EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6631 ISSN 1307-5543 – ejpam.com Published by New York Business Global A New Numerical Solution for Prabhakar Fractional Differential Equations Using the Explicit Fractional Adams Method Mohammad Abdel Aal1,∗, Mohammad S. Ghatasheh2, Hussien Albadawi1, Shaher Momani3 1 Department of Mathematics, Faculty of Arts and Sciences, The World Islamic Sciences & Education University (W.I.S.E), Amman, Jordan 2 Department of Basic Sciences, Faculty of Arts and Educational Sciences, Middle East University, Amman, Jordan 3 Department of Mathematics, Faculty of Sciences, The University of Jordan, Amman, Jordan Abstract. This paper introduces a new way to solve Prabhakar fractional differential equations using an Explicit Fractional Adams Method. These equations are difficult to work with because they have several parameters. The new method, which combines the Explicit Fractional Adams Method with Lagrange interpolation, effectively tackles these difficulties. The paper also includes an analysis to show how well the method works. It provides several examples to demonstrate the method’s effectiveness and compares it with other existing methods. The results show that the proposed method is efficient and easy to use. 2020 Mathematics Subject Classifications: 26A33, 33C45, 34A08, 65L60, 65R10 Key Words and Phrases: Lagrange interpolation, Prabhakar fractional, fractional differential equations, Adams method 1. Introduction Fractional calculus is an extension of traditional calculus that broadens the scope of calculus operations by enabling the differentiation and integration of functions to arbitrary, non-integer orders [1]. This concept has gained significant traction in various scientific and engineering disciplines because it often provides a more accurate representation of real- world systems than classical integer-order calculus [2]. In particular, Fractional calculus has demonstrated its utility in mathematical modelling across diverse fields, including ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6631 Email addresses: mohammad.abdelaal@wise.edu.jo (M. Abdel Aal), mghatasheh@meu.edu.jo (M. S. Ghatasheh), hussien.albadawi@wise.edu.jo (H. Albadawi), S.momani@ju.edu.jo (S. Momani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 2 of 20 science, physics [3], engineering [4], and finance [5]. Its ability to capture memory and hereditary properties of materials and processes makes it especially valuable for mod- elling complex systems that exhibit non-local behaviors. Recent studies, such as the work on new properties of differential transform via difference equations, further illustrate the expanding toolkit and applications of fractional calculus in discrete and hybrid systems [6]. The practical application of fractional derivatives and integrals usually relies on well- established definitions, such as the Riemann–Louville [7] and Caputo approaches [8, 9]. These classical methods have successfully addressed a range of challenges in dynamic sys- tems, such as the well-studied Rössler system, and have been utilized in scenarios involving tempered fractional calculus [10]. However, these traditional techniques primarily empha- size fractional orders and may not adequately address the complexities that arise in certain equations [11]. Given the inherent complexity and non-local nature of fractional differential equations (FDEs), numerical methods are indispensable. Advanced approaches such as the hy- perbolic, rational, fractional, and logarithmic non-polynomial spline methods have been developed to improve accuracy, efficiency, and convergence. Similarly, α-fractional and β- fractional finite difference methods extend the classical FDM by incorporating fractional- order operators to capture memory effects. Analytical–numerical approaches, such as the homotopy perturbation method, have been successfully applied to various functional equa- tions, including fuzzy pantograph problems [13]. Moreover, fixed point theory provides a rigorous framework for proving the existence and uniqueness of solutions to fractional and multidimensional systems [12, 14], further strengthening the mathematical foundation of these techniques. One