Solving Sequential Linear M fractional Differential Equations with Constants Coefficients

TL;DR

This paper introduces a method for solving sequential linear fractional differential equations using the M derivative, discussing existence and uniqueness of solutions with applications to homogeneous and non-homogeneous cases.

Contribution

It presents a novel approach for solving M fractional sequential linear differential equations with constant coefficients, including proofs of existence and uniqueness.

Findings

Method effectively solves M fractional differential equations

Existence and uniqueness of solutions are established

Illustrations demonstrate application to various cases

Abstract

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. The M derivative is the generalization of alternative fractional derivative. This M derivative obey the properties of integer calculus. In this paper, we present the method for solving M fractional sequential linear differential equations with constant coefficients for alpha is greater than or equal to 0 and beta is greater than 0. Existence and Uniqueness of the solutions for the nth order sequential linear M fractional differential equations are discussed in detail. We have present illustration for homogeneous and non homogeneous case.

Equations

$$D_M^{\alpha ,\beta }f(t)=\underset{ϵ\to 0}{\lim }\frac{f(t{\mathbb{E}}_{\beta }(ϵt^{-\alpha })-f(t)}{ϵ}$$

$$\underset{t\to 0^{+}}{\lim }{D}_M^{\alpha ,\beta }f(t)$$

$$D_M^{\alpha ,\beta }f(0)=\underset{t\to 0^{+}}{\lim }{D}_M^{\alpha ,\beta }f(t)$$

$${}_M{I}_a^{\alpha ,\beta }f(t)=\Gamma (\beta +1){\int }_a^t\frac{f(x)}{x^{1-\alpha }}dx$$

$$D_M^{\alpha ,\beta }{(}_M{I}_a^{\alpha ,\beta }f(t))=f(t)$$

$$\int_{a}^{b}f(x)D^{\alpha,\beta}_{M}g(x)d_{\alpha}x=f(x)g(x)\Bigr{|}_{a}^{b}-\int_{a}^{b}g(x)D^{\alpha,\beta}_{M}f(x)d_{\alpha}x$$

$${}^n{D}_M^{\alpha ,\beta }y+p_{n-1}(t{)}^{n-1}{D}_M^{\alpha ,\beta }y+...+p_2(t{)}^2{D}_M^{\alpha ,\beta }y+p_1(t)D_M^{\alpha ,\beta }y+p_0(t)y=0$$

$${}^n{D}_M^{\alpha ,\beta }y+p_{n-1}(t{)}^{n-1}{D}_M^{\alpha ,\beta }y+...+p_2(t{)}^2{D}_M^{\alpha ,\beta }y+p_1(t)D_M^{\alpha ,\beta }y+p_0(t)y=f(t)$$

$$L_{\alpha ,\beta }[y]{=}^n{D}_M^{\alpha ,\beta }y+p_{n-1}(t{)}^{n-1}{D}_M^{\alpha ,\beta }y+...+p_2(t{)}^2{D}_M^{\alpha ,\beta }y+p_1(t)D_M^{\alpha ,\beta }y+p_0(t)y=0$$

$$D_M^{\alpha ,\beta }y+p(t)y=f(t)$$

$$y(t_0)=y_0$$

$$D_M^{\alpha ,\beta }y+p(t)y=f(t)$$

$$\frac{t^{1-\alpha }}{\Gamma (\beta +1)}{y}^{\prime }+p(t)y=f(t)$$

$$y^{\prime }+\Gamma (\beta +1)t^{\alpha -1}p(t)y=\Gamma (\beta +1)t^{\alpha -1}f(t)$$

$${}^n{D}_M^{\alpha ,\beta }y+p_{n-1}(t{)}^{n-1}{D}_M^{\alpha ,\beta }y+...+p_2(t{)}^2{D}_M^{\alpha ,\beta }y+p_1(t)D_M^{\alpha ,\beta }y+p_0(t)y=f(t)$$

$$y(t_0)=y_0,D_M^{\alpha ,\beta }y(t_0)=y_1,...{,}^{n-1}{D}_M^{\alpha ,\beta }y(t_0)=y_{n-1},a<t_0<b$$

