This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach.

The origins of fractal calculus trace back to the 17th century when the philosopher and mathematician Gottfried Leibniz delved into the concept. Leibniz observed that this approach effectively explained unexpected phenomena by examining the properties of self-similar and infinitely complex objects. In the mid-2016s, Atangana1,2 introduced the concept of fractal–fractional calculus. The innovative differential operators can be conceptualized as a convolution of the fractal derivative with the generalized Mittag–Leffler function, the exponential decay law, and the power law. This novel approach has found applications in various scientific disciplines. For applications and related work, we refer to optimal homotopy continuation method.3 Ahmed and Adam constructed a diabetes model with the fractal–fractional of the Atangana–Baleanu derivative.4 Hasib Khan et al. used the fractal–fractional derivative to solve the tuberculosis model in China.5 Atangana et al. obtained numerical solutions for the Chua attractor equation by using fractal–fractional operators.6 Singh et al. derived a tumor growth model with fractal–fractional derivative in the sense of Caputo-Fabrizio.7 Li et al. investigated the bank data using fractal–fractional derivative in the Caputo sense.8

Numerous authors contributed to the development of calculus for fractional and fractal–fractional operators. Abro et al. studied the electrical circuit model in the form of Atangana–Baleanu derivatvie.9 Logeswari et al. obtained mathematical model for COVID-19 using Mittag–Leffler kernel.10 Khater et al. studied the fractional equation of Bogoyavlensky–Konopelchenko in the form of Atangana–Baleanu–Riemann derivative.11 Nisar and co-authors constructed in various fields the Brusselator reaction–diffusion system with the residual power series method12 and epidemic models in sight of fractional calculus.13 The authors have achieved results regarding the asymptotic stability results of fractional difference equations.14 In the field of physics,15 engineering like artificial neural network16 and fractal–fractional operators of Hyers–Ulam stability are used successfully to model real-world processes. Cevikel et al. derived fractional equations with conformable derivative in shallow water17,18 and mathematical physics.19 Altalbe et al. addressed the space-time fractional van der Waals equation employing three analytical methods.20 The authors21 derived partial fractional derivative in the sense of fractional caputo derivative. Ishtiaq et al. used Prabhakar fractional derivative to analyze the magnetized Walter's-B fluid.22 Tai et al. conducted a study on heat transfer of crude oil storage tank based on the fractional-order Maxwell model.23 Burqan et al. constructed ARA-residual model with partial fractional differential equation.24 Baitiche et al. obtained $ψ$-caputo fractional derivative using the partial fractional differential equation.25 In Refs. 26 and 27, the authors investigated YTSF equation and Fitzhugh–Nagumo-type equation. Bekir et al. explored new solitons and periodic solutions applicable to problems in mathematical physics28 and the (2 + 1)-dimensional Boussinesq equation.29 Additionally, the authors formulated a monkeypox model30 and a non-Darcian model31 utilizing fractional derivatives. El-Tantawy et al. used fractional Burger–Fisher equations for the residual power series transform method.32

Ulam began researching the Ulam-type stability theory in 1940. A year later, Hyers solved the problem by assuming that groups are Banach spaces, as defined by Hyers–Ulam stability. ur Rahman et al.33 established the existence and uniqueness of the solution and explored various forms of Hyers–Ulam stability using fixed-point theorems by Schauder and Banach for the piecewise plant-pathogen-herbivore interactions model. Abdullah et al. investigated the Hyers–Ulam stability of fractional differential equations employing the Atangana–Baleanu derivative in conjunction with the $ϕ p$-Laplacian operator.34 The integrodifferential equations play a key role in many fields of modern mathematics. The integrodifferential equations have been studied in various fields by Alharbi et al.35 and Miah et al.36 The authors in Ref. 37 presented the fractional integrodifferential equations in the Atangana–Baleanu derivative, and they analyzed the existence, uniqueness, and Hyers–Ulam stability of the solution derived for the equations. In Ref. 38, the authors solved the Cauchy problem with fractal–fractional operator described by power law, exponential decay, and Mittag–Leffler kernels, and also, the authors explored the numerical and exact solution.

In Ref. 39, Wu et al. investigated the existence and uniqueness of solutions for the Cauchy problem with a Caputo fractional derivative and nonlocal conditions,
$D α g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) ) , 0 < α < 1 , g ( 0 ) + x ( 0 ) = g 0 ,$
where $s ∈ Q : [ 0 , 1 ] , Ψ : Q × X × X → X , x : C ( Q , X ) → X$ defined as $I 1 g ( s ) = ∫ 0 s j ( s , l , g ( l ) ) d l .$ Here, $( X , ‖ . ‖ )$ is a Banach space, and $C = C ( Q , X )$ denotes the Banach space of all bounded continuous functions from Q into X equipped with the norm $‖ . ‖ C .$
Motivated by the above-mentioned works, we study a more general problem of fractal–fractional integrodifferential equation, which is called fractal–fractional Volterra–Fredholm integrodifferential equations in the Riemann–Liouville sense given as follows:
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , ∫ 0 s j ( s , l , g ( l ) ) d l , ∫ 0 S h ( s , l , g ( l ) ) d l ) , 0 < k 1 ≤ 1 , k 2 > 0 ,$
(1.1)
$g ( 0 ) = g 0 ,$
(1.2)
with $s ∈ Q : [ 0 , 1 ]$, where k1 is the fractional order, k2 is the fractal dimension, and $0 FFP D s k 1 , k 2$ denotes the fractal–fractional derivative with power law kernel in the Reimann–Liouville sense. Here, $Ψ ∈ C 2 [ 0 , 1 ] , Ψ ( 0 , g ( 0 ) , 0 , ∫ 0 S h ( 0 , l , g ( l ) ) d l ) = 0.$ Let $I 1 g ( s ) = ∫ 0 s j ( s , l , g ( l ) ) d l$ and $I 2 g ( s ) = ∫ 0 S h ( s , l , g ( l ) ) d l .$ Then, (1.1) and (1.2) become
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , 0 < k 1 ≤ 1 , k 2 > 0 ,$
(1.3)
$g ( 0 ) = g 0 .$
(1.4)
We study the existence and uniqueness of solutions to the proposed equation under consideration and prove the results for Hyers–Ulam stability. Also, Lagrange's interpolation method is used for each instance to compute a numerical solution for the proposed equation. Moreover, we investigate the examples of fractal–fractional integrodifferential equations, and their precise solutions were determined and subsequently compared with the numerical results.

