MPEMathematical Problems in Engineering1563-51471024-123XHindawi10.1155/2020/12509701250970Research ArticleSolution of Singular Integral Equations via Riemann–Liouville Fractional IntegralsAlqudahManar A.1https://orcid.org/0000-0001-6837-8075MohammedPshtiwan Othman2https://orcid.org/0000-0002-8889-3768AbdeljawadThabet345KoksalMehmet Emir1Department of Mathematical SciencesFaculty of SciencesPrincess Nourah Bint Abdulrahman UniversityP.O.Box 84428Riyadh 11671Saudi Arabiapnu.edu.sa2Department of MathematicsCollege of EducationUniversity of SulaimaniSulaimaniKurdistan RegionIraqunivsul.edu.iq3Department of Mathematics and General SciencesPrince Sultan UniversityP.O. Box 66833Riyadh 11586Saudi Arabiapsu.edu.sa4Department of Medical ResearchChina Medical UniversityTaichung 40402Taiwancmu.edu.cn5Department of Computer Science and Information EngineeringAsia UniversityTaichungTaiwanasia.edu.tw2020309202020202672020392020169202030920202020Copyright © 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.

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 Ψ-Riemann–Liouville fractional integrals. Finally, some examples are proposed and they illustrate the rapidness of our new technical method.

Princess Nourah Bint Abdulrahman University
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 . 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 δ¯0,1 and 0<γ<b. Consider the main generalized Abel’s integral equation [4, 5]:(1)uγ=0γutdtgγgtδ¯,and the generalized weakly singular Volterra type integral equation of the second kind [4, 5]:(2)uγ=gγ+0γβ¯utdtgγgtδ¯,where g is a strictly monotonically increasing and differentiable function in the interval 0,b with gγ0 for every γ in 0,b and β¯ is a constant.

Particularly, if δ¯=1/2 and gγ=γ, 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 gγ to move from the highest point of vertical height γ 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 , methods based on wavelets , 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 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 . 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 .

The 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.

2. Preliminaries

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

Definition 1.

Let g:a,b and let a,b, be a finite interval on the real-axis . The left and right-sided Riemann–Liouville fractional integrals of order δ¯>0 are, respectively, defined by (3)a+δ¯gγ=1Γδ¯aγγtδ¯1gtdt,γ>a,bδ¯gγ=1Γδ¯γbtγδ¯1gtdt,γ<b.

Definition 2.

Let a,b, be a finite or infinite interval of the real-axis and δ¯>0. Let ψγ be an increasing and positive function on the interval a,b with a continuous derivative ψγ on the interval a,b. Then, the left and right-sided ψ-Riemann–Liouville fractional integrals of a function f with respect to another function ψγ on a,b are defined by [29, 30](4)a+δ¯:ψgγ=1Γδ¯aγψtψγψtδ¯1gtdt,bδ¯:ψgγ=1Γδ¯γbψtψtψγδ¯1gtdt.

If we set ψγ=γ in (4), then Definition 2 reduces to Definition 1.

The following lemmas hold in [29, 30].

Lemma 1.

Let δ¯>0,μ¯>0, and gγ=γμ¯; then,(5)a+δ¯gγ=Γμ¯+1Γδ¯+μ¯+1γδ¯+μ¯.

Lemma 2.

Let δ¯>0,μ¯>0, and gγ=ψγψaμ¯; then,(6)a+δ¯:ψgγ=Γμ¯+1Γδ¯+μ¯+1ψγψaδ¯+μ¯.

Remark 1.

In this context, δ¯gγ and δ¯:ψgγ stand for 0+δ¯gγ and 0+δ¯:ψgγ, 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 [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B5">5</xref>]).

If gγ is bounded on 0<γ<b, g is strictly monotonically increasing and differentiable function in some interval 0,b and gγ0 for every γ in 0,b. Then, Abel’s integral equation (1) has the following solution:(7)uγ=sinδ¯ππddγ0γgtftgγgt1δ¯dt,0<δ¯<1.

Theorem 1.

If ft is bounded on 0<γ<b, g is strictly monotonically increasing and differentiable function in some interval 0,b and gγ0 for every γ in 0,b. Abel’s integral equation (1) has the following solution:(8)uγ=Γδ¯sinδ¯ππddγδ¯:ψgγ.

Proof.

From Lemma 3 for δ¯0,1, we have(9)uγ=Γδ¯sinδ¯ππddγ1Γδ¯0γgtgγgtδ¯1ftdt.

From this and Definition 2, we get (8). This completes the proof.

Corollary 1.