$$x_1=y,x_2=D_M^{\alpha ,\beta }y,x_3{=}^2{D}_M^{\alpha ,\beta }y,,...,x_n{=}^{n-1}{D}_M^{\alpha ,\beta }y$$

$$D_M^{\alpha ,\beta }{x}_1=x_2$$

$$D_M^{\alpha ,\beta }{x}_2=x_3$$

$$⋮$$

$$D_M^{\alpha ,\beta }{x}_{n-1}=x_n$$

$$D_M^{\alpha ,\beta }{x}_n=-p_{n-1}{x}_n-...-p_2{x}_3-p_1{x}_2-p_0{x}_1+f(t)$$

$$D_M^{\alpha ,\beta }\underset{X(t)}{\underbrace{\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_{n-1}\\ x_n\end{array}\right]}}+\underset{P(t)}{\underbrace{\left[\begin{array}{cccccc}0& -1& 0& 0& \cdots & 0\\ 0& 0& -1& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \cdots & ⋮\\ 0& 0& 0& 0& \cdots & -1\\ p_0& p_1& p_2& p_3& \cdots & p_{n-1}\end{array}\right]}}\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_{n-1}\\ x_n\end{array}\right]=\underset{F(t)}{\underbrace{\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ f(t)\end{array}\right]}}$$

$$D_M^{\alpha ,\beta }X(t)+P(t)X(t)=F(t)$$

$$X^{\prime }(t)+\Gamma (\beta +1)t^{\alpha -1}P(t)X(t)=\Gamma (\beta +1)t^{\alpha -1}F(t)$$

$$i.e,L_{\alpha ,\beta }[c_1{y}_1+c_2{y}_2]=c_1{L}_{\alpha ,\beta }[y_1]+c_2{L}_{\alpha ,\beta }[y_2]$$

$$W_{\alpha ,\beta }(t)=\left|\begin{array}{cccc}{y}_1& y_2& \cdots & y_n\\ D_M^{\alpha ,\beta }{y}_1& D_M^{\alpha ,\beta }{y}_2& \cdots & D_M^{\alpha ,\beta }{y}_n\\ ⋮& ⋮& \cdots & ⋮\\ {}^{n-1}{D}_M^{\alpha ,\beta }{y}_1& {}^{n-1}{D}_M^{\alpha ,\beta }{y}_2& \cdots & {}^{n-1}{D}_M^{\alpha ,\beta }{y}_n\end{array}\right|$$

$$i.e,y=c_1{y}_1+c_2{y}_2+...+c_n{y}_n$$

$$c_1{y}_1(t_0)+c_2{y}_2(t_0)+...+c_n{y}_n(t_0)=y(t_0)$$

$$c_1{D}_M^{\alpha ,\beta }{y}_1(t_0)+c_2{D}_M^{\alpha ,\beta }{y}_2(t_0)+...+c_n{D}_M^{\alpha ,\beta }{y}_n(t_0)=D_M^{\alpha ,\beta }y(t_0)$$

Full text

Mathematics Subject Classification: 26A33, 34AXX.

1. Introduction

While L’Hospital has proposed the idea of fractional derivative in the $17^{th}$ century, several researchers concerted fractional derivative in the recent centuries. Riemann-Liouville, Caputo and other fractional derivatives are defined on the basis of fractional integral form [8, 12, 13].

Recently, Khalil et al.[7] and Katugampola [6] proposed fractional derivatives in the limit form as in usual derivative such as conformable fractional derivative and alternative fractional derivative. Based on these derivative, Sousa and Oliveira [15] introduced M-fractional derivatives which satisfies properties of integer-order calculus.

Theory and applications of the sequential linear fractional differential equations involving Hadamard, Riemann-Liouville, Caputo and Conformable derivatives have been investigated in [1, 2, 3, 4, 9, 10, 11].