area of increasing interest is the Prabhakar fractional differential equations, which incorporate multiple parameters that introduce additional layers of complexity. The chal- lenges often arise from their non-local character and the intricate interplay between pa- rameters, making them difficult to solve with standard techniques. Prabhakar deriva- tives were addressed in [15] using the fractional Fourier transform, demonstrating Mittag- Leffler-Hyers-Ulam stability. In Ref. [17], the invariant subspace method was employed to obtain exact solutions for time-fractional nonlinear differential equations involving the regularized Prabhakar derivative. For commensurate systems, [18] presented closed-form solutions using eigenvectors and eigenvalues, while [19] applied separation of variables to reduce the problem to a Cauchy-type fractional equation, expressible via a two-variable Mittag-Leffler function. The study in [16] explored the Explicit Euler Method for a non- linear fractional oscillation equation using finite-difference schemes. In this paper, we propose a novel application of the Explicit Fractional Adams Method to solve Prabhakar fractional differential equations, known for their complexity due to multiple parameters. By integrating Lagrange interpolation, the proposed method achieves high accuracy and computational efficiency, outperforming existing techniques such as HPTM and FHPTM. A detailed convergence analysis and multiple numerical examples confirm the method’s M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 3 of 20 robustness, with potential extensions using shifted Legendre polynomials. This outline is organized as follows: In Section 2, we discuss basic definitions and important properties related to Prabhakar’s fractional integral. We focus on the basic definitions and key properties of Prabhakar’s fractional integral. It also explain how it relates to other fractional integrals. Section 3 looks at the concept of the Explicit Fractional Adams Method for solving Prabhakar fractional differential equations. This method relates to the Volterra integral equation and addresses the initial value problem. Section 5 provides examples to show how the new explicit fractional Adams method for the Prabhakar derivative works and how accurate it is. We completed all calculations using the software Maple 18. Finally, Section 6 provides the conclusion. 2. Basic Definitions and Important Properties Related to Prabhakar’s Fractional Integral This section provides basic definitions and important properties related to Prabhakar’s fractional integral. This will help readers understand its significance in studying complex problems. 2.1. Some Preliminaries Definition Relating to Prabhakar fractional in- tegral Definition 1. The one-parameter, three-parameter, and five-parameter Mittag-Leffler functions are defined as follows: Mα(ϕ) = ∞∑ n=0 ϕn Γ(αn+ 1) , Re(α) > 0, Mγ α,β(ϕ) = ∞∑ n=0 Γ(γ + n)ϕn Γ(γ)Γ(αn+ β)n! , Re(α) > 0, Mγ α1,α2,β1,β2 (ϕ) = ∞∑ n=0 Γ(γ + n)ϕn Γ(γ)Γ(α1n+ β1)Γ(α2n+ β2)n! , Re(α) > 0, (1) Definition 2. For f ∈ L1(a, b), the Prabhakar fractional integral is defined using a three-parameter Mittag-Leffler function as follows: Iγα,β,τ,a + p(x) = ∫ x a (x− w)β−1Mγ α,β(τ(x− w)α)p(x)dw (2) such that Re(β) > 0, Re(α) > 0 and γ, α, β, τ ∈ C is integer part of α. The definition in (1) can typically be expressed as Iγα,β,τ,a + p(x) = ∫ x a mγ α,β(x− w); τ)p(x)dw, (3) M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 4 of 20 such that mγ α,β(x; τ) = xβ−1Mγ α,β(τx α). (4) Employing equation (1), we can explain the Prabhakar fractional derivative and the regularized Prabhakar derivative Definition 3. For values of β between 0 and 1, (0 < β < 1), if f(x) ∈ M1[a, b] is a function defined on the interval [a, b], we define the Prabhakar fractional derivative using the Riemann-Liouville method in the following way: RDγ α,β,τ,a + p(x) = dν dxν ∫ x a mγ α,β(x− w); τ)p(x)dw, (5) where Re(α) > 0, Re(β) > 0 and γ, α, β, τ ∈ C is integer part of α. Definition 4. If p(x) is a function that belongs to the class ACν(a, b), and if a and b are numbers such that 0 ≤ a < x < b ≤ ∞, then the regularized Prabhakar derivative, using the Caputo definition, is defined