A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties
aa r X i v : . [ m a t h . C A ] A ug A NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYINGSOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICALPROPERTIES J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Abstract.
We introduce a truncated M -fractional derivative type for α -differentiable func-tions that generalizes four other fractional derivatives types recently introduced by Khalilet al., Katugampola and Sousa et al., the so-called conformable fractional derivative, alter-native fractional derivative, generalized alternative fractional derivative and M -fractionalderivative, respectively. We denote this new differential operator by i D α,βM , where the pa-rameter α , associated with the order of the derivative is such that 0 < α < β > M is the notation to designate that the function to be derived involves the truncatedMittag-Leffler function with one parameter.The definition of this truncated M -fractional derivative type satisfies the properties ofthe integer-order calculus. We also present, the respective fractional integral from whichemerges, as a natural consequence, the result, which can be interpreted as an inverse prop-erty. Finally, we obtain the analytical solution of the M -fractional heat equation and presenta graphical analysis. Keywords : Alternative Fractional Derivative, Conformable Fractional Derivative, M -fractionalheat equation, Truncated M -Fractional Derivative type, Truncated Mittag-Leffler Function.MSC 2010 subject classifications. 26A06; 26A24; 26A33; 26A42. Introduction
The non integer-order calculus or fractional calculus, as it is largely diffused, is as importantand ancient as the integer-order calculus, and for many years the scientific community didn’tknow it. Currently, there are numerous and important definitions of fractional derivativestypes, each one of them with its peculiarity and application [1, 2, 3]. Although, to behighlighted only from 1974, after the first international conference on fractional calculus,it has shown to be important and with great applicability in modeling problems, moreprecisely natural phenomena. For this growth, it was necessary the contribution of famousmathematicians, such as Lagrange, Abel, Euler, Liouville, Riemann, as well as recentlyCaputo and Mainardi, among others.It is possible to define various integrals and fractional derivatives. Each definition has itsown peculiarity and thus makes the fractional calculus fruitful in the sense of theory andapplications. We report some types of fractional derivatives that have been introduced sofar, among which we mention: Riemann-Liouville, Caputo, Hadamard, Caputo-Hadamard, AND E. CAPELAS DE OLIVEIRA Riesz, among others [3, 4]. Many of these derivatives are defined from the fractional integralin the Riemann-Liouville sense. Recently, Katugampola [5], has presented a new fractionalintegral unifying six existing fractional integrals, named Riemann-Liouville, Hadamard,Erdlyi-Kober, Katugampola, Weyl and Liouville [3, 6].On the other hand, also recent, Khalil et al. [7] proposed the so-called compatible fractionalderivative of order α with 0 < α < M -fractional derivative involving a Mittag-Leffler function with one parameter [10] that alsosatisfies the properties of integer-order calculus. In this sense, we are going to introduce atruncated M -fractional derivative type that unifies four existing fractional derivative typesmentioned above and which also satisfies the classical properties of integer-order calculus.The study of differential equations has proved very useful over time. One of the main reasonsis that the simplest differential equations have the ability to model more complex naturalsystems [1, 3, 11]. There are a larger number of application involving differential equationsof which we mention: the problem of population dynamics, brachistochronous problem, waveequation, heat equation and others [1, 12]. However, natural systems over time, become morecomplex and more than differential equations, provides a rough and simplified descriptionof the actual process, it is necessary that new and more refined mathematical tools arepresented and studied. In this sense, fractional derivatives are used to propose modelingin order to obtain more precise results in the studies and applications involving differentialequations [11]. Then through the use of properties of a truncated M -fractional derivativetype, we will present an analytical study of the heat equation and through graph, we willanalyze the behavior of the solution in relation