Lately, Gokdogan et al [5] have proved existence and uniqueness theorems for solving sequential linear conformable fractional differential equations. Unal et al [14] provide method to solve sequential linear conformable fractional differential equations with constants coefficients. In this work, We present Existence and Uniqueness theorems and solutions of sequential linear M-fractional differential equations.

The arrangement of this paper is as following: In section 2 we present the concept of M-fractional derivative. In section 3 we provide existence and uniqueness theorems for sequential Linear M-fractional differential equations. In section 4 we propose the solutions of sequential Linear M-fractional differential equations. In section 5 we present solutions of Non-homogeneous case. Finally, Conclusion is present in section 6.

2. M-Fractional Calculus

In this section, we give some necessary definitions and theorems of M – derivative which are explained in [15].

Definition 2.1**.**

Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a function and $t>0$. Then for $0<\alpha<1$, the M-fractional derivative of $f$ of order $\alpha$ is defined as

[TABLE]

Where $\mathbb{E}_{\beta}(.),\beta>0$ is Mittag-Leffler function with one parameter.

If $f$ is M-differentiable in some interval $(0,a),a>0$ and

[TABLE]

⁡ exists, then we have

[TABLE]

Theorem 2.1**.**

Let $0<\alpha\leq 1,\beta>0,a,b\in\mathbb{R}$ and $f,g$ be M - differentiable at a point $t>0$. Then

1. (1)

$D^{\alpha,\beta}_{M}(af+bg)(t)=aD^{\alpha,\beta}_{M}f(t)+bD^{\alpha,\beta}_{M}g(t)$* for all $a,b\in\mathbb{R}$.* 2. (2)

$D^{\alpha,\beta}_{M}(f\cdot g)(t)=f(t)D^{\alpha,\beta}_{M}g(t)+g(t)D^{\alpha,\beta}_{M}f(t)$** 3. (3)

$D^{\alpha,\beta}_{M}(\frac{f}{g})(t)=\frac{g(t)D^{\alpha,\beta}_{M}f(t)-f(t)D^{\alpha,\beta}_{M}g(t))}{[g(t)]^{2}}$** 4. (4)

$D^{\alpha,\beta}_{M}(c)=0$, where $f(t)=c$ is a constant. 5. (5)

$D^{\alpha,\beta}_{M}t^{a}=\frac{a}{\Gamma(\beta+1)}t^{a-\alpha}$** 6. (6)

Moreover, $f$ is differentiable, then $D^{\alpha,\beta}_{M}f(t)=\frac{t^{1-\alpha}}{\Gamma(\beta+1)}\frac{df(t)}{dt}$

Additionally,M-derivatives of certain functions as follows:

1. (1)

$D^{\alpha,\beta}_{M}(sin(\frac{1}{\alpha}t^{\alpha}))=\frac{cos(\frac{1}{\alpha}t^{\alpha})}{\Gamma(\beta+1)}$** 2. (2)

$D^{\alpha,\beta}_{M}(cos(\frac{1}{\alpha}t^{\alpha}))=\frac{-sin(\frac{1}{\alpha}t^{\alpha})}{\Gamma(\beta+1)}$** 3. (3)

$D^{\alpha,\beta}_{M}(e^{\frac{t^{\alpha}}{\alpha}})=\frac{e^{\frac{t^{\alpha}}{\alpha}}}{\Gamma(\beta+1)}$**

Theorem 2.2**.**

Let $a>0$ and $t\geq a$. Then, the M-integral of order $α$ of a function $f$ is defined by

[TABLE]

Theorem 2.3**.**

Let $a\geq 0$ and $0<\alpha<1$. Also let $f$ be a continuous function such that there exists ${}_{M}I^{\alpha,\beta}_{a}f$. Then

[TABLE]

Theorem 2.4**.**

Let $f,g:[a,b]\rightarrow\mathbb{R}$ be two functions such that $f,g$ are differentiable and $0<\alpha<1$. Then

