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

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 integerorder derivatives. Finally, we obtain the analytical solutions of some M-series fractional differential equations.


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.
In the same year, Katugampola [16] introduced the alternative and truncated alternative fractional derivatives for f : [0, ∞) → R as , t > 0, α ∈ (0, 1) and respectively. Here e x i = ∑ i k=0 x k k! is the truncated exponential function. Recently, Sousa and de Oliveira [27,29] introduced the M-fractional and truncated M-fractional derivatives for f : , β ,t > 0, α ∈ (0, 1) , β ,t > 0, α ∈ (0, 1) respectively, by means of one parameter Mittag-Leffler function [12] 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 t 1−α d f dt . In 2009, generalized M-series defined by Sharma and Jain [25,26] β ,γ Here, (α) k is the Pochhammer symbol [1] which given by with the assume (α) 0 = 1. Note that if a j ( 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 p ≤ q; it is convergent for |z| < δ = α α 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.
Note that, if f is M-differentiable in some interval (0, a), a > 0 and exists, then we define 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 a 1 = c 1 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, a n , c m ∈ R and c m = 0, . Also, we use the notation K instead of the constant a 1 ···a p c 1 ···c q Γ(γ) Γ(β +γ) . Now we begin our investigation with an important theorem.
Applying the limit for ε → 0 on both sides, we get Then, f is continuous at t 0 . Besides, using the definition of the truncated M-series, we can write If we apply the limit for ε → 0 on both sides and since f is continuous, we get we can write lim The following theorem is about the basic properties of M-series fractional derivative: 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 For (e): If g is a constant function in a neighborhood of a. Then clearly i D α M f (g(a)) = 0. Now, assume that g is not a constant function, that is, we can find an ε > 0 for any t 1 ,t 2 ∈ (a − ε, a + ε) such that g(t 1 ) = g(t 2 ). Since g is continuous at a and for small enough ε, we have 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 the two limits have opposite sings. So i D α M f (c) = 0. Theorem 5 (Mean value theorem). Let a > 0 and f : [a, b] → R be a function such that: Proof. Consider the following function: The function g provides the conditions of the Rolle's theorem. Then, there exists a point c ∈ (a, b) , such that  (b) f , g are M-differentiable on (a, b) for some α ∈ (0, 1).
Then, there exists c ∈ (a, b), such that: Proof. Consider the following function: g(a)) .
Since f is also continuous in [t 1 ,t 2 ] and M-differentiable in (t 1 ,t 2 ), from Rolle's theorem, there exist a point c ∈ (t 1 ,t 2 ) with . Since t 1 < t 2 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), 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). for some α ∈ (0, 1). Then, for all t ∈ (a, b) Proof. From Theorem 7 we know that for t 1 ,t 2 ∈ [a, b] there exist a c ∈ (t 1 ,t 2 ) such as , so f is decreasing.
Proof. The proof is trivial when you consider the function h(t) = g(t) − f (t).
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 α 1 , α 2 ∈ (0, 1). Then 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 α of f is given as 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 i D α;n M f (t) = Kt n+1−α f (n+1) (t). by using (13), (8) and induction on n.

M-series Fractional Integral
In this section, we defined the corresponding M-series fractional integral operator I α M f (t). We want that our integral operator satisfies i D α be a differentiable function, then from (8) we have the following differential equation which have a solution of the form for a n = 0, (n = 1, 2, . . . , p) 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 where the conditions are same as (7) with a n = 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 I α M f (a) = 0. For the rest of the paper we assume that a n = 0, n = 1, 2, . . . , p.
Theorem 13. Let a ≥ 0, α ∈ (0, 1) and f is a continuous function such that I α M f (t) exists. Then for t ≥ a, 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 Since the function f is differentiable, by using the fundamental theorem of calculus for the integer-order derivatives and (8), we get which gives the result.
which completes the proof.
Corollary 16. Let f : [a, b] → R be a continuous function such that Then, for all t ∈ [a, b] with 0 < a < b, α ∈ (0, 1) and K > 0 we have Proof. From the previous theorem we have which gives the result.
Theorem 17. Let f , g : [a, b] → R be two differentiable functions and α ∈ (0, 1). where Proof. Using the definition of M-series fractional integral (14), (8) and applying fundamental theorem of calculus for integer-order derivatives, we get 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 α order M-series fractional integral of a function f is defined by where the conditions are same as (7) with a n = 0, n = 1, 2, . . . , p.
The following theorem is a generalization of Theorem 14.

Example 20. Consider the heat equation in one dimension
with the initial and boundary conditions 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  (19) is

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.
T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k