to other types of fractional derivatives theso-called local derivatives.This paper is organized as follows: in section 2, our main result, we introduce the conceptof truncated M -fractional derivative type involving a truncated Mittag-Leffler function, aswell as several theorems. Also, we introduce the respective M -fractional integral for whichwe demonstrate the inverse property. In section 3, we present the relationship betweena truncated M -fractional derivative type, introduced here, and the conformable fractionalderivative, generalized and alternative fractional derivative and M -fractional derivative. Insection 4, we perform an analytical study of the M -fractional heat equation in order to obtainthe analytical solution and present some graphs. Concluding remarks close the paper.2. Truncated M -fractional derivative type In this section, we define a truncated M -fractional derivative type and obtain several re-sults that have a great similarity with the results found in the classical calculus. From thedefinition, we present a theorem showing that this truncated M -fractional derivative type NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES3 is linear, obeys the product rule and the composition of two α -differentiable functions, thequotient rule and the chain rule. It is also shown that the derivative of a constant is zero,as well as versions for Rolle’s theorem, the mean value theorem and an extension of themean value theorem. Further, the continuity of this truncated M -fractional derivative typeis shown as in integer-order calculus. Also, we introduce the concept of M -fractional integralof a f function. From the definition, we shown the inverse theorem.We define the truncated Mittag-Leffler function of one parameter by:(2.1) i E β ( z ) = i X k =0 z k Γ ( βk + 1) , with β > z ∈ C .From Eq.(2.1), we define a truncated M -fractional derivative type that unifies other fourfractional derivatives that refer to classical properties of the integer-order calculus.In this work, if a truncated M -fractional derivative type of order α as defined in Eq.(2.2) ofa function f exists, we say that the function f is α -differentiable.Thus, let us begin with the following definition, which is a generalization of the usual defi-nition of integer order derivative. Definition 1.
Let f : [0 , ∞ ) → R . For < α < a truncated M -fractional derivative typeof f of order α , denoted by i D α,βM , is (2.2) i D α,βM f ( t ) := lim ε → f ( t i E β ( εt − α )) − f ( t ) ε , ∀ t > and i E β ( · ) , β > is a truncated Mittag-Leffler function of one parameter, as definedin Eq.(2.1).Note that, if f is α -differentiable in some open interval (0 , a ), a >
0, and lim t → + (cid:16) i D α,βM f ( t ) (cid:17) exist, then we have i D α,βM f (0) = lim t → + (cid:16) i D α,βM f ( t ) (cid:17) . Theorem 1.
If a function f : [0 , ∞ ) → R is α -differentiable for t > , with < α ≤ , β > , then f is continuous at t .Proof. In fact, let us consider the identity(2.3) f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) = f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) ε ! ε. J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Applying the limit ε → ε → f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) = lim ε → f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) − f ( t ) ε ! lim ε → ε = i D α,βM f ( t ) lim ε → ε = 0 . Then, f is continuous at t .Using the definition of truncated Mittag-Leffler function of one parameter, we have(2.4) f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) = f t i X k =0 ( εt − α ) k Γ ( βk + 1) ! . Applying the limit ε → f is a continuous function, wehave lim ε → f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) = lim ε → f t i X k =0 ( εt − α ) k Γ ( βk + 1) ! = f t lim ε → i X k =0 ( εt − α ) k Γ ( βk + 1) ! . (2.5)Further, we have t i E β (cid:0) εt − α (cid:1) = t i X k =0 ( εt − α ) k Γ ( βk + 1)= t + εt − α Γ ( β + 1) + t ( εt − α ) Γ (2 β + 1) + t ( εt − α ) Γ (3 β + 1) + · · · + t ( εt − α ) i Γ ( iβ + 1) . (2.6)Applying the limit ε → ε → i X k =0 ( εt − α ) k Γ ( βk + 1) = 1 . In this way, we conclude that lim ε → f (cid:0) t i E β (cid:0) εt − α (cid:1)(cid:1) = f ( t ) . (2.7)Here, we present the theorem that encompasses the main classical properties of integer ordercalculus. For the chain rule, it is verified through an example, as we will see next. We will NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES5 do here, only the demonstration of the chain rule, for other items, follow the same steps ofTheorem 2 found in the paper by Sousa and Oliveira [9]. Theorem 2.