[TABLE]

where $d_{\alpha}x=\frac{\Gamma(\beta+1)}{x^{1-\alpha}}dx$

3. Existence and Uniqueness Theorem

Let linear sequential M-fractional differential equation of order $n\alpha$

[TABLE]

where ${}^{n}D^{\alpha,\beta}_{M}y=D^{\alpha,\beta}_{M}D^{\alpha,\beta}_{M}...D^{\alpha,\beta}_{M}y,$ ($n$ times)

Similarly, non-homogeneous fractional differential equation with M-derivative is

[TABLE]

We define an $n^{th}$-order differential operator for eqn. (1) as following

[TABLE]

Theorem 3.1**.**

Let $\Gamma(\beta+1)t^{\alpha-1}p(t),\Gamma(\beta+1)t^{\alpha-1}f(t)\in C(a,b)$ and let $y$ be M-differentiable for $0<\alpha\leq 1$ and $\beta>0$. Then the initial value problem

[TABLE]

[TABLE]

has exactly one solution on the interval $(a,b)$ where $t_{0}\in(a,b)$

Proof.

Using property (6) in Theorem 2.1, we have

[TABLE]

[TABLE]

[TABLE]

The proof is clear from classical linear fundamental theorem existence and uniqueness. ∎

Theorem 3.2**.**

If $\Gamma(\beta+1)t^{\alpha-1}p_{n-1}(t),...,\Gamma(\beta+1)t^{\alpha-1}p_{1}(t),\Gamma(\beta+1)t^{\alpha-1}p_{0}(t),\Gamma(\beta+1)t^{\alpha-1}f(t)\in C(a,b)$ and $y$ be $n$ times M-differentiable function, then a solution $y(t)$ of the initial value problem

[TABLE]

[TABLE]

Proof.

The existence of a local solution is obtained by transform our problem into the first order system of differential equations. So, we introduce new variables

[TABLE]

In this, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The above equations can be written as the following

[TABLE]

[TABLE]

[TABLE]

The existence and uniqueness of solution (6)-(7) follows from classical theorems on existence and uniqueness for system equation. ∎

Theorem 3.3**.**

If $y_{1}$ and $y_{2}$ are $n$ times M-differentiable functions and $c_{1},c_{2}$ are arbitrary numbers, then $L_{\alpha,\beta}$ is linear.

[TABLE]

Proof.

We can easily derived the proof of this theorem by applying same procedure in Theorem-4.3 [5] to M- derivative. ∎

Theorem 3.4**.**

If $y_{1},y_{2}...y_{n}$ are the solutions of equation $L_{\alpha,\beta}[y]=0$ and $c_{1},c_{2}...c_{n}$ are arbitrary constants, then the linear combination $y(t)=c_{1}y_{1}+c_{2}y_{2}+...+c_{n}y_{n}$ is also solution of $L_{\alpha,\beta}[y]=0$.

Proof.

We can easily derived the proof of this theorem by applying same procedure in Theorem-4.4[5] to M- derivative. ∎

Definition 3.1**.**

For $n$ functions $y_{1},y_{2}...y_{n}$, we define the M-Wronskain of these function to be the determinant

[TABLE]

Theorem 3.5**.**

Let $y_{1},y_{2}...y_{n}$ be $n$ solutions of $L_{\alpha,\beta}[y]=0$. If there is a $t_{0}\in(a,b)$ such that $W_{\alpha,\beta}(t_{0})\neq 0$, then ${y_{1},y_{2}...y_{n}}$ is a fundamental set of solutions.

Proof.

We need to show that if $y(t)$ is a solution of $L_{\alpha,\beta}[y]=0$, then we can write $y(t)$ as a linear combination of $y_{1},y_{2}...y_{n}$.

[TABLE]

so the problem reduces to finding the constants $c_{1},c_{2},...c_{n}$.These constants are found by solving the following linear system of $n$ equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using Cramer’s rule, we can find

[TABLE]

Since $W_{\alpha,\beta}(t_{0})\neq 0$, it follows that $c_{1},c_{2},...c_{n}$ exist. ∎

Theorem 3.6**.**

Let $y_{1},y_{2}...y_{n}$ be $n$ solutions of $L_{\alpha,\beta}[y]=0$. Then

1. (1)

$W_{\alpha,\beta}(t)$* satisfies the differential equation*$D^{\alpha,\beta}_{M}W_{\alpha,\beta}+p_{(}n-1)W_{\alpha,\beta}=0$** 2. (2)

If $t_{0}$ is any point in $(a,b)$, then

[TABLE]

Further, if $W_{\alpha,\beta}(t_{0})\neq 0$ then $W_{\alpha,\beta}(t)\neq 0$ for all $t\in(a,b)$

Proof.