as follows: RCγ α,β,τ,a + p(x) = ∫ x a mγ α,β(x− w); τ) dνp(x) dxν dw, (6) where Re(α) > 0, Re(β) > 0 and γ, α, β, τ ∈ C is integer part of α, β. Definition 5. This research paper defines Prabhakar fractional differential equations as follows: RCγ α,β,τ,a + p(x) = ∫ x a+ mγ α,β(x− w); τ) dνp(x) dxν dw, (7) where Re(α) > 0, Re(β) > 0 and γ, α, β, τ ∈ C is integer part of α, β. 2.2. Prabhakar fractional integral Key Properties and Relationships with Other Functions The Prabhakar function is closely related to the standard Mittag-Leffler function and its two-parameter version, as well as other special functions. For values of n ≥, the following applies: φn = Γ(φ+ n) Γ(φ) = φ(φ+ 1)....(φ+ n− 1) (8) When φ = 1, the equation M0 α,β = 1 Γ(β) . (9) Equation (9) holds true. When α = β = φ = 1, we get the well-known two-parameter maximum likelihood function, written as; M1 α,β(ϕ) = Mα,β(ϕ) = ∞∑ n=0 ϕn Γ(αn+ β) . (10) M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 5 of 20 we then obtain the classic exponential function, M1 1,1 = eτ . (11) The Prabhakar function is different from the two-parameter Mittag-Leffler function because it has a third parameter, ϕ. This difference means we need to examine the properties related to this additional parameter. An important formula from Prabhakar’s original work in Ref. [23] allows for simplifying the third parameter. It is written as; Mφ+1 α,β (ϕ) = Mα,β−1(ϕ) + (1− β + αφ)Mα,β(ϕ) αφ , (12) Additionally, another simplification formula was derived later in [32] and appears as Mφ+1 α,β (ϕ) = Mα,β−1(ϕ) + (1− β + αφ)Mα,β(ϕ) αϕφ , (13) where φ ̸= 0. We can then utilize both formulas to simplify the value of an additional parameter. These results become clearer when φ is an integer. For instance, let φ = n ∈ N, where N is a natural number. In this scenario, we have Mn+1 α,β (ϕ) = 1 αnN ! n∑ i=0 a (n) i Mα,β−i(ϕ), (14) and Mn+1 α,β (ϕ) = 1 αnϕnN ! n∑ i=0 a (n) i Mα,β−αn−i(ϕ), (15) where φ ̸= 0. This implies that Mn+1 α,β from the right-side of (15) can be expressed as a combination of two-parameter ML functions. The coefficients a (n) i in equations (15) and ((15) are derived from the recursive formula: a (n) i =  (1 + α− β)a (n−1) 0 for i = 0 a (n) i + (1 + α− β + i)a (n−1) i for i = 1, ..., n− 1, 1 for i = n. (16) 3. The Concept of the Explicit Fractional Adams Method for solving Prabhakar fractional differential equations In this section, we discuss the Explicit Fractional Adams Method. This method relates to the Volterra integral equation and addresses the initial value problem. M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 6 of 20 3.1. The Explicit Fractional Adams Method The Volterra integral equation connects to the initial value problem as follows: Definition 5. The explicit fractional Adams method that is related to the Volterra integral equation and addresses the initial value problem is define here as: p(x) = p0 + ∫ x 0 (x− w)β−1Mγ α,β(τ(x− w)α)µ(v; p(x))dw. (17) Then, further break down the integral from equation (17) as follows: p(xk+1) = p(xk) + ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)µ(v; p(x))dw − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α)µ(v; p(x))dw (18) Using three-point Lagrange interpolation, equation (18) can be expressed in a different form. p(xk+1) = p(xk) + ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx. (19) The first integral presented in equation (18) can be expressed in a more concise form as follows: M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 7 of 20 ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx = [ Gk 2h2 ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−2)dx − Gk−1 h2 ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−2)dx + Gk−1 h2 ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−1) ] dx = Gx 2h2 B1 − Gx h2 B2 + Gx−2 2h2 B3. (20) Setting ∫ x 0 (x− w)β−1Mγ α,β(τ(x− w)α)pu−1dp = Γ(u)xβ+u−1Mγ α,β(τ(x− w)α), so that the specific formula is: Z1 = ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xx−2)dx = Γ(3)(xk+1 − w)β+2Mγ α,β+3(τ(xk+1 − w)α)− (xk−2 + xk−1)Γ(2)(xk+1) β+1Mγ α,β(τ(xk+1 − w)α) + xk−1xk−2Γ(1)(xk+1) βMγ α,β(τ(xk+1 − w)α) = (k + 1)βhβ+2 [ 2(k + 1)2Mγ α,β+3(τ(xh+ h)α)− (2k − 3)(k + 1)Mγ α,β+2(τ(xh+ h)α) + (k − 2)(k − 