THE SOLUTIONS OF SOME CERTAIN NON-HOMOGENEOUS FRACTIONAL INTEGRAL EQUATIONS

and I2σ 0+ y(t)+a · I σ 0+y(t)+b · y(t) = t net , where Iσ 0+ is the Riemann-Liouville fractional integral of order σ = 1/2,σ = 1,n ∈ N∪{0}, t ∈ R +, and a,b are constants, by using the Laplace transform technique. We obtain the solutions of these equations are in the form of Mellin-Ross function and in the form of exponential function.

I 2σ 0 + y(t) + a · I σ 0 + y(t) + b · y(t) = t n , and I 2σ 0 + y(t) + a · I σ 0 + y(t) + b · y(t) = t n e t , where I σ 0 + is the Riemann-Liouville fractional integral of order σ = 1/2, σ = 1, n ∈ N ∪ {0},t ∈ R + , and a, b are constants, by using the Laplace transform technique. We obtain the solutions of these equations are in the form of Mellin-Ross function and in the form of exponential function.  Fractional derivative is a part of fractional calculus which has been of interest in recent years.
In addition, of course, the theory of fractional integral has been of interest in recent years, see [24,25,26,27,28,29,30,31,32]. In 1812, P. S. Laplace defined a fractional derivative through an integral. He developed it as a mere mathematical exercise generalizing from a case of integer order. Later, in 1832, J. Liouville recommended a definition based on the formula for differentiating the exponential function known as the first Liouville definition. Next, he presented the second definition formula in terms of an integral, called Liouville, to integrate noninteger order.
After that J. Liouville and G. F. B. Riemann developed an approach to noninteger order derivatives in terms of convergent series, conversely to the Riemann-Liouville approach, that was given an integral. Many researchers focused on developing the theoretical aspects, methods of solution, and applications of fractional integral equations see [30,31,32,33,34,35,36,37,38].
In 2005, T. Morita [6] studied the initial value problem of fractional differential equations by using the Laplace transform. He obtained the solutions to the fractional differential equations with Riemann-Liouville fractional derivative and Caputo fractional derivative or its modification. In 2010, T. Morita and K. Sato [8] studied the initial value problem of fractional differential equations with constant coefficients of the form where 0 D σ m t is the Riemann-Liouville fractional derivative, c l are constants for l = 0, 1, 2, . . . , m − 1, and t ∈ R + . They obtained solutions in terms of the Green's function and distribution theory. Next, they studied the solution of a fractional differential equation of the form where σ = 1, σ = 1/2,t ∈ R + , and a i , b i are constants for i = 0, 1, 2, see [9] for more details. (1) where Γ is the gamma function, λ is a constant, and α ∈ R. Equation (1) can be written in the where I α 0 + is the Riemann-Liouville fractional integral. They applied Babenko's method and fractional integral for solving the above equation.
The linear fractional order integral equations with constant coefficients of the form . . , n}, and f is assumed to be a real valued function of real variable defined on an interval (a, b). The general solution of (2) can be found in [28] for In 2017, D. C. Labora and R. Rodriguez-Lopez [37] showed a new method by applying a suitable fractional integral operator for solving some fractional order integral equations with constant coefficients, and all the integration orders involving are rational. Next, they applied and extended ideas presented in [37] for solving fractional integral equations with Riemann-Liouville definition; see [31] for more details. Moreover, they studied the fractional integral equations with Caputo derivatives and non-rational orders by limiting fractional integral equations with rational orders.
As mentioned in the abstract, we propose the solutions of non-homogeneous fractional integral equations of the form and where I σ 0 + is the Riemann-Liouville fractional integral of order σ = 1/2, σ = 1, n ∈ N ∪ {0},t ∈ R + , and a, b are constants by using the Laplace transform technique and its variants in the classical sense. In Section 2, we introduce definitions of the Riemann-Liouville fractional integral and the Laplace transform which will help us to obtain our main results. In Section 3, we establish our main results and some examples as a consequently of our main results. Finally, we give the conclusions in Section 4.

PRELIMINARIES
Before we proceed to the main results, the following definitions, lemmas, and concepts are required.
Definition 2.1. [23] Let α be a constant, v a real number and t a positive real number. The where Γ * is the incomplete gamma function: in which Γ is the gamma function.
In addition, if v > 0, then E t (v, α) has an integral representation as Example 2.2. Let α be a constant, µ a real number, v and t positive real numbers. Then the following Riemann-Liouville fractional integrals hold: where β ∈ R + and n is an integer that satisfies n − 1 ≤ β < n.
where Re s > v.
Example 2.3. Let α be a constant, n a real number, v and t positive real numbers. Then the following Laplace transforms hold: (iv) L t n e αt = Γ(n + 1) , s > α.

MAIN RESULTS
In this section, we will state our main results and give their proofs.
where I σ 0 + is the Riemann-Liouville fractional integral of order σ = 1/2, σ = 1, n ∈ N ∪ {0}, a, b are constants and t ∈ R + . Then the solutions of (3) are as the follows: (i) If σ = 1/2, and j, k ∈ R \ {0} with j = k such that a = j + k and b = jk, then the solution of (3) is of the form (ii) If σ = 1, and j, k ∈ R \ {0 } with j = k such that a = j + k and b = jk then the solution of (3) is of the form Proof. Applying the Laplace transform to both sides of (3), we have Using Lemma 2.1, Example 2.3 (ii), and denoting the Laplace transform L {y(t)} = Y (s) to (6), we obtain (7) Y (s) = n! s 2σ s n+1 (bs 2σ + as σ + 1) .
For σ = 1/2, equation (7) becomes Y (s) = n! s n bs + as 1/2 + 1 , and turns into with a substitution of u = s 1/2 . Using partial fractions with explicit values of a, b, we can rewrite it as Finally, resubstituting u = s 1/2 and taking the inverse Laplace transform to (8) with the help of Example 2.4 (i), (iv), we obtain a solution of (3) in the form of (4).
Using partial fractions with explicit values of a, b, we can rewrite the above equation as Applying the inverse Laplace transform to (9) and using Example 2.4 (i), and (ii), yield a solution of (3) in the form of (5). In order to include the case n = 0 into the solution formulas of both cases, we adopt the notation 1/Γ(0) = 0. The proof is completed.
where I σ 0 + is the Riemann-Liouville fractional integral of order σ = 1/2, σ = 1, n ∈ N ∪ {0}, a, b are constants and t ∈ R + . Then the solutions of (18) are as the follows: (i) If σ = 1/2, and j, k ∈ R \ {−1, 0, 1} with j = k such that a = j + k and b = jk, then the solution of (18) is of the form (ii) If σ = 1, and j, k ∈ R \ {−1, 0} with j = k such that a = j + k and b = jk, then the solution of (18) is of the form Proof. Performing the Laplace transform to both sides of (18), we have Using Lemma 2.1, Example 2.3 (iv), and denoting the Laplace transform L {y(t)} = Y (s) to (21), we obtain .
Using partial fractions with explicit values of a, b, we can rewrite the above equation as Applying the inverse Laplace transform to (25) with the help of Example 2.4 (iii), and (iv), yield a solution of (18) in the form of (20). In order to include the case n = 0 into the solution formulas of both cases, we adopt the notation 1/Γ(0) = 0. The proof is completed. b · y (t) + a · y (t) + y(t) = t n e t + 2nt n−1 e t + n(n − 1)t n−2 e t .
It is not difficult to verify that (33) satisfies (32).