Let < α ≤ , β > , a, b ∈ R and f, g α -differentiable, at a point t > .Then:(1) i D α,βM ( af + bg ) ( t ) = a i D α,βM f ( t ) + b i D α,βM g ( t ) .(2) i D α,βM ( f · g ) ( t ) = f ( t ) i D α,βM g ( t ) + g ( t ) i D α,βM f ( t ) .(3) i D α,βM (cid:18) fg (cid:19) ( t ) = g ( t ) i D α,βM f ( t ) − f ( t ) i D α,βM g ( t )[ g ( t )] .(4) i D α,βM ( c ) = 0 , where f ( t ) = c is a constant.(5) (Chain rule) If f is differentiable, then i D α,βM ( f ) ( t ) = t − α Γ ( β + 1) df ( t ) dt . Proof.
From Eq.(2.6), we have t i E β (cid:0) εt − α (cid:1) = t + εt − α Γ ( β + 1) + O (cid:0) ε (cid:1) , and introducing the following change, h = εt − α (cid:18)
1Γ ( β + 1) + O ( ε ) (cid:19) ⇒ ε = ht − α (cid:16) β +1) + O ( ε ) (cid:17) , we conclude that i D α,βM f ( t ) = lim ε → f ( t + h ) − f ( t ) ht α − β +1) (1 + Γ ( β + 1) O ( ε ))= t − α Γ ( β + 1) lim ε → f ( t + h ) − f ( t ) h β + 1) O ( ε )= t − α Γ ( β + 1) df ( t ) dt , with β > t > (6) i D α,βM ( f ◦ g ) ( t ) = f ′ ( g ( t )) i D α,βM g ( t ) , for f differentiable at g ( t ) . Now, it is necessary to know if, in addition to the previous Theorem 2 that contains im-portant properties similar to integer-order calculus, this truncated M -fractional derivativetype Eq.(2.2) also has important theorems related to the classical calculus. We shall nowsee that: the Rolle’s theorem, the mean value theorem and its extension coming from theinteger-order calculus can be extended to α -differentiable functions, i.e., that admit trun-cated M -fractional derivative as introduced in Eq.(2.2). J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Theorem 3. (Rolle’s theorem for fractional α -differentiable functions) Let a > , and f :[ a, b ] → R be a function with the properties:(1) f is continuous on [ a, b ] .(2) f is α -differentiable on ( a, b ) for some α ∈ (0 , .(3) f ( a ) = f ( b ) .Then, ∃ c ∈ ( a, b ) , such that i D α,βM f ( c ) = 0 , with β > .Proof. Since f is continuous on [ a, b ] and f ( a ) = f ( b ), there exist c ∈ ( a, b ), at which thefunction has a local extreme. Then, i D α,βM f ( c ) = lim ε → − f ( c i E β ( εc − α )) − f ( c ) ε = lim ε → + f ( c i E β ( εc − α )) − f ( c ) ε . But, the two limits have opposite signs. Hence, i D α,βM f ( c ) = 0.The proof of Theorem 4 and Theorem 5, will be omitted, but follow the same reasoning ofthe respective theorems demonstrated in Sousa and Oliveira [9]. Theorem 4. (Mean-value theorem for fractional α -differentiable functions) Let a > and f : [ a, b ] → R be a function with the properties:(1) f is continuous on [ a, b ] .(2) f is α -differentiable on ( a, b ) for some α ∈ (0 , .Then, ∃ c ∈ ( a, b ) , such that i D α,βM f ( c ) = f ( b ) − f ( a ) b α α − a α α , with β > . Theorem 5. (Extension mean value theorem for fractional α -differentiable functions) Let a > , and f, g : [ a, b ] → R functions that satisfy:(1) f, g are continuous on [ a, b ] .(2) f, g are α -differentiable for some α ∈ (0 , .Then, ∃ c ∈ ( a, b ) , such that (2.8) i D α,βM f ( c ) i D α,βM g ( c ) = f ( b ) − f ( a ) g ( b ) − g ( a ) , with β > . NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES7 Definition 2.
Let α ∈ ( n, n + 1] , for some n ∈ N , β > and f n -differentiable for t > .Then the α -fractional derivative of f is defined by (2.9) i D α,β ; nM f ( t ) := lim ε → f ( n ) ( t i E β ( εt n − α )) − f ( n ) ( t ) ε , since the limit exist. From Definition 2 and the chain rule, that is, from item 5 of Theorem 2, by induction on n , wecan prove that i D α,β ; nM f ( t ) = t n +1 − α Γ ( β + 1) f ( n +1) ( t ), α ∈ ( n, n + 1] and f is ( n + 1)-differentiablefor t > M -fractional derivative type Eq.(2.2) has a corresponding M -fractional integral. Then, we will present the definition and a theorem that correspondsto the inverse property. For other results involving integrals, one can consult [9, 13]. Definition 3.
