EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5924 ISSN 1307-5543 – ejpam.com Published by New York Business Global Computational Methods, Existence and Uniqueness for Solving 2-D Fractional Nonlinear Fredholm Integro-Differential Equation Abeer M. Al-Bugami1,∗, Nuha A. Alharbi1, Amr M. S. Mahdy1 1 Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia Abstract. In this paper, we have investigated the two-dimensional fractional nonlinear Fredholm integro-differential equation (TDFNFI-DE). These equations are used in many fields, including particle dynamics in physics, biology, and control theory. We have developed an effective combined approach in our work that uses Homotopy analysis (HAM) and the Adomian decomposition method (ADM) to solve fractional integro-differential equations in two dimensions. Numerical experiment results show the effectiveness of our recently created technique. We prove the existence and uniqueness of the exact solution. To illustrate the numerical effectiveness of the suggested approach, we provide a number of numerical examples. The suggested approach is accurate and applicable to various nonlinear issues in science, according to theoretical and numerical results. 2020 Mathematics Subject Classifications: 45J05, 45G10, 26A33, 35R11, 65R20, 65L10 Key Words and Phrases: TDFNFI-DE, existence and uniqueness, ADM, HAM 1. Introduction Integral equations are used in many different fields, such as continuous mechanical engineering, potential concept, geophysics, electricity and magnetism, kinetic theory of gasses, hereditary phenomena in biology and physics, quantum mechanics, radiation, op- timisation, which includes optimal system design, theory of communication, economics of mathematics, genetics of populations, medical procedures, computational problems of ra- diative equilibrium, the particle transport problems of the fields of a and reactor principle, acoustics, steady state heat conduction, and radiative heat transfer troubles. Fredholm integral equation is one of the most important integral equations. Singh and Pippal [1] utilized the Shehu transform combined with the Adomian polynomial to solve effectively nonlinear fractional differential equations. Additionally, Ali et al. [2] presented a numeri- cal solution for fractional integro-differential equations with a linear functional argument, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5924 Email addresses: abeer101aa@yahoo.com (A. Al-Bugami), noha1231@hotmail.com (N. Alharbi), amr mahdy85@yahoo.com (A. Mahdy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 2 of 14 employing Chebyshev serious for increased precision. Oyedepo et al. [3] contributed a competitional algorithm specifically designed for fractional Fredholm integro-differential equations, providing a robot and efficacious method for tackling these complex problems. Moreover, Toma and Postavaru [4] proposed a numerical method for solving fractional Fredholm integro-differential equations with a focus on enhancing accuracy. Abdou et al. is performed a comprehensive analysis of the convergence into that Adomian’s method when applied to nonlinear equations in [5]. Abbaoui and Cherruault [6] also performed a comprehensive analysis of the convergence into that Adomian’s method when applied to nonlinear equations. Behiry et al. [7] developed a new algorithm for the decomposi- tion solution of nonlinear differential equations, further advancing the method. Tair et