Research Article | Open Access

Manar A. Alqudah, Pshtiwan Othman Mohammed, Thabet Abdeljawad, "Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals", *Mathematical Problems in Engineering*, vol. 2020, Article ID 1250970, 8 pages, 2020. https://doi.org/10.1155/2020/1250970

# Solution of Singular Integral Equations via Riemann–Liouville Fractional Integrals

**Academic Editor:**Mehmet Emir Koksal

#### Abstract

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.

#### 1. 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 [1–3]. 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 [8], product integration methods [9, 10], fractional multistep methods [11–13], methods based on wavelets [14–16], 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 -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 [20–28].

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.

#### 2. Preliminaries

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 [29]

*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:*

*Proof. *From Lemma 3 for , we haveFrom this and Definition 2, we get (8). This completes the proof.

Corollary 1. *Under the similar assumptions of Theorem 1, if , then the solution of Abel’s integral equation (1) is*

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.

Corollary 2. *Under the similar assumptions of Theorem 2, if , then the solution of the generalized weakly singular Volterra type integral equation (2) is*

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.

Corollary 3. *Under the similar assumptions of Theorem 2, the solution of the generalized weakly singular Volterra type integral equation (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) asThis completes the proof.

Theorem 3. *Under the similar assumptions of Theorem 2, if , then the exact of the generalized weakly singular Volterra type integral equation (2) is given byor equivalently*

*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 with**Moreover, 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

In this section, we consider several test problems corresponding to the equations (1) and (2) to demonstrate the efficiency of our new mechanism.

*Example 1. *Consider the generalized Abel’s integral equation [4]: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 [4]: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

*Example 4. *Consider the weakly singular second kind Volterra integral equation [4, 33, 34]:where , and . Thus, by formula (24), we getHence, the exact solution is .

*Example 5. *Consider the weakly singular second kind Volterra integral equation [9]:In this example, , and . Thus, formula (24) giveswhich is the exact solution of integral equation (54).

*Example 6. *Consider the weakly singular second kind Volterra integral equation [9]:From formula (24) with , and , we getwhich is the exact solution of integral equation (56).

*Example 7. *Consider the Abel integral equation of the second kind [33]:From formula (24) with , and , we getwhich is the exact solution of (58).

#### 5. 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

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

#### Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.