(1)Let us introduce new variables

[TABLE]

From this, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We have

[TABLE]

In our case

[TABLE]

So,

[TABLE]

(2)The above differential equation can be solved by the method of integrating factor, we have

[TABLE]

Thus the proof of theorem is completed. ∎

Theorem 3.7**.**

If $\{y_{1},y_{2}...y_{n}\}$ is a fundamental set of solutions of $L_{\alpha,\beta}[y]=0$ where $\Gamma(\beta+1)t^{\alpha-1}p_{n-1}(t)...\Gamma(\beta+1)t^{\alpha-1}p_{1}(t),\Gamma(\beta+1)t^{\alpha-1}p_{0}(t)\in C(a,b)$,then $W_{\alpha,\beta}(t)\neq 0$ for all $t\in(a,b)$.

Proof.

By applying procedure in Theorem-4.8 [5] to M-derivative, we can easily prove this theorem. ∎

Theorem 3.8**.**

Let $\Gamma(\beta+1)t^{\alpha-1}p_{n-1}(t),...,\Gamma(\beta+1)t^{\alpha-1}p_{1}(t),\Gamma(\beta+1)t^{\alpha-1}p_{0}(t)\in C(a,b)$. The solution set $\{y_{1},y_{2},...,y_{n}\}$ is a fundamental set of solutions to the equation $L_{\alpha,\beta}[y]=0$ if and only if the functions $y_{1},y_{2},...,y_{n}$ are linearly independent.

Proof.

By applying procedure in Theorem-4.9 [5] to M-derivative, we can easily prove this theorem. ∎

Theorem 3.9**.**

Let $y_{1},y_{2},...,y_{n}$ be a fundamental set of solutions of the equation (1) and $y_{p}$ be any particular solution of the non homogeneous equation (2). Then the general solution of the equation is $y=c_{1}y_{1}+c_{2}y_{2}+...+c_{n}y_{n}+y_{p}$

Proof.

Let $L_{\alpha,\beta}$ be the differential operator and $y(t)$ and $y_{p}(t)$ be the solutions of the non homogeneous equation $L_{\alpha,\beta}[y]=f(t)$. If we take $u(t)=y(t)-y_{p}(t)$, then by linearity of $L_{\alpha,\beta}$ we have,

[TABLE]

Then $u(t)$ is a solution of the homogenous equation $L_{\alpha,\beta}[y]=0$. Then by Theorem 3.4

[TABLE]

i.e,

[TABLE]

Then

[TABLE]

∎

4. Solution of Homogeneous Case

Consider the $n$ times M-differentiable function $y$ for $\alpha\in(0,1]$ and $\beta>0$. The homogeneous sequential linear fractional differential equation with M-derivative is

[TABLE]

where ${}^{n}D^{\alpha,\beta}_{M}y=D^{\alpha,\beta}_{M}D^{\alpha,\beta}_{M}...D^{\alpha,\beta}_{M}y$ $n$ times, and the coefficients $p_{0},p_{1},...,p_{n-1}$ are real constants.

We define an $n^{th}$-order differential operator for eqn. (1) as following

[TABLE]

If $y_{1}(t),y_{2}(t),...,y_{n}(t)$ are linearly independent solutions of Eqn.(1), then general solution is

[TABLE]

where $c_{1},c_{2}...c_{n}$ are arbitrary constants.

Lemma 4.1**.**

Suppose that $L_{\alpha,\beta}[.]$ is a linear operator with constant coefficients and $\alpha\in(0,1]$ and $\beta>0$, then for $t>0$

[TABLE]

Where $P_{n}(r)=r^{n}+P_{n-1}r^{n-1}+...+P_{0}$ and $r$ is a real or complex constant

Proof.