al. [8] explored two numerical techniques for solving the linear integro-differential Fredholm equation with a weakly singular kernel, while Al-Bugami [9] focused on two-dimensional Fredholm integro-differential equations with singular kernels, providing accurate numeri- cal method for their solution. Abbaszadeh et al. [10] employed Legendre wavelets to solve fractional Fredholm integro-differential equations, while Rawashdeh et al. [11] provided a detailed convergence analysis for the fractional decomposition method as applied to nonlinear fractional Fredholm integro-differential equations. Li and Pang [12] ] expanded the application of the ADM to nonlinear systems. In addition, Momani and Noor [13] developed numerical techniques to solve fractional integro-differential equations of fourth order. In the realm of two-dimensional nonlinear Fredholm integro-differential equations, Al-Bugami [14] presented effective numerical solutions. Additionally, Hasan [15] applied the (ADM) to nonlinear systems involving fractional Fredholm integro-differential equa- tions. We will use the HAM and ADM for dealing with the (TDFNFI-DE) in this study. The TDFNFI-DE form will evaluate. Dαu(s, t) = f(s, t) + λ ∫ b a ∫ d c K(s, t, τ, ξ)γ(τ, ξ,Dαu(τ, ξ))dτdξ, (1) where u(s, t) is an unknown function , f(s, t),K(s, t, τ, ξ) are given functions, and λ is parameter. Also 0 < α ≤ 1 and (s, t) ∈ Ω = [a, b]× [c, d]. 2. The Existence and Uniqueness Theorem 1. Let (M,d) be a metric space with M̸= ∅. Suppose M ̸= ∅ is complete and let I : M → M be contraction mapping. Then, I has exactly one fixed point. Furthermore, we can express the equation in (1) in the form of an integral operator as follows: T̃Dαu(s, t) = f(s, t) + TDαu(s, t), (2) where TDαu(s, t) = λ ∫ b a ∫ d c K(s, t, τ, ξ)γ(τ, ξ,Dαu(τ, ξ))dτdξ. (3) Also assume the following conditions: A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 3 of 14 (i) K(s, t, τ, ξ) satisfies |K(s, t, τ, ξ)| ≤ κ, where κ is a constant. (ii) f(s, t) is continuous in C[a, b]× C[c, d], with its norm defined as: ||f(s, t)|| = [∫ b a ∫ d c |f(s, t)|2dsdt ] 1 2 = v, where v is a constant. (iii) The continuous function γ(s, t,Dαu(s, t)) satisfies the following conditions for some constant g > g1 > m, g > m: i. [ ∫ b a ∫ d c |γ(s, t,Dαu(s, t))|2dsdt ] 1 2 ≤ g1||Dαu(s, t)||. ii. |γ(s, t,Dαu1(s, t))− γ(s, t,Dαu2(s, t))| ≤ M(s, t) · |Dαu1(s, t)−Dαu2(s, t)|, where ∥M(s, t)∥ = M and M is a constant. (iv) The unknown function Dαu(s, t) behaves similarly to the given function f(s, t) in C[a, b]× C[c, d], with its norm defined as: ∥Dαu(s, t)∥ = [∫ b a ∫ d c |Dαu(s, t)|2dsdt ] 1 2 . Theorem 2. If the first three conditions are satisfied, then eq (1) has a unique solution in C[a, b]× C[c, d]. Lemma 1. The space C[a, b]×C[c, d] is mapped into itself by the operator T̃ defined by (2) under the conditions (1)–(3-i). Proof. Using equation (2) and (3), we have: ∥T̃Dαu(s, t)∥ ≤ ∥f(s, t)∥+ |λ| ∥∥∥∥∫ b a ∫ d c |K(s, t, τ, ξ)||γ(τ, ξ,Dαu(τ, ξ))|dτdξ ∥∥∥∥, (4) by applying condition (2), it follows that: ∥T̃Dαu(s, t)∥ ≤ v + |λ| · [ |K(s, t, τ, ξ)| ][ ∫ b a ∫ d c |γ(s, t,Dαu(s, t))|2dsdt ] 1 2 , (5) using conditions (1) and (3-i), we can estimate: ∥T̃Dαu(s, t)∥ ≤ v + δ∥Dαu(s, t)∥, when δ = |λ|κg. Hence, this inequality (2) confirms that the operator T̃ is bounded, with: ∥T̃Dαu(s, t)∥ ≤ δ∥Dαu(s, t)∥. A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 4 of 14 Lemma 2. The operator T̃ is a contraction on the Banach space