). Then, for all t ∈ (a,b)

  • if i𝒟αf(t)>0 , then f is increasing on [a,b],
  • if i𝒟αf(t)<0 , then f is decreasing on [a,b].

Proof

From Theorem 7 we know that for t1,t2 ∈ [a,b] there exist a c ∈ (t1,t2) such as

i𝒟αf(c)=𝒦f(t2)f(t1)t2ααt1αα.

If i𝒟αf(c)>0 then f (t2) > f (t1) while t2> t1, so f is increasing since t1 and t2 chosen arbitrary. But if i𝒟αf(c)<0 then f (t2) > f (t1) while t2< t1 (or f (t2) < f (t1) while t2> t1), so f is decreasing.

Theorem 10

Let 𝒦 > 0 and f,g : [a,b] → ℝ be functions which continuous on [a,b], ℳ-differentiable on (a,b) for some α ∈ (0,1) and for all t ∈ [a,b], i𝒟αf(t)i𝒟αg(t). Then,

  • if f (a) = g(a), then f (t) ≤ g(t) for all t ∈ [a,b],
  • if f (b) = g(b), then f (t) ≥ g(t) for all t ∈ [a,b].

Proof

The proof is trivial when you consider the function h(t) = g(t) − f (t).

Theorem 11

Let f : [0,∞) → ℝ be a two times differentiable function with t > 0 and α1,α2 ∈ (0,1). Then

i𝒟α1+α2f(t)i𝒟α1(i𝒟α2f)(t).

Proof

From the equality (8) we have

i𝒟α1+α2f(t)=𝒦t1α1α2f(t),
but for the other side we have
i𝒟α1(i𝒟α2f)(t)=i𝒟α1(𝒦t1α2f(t))=𝒦2t1α1(t1α2f(t))=𝒦2t1α1α2((1α2)f(t)+tf(t)).

The proof is clear from (11) and (12).

The following result is the direct consequences of the previous theorem.

Corollary 12

Let f : [0,∞) → ℝ be a two times differentiable function with t > 0 and α1,α2 ∈ (0,1). Then

i𝒟α1(i𝒟α2f)(t)i𝒟α2(i𝒟α1f)(t).

The following definition is about the ℳ-series fractional derivative operator for α ∈ (n,n + 1], n ∈ ℕ.

Definition 3

Let α ∈ (n,n + 1], n ∈ ℕ and for t > 0, f be a n times differentiable function. The truncated ℳ-series fractional derivative of order a of f is given as

i𝒟α;nf(t):=limɛ0f(n)(Γ(γ)tip,qβ,γ(ɛtnα))f(n)(t)ɛ,
if and only if the limit exists.

Remark 1

For t > 0, α ∈ (n,n + 1] and for (n + 1) times differentiable function f, it is easy to show that

i𝒟α;nf(t)=𝒦tn+1αf(n+1)(t).
by using (13), (8) and induction on n.

3 ℳ-series Fractional Integral

In this section, we defined the corresponding ℳ-series fractional integral operator αf(t) . We want that our integral operator satisfies i𝒟α(αf(t))=f(t) . Let F(t)=αf(t) be a differentiable function, then from (8) we have the following differential equation

f(t)=i𝒟α(F(t))=𝒦t1αdF(t)dt,
which have a solution of the form for an ≠ 0, (n = 1,2,..., p)
F(t)=𝒦1f(t)t1αdt.

This yields the following definition.

Definition 4

Let a ≥ 0 and t ≥ a, and f is defined in (a,t]. If the following improper Riemann integral exists, then for α ∈ (0,1), the α order ℳ-series fractional integral of a function f is defined by

αf(t):=α[a1apc1cq;β,γ]f(t)=𝒦1atf(t)t1αdt,
where the conditions are same as (7) with an ≠ 0, n = 1,2,..., p.

Remark 2

It can easily seen from the definition of ℳ-series fractional integral that, the integral operator is linear and αf(a)=0 .

For the rest of the paper we assume that an ≠ 0, n = 1,2,..., p.

Theorem 13

Let a ≥ 0, α ∈ (0,1) and f is a continuous function such thatαf(t)exists. Then for t ≥ a,

i𝒟α(αf(t))=f(t).

Proof

Since f is continuous, αf(t) is differentiable. Then from (8) we have

i𝒟α(αf(t))=𝒦t1αddtαf(t)=t1αddt(atf(t)t1αdt)=f(t),
which completes the proof.