M-derivatives of $y=e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}$ are

[TABLE]

We substitute $y=e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}$ and Eqn.(10) in $L_{\alpha,\beta}[y]$

[TABLE]

[TABLE]

[TABLE]

Hence, the proof is completed. ∎

The solution to the equation (8) is $y=e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}$.

It follows from Eqn.(9) and Lemma 3.1 that

[TABLE]

Where $P_{n}(r)=r^{n}+P_{n-1}r^{n-1}+...+P_{0}$ is called as the characteristic polynomial. For all $r$, we have $e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}\neq 0$. Hence $P_{n}(r)=0$.

Here

[TABLE]

is called as the characteristic equation.

Lemma 4.2**.**

Let $r$ be a root of the characteristic equation (11), then

[TABLE]

and $\frac{\partial^{l}}{\partial r^{l}}e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}=(\frac{\Gamma(\beta+1)}{\alpha}t^{\alpha})^{l}e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}$ where $l$ is integer.

Proof.

From Theorem 3.3 it follows that $L_{\alpha,\beta}[.]$ is linear and also $\frac{\partial}{\partial r}$ is linear by property of classical derivative. Hence

[TABLE]

Additionally, from classical derivative, it follows that

[TABLE]

∎

Lemma 4.3**.**

If $r_{1}$ is a root of multiplicity of $\mu_{1}$of the characteristic equation (11), then the functions $y_{1,l}(t)$ , where $l=0,1,...,\mu_{l-1}$ such that

[TABLE]

are solutions of Eq.(8).

Proof.

Consider $L_{\alpha,\beta}[e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}]=p_{n}(r)e^{\frac{r\Gamma(\beta+1)}{\alpha}t^{\alpha}}$. From Lemma 4.2 and applying classical Leibniz rule it follows that

[TABLE]

[TABLE]

Since \frac{\partial^{j}}{\partial r^{j}}\big{[}P_{n}(r)\big{]}_{r=r_{1}}=0 for $j=0,1,...,\mu_{1}-1$

[TABLE]

From Lemma 4.2

[TABLE]

[TABLE]

Hence $y_{1,l}(t)$ are solutions of Eq.(8). ∎

Corollary 4.1**.**

Let $r_{j},j=1,2,...,k$ are distinct roots of multiplicity $\mu_{j},j=1,2,...,k$ of the characteristic Eq.(5). Then the following functions

[TABLE]

Proof.

Corollary 4.1 follows from Lemma 4.3 and Theorem 3.5. ∎

Lemma 4.4**.**

If $r_{1}$ and $\bar{r_{1}}$ $(r_{1}=a+ib,b\neq 0)$ are complex roots of multiplicity $\sigma_{1}$ of the characteristic equation (11), then for $l=0,1,...,\sigma_{1}-1$, the functions

[TABLE]

and

[TABLE]

are linearly independent solutions of Eq.(8).

Proof.

Since $r_{1}=a+ib$ is a root of multiplicity $\sigma_{1}$ of the characteristic equation (11), From Lemma 4.3 and using Euler’s identity it follows that, the functions

[TABLE]

i.e

[TABLE]

are solutions of the Eq.(8). Similarly, for $\bar{r_{1}}=a-ib$, the functions

[TABLE]

i.e

[TABLE]

are solutions of the Eq.(8). Hence proof is completed. ∎

Corollary 4.2**.**

If \big{\{}r_{j},\bar{r_{j}}\big{\}}_{j=1}^{m},r_{j}=a_{j}+ib_{j},b_{j}\neq 0 distinct $2m$ roots of multiplicity \big{\{}\sigma_{j}\big{\}}_{j=1}^{m} of the characteristic equation (11),then, the functions

[TABLE]

and

[TABLE]

Proof.