We recall to some important definitions related to fractal–fractional differential operators. Also, in this work, we assume the space ${ g ( s ) ∈ C ( [ 0 , 1 ] ) → ℝ }$ with the norm $|| g || = m a x s ∈ [ 0 , 1 ] | g ( s ) |$.

Definition II.1. Let the function $g ∈ ℂ ( ( a , b ) , ℝ )$ be fractal differentiable on $( a , b ) ,$ with order $0 < k 2 ≤ 1$, then the fractal–fractional derivative of g of order $0 < k 1 ≤ 1$ in Riemann–Liouville sense with power law is given as1,2,5
$0 FFP D s k 1 , k 2 g ( s ) = 1 Γ ( 1 − k 1 ) d d s k 2 ∫ 0 s g ( η ) ( s − η ) − k 1 d η .$
Definition II.2. Let $g ∈ ℂ ( ( a , b ) , ℝ )$ be fractal differentiable on $( a , b ) ,$ with order $0 < k 2 ≤ 1$, then the fractal–fractional derivative of g of order $0 < k 1 ≤ 1$ in Riemann–Liouville sense with exponential decay kernel is given as1,2,5
$0 FFE D s k 1 , k 2 g ( s ) = M ( k 1 ) Γ ( 1 − k 1 ) d d s k 2 ∫ 0 s g ( η ) exp [ − k 1 1 − k 1 ( s − η ) ] d η .$
Definition II.3. Let the function $g ∈ ℂ ( ( a , b ) , ℝ )$ be fractal differentiable on $( a , b ) ,$ with order $0 < k 2 ≤ 1$, then the fractal–fractional derivative of g of order $0 < k 1 ≤ 1$ in Riemann–Liouville sense with Mittag–Leffler kernel is given as1,2,5
$0 FFM D s k 1 , k 2 g ( s ) = A B ( k 1 ) Γ ( 1 − k 1 ) d d s k 2 ∫ 0 s g ( η ) E k 1 [ − k 1 1 − k 1 ( s − η ) 1 k ] d η .$
The hypothesis37 presented below is utilized to establish both the existence and uniqueness outcomes for Eq. (1.3).
• $H 1$ Let $u ∈ C [ 0 , 1 ]$ and suppose that $Ψ ∈ ( C [ 0 , 1 ] × Q × Q × Q , Q )$ is a continuous function and there exist non-negative constants Z1, Z2, and Z such that
$| | Ψ ( s , u 1 , v 1 , w 1 ) − Ψ ( s , u 2 , v 2 , w 2 ) | | ≤ Z 1 ( | | u 1 − u 2 | | + | | v 1 − v 2 | | + | | w 1 − w 2 | | ) ,$

for all $u 1 , v 1 , w 1$ in V, $Z 2 = m a x s ∈ Q | | Ψ ( s , 0 , 0 , 0 ) | |$ and $Z = max { Z 1 , Z 2 } .$ Let V=C[Q,U] be the set continuous functions on Q with values in the Banach spaces U.

• $H 2$ There exist non-negative constants n1, n2, and n such that
$| | I 1 ( s , l , v 1 ) − I 1 ( s , l , v 2 ) | | ≤ n 1 ( | | v 1 − v 2 | | ) ,$

for all v1, v2 in V, $n 2 = m a x ( s , l ) ∈ D | | I 1 ( s , l , 0 ) | |$ and $n = max { n 1 , n 2 } .$

• $H 3$ There exist non-negative constants c1, c2, and c such that
$| | I 2 ( s , l , v 1 ) − I 2 ( s , l , v 2 ) | | ≤ c 1 ( | | v 1 − v 2 | | ) ,$

for all v1, v2 in V, $c 2 = max ( s , l ) ∈ D | | I 1 ( s , l , 0 ) | |$, and $c = max { c 1 , c 2 } .$

We are examining the integrodifferential problem presented as
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , if s > 0 ,$
(3.1)
$g ( 0 ) = g 0 , if s = 0 ,$
(3.2)
where $I 1 g ( s ) = ∫ 0 s j ( s , l , g ( l ) d l , I 2 g ( s ) = ∫ 0 S h ( s , l , g ( l ) ) d l .$
We define the norm as
$| | δ | | 2 = s u p s ∈ D δ | δ ( s ) | ,$
(3.3)
$Ψ ∈ C 2 [ 0 , S ] , 0 < k 1 ≤ 1 , k 2 > 0.$
It is assumed that $Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) )$ holds the following conditions:
1. For all $s ∈ [ 0 , S ] , | Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) | 2 < K ( 1 + | g | 2 + | I 1 g | 2 + | I 2 g | 2 ) .$