1)Mγ α,β+1(τ(xh+ h)α ] . (21) By employing the same established procedure, we successfully obtained. Z2 = (k + 1)βhβ+2 [ 2(k + 1)2Mγ α,β+3(τ(xh+ h)α)− (2k − 2)(k + 1)Mγ α,β+2(τ(xh+ h)α) + (k − 2)(k − 1)Mγ α,β+1(τ(xh+ h)α ] . (22) and ditto to; M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 8 of 20 Z3 = (k + 1)βhβ+2 [ 2(k + 1)2Mγ α,β+3(τ(xh+ h)α)− (2k − 1)(k + 1)Mγ α,β+2(τ(xh+ h)α) + (k − 2)(k − 1)Mγ α,β+1(τ(xh+ h)α ] . (23) Using the established procedure, we successfully obtained the second integral in equa- tion (18), which can be rewritten as: ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx = [ Gk 2h2 ∫ xk+1 0 (xk+1 − x)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−2)dx − Gk−1 h2 ∫ xk+1 0 (xk+1 − x)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−2)dx + Gk−1 2h2 ∫ xk+1 0 (xk+1 − x)β−1Mγ α,β(τ(xk+1 − w)α)(x− xk−1)(x− xk−1) ] dx = Gx 2h2 B1 − Gx h2 B2 + Gx−2 2h2 B3. (24) Z4 = (k)βhβ+2 [ 2k2Mγ α,β+3(τ(xh) α)− (2k − 3)kMγ α,β+2(τ(xh) α) + (k − 2)(k − 1)Mγ α,β+1(τ(xh) α ] . (25) Z5 = (k)βhβ+2 [ 2k2Mγ α,β+3(τ(xh) α)− (2k − 2)kMγ α,β+2(τ(xh) α) + (k − 2)(k − 1)Mγ α,β+1(τ(xh) α ] . (26) and ditto to; Z6 = (k)βhβ+2 [ 2k2Mγ α,β+3(τ(xh) α)− (2k − 2)kMγ α,β+2(τ(xh) α) + (k − 1)(k − 1)Mγ α,β+1(τ(xh) α ] . (27) M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 9 of 20 A new method for solving the Prabhakar fractional differential equation is presented below: p(xk+1) = p(xk) + Gx 2h2 B1 − Gx h2 B2 + Gx−2 2h2 B3 − Gx−2 2h2 B4 + Gx−2 h2 B5 − Gx−2 2h2 B6. (28) In this paper, we seek to address the Prabhakar fractional differential equation, which is defined as follows: Dγ α,β,τp(x) = f(x, p(x)), (29) subject to the initial condition p(0) = p0, where τ ∈ R and α, β, γ ∈ (0, 1). 4. Convergence Analysis In this subsection, we present the convergence analysis for the proposed numerical methods addressing the Prabhakar fractional differential equation. The following theorem summarizes our findings. Theorem 1. Let p(x) represent the solution to the initial value problem outlined in equa- tion (29). The error function associated with this solution is denoted as ℜ(t, β, k). Using the explicit fractional Adams method, we can calculate the numerical solution for the prob- lem described in (29)). This approach involves approximating p(x) based on previous values and using the error function to assess the accuracy of the solution. p(xk+1) = p(xk) + ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx+ ℜ(t, β, k), (30) such that ||ℜ(t, β, k)||∞ < ν. Proof. Based on the (18) and (19), we arrived at; M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 10 of 20 p(xk+1) = p(xk) + ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)µ(v; p(x))dw − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α)µ(v; p(x))dw = p(xk) + ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 + G(k+1)(x) (k + 1)! k∏ i=1 (x− xi)dx ] − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 + G(k+1)(x) (k + 1)! k−1∏ i=1 (x− xi)dx ] = x(xk) + L(x, β, x) + ℜ(x, β, x). (31) The functions L(x, β, x) and ℜ(x, β, x) are defined as follows: L(x, β, x) = ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx − ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α) [ (x− xk−1)(x− xk−2) (xk − xk−1)(xk − xk−2) Gk + (x− xk)(x− xk−2) (xk−1 − xk)(xk−1 − xk−2) Gk−1 + (x− xk−1)(x− xk−1) (xk−2 − xk)(xk−2 − xk−1) Gk−2 ] dx, (32) and ℜ(x, β, x) = ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ G(k+1)(x) (k + 1)! k∏ i=1 (x− xi) ] dx − ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ G(k+1)(x) (k + 1)! k−1∏ i=1 (x− xi) ] dx. (33) It now remains for us to show that the proposed scheme is convergent. We do this by ensuring that ||ℜ(t, β, k)||∞ < ν. M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 11 of 20 ||ℜ(t, β, k)||∞ = ∣∣∣∣∣ ∣∣∣∣∣ ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ G(k+1)(x) (k + 1)! k∏ i=1 (x− xi) ] dx − ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ G(k+1)(x) (k + 1)! k−1∏ i=1 (x− xi) ] dx ∣∣∣∣∣ ∣∣∣∣∣ ∞ < ∣∣∣∣∣ ∣∣∣∣∣ ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α) [ G(k+1)(x) (k + 1)! k∏ i=1 (x− xi) ] dx ∣∣∣∣∣ ∣∣∣∣∣ ∞ + ∣∣∣∣∣ ∣∣∣∣∣ ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α) [ G(k)(x) (k)! k−1∏ i=1 (x− xi) ] dx ∣∣∣∣∣ ∣∣∣∣∣ ∞ < max