#### References

- Y. Guo and B. Ma, “Stability analysis for a class of caputo fractional time-varying systems with nonlinear dynamics,” in
*Lecture Notes in Electrical Engineering*, Y. Jia, J. Du, and W. Zhang, Eds., vol. 459, Springer, Singapore, 2018. View at: Google Scholar - M. Eckert, K. Nagatou, F. Rey, O. Stark, and S. Hohmann, “Solution of time-variant fractional differential equations with a generalized peano–baker series,”
*IEEE Control Systems Letters*, vol. 3, no. 1, pp. 79–84, 2019. View at: Publisher Site | Google Scholar - T. Kaczorek and D Idczak, “Cauchy formula for the time-varying linear systems with caputo derivative,”
*Fractional Calculus and Applied Analysis*, vol. 20, no. 2, 2017. View at: Publisher Site | Google Scholar - A. M. Wazwaz,
*Linear and Nonlinear Integral Equations: Methods and Applications*, Springer, Berlin, Germany, 2011. - A. M. Wazwaz,
*A First Course in Integral Equations*, World Scientific Publishing, , Singapore, 2 edition, 2015. - R. Gorenflo and S. Vessella, “Abel integral equations,” in
*Lecture Notes In Mathematics*, Springer, Berlin, Germany, 1991. View at: Google Scholar - R. Gorenflo, “Abel integral equations with special emphasis on applications,” in
*Lecture Notes In Mathematical Sciences*, vol. 13, The University of Tokyo, Graduate School of Mathematical Sciences, Tokyo, Japan, 1996. View at: Google Scholar - H. Brunner, “Collocation methods for volterra integral and related functional differential equations,” in
*Cambridge Monographs on Applied and Computational Mathematics*, vol. 15, Cambridge University Press, Cambridge, UK, 2004. View at: Google Scholar - C. T. H. Baker,
*The Numerical Treatment of Integral Equations*, Clarendon Press, Oxford, UK, 1977. - P. Baratella and A. P. Orsi, “A new approach to the numerical solution of weakly singular Volterra integral equations,”
*Journal of Computational and Applied Mathematics*, vol. 163, no. 2, pp. 401–418, 2004. View at: Publisher Site | Google Scholar - C. Lubich, “Fractional linear multistep methods for Abel–Volterra integral equations of the second kind”,”
*Math. Comp.*, vol. 45, no. 172, pp. 463–469, 1985. View at: Google Scholar - C. Lubich, “Discretized fractional calculus,”
*SIAM Journal on Mathematical Analysis*, vol. 17, no. 3, pp. 704–719, 1986. View at: Google Scholar - R. Plato, “Fractional multistep methods for weakly singular volterra integral equations of the first kind with perturbed data,”
*Numerical Functional Analysis and Optimization*, vol. 26, no. 2, pp. 249–269, 2005. View at: Publisher Site | Google Scholar - Ü. Lepik, “Solving fractional integral equations by the Haar wavelet method,”
*Applied Mathematics and Computation*, vol. 214, no. 2, pp. 468–478, 2009. View at: Publisher Site | Google Scholar - H. Saeedi, M. M. Moghadam, N. Mollahasani, and G. N. Chuev, “A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 3, pp. 1154–1163, 2011a. View at: Publisher Site | Google Scholar - H. Saeedi, N. Mollahasani, M. Moghadam, and G. Chuev, “An operational Haar wavelet method for solving fractional Volterra integral equations,”
*International Journal of Applied Mathematics and Computer Science*, vol. 21, no. 3, pp. 535–547, 2011b. View at: Publisher Site | Google Scholar - L. Bougoffa, A. Mennouni, and R. C. Rach, “Solving Cauchy integral equations of the first kind by the Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4423–4433, 2013. View at: Publisher Site | Google Scholar - S. K. Vanani and F. Soleymani, “Tau approximate solution of weakly singular Volterra integral equations,”
*Mathematical Models and Computer Simulations*, vol. 57, pp. 494–502, 2013. View at: Google Scholar - F. Jarad, T. Abdeljawad, and T. Abdeljawad, “Generalized fractional derivatives and Laplace transform,”
*Discrete & Continuous Dynamical System*, vol. 13, no. 3, pp. 709–722, 2020. View at: Publisher Site | Google Scholar - L. Akinyemi and O. S. Iyiola, “Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform,”
*Mathematical Methods in the Applied Sciences*, vol. 43, no. 12, pp. 7442–7464, 2020. View at: Publisher Site | Google Scholar - L. Akinyemi, O. S. Iyiola, and U. Akpan, “Iterative methods for solving fourth- and sixth-order time-fractional Cahn-Hillard equation,”
*Mathematical Methods in the Applied Sciences*, vol. 43, pp. 4050–4074, 2020. View at: Publisher Site | Google Scholar - M. Şenol, O. S. Iyiola, H. Daei Kasmaei, and L. Akinyemi, “Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent-Miodek system with energy-dependent Schr odinger potential,”
*Advances in Difference Equations*, vol. 2019, Article ID 160681, 462 pages, 2019. View at: Publisher Site | Google Scholar - M. Martinez, P. O. Mohammed, and J. E. N. Valdes, “Non-conformable fractional Laplace transform,”
*Kragujevac Journal of Mathematics*, vol. 46, no. 3, pp. 341–354, 2022. View at: Google Scholar - F. K. Hamasalh and P. O. Muhammed, “Computational method for fractional differential equations using nonpolynomial fractional spline,”
*Mathematical Sciences Letters*, vol. 5, no. 2, pp. 131–136, 2016. View at: Publisher Site | Google Scholar - F. K. Hamasalh and P. O. Muhammad, “Generalized quartic fractional spline interpolation with applications,”
*International Journal of Open Problems in Computer Science and Mathematics*, vol. 8, no. 1, pp. 67–80, 2015. View at: Publisher Site | Google Scholar - K. Diethelm,
*The Analysis of Fractional Differential Equations*, Springer, Berlin, Germany, 2010. - M. E. Koksal, “Stability analysis of fractional differential equations with unknown parameters,”
*Nonlinear Analysis: Modelling and Control*, vol. 24, no. 2, pp. 224–240, 2019. View at: Publisher Site | Google Scholar - M. E. Koksal, “Time and frequency responses of non-integer order RLC circuits,”
*AIMS Mathematics*, vol. 4, no. 1, pp. 61–75, 2019. View at: Google Scholar - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, Netherlands, 2006. - J. V. C. Sousa and E. C. Oliveira, “On the Hilfer fractional derivative,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 60, pp. 72–91, 2018. View at: Google Scholar - R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin,
*Mittag-Leffler Functions, Related Topics and Applications*, Springer, Berlin, Germay, 2014. - A. Fernandez, D. Baleanu, and H. M. Srivastava, “Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 67, pp. 517–527, 2019. View at: Publisher Site | Google Scholar - S. Kumar, A. Kumar, D. Kumar, J. Singh, and A. Singh, “Analytical solution of Abel integral equation arising in astrophysics via Laplace transform,”
*Journal of the Egyptian Mathematical Society*, vol. 23, no. 1, pp. 102–107, 2015. View at: Publisher Site | Google Scholar - R. K. Pandey, O. P. Singh, and V. K. Singh, “Efficient algorithms to solve singular integral equations of Abel type,”
*Computers & Mathematics with Applications*, vol. 57, no. 4, pp. 664–676, 2009. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 Manar A. Alqudah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.