Theorem 14

Let f : (a,b) → ℝ be a differentiable function and α ∈ (0,1]. Then, for all t > a, we have

α(i𝒟αf(t))=f(t)f(a).

Proof

Since the function f is differentiable, by using the fundamental theorem of calculus for the integer-order derivatives and (8), we get

α(i𝒟αf(t))=𝒦1ati𝒟αf(t)t1αdx=atdf(t)dtdx=f(t)f(a),
which gives the result.

Remark 3

If f (a) = 0 then α(i𝒟αf(t))=i𝒟α(αf(t))=f(t) .

Theorem 15

Let f : [a,b] → ℝ be a continuous function with 0 < a < b and α ∈ (0,1). Then for 𝒦 > 0 we have

|αf|(t)α|f|(t).

Proof

From the definition of ℳ-series fractional integral we have

|αf(t)|=|𝒦1atf(x)x1αdx||𝒦1||atf(x)x1αdx|𝒦1at|f(x)x1α|dx=𝒦1at|f(x)|x1αdx,
which completes the proof.

Corollary 16

Let f : [a,b] → ℝ be a continuous function such that

N=supt[a,b]|f(t)|.

Then, for all t ∈ [a,b] with 0 < a < b, α ∈ (0,1) and 𝒦 > 0 we have

|αf(t)|𝒦1N(tααaαα).

Proof

From the previous theorem we have

|αf|(t)α|f|(t)=𝒦1at|f(x)|x1αdx=𝒦1Natxα1dx,
which gives the result.

Theorem 17

Let f,g : [a,b] → ℝ be two differentiable functions and α ∈ (0,1). Then

abf(t)i𝒟αg(t)dαt=f(t)g(t)|ababg(t)i𝒟αf(t)dαt,
where dαt = 𝒦−1tα −1dt.

Proof

Using the definition of ℳ-series fractional integral (14), (8) and applying fundamental theorem of calculus for integer-order derivatives, we get

abf(t)i𝒟αg(t)dαt=𝒦1abf(t)t1αi𝒟αg(t)dt=abf(t)dg(t)dtdt=f(t)g(t)|ababg(t)df(t)dtdt=f(t)g(t)|ababg(t)i𝒟αf(t)dαt,
which completes the proof.

Now we define the ℳ-series fractional integral for α ∈ (n,n + 1] as follows.

Definition 5

Let a ≥ 0 and t ≥ a, and f is defined in (a,t]. If the following improper Riemann integral exists, then for α ∈ (n,n + 1 ), the a order ℳ-series fractional integral of a function f is defined by

α;nf(t):=α;n[a1apc1cq;β,γ]f(t)=𝒦1atdtatdtatn+1timesf(t)tn+1αdt,
where the conditions are same as (7) with an ≠ 0, n = 1,2,..., p.

The following theorem is a generalization of Theorem 14.

Theorem 18

Let α ∈ (n,n + 1] and f : [a,∞) → ℝ be (n + 1) times differentiable function for t > a. Then we have

α;n(i𝒟α;nf)(t)=f(t)k=0nf(k)(a)(ta)kk!.

Proof

From (7) and (15) we have

α;n(i𝒟α;nf)(t)=𝒦1atdtatdtatn+1timesi𝒟α;nf(t)tn+1αdt=atdtatdtatn+1timesf(n+1)(t)dt,
which gives the result.

4 Applications to ℳ-series Fractional Differential Equations

In this section, we obtained the general solutions of linear fractional differential equations including the ℳ-series fractional derivative operator.

Example 19

Let u = u(t) is a ℳ-differentiable function and assume that for α ∈ (0,1] the linear ℳ-series fractional differential equation

i𝒟αu(t)+p(t)u(t)=q(t)
is given. If u is also a differentiable function then by using(8), we get a linear ordinary differential equation
du(t)dt+𝒦1tα1p(t)u(t)=𝒦1tα1q(t).

The integrating factor of the equation can be found as µ(t) = e𝒦 ʃtα−1p(t)dt, which yields the solution as

u(t)=e𝒦1p(t)t1αdt[𝒦1q(t)t1αe𝒦1p(t)t1αdtdt+C],
where C is a constant. By definition of the ℳ-series integral operator we can write the last equality as
u(t)=eαp(t)[α(q(t)eαp(t))+C].

If we choose p(t) = −λ, q(t) = 0, then the linear ℳ-series fractional differential equation(16)turns to

i𝒟αu(t)=λu(t),
and the general solution can be found from(17)as
u(t)=Ce𝒦1λαtα.
Since et=1,11,1(t) , we can write the solution by means of truncated ℳ-series as
u(t)=C1,11,1(𝒦1λαtα).