x∈[0,xk] |G(k+1)(x)| (k + 1)! ∣∣∣∣∣ ∣∣∣∣∣ k∏ i=1 (x− xi) ∣∣∣∣∣ ∣∣∣∣∣ ∞ ∫ xk+1 0 (xk+1 − w)β−1Mγ α,β(τ(xk+1 − w)α)dx + max x∈[0,xk] |G(k)(x)| k! ∣∣∣∣∣ ∣∣∣∣∣ k∏ i=1 (x− xi) ∣∣∣∣∣ ∣∣∣∣∣ ∞ ∫ xk 0 (xk − w)β−1Mγ α,β(τ(xk − w)α)dx = max x∈[0,xk] |G(k+1)(x)| (k + 1)! ∣∣∣∣∣ ∣∣∣∣∣ k∏ i=1 (x− xi) ∣∣∣∣∣ ∣∣∣∣∣ ∞ x β k+1M γ α,β(τ(x k+1 β )) + max x∈[0,xk] |G(k)(x)| (k)! ∣∣∣∣∣ ∣∣∣∣∣ k∏ i=1 (x− xi) ∣∣∣∣∣ ∣∣∣∣∣ ∞ x β kM γ α,β(τ(x K β )) < sup x∈[0,xk+1] ( max x∈[0,xk] |G(k+1)(x)| (k + 1)! , max x∈[0,xk+1] |G(k)(x)| (k)! ) × [ k!hk+1 4 x β k+1M γ α,β+1(τ(x α K+1)) + (k − 1)!hk 4 x β kM γ α,β+1(τ(x α K)) ] < ν. (34) 5. Numerical Application of Tempered Fractional Differential Equations with Respect to Another Function This section provides examples to show how the new explicit fractional Adams method for the Prabhakar derivative works and how accurate it is. We completed all calculations using the software Maple 18. Example 1. We look at the simple nonlinear fractional differential equation defined using the Prabhakar derivative [22]: RDγ α,β,τ,a + p(x) = g(x)− u2(x). (35) Given the initial condition that M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 12 of 20 p(0) = 0, (36) so that g(t) = 2 t− 3 t2 − t3 + ( 2 t ( 1− β + β tβ Γ (β + 2) ) − 3 t2 ( 2 t ( 1− β + 2 β tβ Γ (β + 3) ) + t3 ( 1− β + 6 β tβ Γ (β + 4) )))2 , (37) The exact solution is; p(t) = ( 2 t ( 1− β + β tβ Γ (β + 2) ) − 6 t3 ( 1− β + 2 β tβ Γ (β + 3) ) + t3 ( 1− β + 6 β tβ Γ (β + 4) ))2 . (38) Solution: To check how accurate the explicit fractional Adams method from Theorem 1 is, we use it to estimate the Prabhakar derivative derivative for Example 1. We applied the five-parameter Mittag-Leffler functions as defined: For RDγ α,β,τ,a + p(t) = ∞∑ k=0 Γ (−r + k)wkΓ (p+ 1) tαk−β+p Γ (αk − β + p+ 1)Γ (−r) Γ (k) k (39) where β = 0.99, α = 1, w = 1, r = 1 So that, RDγ α,β,τ,a+p(x) = − 32 t 7 4 315Γ ( 3 4 ) π ( 7 √ t √ 2 ( Γ ( 3 4 ))2 2F2( 1 4 , 3 4 ; 3 2 , 13 4 ; t)− 15 2F2(− 1 4 , 1 4 ; 1 2 , 11 4 ; t)π ) (40) Table 1: The comparison between the Homotopy Perturbation Transform Method (HPTM) and proposed solutions at when β = 0.99 t Exact HPTM solutions Absolute Error 0.01 0.0001 0.0003 0.0000 0.00000E+00 0.02 0.0004 0.0008 0.0005 1.69000E-01 0.03 0.0009 0.0015 0.0012 2.72000E-01 0.04 0.0015 0.0023 0.0021 3.03000E-01 0.05 0.0024 0.0034 0.0031 2.56000E-01 0.06 0.0034 0.0046 0.0044 1.25000E-01 0.07 0.0046 0.0059 0.0058 9.60000E-0 0.08 0.0059 0.0075 0.0073 4.13000E-01 0.09 0.0074 0.0091 0.0090 8.32000E-01 M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 13 of 20 We used the proposed method to get the results shown in Table 1. This Table compares our results with the iteration method (HPTM) in Ref. [22], using a step size of h = 0.01. Our numerical results show that even with just a few terms (N = 2), we achieved good results compared to the method in Ref. [22]. Figure 1 shows a comparison between the exact equation and the approximate solution. This comparison demonstrates that the proposed method is accurate, as the approximate solution closely matches the exact values. Also, Figure 1 shows the results for different values of the parameter β: 0.95, 0.75, and 0.55, as explained in Example 1. Changing the value of β has a significant effect on the process. This demonstrates how adjustments in this parameter can influence the results and confirms that the proposed method is strong enough to work in various situations. Figure 1: The results of the comparison between the approximate and exact solutions for Example 1 Table 2 compares our results with those obtained from the Fractional Homotopy Per- turbation Transform Method (FHPTM) when β = 0.95. Our findings indicate that we achieved favourable results compared to the method described in Ref. [22]. The numerical results demonstrate that our proposed method is an efficient algorithm when compared to the HPTM and FHPTM methods discussed in Ref. [22]. Example 2. We look at nonlinear fractional differential equation defined using the Prabhakar derivative; RDγ α,β,τ,a + p(x) = g(x)− u2(x). (41) Given the initial condition that M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 14 of 20 Figure 2: The results of the comparison between the approximate and exact solutions