2. For all $s ∈ [ 0 , S ] , g 1 , g 2 ∈ c 2 [ 0 , S ] ,$
$| Ψ ( s , g 1 , I 1 g 1 , I 2 g 1 ) − Ψ ( s , g 2 , I 1 g 2 , I 2 g 2 ) | 2 < K ( | g 1 − g 2 | + | I 1 g 1 − I 1 g 2 | + | I 2 g 1 − I 2 g 2 | ) 2 .$
Note that Eqs. (3.1) and (3.2) can be transformed to
$0 FFP D s k 1 , k 2 g ( s ) = k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , s > 0 ,$
(3.4)
$g ( 0 ) = g 0 , if s = 0.$
(3.5)
Thus, Eqs. (3.4) and (3.5) are expressed as
$g ( s ) = k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η ,$
(3.6)
$g ( 0 ) = g 0 .$
(3.7)
The defined mapping below is utilized to investigate the existence and uniqueness of the specific requirements,
$χ ( g ( s ) ) = k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η ,$
(3.8)
$| χ ( g ( s ) ) | 2 = k 2 2 ( Γ ( 1 − k 1 ) ) 2 | ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η | 2 < 2 k 2 2 Γ 2 ( 1 − k 1 ) ∫ 0 s η 2 k 2 − 2 ( s − η ) − 2 k 1 d η ∫ 0 s | Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) | 2 d η .$
(3.9)
First, we evaluate $∫ 0 s η 2 k 2 − 2 ( s − η ) − 2 k 1 d η ,$ and substitute $s p = η$,
$∫ 0 s ( s p ) 2 k 2 − 2 ( s − p s ) − 2 k 1 s d p = s 2 k 2 − 2 k 1 − 1 ∫ 0 s p 2 k 2 − 2 ( 1 − p ) − 2 k 1 d p = s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ,$
(3.10)
$B ( p 1 , p 2 ) = ∫ 0 s s p 1 − 1 ( 1 − s ) p 2 − 1 d s ,$
(3.11)
where $R e ( p 1 ) , R e ( p 2 ) > 0$ and
$k 2 − 1 > 0 and 1 − 2 k 1 > 0 , k 2 > 1 2 and 1 > 2 k 1 ≥ k 1 < 1 2 .$
Thus,
$| χ ( g ( s ) ) | 2 < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ∫ 0 s | Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) | 2 d η < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K ∫ 0 s ( 1 + | g ( s ) | 2 + | I 1 g ( s ) | 2 + | I 2 g ( s ) | 2 ) d η < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K ∫ 0 s ( 1 + | g ( s ) | 2 + n 2 | g ( s ) | 2 + c 2 | g ( s ) | 2 ) d η < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K ∫ 0 s ( 1 + | g ( s ) | 2 ( 1 + n 2 + c 2 ) ) d η < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K ( 1 + n 2 + c 2 ) s ( 1 ( 1 + n 2 + c 2 ) + | g ( s ) | 2 ) < 2 k 2 2 s 2 k 2 − 2 k 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K 1 ( 1 + ‖ g ‖ 2 ) ,$
(3.12)
where $K 1 = K ( 1 + n 2 + c 2 )$ and $1 ( 1 + n 2 + c 2 ) < 1 ,$
$| χ ( g ( s ) ) | 2 < K 1 ¯ ( 1 + ‖ g ‖ 2 ) ,$
(3.13)
since
$K 1 ¯ = 2 k 2 2 s 2 k 2 − 2 k 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K 1 .$
This implies that the mapping χ fulfills the linear growth condition. Subsequently, we proceed to address the Lipschitz condition,
$| χ 1 g 1 − χ 2 g 2 | 2 = | k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 ( s − η ) − k 1 Ψ ( η , g 1 ( η ) , I 1 g 1 ( η ) , I 2 g 1 ( η ) ) d η − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 ( s − η ) − k 1 Ψ ( η , g 2 ( η ) , I 1 g 2 ( η ) , I 2 g 2 ( η ) ) d η | 2 = | k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 ( s − η ) − k 1 [ Ψ ( η , g 1 ( η ) , I 1 g 1 ( η ) , I 2 g 1 ( η ) ) − Ψ ( η , g 2 ( η ) , I 1 g 2 ( η ) , I 2 g 2 ( η ) ) ] d η | 2 ,$
(3.14)
$| χ 1 g 1 − χ 2 g 2 | 2 < 2 k 2 2 Γ ( 1 − k 1 ) 2 ∫ 0 s η 2 k 2 − 2 ( s − η ) − 2 k 1 d η ∫ 0 s | Ψ ( η , g 1 ( η ) , I 1 g 1 ( η ) , I 2 g 1 ( η ) ) − Ψ ( η , g 2 ( η ) , I 1 g 2 ( η ) , I 2 g 2 ( η ) | 2 d η < 2 k 2 2 Γ ( 1 − k 1 ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ∫ 0 s | Ψ ( η , g 1 ( η ) , I 1 g 1 ( η ) , I 2 g 1 ( η ) ) − Ψ ( η , g 2 ( η ) , I 1 g 2 ( η ) , I 2 g 2 ( η ) | 2 d η ,$
(3.15)
Therefore, Eq. (3.15) can be written as
$| χ 1 g 1 − χ 2 g 2 | 2 < 2 k 2 2 s 2 k 2 − 2 k 1 − 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ∫ 0 s K ( ‖ g 1 − g 2 ‖ + ‖ I 1 g 1 − I 1 g 2 ‖ + ‖ I 2 g 1 − I 2 g 2 ‖ ) 2 d η < 2 k 2 2 s 2 k 2 − 2 k 1 − 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ∫ 0 s K ( ‖ g 1 − g 2 ‖ + n ‖ g 1 − g 2 ‖ + c ‖ g 1 − g 2 ‖ ) 2 d η < 2 k 2 2 s 2 k 2 − 2 k 1 − 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ∫ 0 s K ( ‖ g 1 − g 2 ‖ ( 1 + n + c ) ) 2 d η < 2 k 2 2 s 2 k 2 − 2 k 1 − 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K s ( ‖ g 1 − g 2 ‖ ( 1 + n + c ) ) 2 = 2 k 2 2 s 2 k 2 − 2 k 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) K 1 ( ‖ g 1 − g 2 ‖ 2 ) ,$
(3.16)
where $K 1 = K ( 1 + n + c ) 2 ,$
$| χ 1 g 1 − χ 2 g 2 | 2 < K 1 ¯ | | g 1 − g 2 | | 2 ,$
(3.17)
finally,
$K 1 ¯ = 2 k 2 2 s 2 k 2 − 2 k 1 Γ ( 1 − k 1 ) B ( 2 k 2 − 1 , 1 − 2 k 1 ) K 1 .$
The presence of the linear growth condition leads to the conclusion that our equation possesses a unique solution.