C[a, b] × C[c, d], if conditions (1) and (3-ii) are satisfied. Proof. Suppose we have two unknown function Dαu1(s, t) and Dαu2(s, t) in C[a, b]× C[c, d], we can apply equations (2) and (3) to obtain: ∥(T̃Dαu1 − T̃Dαu2)(s, t)∥ ≤ |λ| ∥∥∥∥∫ b a ∫ d c |K(s, t, τ, ξ)|· |γ(τ, ξ,Dαu1(τ, ξ))− γ(τ, ξ,Dαu2(τ, ξ))|dτdξ ∥∥∥∥, by using condition (3-ii), we get: ∥(T̃Dαu1 − T̃Dαu2)(s, t)∥ ≤ |λ| · [ |K(s, t, τ, ξ)| ] ·[ ∫ b a ∫ d c M2(s, t)|Dαu1(τ, ξ)−Dαu1(τ, ξ)|2dτdξ ] 1 2 . Thus, we conclude: ∥(T̃Dαu1 − T̃Dαu2)(s, t)∥ ≤ δ∥Dαu1(s, t)−Dαu2(s, t)∥. 3. ADM In this section, we will address the TDFNFI-DE using ADM. Let’s consider (1), in witch f(s, t) is a function that is bounded, and for all (s, t) belongs to the set Ω = [a, b] × [c, d]. The kernel |K(s, t, τ, ξ)| ≤ κ. The nonlinear term γ(τ, ξ,Dαu(τ, ξ)) is Lipchitz continuous and satisfies the following condition: |γ(Dαu)− γ(Dαũ)| ≤ L|Dαu−Dαũ|. We define the space C[a, b] × C[c, d], continuous functions on the rectangle [a, b] × [c, d], with the associated distance function d∗(Dαũ, Dαu) defined by: d∗(Dαũ, Dαu) = max (s,t)∈Ω |Dαũ(s, t)−Dαu(s, t)|. (6) The unknown function Dαu(s, t) is assumed to have the series form: Dαu(s, t) = ∞∑ n=0 Dαun(s, t). (7) The nonlinear term γ(τ, ξ,Dαu(τ, ξ)) in (1) is similarly decomposed into an infinite series: γ(τ, ξ,Dαu(s, t)) = ∞∑ n=0 Anun(s, t), (8) A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 5 of 14 where the traditional form of the Adomian polynomials An is [16][17] given by: An = 1 n! [ dn dλn γ( ∞∑ i=0 λiDαui)]λ=0. (9) Another formulation of the Adomian polynomials is expressed as: An = γ(Sn)− n−1∑ i=0 Ai, (10) where Sn is defined by: Sn = n∑ i=0 Dαui(s, t). (11) By applying the Adomian decomposition method to equation (1), we obtain the solution in series form: Dαu(s, t) = ∞∑ n=0 Dαun(s, t), (12) where Dαu0(s, t) = f(s, t), Dαui(s, t) = λ ∫ b a ∫ d c K(s, t, τ, ξ)Ai−1dτdξ, i ≥ 1. (13) 4. HAM In this section, we aim to solve the nonlinear (FNT-DFIDE) using the Homotopy Analysis Method (HAM). This approach is novel because, to the best of our knowledge, previous studies have primarily applied HAM to solve two-dimensional fractional integro- differential equations. Consider (1), where K(s, t, τ, ξ) and f(s, t) are known functions, and γ(τ, ξ,Dαu(τ, ξ)) is a known nonlinear function of Dαu. To describe the method, we define the nonlinear operator N : N [Dαu] = Dαu(x, t)− f(x, t)− ∫ b a ∫ d c K(x, t, τ, ξ)γ(τ, ξ,Dαu(τ, ξ))dτdξ = 0. (14) Let Dαu0(s, t) denote an initial guess for the exact solution u(s, t), and choose a non-zero auxiliary parameter h ̸= 0, an auxiliary function H(s, t), and linear operator L, such that L[e(s, t)] = 0 whenever e(s, t) = 0. For p ∈ [0, 1], we construct the following homotopy: (1− p)[ϕ(s, t; p)−Dαu0(x, t)]− phH(s, t)N [ϕ(s, t; p)] = H[ϕ(s, t; p);Dαu0(s, t), H(s, t), h, p]. (15) A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 6 of 14 it is important to emphasize that we have significant flexibility in choosing the initial guess Dαu0(s, t), the auxiliary linear operator L, the non-zero parameter h, and the auxiliary function H(s, t). Additionally, let H̃ be a secondary auxiliary function that ensures the homotopy satisfies the condition: H̃[ϕ(s, t; p);Dαu0(s, t), H(s, t), h, p] = 0. (16) Thus, the homotopy equation can be written as: (1− p)[ϕ(s, t; p)−Dαu0(x, t)] = phH(s, t)N [ϕ(s, t; p)], (17) when p = 0, from equation (17), we get: ϕ(s, t; p) = Dαu0(s, t). (18) Furthermore, when p = 1 and h ̸= 0, H(s, t) ̸= 