Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan


Introduction
Fractional dynamic varying systems with singular kernels either in the Riemann-Liouville sense or in the Caputo sense have been investigated in the literature [1][2][3]. To solve a fractional dynamic equation, we always apply a corresponding fractional integral operator. e action of this integral operator will transform the fractional dynamic equation into its corresponding integral equation whose singularity is reflected in the kernel. Motivated by this fact, in this article, we introduce a new technique to solve main generalized Abel's integral equations and generalized weakly singular Volterra integral equations analytically.
Let δ∈ (0, 1) and 0 < c < b. Consider the main generalized Abel's integral equation [4,5]: and the generalized weakly singular Volterra type integral equation of the second kind [4,5]: where g is a strictly monotonically increasing and differentiable function in the interval (0, b) with g(c) ≠ 0 for every c in (0, b) and β is a constant. Particularly, if δ � (1/2) and g(c) � c, then integral equation (1) reduces to the classical Abel's integral equation in which Abel, in 1823, investigated the motion of a particle that slides down along a smooth unknown curve under the influence of the gravity in a vertical plane. e particle takes the time g(c) to move from the highest point of vertical height c to the lowest point 0 on the curve. is problem is derived to find the equation of that curve. Indeed, Abel's integral equation is one of the most famous equations that frequently appear in many engineering problems and physical properties such as heat conduction, semiconductors, chemical reactions, and metallurgy (see, e.g., [6,7]). Besides, over the past few years, many numerical methods for solving Abel's integral equation have been developed, such as collocation methods [8], product integration methods [9,10], fractional multistep methods [11][12][13], methods based on wavelets [14][15][16], backward Euler methods [9], Adomian decomposition method [17], and Tau approximation method [18].
Recently, Wazwaz [4,5] solved integral equations (1) and (2). Unlike [4,5], in the present paper, we are going to introduce a new technique, which is based on the Adomian decomposition method, Laplace transform method, and Ψ-Riemann-Liouville fractional integrals, for solving main generalized Abel's integral equations and generalized weakly singular Volterra integral equations. For related works on generalized fractional derivatives in the Riemann-Liouville and Caputo senses and their Laplace transforms, we refer to [19]. Moreover, there are several methods used in obtaining approximate solutions to linear and nonlinear fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs) and their real-world applications. For this reason, we advise the readers to visit [20][21][22][23][24][25][26][27][28]. e paper is organized as follows. In Section 2, we recall the definitions of Riemann-Liouville fractional integrals, Ψ-Riemann-Liouville fractional integrals, and some essential properties. Section 3 is devoted to deliver the main results for the generalized Abel's integral equations and generalized weakly singular Volterra integral equations. In Section 4, several examples are considered to illustrate the applicability of our main results.

Preliminaries
Here, we give the definitions of Riemann-Liouville fractional integrals, Ψ-Riemann-Liouville fractional integrals, and some essential properties.

Main Results
Now, we give our main results.
Lemma 3 (see [4,5]). If g(c) is bounded on 0 < c < b, g is strictly monotonically increasing and differentiable function in some interval (0, b) and g(c) ≠ 0 for every c in (0, b). en, Abel's integral equation (1) has the following solution: (1) has the following solution: Proof. From Lemma 3 for δ ∈ (0, 1), we have From this and Definition 2, we get (8). is completes the proof.  (1) is Now, the solution of (2) can be obtained in the following theorem. 0)) μ , g is strictly monotonically increasing and differentiable function in some interval (0, b) and g(c) ≠ 0 for every c in (0, b). e generalized weakly singular Volterra type integral equation (2) has the following solution: where E δ,μ is the 2-parameter Mittag-Leffler function which is defined by [31,32] Proof. From the Adomian decomposition method, we substitute the decomposition series into both sides of (2) to obtain e components u 0 (c), u 1 (c), u 2 (c), . . . are determined by using (3) in the Adomian recurrence relation: us, the exact solution is  (2) is e noise terms may appear between components of u 0 (c) and u 1 (c) of (15) with opposite signs. Hence, by canceling these noise terms between these components, we may give the exact solution that should be justified through substitution and thus minimize the size of the calculations. In this situation, we use the following corollary.  (2) can be obtained as Proof. To prove this, we use one of the Mittag-Leffler function's properties, that is [31,32], From this, we can write (17) as is completes the proof. □ Theorem 3. Under the similar assumptions of eorem 2, if g(c) � ∞ j�1 θ j c μ j , then the exact of the generalized weakly singular Volterra type integral equation (2) is given by or equivalently Proof. By the same manner of eorem 2, we get is completes the proof of the first part of this theorem. By using property (19), we easily obtain the proof of the second part of this theorem. us, the proof of eorem 3 is completed.
□ Remark 2. Due to the occurrence of the noise terms between u 0 (c) and u 1 (c), we can write u(c) as which is obtained from (21) and (22). Now, we introduce some spaces of the continuous functions in order to obtain the boundness of the above solution.

Mathematical Problems in Engineering
Let g ∈ C[a, b]; then, by using Definition 3 and Lemma 2 and since ψ is an increasing function, we have is completes the proof of the first inequality of (ii). By making use of eorem 1 and inequality (37) with a � 0, we can deduce the second part inequality in (ii). □ Theorem 5. For any δ ∈ (0, 1], μ, a ≥ 0, we have Proof. e proof is similar to Lemma 2, so it is omitted. □ Theorem 6. Let 0 ≤ a < b < ∞, δ ∈ (0, 1), μ > 0 and g ∈ C ψ μ [a, b]. If μ < δ, then we have Moreover, the solution u(c) in eorem 1 vanishes at c � 0. C[a, b] and there exists some M > 0 such that and (41) en, by making use of Remark 3 and eorem 5, it follows that Taking the limit on both sides, it follows that which completes the proof of the first part. By making use of eorem 1 and formula (42) with a � 0, we can deduce the second part of theorem.

Conclusion
In the present article, a new technique involving Riemann-Liouville fractional integrals has been used to solve main generalized Abel's integral equations and generalized weakly singular Volterra integral equations. Also, we have solved several examples with our proposed technique. We can observe that our developed technique is easy and straightforward to apply.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.