To prove corollary 4.2, it is sufficient to apply the Lemma 4.4 and Theorem 3.5. ∎

Theorem 4.1**.**

If \big{\{}r_{j}\big{\}}_{j=1}^{k} are distinct $k$ roots of multiplicity \big{\{}\mu_{j}\big{\}}_{j=1}^{k} and \big{\{}\lambda_{j},\bar{\lambda_{j}}\big{\}}_{j=1}^{m},\lambda_{j}=a_{j}+ib_{j},b_{j}\neq 0 are distinct $2m$ roots of multiplicity \big{\{}\sigma_{j}\big{\}}_{j=1}^{m} of the characteristic equation (11) such that $\sum_{j=1}^{k}\mu_{j}+2\sum_{j=1}^{m}\sigma_{j}=n$, then the functions

[TABLE]

[TABLE]

and

[TABLE]

are the fundamental set of solutions of the equation (8).

Proof.

The proof of the Theorem 4.1 is follows from Corollary 4.1, Corollary 4.2 and Theorem 3.5. ∎

Example 4.1**.**

[TABLE]

The characteristic equation of (12) is

[TABLE]

*Therefore, the roots are $r=-3$ and $r=-1$

Hence, the general solution is*

[TABLE]

Example 4.2**.**

[TABLE]

The characteristic equation of (13) is

[TABLE]

*The roots are $r_{1,2}=2$

Hence, the general solution is*

[TABLE]

Example 4.3**.**

[TABLE]

The characteristic equation of (14) is

[TABLE]

*The roots are $r_{1}=-2+i$ and $r_{2}=-2-i$

Hence, the general solution is*

[TABLE]

5. Solution of Non-Homogeneous Case

In this section, Method of variation of parameters is applied to derive the particular solution of the equation.

[TABLE]

where $y$ is $n$ times M-differentiable function for $\alpha\in(0,1]$ and $\beta>0$.

Theorem 5.1**.**

If $u(t)$ is a solution of homogeneous case of the equation (15) such that

[TABLE]

then particular solution of the equation (15) is

[TABLE]

Where $c_{1}(t),c_{2}(t),...,c_{n}(t)$ provide following system of equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

The solution of the equation (15) is in the form

[TABLE]

The M-derivative of $v(t)$ for $\alpha\in(0,1]$ and $\beta>0$ will be

[TABLE]

Applying the first condition $\sum_{i=1}^{n}D^{\alpha,\beta}_{M}c_{i}(t)y_{i}(t)=0$, we obtain

[TABLE]

If we calculate the M-derivative of $D^{\alpha,\beta}_{M}v(t)$ for $\alpha\in(0,1]$ and $\beta>0$, then we get

[TABLE]

Apply second condition $\sum_{i=1}^{n}D^{\alpha,\beta}_{M}c_{i}(t)D^{\alpha,\beta}_{M}y_{i}(t)=0$, we obtain

[TABLE]

By continuing in this way, we get

[TABLE]

We substitute $v(t),^{2}D^{\alpha,\beta}_{M}v(t),...,^{n}D^{\alpha,\beta}_{M}v(t)$ in the equation (15), we have

[TABLE]

[TABLE]

Since $y_{1}(t),y_{2}(t),...,y_{n}(t)$ are solutions of homogeneous case of equation (8), then

[TABLE]

We obtain $n^{th}$ condition as

[TABLE]

Hence we obtain the following system

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Solving the above system (17) provides $D^{\alpha,\beta}_{M}c_{i}(t)$, $i=1,2,...,n$. Therefore we can write the particular solution of equation (15) as $v(t)=\sum_{i=1}^{n}c_{i}(t)y_{i}(t)$ ∎

Example 5.1**.**

[TABLE]

(a) Let $f(t)=e^{2t^{\alpha}}$. For $v(t)=c_{1}(t)e^{\frac{-3\Gamma(\beta+1)}{\alpha}t^{\alpha}}+c_{2}(t)e^{\frac{-\Gamma(\beta+1)}{\alpha}t^{\alpha}}$, the system of equations are built by the conditions as following

[TABLE]

[TABLE]

