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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

Fractional dynamic varying systems with singular kernels either in the Riemann–Liouville sense or in the Caputo sense have been investigated in the literature [

Let

Particularly, if

Recently, Wazwaz [

The paper is organized as follows. In Section

Here, we give the definitions of Riemann–Liouville fractional integrals,

Let

Let

If we set

The following lemmas hold in [

Let

Let

In this context,

In this paper, using the Adomian decomposition method and Laplace transform method combined with Lemma

Now, we give our main results.

If

If

From Lemma

From this and Definition

Under the similar assumptions of Theorem

Now, the solution of (

If

From the Adomian decomposition method, we substitute the decomposition series

The components

Thus, the exact solution is

Under the similar assumptions of Theorem

The noise terms may appear between components of

Under the similar assumptions of Theorem

To prove this, we use one of the Mittag–Leffler function’s properties, that is [

From this, we can write (

This completes the proof.

Under the similar assumptions of Theorem

By the same manner of Theorem

This completes the proof of the first part of this theorem.

By using property (

Due to the occurrence of the noise terms between

Now, we introduce some spaces of the continuous functions in order to obtain the boundness of the above solution.

Let

We denote by

We define the weighted space

Note that

Let

The proof follows directly from the substitution

Let

If

Moreover, if

For any

Moreover, for any

From Definition

Then, by making use of Remark

Since

This completes the proof of the first part inequality in (i). By making use of Theorem

Let

This completes the proof of the first inequality of (ii). By making use of Theorem

For any

The proof is similar to Lemma

Let

Moreover, the solution

Let

Then, by making use of Remark

Taking the limit on both sides, it follows that

In this section, we consider several test problems corresponding to the equations (

Consider the generalized Abel’s integral equation [

Consider the generalized Abel’s integral equation [

Here, equation (

From this, we have

Consider the weakly singular second kind Volterra integral equation [

Using Corollary

Consider the weakly singular second kind Volterra integral equation [

Hence, the exact solution is

Consider the weakly singular second kind Volterra integral equation [

In this example,

Consider the weakly singular second kind Volterra integral equation [

From formula (

Consider the Abel integral equation of the second kind [

From formula (

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.

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.