Let a ≥ and t ≥ a . Also, let f be a function defined in ( a, t ] and < α < .Then, the M -fractional integral of order α of function f is defined by [9](2.10) M I α,βa f ( t ) = Γ ( β + 1) Z ta f ( x ) x − α dx, with β > Theorem 6. (Inverse) Let a ≥ < α <
1. Also, let f be a continuous function suchthat exist M I α,βa f . Then(2.11) i D α,βM ( M I α,βa f ( t )) = f ( t ) , with t ≥ a and β > Proof.
In fact, using the chain rule as seen in Theorem 2, we have i D α,βM (cid:0) M I α,βa f ( t ) (cid:1) = t − α Γ ( β + 1) ddt ( M I α,βa f ( t ))= t − α Γ ( β + 1) ddt (cid:18) Γ ( β + 1) Z ta f ( x ) x − α dx (cid:19) = t − α Γ ( β + 1) (cid:18) Γ ( β + 1) t − α f ( t ) (cid:19) = f ( t ) . (2.12)With the condition f ( a ) = 0, by Theorem 6, that is, Eq.(2.12), we have M I α,βa h i D α,βM f ( t ) i = f ( t ). J. VANTERLER DA C. SOUSA AND E. CAPELAS DE OLIVEIRA Relation with other fractional derivatives types
In this section, we will discuss the relationship between the fractional conformable derivativeproposed by Khalil et al. [7], the alternative fractional derivative and the generalized alter-native fractional derivative proposed by Katugampola [8] and the M -fractional derivativeproposed by Sousa and Oliveira [9], with our truncated M -fractional derivative type.Khalil et al. [7] proposed a definition of a fractional derivative, called conformable fractionalderivative that refers to the classical properties of integer order calculus, given by(3.1) f ( α ) ( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε , with α ∈ (0 ,
1) and t > D α f ( t ) = lim ε → f (cid:16) te εt − α (cid:17) − f ( t ) ε , with α ∈ (0 ,
1) and t > k e x , proposed another generalized fractional derivative, given by(3.3) D αk f ( t ) = lim ε → f (cid:16) k e εt − α t (cid:17) − f ( t ) ε , with α ∈ (0 ,
1) and t > M -fractional derivative D α,βM where theparameter β > M is the notation to designate that the function to be derived involvesthe Mittag-Leffler function of one parameter, given by(3.4) D α,βM f ( t ) := lim ε → f ( t E β ( εt − α )) − f ( t ) ε , with α ∈ (0 ,
1) and t > M -fractional derivative type Eq.(2.2) is moregeneral than the fractional derivatives Eq.(3.1), Eq.(3.2), Eq.(3.3) and Eq.(3.4). We willnow study particular cases involving such fractional derivatives.Choosing β = 1 and applying the limit i → D α,βM f ( t ) = lim ε → f ( t E ( εt − α )) − f ( t ) ε . NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES9 But, it is know that(3.6) E (cid:0) εt − α (cid:1) = X k =0 ( εt − α ) k Γ ( k + 1) = 1 + εt − α . Thus, we conclude that(3.7) D α,βM f ( t ) = lim ε → f ( t + εt − α ) − f ( t ) ε = f ( α ) ( t ) , which is exactly the conformable fractional derivative Eq.(3 . β = 1 and applying the limit i → ∞ on both sides of Eq.(2.2), we have(3.8) ∞ D α,βM f ( t ) = lim ε → f ( t ∞ E ( εt − α )) − f ( t ) ε . But, as we have(3.9) ∞ E (cid:0) εt − α (cid:1) = ∞ X k =0 ( εt − α ) k Γ ( k + 1) = e εt − α , we conclude that(3.10) ∞ D α,βM f ( t ) = lim ε → f (cid:16) te εt − α (cid:17) − f ( t ) ε = D α f ( t ) , which is exactly the alternative fractional derivative, Eq.