for Example 1 Table 2: The comparison between the Homotopy Perturbation Transform Method (HPTM), Fractional Homotopy Perturbation Transform Method (FHPTM) and proposed solutions at when β = 0.95 t Exact HPTM FHPTM Approximate solution Absolute Error 0.01 0.0001 0.0011 0.0008 0.000001 1.33339E+00 0.02 0.0004 0.0024 0.0017 0.000006 9.18072E-01 0.03 0.0009 0.0039 0.0027 0.000016 5.18988E-01 0.04 0.0015 0.0056 0.0039 0.000034 1.57268E-01 0.05 0.0024 0.0074 0.0052 0.000062 1.59033E-01 0.06 0.0034 0.0093 0.0065 0.000101 4.25645E-01 0.07 0.0046 0.0115 0.0080 0.000155 6.39892E-01 0.08 0.0059 0.0137 0.0096 0.000225 7.99914E-01 0.09 0.0074 0.0161 0.0113 0.000316 9.04328E-01 p(0) = 0 (42) So that g(x) = (2t− 3t2 + t3)2 + 2t1−β Γ(2− β) − 6t2−β Γ(3− β) + 6t3−β Γ(4− β) . (43) The exact solution is; M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 15 of 20 p(t) = (−t3 − 3t2 + 2t)2. (44) Solution: To check how accurate the explicit fractional Adams method from Theorem 1 is, we use it to estimate the Prabhakar derivative derivative for Example 2. We applied the five-parameter Mittag-Leffler functions as defined, where β = 0.95, α = 1, w = 1, r = 1 So that, RDγ α,β,τ,a+p(x) = ( −t3 − 3 t2 + 2 t )2 +2.054433730 t0.05−5.869810658 t1.05+2.863322272 t2.05 (45) Figure 3: The results of the comparison between the approximate and exact solutions for Example 2 Figure 3 presents a comparison between the exact equation and the approximate solu- tion. This analysis demonstrates the accuracy of the proposed method, as the approximate solution shows a close alignment with the exact values. Table 4 presents a comprehensive comparison between the approximate and exact solutions, with particular emphasis on the associated error rates. A thorough analysis, alongside established methodologies in the field, reveals that the proposed approach not only markedly reduces computational costs but also delivers improved efficiency when compared to solutions within both the approximate and exact frameworks. This reduction in computational expense is vital for applications where resources and time are constrained. Example 3. We look at nonlinear fractional differential equation defined using the M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 16 of 20 Table 3: The comparison between the exact solutions and the absolute error for Example 2 t Exact solutions Absolute error Absolute error 0.1 1.3619489510 0.0285610000 1.33339E+00 0.2 0.9920555733 0.0739840000 9.18072E-01 0.3 0.6107966552 0.0918090000 5.18988E-01 0.4 0.2228042575 0.0655360000 1.57268E-01 0.5 -0.1434077531 0.0156250000 1.59033E-01 0.6 -0.4164291000 0.0092160000 4.25645E-01 0.7 -0.4693227850 0.1705690000 6.39892E-01 0.8 -0.1076897890 0.6922240000 7.99914E-01 0.9 0.9425527730 1.8468810000 9.04328E-01 1.0 3.0479453440 4.0000000000 9.52055E-01 Prabhakar derivative in Ref. [25]; RDγ α,β,τ,a + p(x) = g(x)− u2(x). (46) Given the initial condition that p(0) = 0 (47) So that g(x) = 2t2−β Γ(3− β) + t2. (48) The exact solution is; p(t) = t2. (49) Solution: (48) is solved to be; 1.956603553t1.05 + t2. (50) Figure 4 illustrates a comparative analysis between the exact equation and the ap- proximate solution derived through the proposed method. This analysis demonstrates the accuracy of the method, as there is a strikingly close alignment between the approximate solutions and the exact values, indicating that the approach effectively captures the un- derlying mathematical behaviour. In addition, Figure 5 showcases the results obtained for varying values of the parameter β, specifically 0.95, 0.75, and 0.55, as detailed in Example 3. The variations in β significantly influence the overall process, revealing how changes in this parameter can impact the outcome and further validating the robustness of the proposed method in accommodating different scenarios. M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 17 of 20 6. Conclusion In our study, we present an approach to solve Prabhakar fractional differential equa- tions through the Explicit Fractional Adams Method. The inherent complexity of these equations, due to the multitude of parameters involved, poses significant challenges. How- ever, our innovative numerical scheme effectively addresses