In this segment, our focus revolves around exploring the concept of Hyers–Ulam stability as it pertains to Eqs. (1.3) and (1.4). To begin, we provide an elucidation of the precise definition of this particular form of stability.

Definition IV.1. Equations (1.3) and (1.4) has Hyers–Ulam stability, if there exists a non-negative constant $M$ such that for all $ε > 0$, and every h(s) satisfying37
$| 0 FFP D s k 1 , k 2 h ( s ) − Ψ ( s , h ( s ) , I 1 h ( s ) , I 2 h ( s ) ) | 2 ≤ ε ,$
(4.1)
there exists a solution g(s) of Eqs. (1.3) and (1.4) such that $| h ( s ) − g ( s ) | 2 ≤ M ε$. We call such $M$ a Hyers–Ulam stability constant for Eqs. (1.3) and (1.4).

Theorem IV.2. If all the hypotheses for the existence of solutions to Eqs. (1.3) and (1.4) be satisfied. Then, it follows that both (1.3) and (1.4) are Hyers–Ulam stability.

Proof: If h(s) satisfies (4.1), then h(s) satisfies the following equation:
$| h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η | 2 ≤ 2 k 2 2 ( Γ ( 1 − k 1 ) ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ε .$
(4.2)
If h(s) satisfies (4.1), then h(s) satisfies $0 FFP D s k 1 , k 2 h ( s ) = Ψ ( s , h ( s ) , I 1 h ( s ) , I 2 h ( s ) ) + ζ ( s ) ,$ where $| ζ ( s ) | 2 ≤ ε .$
This equation can be transformed to the Riemann–Liouville equation as follows:
$h ( s ) = k 2 s k 2 − 1 Ψ ( s , h ( s ) , I 1 h ( s ) , I 2 h ( s ) ) + k 2 s k 2 − 1 ζ ( s ) = k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η + k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 ζ ( s ) ( s − η ) − k 1 d η ,$
(4.3)
$| h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η | 2 = | k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 ζ ( s ) ( s − η ) − k 1 d η | 2 < 2 k 2 2 ( Γ ( 1 − k 1 ) ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) | ζ ( s ) | 2 ≤ 2 k 2 2 ( Γ ( 1 − k 1 ) ) 2 s 2 k 2 − 2 k 1 − 1 B ( 2 k 2 − 1 , 1 − 2 k 1 ) ε ≤ K ε .$
(4.4)
Therefore,
$| h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η | 2 ≤ K ε .$
(4.5)
Now, we have to prove $| h ( s ) − g ( s ) | 2 ≤ M ε ,$
$| h ( s ) − g ( s ) | 2 = | h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η | 2 = | h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η + k 2 Γ ( 1 − k 1 ) ∫ 0 s τ k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η | 2 ≤ 2 | h ( s ) − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η | 2 + 2 | k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , h ( η ) , I 1 h ( η ) , I 2 h ( η ) ) ( s − η ) − k 1 d η − k 2 Γ ( 1 − k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) − k 1 d η | 2 ≤ 2 K ε + 2 { 2 k 2 2 s 2 k 2 − 2 k 1 Γ ( 1 − k 1 ) 2 B ( 2 k 2 − 1 , 1 − 2 k 1 ) | | h − g | | 2 } ≤ 2 K ε + 2 K ¯ | | h − g | | 2 .$
(4.6)
This equation can be written as
$| h ( s ) − g ( s ) | 2 ≤ M ε ,$
(4.7)
where $M = 2 K 1 − 2 K ¯$. Therefore, Eqs. (1.3) and (1.4) are Hyers–Ulam stable.
Employing the fractal–fractional derivative and power law, we present a numerical solution for the integrodifferential problem,
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , if s > 0 , g ( 0 ) = g 0 , if s = 0.$
Let us consider $0 < s 1 < s 2 < … < s n + 1 = S .$
We have that for all $s ∈ ( 0 , S )$,
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) )$
(5.1)
can be converted to
$g ( s ) = k 2 Γ ( k 1 ) ∫ 0 s η k 2 − 1 ( s − η ) k 1 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η .$
(5.2)
Putting $s = s n + 1$ in Eq. (5.2), we have
$g ( s n + 1 ) = k 2 Γ ( k 1 ) ∫ 0 s n + 1 η k 2 − 1 ( s n + 1 − η ) k 1 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η = k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 η k 2 − 1 ( s n + 1 − η ) k 1 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η .$