Under the similar assumptions of Theorem 1, if gγ=gγg0μ¯, then the solution of Abel’s integral equation (1) is(10)uγ=sinδ¯ππΓδ¯Γμ¯+1Γδ¯+μ¯+1ddγgγg0δ¯+μ¯.

Now, the solution of (2) can be obtained in the following theorem.

Theorem 2.

If gγ=θ¯gγg0μ¯, g is strictly monotonically increasing and differentiable function in some interval 0,b and gγ0 for every γ in 0,b. The generalized weakly singular Volterra type integral equation (2) has the following solution:(11)uγ=θ¯Γμ¯+1gγg0μ¯E1δ¯,μ¯+1β¯Γ1δ¯gγg01δ¯,where Eδ¯,μ¯ is the 2-parameter Mittag–Leffler function which is defined by [31, 32](12)Eδ¯,μ¯γ=n=0γnΓδ¯n+μ¯.

Proof.

From the Adomian decomposition method, we substitute the decomposition series(13)uγ=n=0unγ,into both sides of (2) to obtain(14)n=0unγ=gγ+0γβ¯gγgtδ¯n=0untdt.

The components u0γ,u1γ,u2γ, are determined by using (3) in the Adomian recurrence relation:(15)u0γ=gγ=θ¯gγg0μ¯,u1γ=β¯0γgtg0μ¯gγgtδ¯dt=θ¯β¯Γ1δ¯Γμ¯+1Γ2δ¯+μ¯gγg01δ¯+μ¯,u2γ=θ¯β¯20γu1tgγgtδ¯dt=θ¯β¯2Γ1δ¯2Γμ¯+1Γ32δ¯+μ¯gγg021δ¯+μ¯,u3γ=θ¯β¯30γu2tgγgtδ¯dt=θ¯β¯3Γ1δ¯3Γμ¯+1Γ43δ¯+μ¯gγg031δ¯+μ¯.

Thus, the exact solution is(16)uγ=θ¯gγg0μ¯+β¯Γ1δ¯Γμ¯+1Γ1δ¯+μ¯+1gγg01δ¯+μ¯+β¯2Γ1δ¯2Γμ¯+1Γ21δ¯+μ¯+1gγg021δ¯+μ¯+=θ¯Γμ¯+1gγg0μ¯n=0β¯Γ1δ¯gγg01δ¯nΓ1δ¯n+μ¯+1=θ¯Γμ¯+1gγg0μ¯E1δ¯,μ¯+1β¯Γ1δ¯gγg01δ¯,which completes the proof.

Corollary 2.

Under the similar assumptions of Theorem 2, if gγ=γ, then the solution of the generalized weakly singular Volterra type integral equation (2) is(17)uγ=θ¯Γμ¯+1γμ¯E1δ¯,μ¯+1β¯Γ1δ¯γ1δ¯.

The noise terms may appear between components of u0γ and u1γ 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(18)uγ=θ¯Γμ¯+1β¯Γ1δ¯γδ¯+μ¯1E1δ¯,δ¯+μ¯β¯Γ1δ¯γ1δ¯1Γδ¯+μ¯.

Proof.

To prove this, we use one of the Mittag–Leffler function’s properties, that is [31, 32],(19)Eδ¯,μ¯γ=1γEδ¯,μ¯δ¯γ1xΓμ¯δ¯.

From this, we can write (17) as(20)uγ=θ¯Γμ¯+1γμ¯β¯Γ1δ¯γ1δ¯E1δ¯,δ¯+μ¯β¯Γ1δ¯γ1δ¯1Γδ¯+μ¯.

This completes the proof.

Theorem 3.

Under the similar assumptions of Theorem 2, if gγ=j=1θ¯jγμ¯j, then the exact of the generalized weakly singular Volterra type integral equation (2) is given by(21)uγ=j=1θ¯jΓμ¯j+1γμ¯jE1δ¯,μ¯j+1β¯Γ1δ¯γ1δ¯,or equivalently(22)uγ=j=1θ¯jΓμ¯j+1β¯Γ1δ¯γδ¯+μ¯j1E1δ¯,δ¯+μ¯jβ¯Γ1δ¯γ1δ¯1Γδ¯+μ¯j.

Proof.

By the same manner of Theorem 2, we get(23)uγ=θ¯1Γμ¯1+1γμ¯1E1δ¯,μ¯1+1β¯Γ1δ¯γ1δ¯+θ¯2Γμ¯2+1γμ¯2E1δ¯,μ¯2+1β¯Γ1δ¯γ1δ¯+θ¯3Γμ¯3+1γμ¯3E1δ¯,μ¯3+1β¯Γ1δ¯γ1δ¯+.