these issues by utilizing La- grange interpolation within the Explicit Fractional Adams Method. We provide a conver- gence analysis of our newly developed scheme, supported by several illustrative examples that clearly demonstrate its superior effectiveness compared to existing solutions. Our findings confirm that this method is not only highly efficient but also remarkably straight- forward to implement, revealing its considerable potential for practical applications. Fur- thermore, we strongly advocate for further research to explore additional enhancements to Figure 4: The results of the comparison between the approximate and exact solutions for Example 3 Figure 5: The results of the comparison for different value of β for Example 3 M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 18 of 20 Table 4: The comparison between the exact solutions and the absolute error for Example 3 t Exact solutions Absolute error Absolute Error 0.1 0.210 0.010 2.000E-01 0.2 0.440 0.040 4.000E-01 0.3 0.690 0.090 6.000E-01 0.4 0.960 0.160 8.000E-01 0.5 1.250 0.250 1.000E+00 0.6 1.560 0.360 1.200E+00 0.7 1.890 0.490 1.400E+00 0.8 2.240 0.640 1.600E+00 0.9 2.610 0.810 1.800E+00 1.0 3.000 1.000 2.000E+00 this method, particularly in conjunction with the shifted Legendre polynomials technique. Such advancements will undoubtedly lead to a more robust and comprehensive solution for the tempered fractional problem, with far-reaching implications across diverse fields such as physics, engineering, and finance. Acknowledgements The authors are grateful to the Middle East University, Amman, Jordan, for financial support granted to cover the publication fees of this research article. References [1] P. Cambeses-Franco, R. Rial, & J. M. Ruso, Integrating classical and fractional calculus rheological models in developing hydroxyapatite-enhanced hydrogels, Physics of Fluids, (2024) 36(7). https://doi.org/10.1063/5.0213561. [2] D. Xue & I. Bai, Introduction to Fractional Calculus. In Fractional Calculus: High- Precision Algorithms and Numerical Implementations Singapore: Springer Nature Sin- gapore, (2024) (pp. 1-17). https : //doi.org/10.1007/978− 981− 99− 2070− 91. [3] G. Barbero, L. R. Evangelista, R. S. Zola, E. K.Lenzi, & A. M. Scarfone, A Brief Review of Fractional Calculus as a Tool for Applications in Physics: Adsorption Phenomena and Electrical Impedance in Complex Fluids, Fractal and Fractional, (2024), 8(7), 369. https://doi.org/10.3390/fractalfract8070369. [4] S. A. Elsamani, A. A. Mohmmed, K. M. Kogan, S. Y. M. Salih, & I. A. Ahmed, Frac- tional Calculus Applications in Engineering Insights into Four-Dimensional Chaotic Systems. Journal of Theory, Mathematics and Physics, (2024), 3(5), 75-84: Available online: https://jtmp.innovascience.uz/index.php/journal/article/view/147. [5] M. D. Johansyah, A. Sambas, S. Qureshi, S. Zheng, T. M. Abed-Elhameed, S. Vaidyanathan, & I. M. Sulaiman, Investigation of the hyperchaos and control in the M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 19 of 20 fractional order financial system with profit margin, Partial Differential Equations in Applied Mathematics, (2024), 9, 100612. https://doi.org/10.1016/j.padiff.2023.100612. [6] Al-Ahmad, S., Mamat, M., AlAhmad, R., Sulaiman, I. M., Ghazali, P. L., & Mohamed, M. A. On new properties of differential transform via difference equations. International Journal of Engineering & Technology, (2018), 7(3.28), 321-324. [7] H. M. Fahad & A. Fernandez, Operational calculus for the Riemann–Liouville frac- tional derivative with respect to a function and its applications, Fractional Calculus and Applied Analysis, (2021), 24(2), 518-540. DOI: doi.org/10.1515/fca-2021-0023. [8] H. M. Fahad & A.Fernandez, Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differen- tial equations, Applied Mathematics and Computation, (2021), 409, 126400. https://doi.org/10.1016/j.amc.2021.126400. [9] A. E. Owoyemi, I. M. Sulaiman, P. Kumar, V. Govindaraj & M. Mamat, Some novel mathematical analysis on the fractional-order 2019-nCoV dynamical model, Mathematical Methods in the Applied Sciences, (2023). 