(5.3)
Within $[ s γ , s γ + 1 ] ,$ we approximate
$Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ≈ P γ ( η ) , = Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) + ( η − s γ ) h ( Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) − Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ) .$
(5.4)
Substituting the previously mentioned values in the general equations leads to
$g ( s n + 1 ) ≈ k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 η k 2 − 1 [ Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ( τ − s γ ) h ( Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) − Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ) ] ( s n + 1 − η ) k 1 − 1 d η = k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 [ Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) η k 2 − 1 ( s n + 1 − η ) k 1 − 1 d η ] + k 2 Γ ( k 1 ) ∑ γ = 0 n − 1 ∫ s γ s γ + 1 η k 2 − 1 [ Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) − Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ] η − s γ h ( s n + 1 − η ) k 1 − 1 d η + k 2 Γ ( k 1 ) ∫ s n s n + 1 ( s n + 1 − η ) k 1 − 1 η k 2 − 1 ( η − s n ) h [ Ψ ( s n + 1 , g p ( s n + 1 ) , I 1 g p ( s n + 1 ) , I 2 g p ( s n + 1 ) ) − Ψ ( s n , g ( s n ) , I 1 g ( s n ) , I 2 g ( s n ) ) ] d τ ,$
(5.5)
$g n + 1 = k 2 Γ ( k 1 ) ∑ γ = 0 n Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ∫ s γ s γ + 1 η k 2 − 1 ( s n + 1 − η ) k 1 − 1 d η + k 2 Γ ( k 1 ) ∑ γ = 0 n − 1 ( Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) − Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) ) h × ∫ s γ s γ + 1 ( η − s γ ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 d η + k 2 Γ ( k 1 ) ∫ s n s n + 1 ( s n + 1 − η ) k 1 − 1 × [ Ψ ( s n + 1 , g p ( s n + 1 ) , I 1 g p ( s n + 1 ) , I 2 g p ( s n + 1 ) ) − Ψ ( s n , g ( s n ) , I 1 g ( s n ) , I 2 g ( s n ) ) ] η − s n h η k 2 − 1 d η .$
(5.6)
Let us examine the fractal–fractional derivative with exponential decay,
$0 FCP D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , s > 0 ,$
(5.7)
$g ( 0 ) = g 0 , if s = 0.$
(5.8)
We convert system into the following:
$0 C F D s k 1 , k 2 g ( s ) = k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , s > 0 ,$
(5.9)
$g ( 0 ) = g 0 , if s = 0.$
(5.10)
Thus, we obtain
$g ( s ) = ( 1 − k 1 ) k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) + k 1 k 2 ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η , if s > 0 ,$
(5.11)
$g ( 0 ) = g 0 .$
(5.12)
At $s = s n + 1 ,$
$g ( s n + 1 ) = ( 1 − k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g ( s n + 1 ) , I 1 g ( s n + 1 ) , I 2 g ( s n + 1 ) ) + k 1 k 2 ∫ 0 s n + 1 η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η ,$
(5.13)
$g ( s n ) = ( 1 − k 1 ) k 2 s n k 2 − 1 Ψ ( s n , g ( s n ) , I 1 g ( s n ) , I 2 g ( s n ) ) + k 1 k 2 ∫ 0 s n η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η .$
(5.14)
The difference produces
$g ( s n + 1 ) − g ( s n ) = ( 1 − k 1 ) k 2 ( s n + 1 k 2 − 1 Ψ ( s n + 1 , g ( s n + 1 ) , I 1 g ( s n + 1 ) , I 2 g ( s n + 1 ) ) − s n k 2 − 1 Ψ ( s n , g ( s n ) , I 1 g ( s n ) , I 2 g ( s n ) ) ) + k 1 k 2 ∫ s n s n + 1 η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η ,$
(5.15)
within $[ s n , s n + 1 ] ,$ we approximate $Ψ ( τ , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) d η$ as follows:
$Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ≈ P n ( η ) = ( s γ + 1 − η ) s γ + 1 − t γ Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) + ( η − s γ ) s γ + 1 − s γ Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) .$
(5.16)
We substitute the approximate function to acquire