For the fixed values an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q), this result coincides with the results given in [27] when λ = 1 and coincides with the corresponding integer-order result when α = β = λ = 1.

Fig. 1
Fig. 1

The graphs of (18) from α = 0.25 (green) to α = 1.00 (black) by step size 0.25.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00016

In the following, the reader can find the graphs of the solution function (18) for different α,β and γ values with the fixed values C = l = 1 and an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q).

Example 20

Consider the heat equation in one dimension

αu(x,t)tα=k2u(x,t)x2,0<x<L,t>0,
with the initial and boundary conditions
u(0,t)=0,u(L,t)=0,u(x,0)=f(x),t0,0xL.

Here αtα=i𝒟α , u = u(x,t) is a ℳ-differentiable function, α ∈ (0,1] and k is a positive constant. Suppose that u(x,t) = P(x)Q(t). Using separation of variables method we get a system of differential equations

dαdtαQ(t)kξQ(t)=0,d2dx2P(x)ξP(x)=0.
From the above example and the ordinary differential equations theory, we know that these equations have solutions of the form
Qn(t)=e𝒦1(nπL)2kαtα,n=1,2,3,Pn(x)=sin(nπxL),n=1,2,3,
So, the formal solution of the heat equation(19)is
u(x,t)=n=0bnsin(nπxL)e𝒦1(nπL)2kαtα,
where bn=2L0Lf(x)sin(nπxL)dx.

Let us fixed the values an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q) in(20). Choosing γ = 1 yields us the same result in [29]; γ = β = 1 yields us the same result in [7], and α = β = γ = 1 yields us the same result with the integer-order heat equation.

If we choose f (x) = sin(x), L = π, k = 1 in(19)we have

u(x,t)=2πn=00πsin(x)sin(nx)dxsin(nx)e𝒦1n2αtα,
which differ from 0 only for n = 1. So, the solution of the problem is
u(x,t)=sin(x)e𝒦1tαα.
This result is the same as the corresponding integer-order problem when an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q) and α = β = γ = 1.

In the following, the reader can find the graphs which obtained by (21), for different values of α, β and γ with the fixed values an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q).

Fig. 2
Fig. 2

The graphs of (21) from α = 0.25 (bottom) to α = 1.00 (top) by step size 0.25.

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00016

Example 21

Let f : [0,∞) → ℝ, t > a > 0. Consider the following ℳ-series fractional differential equation

i𝒟α(i𝒟αf)+p(t)i𝒟αf+q(t)f=0
where p and q are ℳ-differentiable functions of t. Assume that(22)has a solution, say f1. To find the second linearly independent solutions of(22), we start by assuming that f2(t) = v(t) f1(t) where v is an ℳ-differentiable function. So, from the chain rule, we have
i𝒟αf2(t)=i𝒟α(vf1)(t)=v(t)i𝒟αf1(t)+f1(t)i𝒟αv(t),i𝒟α(i𝒟αf2)(t)=i𝒟α(v(t)i𝒟αf1(t)+f1(t)i𝒟αv(t))=v(t)i𝒟α(i𝒟αf1)(t)+i𝒟αf1(t)i𝒟αv(t)+f1(t)i𝒟α(i𝒟αv)(t)+i𝒟αf1(t)i𝒟αv(t).
Substituting these in(22)and remembering that f1is a solution of it, we get
f1(t)i𝒟α(i𝒟αv)(t)+2i𝒟αf1(t)i𝒟αv(t)+p(t)f1(t)i𝒟αv(t)=0.
Now, if we let w(t)=i𝒟αv(t) , then it becomes
i𝒟αw(t)+(p(t)+2i𝒟αf1(t)f1(t))w(t)=0.
From Example 19, the solution of this equation can be found as
w(t)=Ceα(p(t)+2i𝒟αf1(t)f1(t))=Ceαp(t)f12(t),(C),
which yields
v(t)=Cα(eαpf12(t)).
Then we find the second solution as
f2(t)=Cf1(t)α(eαpf12(t)).

Example 22

Consider the following differential equation for f : [0,∞) → ℝ, t > a > 0:

i𝒟23i𝒟23ft13i𝒟23f=0.
Clearly, f1(t) = 1 is a solution of this equation and p(t)=t13 . Using formula(23)we obtain the second solution as
f2(t)=C23(e23(t13)).

