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Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals
In this attempt, we introduce a new technique to solve main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations analytically. This technique is based on the Adomian decomposition method, Laplace transform method, and -Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.
Fractional dynamic varying systems with singular kernels either in the Riemann–Liouville sense or in the Caputo sense have been investigated in the literature [1–3]. To solve a fractional dynamic equation, we always apply a corresponding fractional integral operator. The 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 and . 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 is a strictly monotonically increasing and differentiable function in the interval with for every in and is a constant.
Particularly, if and , 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. The particle takes the time to move from the highest point of vertical height to the lowest point 0 on the curve. This 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 , product integration methods [9, 10], fractional multistep methods [11–13], methods based on wavelets [14–16], backward Euler methods , Adomian decomposition method , and Tau approximation method .
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 -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 . 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 [20–28].
The paper is organized as follows. In Section 2, we recall the definitions of Riemann–Liouville fractional integrals, -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.
Here, we give the definitions of Riemann–Liouville fractional integrals, -Riemann–Liouville fractional integrals, and some essential properties.
Definition 1. Let and let be a finite interval on the real-axis . The left and right-sided Riemann–Liouville fractional integrals of order are, respectively, defined by 
Definition 2. Let be a finite or infinite interval of the real-axis and . Let be an increasing and positive function on the interval with a continuous derivative on the interval . Then, the left and right-sided -Riemann–Liouville fractional integrals of a function with respect to another function on are defined by [29, 30]If we set in (4), then Definition 2 reduces to Definition 1.
The following lemmas hold in [29, 30].
Lemma 1. Let , and ; then,
Lemma 2. Let , and ; then,
Remark 1. In this context, and stand for and , respectively.
In this paper, using the Adomian decomposition method and Laplace transform method combined with Lemma 1, we produce a new powerful technique. By using this technique, we obtain exact solution for main generalized Abel’s integral equations and generalized weakly singular Volterra integral equations.
3. Main Results
Now, we give our main results.
Lemma 3 (see [4, 5]). If is bounded on , is strictly monotonically increasing and differentiable function in some interval and for every in . Then, Abel’s integral equation (1) has the following solution:
Theorem 1. If is bounded on , is strictly monotonically increasing and differentiable function in some interval and for every in . Abel’s integral equation (1) has the following solution:
Now, the solution of (2) can be obtained in the following theorem.
Theorem 2. If , is strictly monotonically increasing and differentiable function in some interval and for every in . The generalized weakly singular Volterra type integral equation (2) has the following solution:where is the 2-parameter Mittag–Leffler function which is defined by [31, 32]
Proof. From the Adomian decomposition method, we substitute the decomposition seriesinto both sides of (2) to obtainThe components are determined by using (3) in the Adomian recurrence relation:Thus, the exact solution iswhich completes the proof.
The noise terms may appear between components of and 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.
Proof. By the same manner of Theorem 2, we getThis 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. Thus, the proof of Theorem 3 is completed.
Remark 2. Due to the occurrence of the noise terms between and , we can write aswhich 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.
Definition 3. Let and be a finite interval on the half-axis with . Then,(i)We denote by the space of continuous functions on with the norm(ii)We define the weighted space of functions with respect to an increasing function on bywith the normNote that .
Remark 3. Let ; then, we have
Proof. The proof follows directly from the substitution and the definition of the beta function.
Theorem 4. Let , and . Then,
If , the fractional operator is bounded from into with(i)Moreover, if , the solution in Theorem 1 is bounded from into with(ii)For any , the fractional operator is a mapping from into withMoreover, for any , the solution in Theorem 1 is bounded from into with
Proof. (i)From Definition 3, we get Then, by making use of Remark 3, it follows that Since is an increasing function, it follows that This completes the proof of the first part inequality in (i). By making use of Theorem 1 and inequality (34) with , we obtain Let ; then, by using Definition 3 and Lemma 2 and since is an increasing function, we haveThis completes the proof of the first inequality of (ii). By making use of Theorem 1 and inequality (37) with , we can deduce the second part inequality in (ii).
Theorem 5. For any , we have
Proof. The proof is similar to Lemma 2, so it is omitted.
Theorem 6. Let and . If , then we have
Moreover, the solution in Theorem 1 vanishes at .
Proof. Let ; then, and there exists some such thatandThen, by making use of Remark 3 and Theorem 5, it follows thatTaking the limit on both sides, it follows thatwhich completes the proof of the first part. By making use of Theorem 1 and formula (42) with , we can deduce the second part of theorem.
4. Test Examples
Example 1. Consider the generalized Abel’s integral equation :where and . It is clear that is strictly monotonically increasing in and for each in . Using Corollary 1 with , we getwhich is the exact solution, where we have used the fact that .
Example 2. Consider the generalized Abel’s integral equation :Here, equation (7) takes the following form:From this, we have , and . Thus, Corollary 1 gives the exact solution:where we have used the fact that .
Example 3. Consider the weakly singular second kind Volterra integral equation [4, 33, 34]:Using Corollary 3 with , and , we can easily obtainwhere is the complementary error function which is defined as
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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.
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