$g n + 1 = g n + k 2 ( s n + 1 k 2 − 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) − s n k 2 − 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) ( 1 − k 1 ) + k 1 k 2 ∫ s n s n + 1 η k 2 − 1 { s γ + 1 − η h Ψ ( s γ , g ( s γ ) , I 1 g ( s γ ) , I 2 g ( s γ ) ) + η − s γ h Ψ ( s γ + 1 , g ( s γ + 1 ) , I 1 g ( s γ + 1 ) , I 2 g ( s γ + 1 ) ) } d η ,$
(5.17)
$g n + 1 = g n + k 2 ( 1 − k 1 ) ( s n + 1 k 2 − 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) − s n k 2 − 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) + k 1 k 2 Ψ ( s n , g n , I 1 g n , I 2 g n ) h ∫ s n s n + 1 η k 2 − 1 ( s γ + 1 − η ) d η + k 1 k 2 Ψ ( s n + 1 , g n + 1 , I 1 g n + 1 , I 2 g n + 1 ) h ∫ s n s n + 1 η k 2 − 1 ( η − s n ) d η ,$
(5.18)
and we have
$∫ s n s n + 1 ( η k 2 − 1 s n + 1 − η k 2 − 1 ) d η = ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 − ( n + 1 ) k 2 + 1 k 2 + 1 − n 2 k ( n + 1 ) k 2 + n k 2 + 1 k 2 + 1 } ,$
(5.19)
$∫ s n s n + 1 ( η k 2 − 1 η − η k 2 − 1 s n ) d η = ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 + 1 − n ( n + 1 ) k 2 + 1 k 2 + 1 − n k 2 + 1 k 2 + 1 + n k 2 + 1 k 2 } .$
(5.20)
Thus,
$g n + 1 = g n + k 2 ( 1 − k 1 ) ( Δ s ) k 2 − 1 { ( n + 1 ) k 2 + 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) − n k 2 + 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) } + k 2 k 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) h ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 − ( n + 1 ) k 2 + 1 k 2 + 1 − n k 2 ( n + 1 ) k 2 + n k 2 + 1 k 2 + 1 } + k 1 k 2 ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 + 1 − n ( n + 1 ) k 2 + 1 k 2 + 1 − n k 2 + 1 k 2 + 1 + n k 2 + 1 k 2 } Ψ ( s n + 1 , g n + 1 , I 1 g n + 1 , I 2 g n + 1 ) h ,$
(5.21)
where $g n + 1 p$ is the predictor,
$∫ s n s n + 1 η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ≈ k 2 Ψ ( s n , g n , I 1 g n , I 2 g n ) ∫ s n s n + 1 η k 2 − 1 d η ≈ k 2 Ψ ( s n , g n , I 1 g n , I 2 g n ) Δ s k 2 k 2 ( ( n + 1 ) k 2 − n k 2 ) , g n + 1 p ≈ Ψ ( s n , g n , I 1 g n , I 2 g n ) Δ s k 2 ( ( n + 1 ) k 2 − n k 2 ) ,$
(5.22)
$g n + 1 = g n + k 2 ( 1 − k 1 ) ( Δ s ) k 2 − 1 { ( n + 1 ) k 2 + 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) − n k 2 + 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) } + k 2 k 1 Ψ ( s n , g n , I 1 g n , I 2 g n ) ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 − ( n + 1 ) k 2 + 1 k 2 + 1 − n k 2 ( n + 1 ) k 2 + n k 2 + 1 k 2 + 1 } + k 1 k 2 ( Δ s ) k 2 + 1 { ( n + 1 ) k 2 + 1 k 2 + 1 − n ( n + 1 ) k 2 + 1 k 2 + 1 − n k 2 + 1 k 2 + 1 + n k 2 + 1 k 2 } Ψ ( s n + 1 , g n + 1 , I 1 g n + 1 , I 2 g n + 1 ) ,$
(5.23)
since
$g n + 1 p = Ψ ( s n , g n , I 1 g n , I 2 g n ) Δ s k 2 ( ( n + 1 ) k 2 − n k 2 ) .$
We continue our analysis by considering the following integrodifferential problem:
$0 FFM D s k 1 , k 2 g ( s ) = Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , s > 0 ,$
(5.24)
$g ( 0 ) = g 0 , s = 0.$
(5.25)
It is transformed into the following form:
$0 ABR D s k 1 g ( s ) = k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) , s > 0 ,$
(5.26)
$g ( 0 ) = g 0 , if s = 0.$
(5.27)
Thus,
$g ( s ) = ( 1 − k 1 ) k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) + k 2 Γ ( k 1 ) ∫ 0 s η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s − η ) k 1 − 1 d η , s > 0 ,$
(5.28)
$g ( 0 ) = g 0 , s = 0.$
(5.29)
At $s = s n + 1 ,$
$g ( s n + 1 ) = ( 1 + k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g ( s n + 1 ) , I 1 g ( s n + 1 ) , I 2 g ( s n + 1 ) ) + k 1 k 2 Γ ( k 1 ) ∫ 0 s n + 1 η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s n + 1 − η ) k 1 − 1 d η ,$
(5.30)
$g ( 0 ) = g 0 .$
(5.31)
Then,