The ℳ-series fractional integral

23(t13)=𝒦1(ta),
can be found by using the definition of 23 . From here we get,
f2(t)=C23(e𝒦1(ta))=C𝒦1e𝒦1aatx13e𝒦1xdx.

With transformation u = 𝒦−1x we have,

f2(t)=C𝒦13e𝒦1a[𝒦1a𝒦1tu13eudu]=C𝒦13e𝒦1a[𝒦1au13eudu𝒦1tu13eudu],(t>a>0)=C𝒦13e𝒦1a[Γ(2/3,𝒦1a)Γ(2/3,𝒦1t)],
where γ is the incomplete gamma function which defined as
Γ(δ,ν)=νtδ1etdt,
for δ > 0. From here we get the solution as
f(t)=1+C𝒦13e𝒦1a[Γ(2/3,𝒦1a)Γ(2/3,𝒦1t)].

For the fixed values an = 1, cm = 1, (n = 1,2,..., p; m = 1,2,...,q), this result coincides with the results given in [15] when c = β = γ = 1.

In the following, we plotted the graphs of solution function which obtained by (24), for different values of β and γ with the fixed values a = 0, c = 1, α=23 and c1cqa1ap=1 .

Fig. 3
Fig. 3

The graphs of (24) for the values β = γ = 1 (black); β = 0.5,γ = 1 (blue); β = 1,γ = 0.5 (red) and β = γ = 0.5 (green).

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00016

5 Concluding Remarks and Observations

In this paper, we first presented a fractional derivative operator, which is also a generalization of truncated M-fractional derivative, by using generalized M-series. Then we gave a definition of corresponding integral operator. Unlike fractional operators with different kernels, we showed that there are many common properties provided by both these and the corresponding integer-order operators. We also used these operators in differential equation problems as application and we plotted the graphs of the solutions for various values of α,β and γ. These problems are hard to solve by means of the classical definitions of fractional derivatives.

Besides, from equality (e) of Example 1, we observed that, for polynomials, truncated ℳ-series fractional derivative coincides with the Riemann-Liouville and Caputo fractional derivatives [20] up to a constant multiple. In this case, we can say that the truncated ℳ-series fractional derivative operator can be used instead of Riemann-Liouville or Caputo type derivatives (and also their generalizations) to solve some difficult problems.

Our definition is also a generalization of the 𝒱-fractional derivative for p = q = 1 which defined in [28]. It is also possible to define new fractional derivatives by using other special functions instead of M-series. Since M-series is a general class of special functions, all future definitions have chance to be the special cases of our definition. Further properties and applications of ℳ-series fractional operators will be discussed in forthcoming papers.

Acknowledgements

Some parts of this work was presented in the 4th International Conference on Computational Mathematics and Engineering Sciences which organized by Akdeniz University on April 20–22, 2019 in Antalya-Turkey.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Andrews L.C., (1985), Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York.

  • [2]

    P. Agarwal, (2012), Further results on fractional calculus of Saigo operators, Applications and Applied Mathematics, 7 (2), pp. 585–594.

  • [3]

    P. Agarwal, J. Choi, R. B. Paris, (2015), Extended Riemann-Liouville fractional derivative operator and its applications. Journal of Nonlinear Science and Applications, 8 (5), pp.451–466.

  • [4]

    A. Atangana, D. Baleanu, (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2), pp. 763–769.

  • [5]

    A. Atangana, J. F. Gómez-Aguilar, (2017), Hyperchaotic behavior obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos, Solitons and Fractals, 102, pp. 285–294.

  • [6]

    M. Caputo, M. Fabrizio, (2015), A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2), pp. 73–85.

  • [7]

    Y. Çenesiz, (2015), A. Kurt, The solutions of time and space conformable fractional heat equations with conformable Fourier Transform, Acta Univ. Sapientiae, Mathematica, 7, pp. 130–140.

  • [8]

    V.B.L. Chaurasia and D. Kumar, (2010), On the solutions of generalized fractional kinetic equations, Adv. Studies Theor. Phys, 4 (16), pp. 773–780.

  • [9]

    A. Chouhan and A. M. Khan, (2015), Unified integrals associated with H-functions and M-series, Journal of Fractional Calculus and Applications, 6 (2), pp. 11–16.

  • [10]

    A. Chouhan and S. Sarswat, (2012), Certain properties of fractional calculus operators associated with M-series, Scientia: Series A: Mathematical Sciences, 22, pp. 25–30.