(3 . β = 1 in Eq.(2.2), we have(3.11) i D α,βM f ( t ) = lim ε → f ( t i E ( εt − α )) − f ( t ) ε . Remembering that(3.12) i E (cid:0) εt − α (cid:1) = i X k =0 ( εt − α ) k Γ ( k + 1) = e εt − α i , we have(3.13) i D α,βM f ( t ) = lim ε → f (cid:16) e εt − α i t (cid:17) − f ( t ) ε = D αi f ( t ) , exactly the generalized fractional derivative, Eq.(3 . i → ∞ on both sides of Eq.(2.2), we have(3.14) ∞ D α,βM f ( t ) = lim ε → f ( t ∞ E β ( εt − α )) − f ( t ) ε , AND E. CAPELAS DE OLIVEIRA since(3.15) ∞ E β (cid:0) εt − α (cid:1) = ∞ X k =0 ( εt − α ) k Γ ( k + 1) = E β (cid:0) εt − α (cid:1) , we conclude that(3.16) ∞ D α,βM f ( t ) = lim ε → f ( t E β ( εt − α )) − f ( t ) ε = D α,βM f ( t ) , exactly the M -fractional derivative, Eq.(3 . Application
In this section, we obtain the solution of the heat equation using a truncated M -fractionalderivative type with 0 < α < ∂u ( x, t ) ∂t = k ∂ u ( x, t ) ∂x , < x < L, t > , where k is a positive constant.Using a M -fractional derivative type, we propose an M -fractional heat equation given by(4.1) ∂ α u ( x, t ) ∂t α = k ∂ u ( x, t ) ∂x , < x < L, t > , where 0 < α < u (0 , t ) = 0 , t ≥ , (4.2) u ( L, t ) = 0 , t ≥ ,u ( x,
0) = f ( x ) , ≤ x ≤ L. We start, considering the so-called M -fractional linear differential equation with constantcoefficients(4.3) ∂ α v ( x, t ) ∂t α ± µ v ( x, t ) = 0 , where µ is a positive constant.Using the item 5 in Theorem 2, the Eq.(4.1) can be written as follows t − α Γ ( β + 1) dv ( x, t ) dt ± µ v ( x, t ) = 0 , whose solution is given by(4.4) v ( t ) = ce ± Γ( β +1) µ tαα , with 0 < α < β > NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES11 Now, we will use separation of variables method to obtain the solution of the M -fractionalheat equation. Then, considering u ( x, t ) = P ( x ) Q ( t ) and replacing in Eq.(4.1), we get d α dt α Q ( t ) P ( x ) = k d dx P ( x ) Q ( t )which implies(4.5) 1 kQ ( t ) d α dt α Q ( t ) = 1 P ( x ) d dx P ( x ) = ξ. From Eq.(4.5), we obtain a system of differential equations, given by(4.6) d α dt α Q ( t ) − kξQ ( t ) = 0and(4.7) d dx P ( x ) − ξP ( x ) = 0 . First, let’s find the solution of Eq.(4.7). For this, we must study three cases, that is, ξ = 0 ,ξ = − µ e ξ = µ . Case 1: ξ = 0.Substituting ξ = 0 into Eq.(4.7), we have(4.8) d dx P ( x ) − ξP ( x ) = 0 , whose solution is given by P ( x ) = c x + c , with c and c arbitrary constant. Using theinitial conditions given by Eq.(4.2), we obtain that c = c = 0. Like this, P ( x ) = 0, whichimplies u ( x, t ) = 0 trivial solution.Case 2: ξ = − µ .Substituting ξ = − µ into Eq.(4.7), we get d dx P ( x ) + µ P ( x ) = 0 , whose solution is given by P ( x ) = c sin ( µx ) + c cos ( µx ), with c and c arbitrary constant.Using the initial conditions Eq.(4.2), we obtain c = 0 and 0 = c sin ( µx ) which implies that µ = nπL , with n = 1 , , ... . Then, we obtain P n ( x ) = a n sin (cid:16) nπxL (cid:17) and µ = nπL . Case 3: ξ = µ . AND E. CAPELAS DE OLIVEIRA Substituting ξ = µ into Eq.