$g ( s n + 1 ) = ( 1 + k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g ( s n + 1 ) , I 1 g ( s n + 1 ) , I 2 g ( s n + 1 ) ) + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 η k 2 − 1 Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ( s n + 1 − η ) k 1 − 1 d η ,$
(5.32)
$g ( 0 ) = g 0 .$
(5.33)
Within $[ s γ , s γ + 1 ] ,$ we approximate $Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) )$ as follows:
$Ψ ( τ , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) ≈ s γ + 1 − η h Ψ ( s γ , g γ , I 1 g γ , I 2 g γ ) + η − s γ h Ψ ( s γ + 1 , g γ + 1 , I 1 g γ + 1 , I 2 g γ + 1 ) ,$
(5.34)
$g n + 1 = ( 1 + k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 { s γ + 1 − η h Ψ ( s γ , g γ , I 1 g γ , I 2 g γ ) + η − s γ h Ψ ( s γ + 1 , g γ + 1 , I 1 g γ + 1 , I 2 g γ + 1 ) } ( s n + 1 − η ) k 1 − 1 η k 2 − 1 d η = ( 1 − k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n ∫ s γ s γ + 1 { s γ + 1 − η h Ψ ( s γ , g γ , I 1 g γ , I 2 g γ ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 } d η + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n − 1 ∫ s γ s γ + 1 { η − s γ h Ψ ( s γ + 1 , g γ + 1 , I 1 g γ + 1 , I 2 g γ + 1 ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 } d η + k 1 k 2 Γ ( k 1 ) ∫ s n s n + 1 { η − s γ h Ψ ( s γ + 1 p , g γ + 1 p , I 1 g γ + 1 , I 2 g γ + 1 p ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 } d η ,$
(5.35)
$g n + 1 = ( 1 − k 1 ) k 2 s n + 1 k 2 − 1 Ψ ( s n + 1 , g n + 1 p , I 1 g n + 1 p , I 2 g n + 1 p ) + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n Ψ ( s γ , g γ , I 1 g γ , I 2 g γ ) h ∫ s γ s γ + 1 ( s n + 1 − η ) k 1 − 1 η k 2 − 1 d η + k 1 k 2 Γ ( k 1 ) ∑ γ = 0 n − 1 Ψ ( s γ + 1 , g γ + 1 , I 1 g γ + 1 , I 2 g γ + 1 ) h ∫ s γ s γ + 1 ( η − s γ ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 d η + k 1 k 2 Γ ( k 1 ) ∫ s n s n + 1 { η − s γ h Ψ ( s γ + 1 p , g γ + 1 p , I 1 g γ + 1 , I 2 g γ + 1 p ) ( s n + 1 − η ) k 1 − 1 η k 2 − 1 } d η .$
(5.36)
Some instances of fractal–fractional equations are as follows:
$0 FFP D s k 1 , k 2 g ( s ) = s ℓ ,$
(6.1)
$0 FFE D s k 1 , k 2 g ( s ) = s ℓ ,$
(6.2)
$0 FFM D s k 1 , k 2 g ( s ) = s ℓ .$
(6.3)
1. We begin with the power law kernel,
$0 FFP D s k 1 , k 2 g ( s ) = Ψ ( η , g ( η ) , I 1 g ( η ) , I 2 g ( η ) ) = s ℓ ,$
(6.4)
where
$I 1 g ( η ) = ∫ 0 s s ℓ d η = s ℓ + 1 ℓ + 1 , I 2 g ( η ) = ∫ 0 S s ℓ d η = S ℓ + 1 ℓ + 1 .$
Let $0 FFP D s k 1 , k 2 g ( s ) = s ℓ ,$
$0 R L D s k 1 , k 2 g ( s ) = k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) = k 2 s k 2 − 1 ( s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ) ,$
(6.5)
$g ( s ) = k 2 Γ ( k 1 ) ∫ 0 s η k 2 − 1 ( η ℓ + η ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ) ( s − η ) k 1 − 1 d η ,$
(6.6)
put $s p = η , d η = s d p ,$
$g ( s ) = k 2 Γ ( k 1 ) ∫ 0 s ( η ℓ + k 2 − 1 + η k 2 + ℓ ℓ + 1 + η k 2 − 1 S ℓ + 1 ℓ + 1 ) ( s − η ) k 1 + 1 d η = k 2 ℓ ( k 1 ) ∫ 0 s ( ( s p ) ℓ + k 2 + 1 + ( s p ) k 2 + ℓ ℓ + 1 + ( s p ) k 2 − 1 S ℓ + 1 ℓ + 1 ) ( s − s p ) k 1 − 1 s d p = k 2 Γ ( k 1 ) { s k 1 + k 2 + ℓ − 1 ∫ 0 s p ℓ + k 2 − 1 ( 1 − p ) k 1 − 1 d p + s k 1 + k 2 + ℓ ℓ + 1 ∫ 0 s p k 2 + ℓ ( 1 − p ) k 1 − 1 d p + s k 1 + k 2 − 1 S ℓ + 1 ℓ + 1 ∫ 0 s p k 2 − 1 ( 1 − p ) k 1 − 1 d p } = k 2 Γ ( k 1 ) { s k 1 + k 2 + ℓ − 1 B ( k 2 + ℓ , k 1 ) + s k 1 + k 2 + ℓ ℓ + 1 B ( ℓ + k 2 + 1 , k 1 ) + s k 1 + k 2 − 1 S ℓ + 1 ℓ + 1 B ( k 2 , k 1 ) } ,$
(6.7)
where
$B ( p 1 , p 2 ) = ∫ 0 s s p 1 − 1 ( 1 − s ) p 2 − 1 d s .$
Therefore, the exact solution for this equation is
$g ( s ) = k 2 Γ ( k 1 ) { s k 1 + k 2 + ℓ − 1 B ( k 2 + ℓ , k 1 ) + s k 1 + k 2 + ℓ ℓ + 1 B ( ℓ + k 2 + 1 , k 1 ) + s k 1 + k 2 − 1 S ℓ + 1 ℓ + 1 B ( k 2 , k 1 ) } .$
(6.8)