This 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 u0γ and u1γ, we can write uγ as(24)uγ=θ¯1Γμ¯1+1γμ¯1E1δ¯,μ¯1+1β¯Γ1δ¯γ1δ¯+θ¯2Γμ¯2+1β¯Γ1δ¯γδ¯+μ¯21E1δ¯,δ¯+μ¯2β¯Γ1δ¯γ1δ¯1Γδ¯+μ¯2+,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.

Definition 3.

Let 0<μ¯1 and J=a,b be a finite interval on the half-axis + with 0a<b<. Then,

We denote by Ca,b the space of continuous functions g on J with the norm(25)gC=maxgγ;γa,b.

We define the weighted space Cμ¯ψa,b of functions g with respect to an increasing function ψ on a,b by(26)Cμ¯ψa,bmaxg:a,b,ψγψaμ¯gγCa,b,

with the norm(27)gCμ¯ψmaxψγψaμ¯gγ;γa,b.

Note that C0ψa,b=Ca,b.

Remark 3.

Let δ¯,β¯,τ>0; then, we have(28)τtgtgsδ¯gsgτβ¯gsds=gtgτδ¯+β¯+1011zδ¯zβ¯dz=Γδ¯+1Γβ¯+1Γδ¯+β¯+2gtgτδ¯+β¯+1.

Proof.

The proof follows directly from the substitution gs=gτ+zgtgτ and the definition of the beta function.

Theorem 4.

Let 0a<b<,δ¯0,1, and μ¯>0. Then,

If μ¯δ¯, the fractional operator a+δ¯:ψ is bounded from Cμ¯ψa,b into Ca,b with(29)a+δ¯:ψgCgCμ¯ψΓ1μ¯Γδ¯μ¯+1ψbψaδ¯μ¯.

Moreover, if μ¯δ¯1, the solution ux in Theorem 1 is bounded from Cμ¯ψa,b into Ca,b with(30)uCgCμ¯ψsinδ¯πΓδ¯Γ1μ¯πΓδ¯μ¯ψbψ0δ¯μ¯1ψb.

For any gCa,b, the fractional operator a+δ¯:ψ is a mapping from Ca,b into Ca,b with(31)a+δ¯:ψgCgC1Γδ¯+1ψbψaδ¯.

Moreover, for any gCa,b, the solution ux in Theorem 1 is bounded from Ca,b into Ca,b with(32)uCgCsinδ¯ππψbψ0δ¯1ψb.

Proof.

From Definition 3, we get

(33)a+δ¯:ψgC=maxa+δ¯:ψgx;γa,b=maxγa,b1Γδ¯aγψsψγψsδ¯1gsds=maxγa,b1Γδ¯aγψsψγψsδ¯1ψsψaμ¯ψsψaμ¯gsdsgCμ¯ψ1Γδ¯aγψsψγψsδ¯1ψsψaμ¯ds.

Then, by making use of Remark 3, it follows that(34)a+δ¯:ψgCgCμ¯ψΓ1μ¯Γδ¯μ¯+1ψγψaδ¯μ¯.

Since ψ is an increasing function, it follows that(35)a+δ¯:ψgCgCμ¯ψΓ1μ¯Γδ¯μ¯+1ψbψaδ¯μ¯.

This completes the proof of the first part inequality in (i). By making use of Theorem 1 and inequality (34) with a=0, we obtain(36)uCsinδ¯πΓδ¯πgCμ¯ψΓ1μ¯Γδ¯μ¯+1ddγψγψ0δ¯μ¯=gCμ¯ψsinδ¯πΓδ¯Γ1μ¯πΓδ¯μ¯ψγψ0δ¯μ¯1ψxgCμ¯ψsinδ¯πΓδ¯Γ1μ¯πΓδ¯μ¯ψbψ0δ¯μ¯1ψb.

Let gCa,b; then, by using Definition 3 and Lemma 2 and since ψ is an increasing function, we have(37)a+δ¯:ψgC=maxγa,b1Γδ¯aγψsψγψsδ¯1gsds=gCa+δ¯:ψ1=gC1Γδ¯+1ψγψaδ¯gC1Γδ¯+1ψbψaδ¯.

This completes the proof of the first inequality of (ii). By making use of Theorem 1 and inequality (37) with a=0, we can deduce the second part inequality in (ii).

Theorem 5.

For any δ¯0,1,μ¯,a0, we have(38)a+δ¯:ψψsψaμ¯γ=Γμ¯+1Γδ¯+μ¯+1ψγψaδ¯+μ¯.

Proof.

The proof is similar to Lemma 2, so it is omitted.