(4.7), we get d dx P ( x ) − µ P ( x ) = 0whose solution is given by P ( x ) = c e µx + c e − µx = A cosh ( µx ) + B sinh ( µx ), with c , c , A , B arbitrary constant. Using the boundary conditions Eq.(4.2), we have A = 0 and0 = B sinh ( µx ). As λ = − µ < λL = 0 then sinh ( µx ) = 0 . Like this, we get B = 0and then P n ( x ) = 0, which implies u ( x, t ) = 0, trivial solution.Therefore, the solution of Eq.(4.7) is given by(4.9) P n ( x ) = a n sin (cid:16) nπxL (cid:17) and µ = nπL . Using the Eq.( 4.3) and Eq.(4.4), we have(4.10) Q n ( t ) = b n exp (cid:18) − Γ ( β + 1) (cid:16) nπL (cid:17) kα t α (cid:19) , where b n are constant coefficients.So, using the Eq.(4.9) and Eq.(4.10), the partial solutions of Eq.(4.1), is given by u β ( x, t ) = ∞ X n =1 c n sin (cid:16) nπxL (cid:17) exp (cid:18) − Γ ( β + 1) (cid:16) nπL (cid:17) kα t α (cid:19) . Using Eq.(4.2), we get u ( x,
0) = f ( x ) = ∞ X n =1 c n sin (cid:16) nπxL (cid:17) which provides c n through c n = 2 L Z L f ( x ) sin (cid:16) nπxL (cid:17) dx. So, we conclude that the solution of M -fractional heat equation Eq.(4.1), satisfying theconditions Eq.(4.2), is given by(4.11) u β ( x, t ) = ∞ X n =1 sin (cid:16) nπxL (cid:17) exp (cid:18) − Γ ( β + 1) (cid:16) nπL (cid:17) kα t α (cid:19) (cid:18) L Z L f ( x ) sin (cid:16) nπxL (cid:17) dx (cid:19) . Taking the limit β → u ( x, t ) = ∞ X n =1 sin (cid:16) nπxL (cid:17) exp (cid:18) − (cid:16) nπL (cid:17) kα t α (cid:19) (cid:18) L Z L f ( x ) sin (cid:16) nπxL (cid:17) dx (cid:19) , which is exactly the solution of the fractional heat equation proposed by enesiz et al. [14]. NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES13 On the other hand, taking the limit β → α →
1, using Eq.(2.2), we recover the solutionof heat equation of integer order.(4.13) u ( x, t ) = ∞ X n =1 sin (cid:16) nπxL (cid:17) exp (cid:18) − (cid:16) nπL (cid:17) kt (cid:19) (cid:18) L Z L f ( x ) sin (cid:16) nπxL (cid:17) dx (cid:19) . Next, we will present some plots by choosing values for the parameters β and α , to see thebehavior of the solution presented in Eq (2.2). The graphics were plotted using MATLAB7:10 software (R2010a). For the elaboration of the following plots, we choose the function f ( x ) = 50 x (1 − x ) and for each fixed β , we vary the α parameter. Figure 1.
Analytical solution of the M -fractional heat equation Eq.(4.11).We consider the values β = 0 . L = 1, k = 0 .
003 and at time t = 150. u ( x ,t ) α =0.1 α =0.6 α =1.0 AND E. CAPELAS DE OLIVEIRA Figure 2.
Analytical solution of the M -fractional heat equation Eq.(4.11).We take the values β = 1 . L = 1, k = 0 .
003 and at time t = 150. u ( x ,t ) α =0.1 α =0.6 α =1.0 NEW TRUNCATED M -FRACTIONAL DERIVATIVE TYPE UNIFYING SOME FRACTIONAL DERIVATIVE TYPES WITH CLASSICAL PROPERTIES15 Figure 3.
Analytical solution of the M -fractional heat equation Eq.(4.11).We chose the values β = 2 . L = 1, k = 0 .
003 and at time t = 150. u ( x ,t ) α =0.1 α =0.6 α =1.0 Concluding remarks