  • [11]

    A. Çetinkaya, İ. O. Kıymaz, P. Agarwal, R. Agarwal, (2018), A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Advances in Difference Equations, 2018:156.

  • [12]

    R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, (2014), Mittag-Leffler Functions, Related Topic and Applications, Springer, Berlin.

  • [13]

    J. F. Gómez-Aguilar, A. Atangana, (2017), New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132:13, pp. 1–21.

  • [14]

    A. Faraj, T. Salim, S. Sadek and J. Ismail, (2014), A Generalization of M-Series and Integral Operator Associated with Fractional Calculus, Asian Journal of Fuzzy and Applied Mathematics, 2 (5), pp. 142–155.

  • [15]

    O. S. Iyiola, E. R. Nwaeze, (2016), Some new results on the new conformable fractional calculus with application using D’Alambert approach, Progr. Fract. Differ. Appl., 2 (2), pp. 115–122.

  • [16]

    U.R. Katugampola, (2014), A new fractional derivative with classical properties, arXiv:1410.6535v2.

  • [17]

    R. Khalil, M.A. Horani, A. Yousef and M. Sababheh, (2014), A new definition of fractional derivative, Jour. of Comp. and App. Math., 264, pp. 65–70.

  • [18]

    İ. O. Kıymaz, A. Çetinkaya, P. Agarwal, (2016), An extension of Caputo fractional derivative operator and its applications, Journal of Nonlinear Science and Applications, 9 (6), pp. 3611–3621.

  • [19]

    İ. O. Kıymaz, A. Çetinkaya, P. Agarwal, S. Jain, (2017), On a New Extension of Caputo Fractional Derivative Operator, Advances in Real and Complex Analysis with Applications, Birkhäuser Basel, pp. 261–275.

  • [20]

    A.A. Kilbas, H.M. Srivastava, J.J. Trujilli, (2006), Theory and Applications of Fractional Differential Equations, El-sever, Amsterdam, The Netherlands.

  • [21]

    D. Kumar and R.K. Saxena, (2015), Generalized fractional calculus of the M-Series involving F3 hypergeometric function, Sohag J. Math, 2 (1), pp. 17–22.

  • [22]

    J. Losada, J. J. Nieto, (2015), Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2), pp. 87–92.

  • [23]

    G.M. Mittag-Leffler, (1903), Sur la nouvelle fonction E(x), C. R. Acad. Sci. Paris, (Ser. II) 137, pp. 554–558.

  • [24]

    T.R. Prabhakar, (1971), A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19, pp. 7–15.

  • [25]

    M. Sharma, (2008), Fractional integration and fractional differentiation of the M-series, Fract. Calc. Appl. Anal., 11:2, pp. 187–191.

  • [26]

    M. Sharma and R. Jain, (2009), A note on a generalized M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal., 12:4, pp. 449–452.

  • [27]

    J.V.C. Sousa and E.C. de Oliveira, (2017), On the local M-derivative, arXiv:1704.08186v3.

  • [28]

    J.V.C. Sousa and E.C. de Oliveira, (2017), Mittag-Leffler Functions and the Truncated 𝒱-fractional Derivative, Mediterr. J. Math., 14:244.

  • [29]

    J.V.C. Sousa and E.C. de Oliviera, (2018), A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), pp. 83–96.

  • [30]

    J.V.C. Sousa and E.C. de Oliviera, (2018), Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation, Computational and Applied Mathematics, 37 (4), pp. 5375–5394.

  • [31]

    J.V.C. Sousa and E.C. de Oliviera, (2018), A Truncated 𝒱-Fractional Derivative in R n, Turk. J. Math. Comput. Sci., 8, pp. 49–64.

  • [32]

    A. Wiman, (1905), Uber den Fundamental satz in der Theorie de Funktionen E(x), Acta Math., 29, pp. 191–201.

  • [33]

    X. J. Yang, H. M. Srivastava, J. T. Machado, (2016), A new fractional derivative without singular kernel: Application to the modeling of the steady heat flow, Thermal Science, 20 (2), pp. 753–756.

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  • View in gallery

    The graphs of (18) from α = 0.25 (green) to α = 1.00 (black) by step size 0.25.

  • View in gallery

    The graphs of (21) from α = 0.25 (bottom) to α = 1.00 (top) by step size 0.25.

  • View in gallery

    The graphs of (24) for the values β = γ = 1 (black); β = 0.5,γ = 1 (blue); β = 1,γ = 0.5 (red) and β = γ = 0.5 (green).