46(4), 4466-4474. https://doi.org/10.1002/mma.8772. [10] I. M. Sulaiman, A. E. Owoyemi, M. A. A Nawi, S. S. Muhammad, U. R. Muhammad, A. F. Jameel & Nawawi, M. K. M., Stability and Bifurcation Analysis of Rössler System in Fractional Order, In Advances in Intelligent Manufacturing and Mechatronics: Se- lected Articles from the Innovative Manufacturing, Mechatronics and Materials Forum (iM3F 2022), Pahang, Malaysia (pp. 239-250). Singapore: Springer Nature Singapore. https : //doi.org/10.1007/978− 981− 19− 8703− 820. [11] Aal, M. A.& Arafah, A. (2025). An Efficient Symmetric Operational Matrix Method for Solving Tempered Fractional Differential Equations with Respect to Another Func- tion. International Journal of Neutrosophic Science (IJNS), 26(1). [12] lsamir, H., Noorani, M. S., Shatanawi, W., Aydi, H., Akhadkulov, H., Qawaqneh, H., & Alanazi, K. (2019). Fixed Point Results in Metric-like Spaces via Sigma-simulation Functions. European Journal of pure and applied mathematics, 12(1), 88-100. [13] Jameel, A. F., Saaban, A., Ahadkulov, H., & Alipiah, F. M. (2017, September). Approximate solution fuzzy pantograph equation by using homotopy perturbation method. In Journal of Physics: Conference Series (Vol. 890, No. 1, p. 012023). IOP Publishing. [14] Akhadkulov, H., Saaban, A. B., Alipiah, M. F., & Jameel, A. F. (2018). On appli- cations of multidimensional fixed point theorems. Nonlinear Functional Analysis and Applications, 585-593. [15] S. Deepa, A. Ganesh, V. Ibrahimov, S. S. Santra, V. Govindan, K. M.Khedher & S. Noeiaghdam, Fractional fourier transform to stability analysis of fractional differen- tial equations with prabhakar derivatives, Azerbaijan Journal of Mathematics, (2022), 12(2), 1-23: Available online: https://www.azjm.org/. [16] V. A. Kim & R. I. Parovik, Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation, Fractal and Fractional, (2022) 6(5), 274. https://doi.org/10.3390/fractalfract6050274. [17] Y. M. Chu, M. Inc, M. S. Hashemi & S. Eshaghi, Analytical treatment of regularized M. Abdel Aal et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6631 20 of 20 Prabhakar fractional differential equations by invariant subspaces., Computational and Applied Mathematics, (2022), 41(6), 271. https://doi.org/10.1007/s40314-022-01977- 1. [18] T. E. Namarneh & M. Al-Refai, Analytical study to systems of fractional differen- tial equations with Prabhakar derivative, IFAC-PapersOnLine, (2024), 58(12), 155-160. https://doi.org/10.1016/j.ifacol.2024.08.182. [19] E. T. Karimov & A. Hasanov, On a boundary-value problem in a bounded do- main for a time-fractional diffusion equation with the Prabhakar fractional deriva- tive, Bulletin of the Karaganda university. Mathematics series, (2023), 111(3), 39-46, https://doi.org/10.31489/2023m3/39-46. [20] F. Laxmi, S. Jain, P. Agarwal & G. V. Milovanović, Numerical calculation of the extension of k-beta function and some new extensions by using two parameter k- Mittag-Leffler function, Applied Mathematics and Computation, (2024), 479, 128857. https://doi.org/10.1016/j.amc.2024.128857. [21] M. A. Özarslan, & A. Fernandez, On a five-parameter Mittag-Leffler function and the corresponding bivariate fractional operators Fractal and Fractional, (2021), 5(2), 45. https://doi.org/10.3390/fractalfract5020045. [22] M. H. Derakhshan & A. Aminataei, Comparison of homotopy perturbation trans- form method and fractional Adams–Bashforth method for the Caputo–Prabhakar nonlinear fractional differential equations, Iranian Journal of Numerical Anal- ysis and Optimization, (2020), 10(2), 63-85: Available online: https : //ijnao.um.ac.ir/article25399.html. [23] R. Garra & r. Garrappa, The Prabhakar or three parameter Mittag–Leffler function: Theory and application, Communications in Nonlinear Science and Numerical Simula- tion, (2018), 56, 314-329. https://doi.org/10.1016/j.cnsns.2017.08.018. [24] A. Saadatmandi & M. Dehghan, A new operational matrix for solving fractional-order differential equations, Computers and mathematics with applications,(2010), 59(3), 1326-1336, https://doi.org/10.1016/j.camwa.2009.07.006. [25] S. Qureshi & P. Kumar, Using Shehu integral transform to solve fractional order Ca- puto type initial value problems, Journal of Applied Mathematics and Computational Mechanics, (2019), 18(2), 75-83. DOI: 10.17512/jamcm.2019.2.07.