Applying Lagrange's interpolation technique, we acquired numerical solutions for the fractal–fractional integrodifferential equation. The resulting numerical results are depicted in Figs. 1 and 2. Through this process, we derived a system of equations with unknown coefficients. The solution obtained using the suggested method is in excellent agreement with the exact solution and shows that this approach can solve the problem effectively. The comparisons are made between numerical solutions and exaction solutions to illustrate the validity and great potential of the proposed technique.

A comparison of exact and corresponding numerical solutions for time variation and different values of fractional $( k 1 )$ order and fixed value of fractal $( k 2 )$ order is shown in Fig. 1.

2. With the exponential kernel, we have
$0 C F D s k 1 g ( s ) = k 2 s k 2 − 1 [ s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ] ,$
(6.9)
$g ( s ) = ( 1 − k 1 ) k 2 s k 2 − 1 [ s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 ∫ 0 s η k 2 − 1 [ s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ] d η = ( 1 − k 1 ) k 2 [ s ℓ + k 2 − 1 + s ℓ + k 2 ℓ + 1 + s k 2 − 1 S ℓ + 1 ℓ + 1 ] + k 1 k 2 [ s k 2 + ℓ k 2 + ℓ + s ℓ + k 2 + 1 ( k 2 + ℓ + 1 ) ( ℓ + 1 ) + s k 2 S ℓ + 1 ( k 2 ) ( ℓ + 1 ) ] .$
(6.10)
3. With the Mittag–Leffler kernel, we have
$0 ABR D s k 1 , k 2 g ( s ) = k 2 s k 2 − 1 Ψ ( s , g ( s ) , I 1 g ( s ) , I 2 g ( s ) ) = k 2 s k 2 − 1 ( s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ) ,$
(6.11)
$g ( s ) = ( 1 − k 1 ) k 2 s k 2 − 1 [ s ℓ + s ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 Γ ( k 1 ) ∫ 0 s η k 2 − 1 [ η ℓ + η ℓ + 1 ℓ + 1 + S ℓ + 1 ℓ + 1 ] ( s − η ) k 1 − 1 d η = ( 1 − k 1 ) k 2 [ s ℓ + k 2 − 1 + s ℓ + k 2 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 Γ ( k 1 ) ∫ 0 s η k 2 k 2 − 1 [ η ℓ + k 2 − 1 + η ℓ + k 2 ℓ + 1 + η k 2 − 1 S ℓ + 1 ℓ + 1 ] ( s − η ) k 1 − 1 d η ,$
(6.12)
put $s p = η , d η = s d p ,$
$g ( s ) = ( 1 − k 1 ) k 2 [ s ℓ + k 2 − 1 + s ℓ + k 2 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 Γ ( k 1 ) ∫ 0 s η k 2 k 2 − 1 [ ( s p ) ℓ + k 2 − 1 + ( s p ) ℓ + k 2 ℓ + 1 + ( s p ) k 2 − 1 S ℓ + 1 ℓ + 1 ] ( s − η ) k 1 − 1 d η = ( 1 − k 1 ) k 2 [ s ℓ + k 2 − 1 + s ℓ + k 2 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 Γ ( k 1 ) [ s ℓ + k 2 + k 1 − 1 B ( ℓ + k 2 , k 1 ) + s ℓ + k 2 + k 1 k 1 + 1 B ( ℓ + k 2 + 1 , k 1 ) + s k 1 + k 2 − 1 S ℓ + 1 ℓ + 1 B ( k 2 , k 1 ) ] .$
(6.13)
FIG. 1.

Exact and numerical solution with the power-law kernel.

FIG. 1.

Exact and numerical solution with the power-law kernel.

Close modal
FIG. 2.

Exact and numerical solution with the Mittag–Leffler kernel.

FIG. 2.

Exact and numerical solution with the Mittag–Leffler kernel.

Close modal
Therefore, the exact solution will be
$g ( s ) = ( 1 − k 1 ) k 2 [ s ℓ + k 2 − 1 + s ℓ + k 2 ℓ + 1 + S ℓ + 1 ℓ + 1 ] + k 1 k 2 Γ ( k 1 ) [ s ℓ + k 2 + k 1 − 1 B ( ℓ + k 2 , k 1 ) + s ℓ + k 2 + k 1 k 1 + 1 B ( ℓ + k 2 + 1 , k 1 ) + s k 1 + k 2 − 1 S ℓ + 1 ℓ + 1 B ( k 2 , k 1 ) ] .$
(6.14)
A comparison of exact and corresponding numerical solutions for time variation and different values of fractional $( k 1 )$ and fractal $( k 2 )$ order is shown in Fig. 2.

In this work, the existence and uniqueness of fractal–fractional integrodifferential equation are investigated using linear growth and Lipschitz conditions. Further, we established the Hyers–Ulam stability conditions for the model. We presented the numerical and exact solutions graphically. From Figs. 1 and 2, we noticed that the exact and numerical solutions coincide for different values of k1 and k2 indicating that they are stable.

A fractal–fractional integrodifferential equation is applied to model processes in practical disciplines such as physics, engineering, finance, and biology. This equation can be used to describe problems in various fields, including viscoelasticity, acoustics, electromagnetism, and hydrology. In the future, this type of fractal–fractional differential equation and its solutions can be used in various domains of engineering and science.

The authors have no conflicts to disclose.

G. Gokulvijay: Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). S. Sabarinathan: Conceptualization (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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