Theorem 6.

Let 0a<b<,δ¯0,1,μ¯>0 and gCμ¯ψa,b. If μ¯<δ¯, then we have(39)a+δ¯:ψga=limxaa+δ¯:ψgγ=0.

Moreover, the solution uγ in Theorem 1 vanishes at γ=0.

Proof.

Let gCμ¯ψa,b; then, ψγψaμ¯gγCa,b and there exists some M>0 such that(40)ψγψaμ¯gγM,γa,b,and(41)a+δ¯:ψgγ=1Γδ¯aγψtψγψtδ¯1gtdt=1Γδ¯aγψsψγψsδ¯1ψsψγμ¯ψsψγμ¯gsds.

Then, by making use of Remark 3 and Theorem 5, it follows that(42)a+δ¯:ψgMΓδ¯aγψsψγψsδ¯1ψsψγμ¯ds=Ma+δ¯:ψψsψγμ¯=Γ1μ¯Γδ¯μ¯+1ψγψaδ¯μ¯.

Taking the limit on both sides, it follows that(43)limγaa+δ¯:ψg=0,which completes the proof of the first part. By making use of Theorem 1 and formula (42) with a=0, 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 :(44)43sinγ3/4=0γutdtsinγsint1/4,0<γ<π2,where δ¯=1/4,gγ=4/3sinγ3/4 and gγ=sinγ. It is clear that gγ is strictly monotonically increasing in 0<γ<π/2 and gγ0 for each γ in 0<γ<π/2. Using Corollary 1 with μ¯=3/4, we get(45)uγ=12π433/4Γ1/4Γ3/4Γ2ddγsinγ=cosγ,which is the exact solution, where we have used the fact that Γ1/4Γ3/4=2π.

Example 2.

Consider the generalized Abel’s integral equation :(46)625γ25/6=0γutdtγ5t51/6,0<γ<2.

Here, equation (7) takes the following form:(47)uγ=sinδ¯ππddγ0x5t46/25t25/6γ5t51/6dt=12π625ddγ0x5t4t55/6γ5t51/6dt.

From this, we have δ¯=1/6,gγ=6/25γ25/6,gγ=γ5, and μ¯=5/6. Thus, Corollary 1 gives the exact solution:(48)uγ=12πddγ25πγ5=γ4,where we have used the fact that Γ1/6Γ5/6=2π.

Example 3.

Consider the weakly singular second kind Volterra integral equation [4, 33, 34]:(49)uγ=2γ0γutdtγt,x0,2.

Using Corollary 3 with θ¯=2,μ¯=1/2,β¯=1, and δ¯=1/2, we can easily obtain(50)uγ=1E1/2,1πγ=1eπxerfcπγ,where erfc is the complementary error function which is defined as(51)erfcw=2πwez2dz.

Example 4.

Consider the weakly singular second kind Volterra integral equation [4, 33, 34]:(52)uγ=γ2+1615γ52/0γutdtγt,x0,1,where θ¯1=1,θ¯2=16/15,μ¯1=2,μ¯2=5/2,β¯=1, and δ¯=1/2. Thus, by formula (24), we get(53)uγ=Γ2+1γ2E1/2,3πγ1615Γ5/2+1γ5/2Γ1/2γ1/2E1/2,3πγ1Γ3=2γ2E1/2,3πγ2γ2E1/2,3πγ+γ2=γ2.

Hence, the exact solution is uγ=γ2.

Example 5.

Consider the weakly singular second kind Volterra integral equation :(54)uγ=γπγ+20γutdtγt.

In this example, θ¯1=1,θ¯2=π,μ¯1=1/2,μ¯2=1,β¯=2, and δ¯=1/2. Thus, formula (24) gives(55)uγ=12πγE1/2,3/22πγ12πγE1/2,3/22πγ+γ=γ,which is the exact solution of integral equation (54).

Example 6.

Consider the weakly singular second kind Volterra integral equation :(56)uγ=12γ+0γutdtγt.

From formula (24) with θ¯1=1/2,θ¯2=1,μ¯1=0,μ¯2=1/2,β¯=1, and δ¯=1/2, we get(57)uγ=12E1/2,1πγ12E1/2,1πγ+12=12,which is the exact solution of integral equation (56).

Example 7.

Consider the Abel integral equation of the second kind :(58)uγ=γ+43γ3/20γutdtγt.

From formula (24) with θ¯1=1,θ¯2=4/3,μ¯1=1,μ¯2=3/2,β¯=1, and δ¯=1/2, we get(59)uγ=γE1/2,2πγγE1/2,2πγ+γ=γ,which 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.

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