Solving the above system of equations and using M-integral we obtain $c_{1}(t)=\frac{-\Gamma(\beta+1)}{4\alpha+6\Gamma(\beta+1)}e^{\frac{2\alpha+3\Gamma(\beta+1)}{\alpha}t^{\alpha}},c_{2}(t)=\frac{\Gamma(\beta+1)}{4\alpha+2\Gamma(\beta+1)}e^{\frac{2\alpha+\Gamma(\beta+1)}{\alpha}t^{\alpha}}$. Then particular solution $v(t)$ is

[TABLE]

(b) Let $f(t)=2t^{2\alpha}+t^{\alpha}-3$. The system of equations for this case is

[TABLE]

[TABLE]

Solve this system of equations, we have

[TABLE]

[TABLE]

Hence, particular solution $v(t)$ is obtained by

[TABLE]

(c) Let $f(t)=sin2t^{\alpha}$. The system of equations for this case is

[TABLE]

[TABLE]

Solve this system of equations, we have

[TABLE]

[TABLE]

Hence, particular solution $v(t)$ is obtained by

[TABLE]

(d) Let $f(t)=e^{2t^{\alpha}}t^{\alpha}$. The system of equations for this case is

[TABLE]

[TABLE]

Solve this system of equations, we have

[TABLE]

[TABLE]

Hence, particular solution $v(t)$ is obtained by

[TABLE]

(e) Let $f(t)=e^{-4t^{\alpha}}$. Take $\alpha\neq\frac{3}{4}$ and $\alpha\neq\frac{1}{4}$, the system of equations for this case is

[TABLE]

[TABLE]

Solve this system of equations, we have

[TABLE]

[TABLE]

Hence, we obtain particular solution $v(t)$ as following:

[TABLE]

*Take $\alpha=\frac{3}{4}$ and $v(t)=c_{1}(t)e^{-4\Gamma(\beta+1)t^{\frac{3}{4}}}+c_{2}(t)e^{\frac{-4\Gamma(\beta+1)}{3}t^{\frac{3}{4}}}$

The system of equations is*

[TABLE]

[TABLE]

*We solve the above equation, c_{1}(t)=-\frac{\Gamma(\beta+1)}{-6+6\Gamma(\beta+1)}e^{\big{(}-4+4\Gamma(\beta+1)\big{)}t^{\frac{3}{4}}} and c_{2}(t)=\frac{\Gamma(\beta+1)}{-6+2\Gamma(\beta+1)}e^{\big{(}-12+4\Gamma(\beta+1)\big{)}t^{\frac{3}{4}}} is obtained.

The particular solution is $v(t)=\frac{\Gamma(\beta+1)^{2}}{9-12\Gamma(\beta+1)+3\Gamma(\beta+1)^{2}}e^{-4t^{\frac{3}{4}}}$*

*Take $\alpha=\frac{1}{4}$ and $v(t)=c_{1}(t)e^{-12\Gamma(\beta+1)t^{\frac{1}{4}}}+c_{2}(t)e^{-4\Gamma(\beta+1)t^{\frac{1}{4}}}$

The system of equations is*

[TABLE]

[TABLE]

*We solve the above equation,c_{1}(t)=-\frac{\Gamma(\beta+1)}{-2+6\Gamma(\beta+1)}e^{\big{(}-4+12\Gamma(\beta+1)\big{)}t^{\frac{1}{4}}} and c_{2}(t)=\frac{\Gamma(\beta+1)}{-2+2\Gamma(\beta+1)}e^{\big{(}-4+4\Gamma(\beta+1)\big{)}t^{\frac{1}{4}}} is obtained.

The particular solution is $v(t)=\frac{4\Gamma(\beta+1)^{2}}{4-16\Gamma(\beta+1)+12\Gamma(\beta+1)^{2}}e^{-4t^{\frac{1}{4}}}$*

6. Conclusion

In this paper, Existences and Uniqueness theorems for sequential linear M-fractional differential equations are presented. We give solution of M-fractional differential equations with constants for homogeneous case using fractional exponential function and for non homogeneous case, we applied method of variation of parameters.