0, equation (17) becomes: ϕ(s, t; 1) = Dαu(s, t). (19) If we observe the value of p from 0 to 1m the embedding parameter gradually increases for ϕ(s, t; p) in equations (1) and (19), such that it begins from the initial approximation and reaches the exact solution. We refer to this change in homotopy deformation, and we can be expressed in the following form: ϕ(s, t; p) = Dαu0(x, t) + ∞∑ r=1 Dαur(s, t)p r. (20) Here, ur(s, t) is given by: Dαur(s, t) = 1 r! ∂ϕ(s, t; p) ∂pr ∣∣∣∣ p=0 (21) Based on these assumptions, we can write the exact solution as: Dαu(s, t) = ϕ(s, t; 1) = Dαu0(s, t) + ∞∑ r=1 Dαur(s, t)p (22) Thus, the sequence of approximations for Dαu(s, t) can be expressed as: Dαūn(s, t) = {Dαu0(s, t), D αu1(s, t), ...., D αun(s, t)} (23) The zero-order deformation equation (21), in accordance with equation (17), can be used to construct the governing equation of Dαum(s, t). The called mth-order deformation equation gets by differentiating the zero-order deformation equation (17) r times with regard to p, dividing the result by r!, and setting r = 0. Besed on equation (21), the governing equation for Dαum(s, t) can be derived from the zero-order deformation equation (17). By differentiating equation (17) r times with A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 7 of 14 respect to r, then dividing by r! and setting p = 0, we obtain the mth-order deformation equation as follows: L[Dαur(s, t)− σrD αur−1(s, t)] = hH(s, t)J (Dαūr−1(s, t)) Dαur(0, 0) = 0 (24) where J (Dαūr−1(s, t)) = 1 (r − 1)! ∂r−1N [ϕ(s, t; p)] ∂pr−1 ∣∣∣∣ p=0 (25) and the condition σr is defined as: σr =  0 (r ≤ 0) 1 (r > 0) (26) 5. Numerical Applications Application 1. Consider the following Caputo fractional linear Fredholm integro-differential equation D0.5u(s, t) = 2.256758334 s √ t− 0.6790610904 + ∫ 1 0 ∫ 1 0 √ τξ D0.5u(τ, ξ)dτdξ (27) The exact solution u(s, t) = 2.256758334 s √ t . Table 1: Numerical results and absolute error values using (ADM) and (HAM), N = 10, at fractional order α = 0.5 in Application 1. s t uExact ADM HAM uADM ErrorADM uHAM ErrorHAM 0.0 0.0 0 0 0 4× 10−8 4× 10−8 0.1 0.1 0.07136496464 0.07136496464 0 0.07136500464 4.000× 10−8 0.2 0.2 0.2018506018 0.2018506017 1× 10−10 0.2018506417 3.99× 10−8 0.3 0.3 0.3708232338 0.3708232338 0 0.3708232738 4.00× 10−8 0.4 0.4 0.5709197172 0.5709197171 1× 10−10 0.5709197571 3.99× 10−8 0.5 0.5 0.797884561 0.7978845608 2× 10−10 0.7978846008 3.98× 10−8 0.6 0.6 1.048846493 1.048846493 0 1.048846533 4.0× 10−8 0.7 0.7 1.321697642 1.321697641 1× 10−9 1.321697681 3.9× 10−8 0.8 0.8 1.614804814 1.614804814 0 1.614804854 4.0× 10−8 0.9 0.9 1.926854045 1.926854045 0 1.926854085 4.0× 10−8 1.0 1.0 2.256758334 2.256758334 0 2.256758374 4.0× 10−8 A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 8 of 14 Figure 1: Exact Solution of (ADM) for α = 0.5 in Application 1 Figure 2: Approximate Solution of (ADM) for α = 0.5 in Application 1 Figure 3: Exact Solution of (HAM) for α = 0.5 in Application 1 Figure 4: Approximate Solution of (HAM) for α = 0.5 in Application 1 Application 2. Consider the following Caputo fractional linear Fredholm integro-differential equation D0.8u(s, t) = 2.178248842 st1/5 − 0.8324154416 + ∫ 1 0 ∫ 1 0 √ τξ D0.8u(τ, ξ)dτdξ (28) The exact solution is u(s, t) = 2.178248842 st1/5. A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 9 of 14 Table 2: Numerical results and absolute error values using (ADM) and (HAM), N = 10, at fractional order α = 0.8 in Application 2. s t uExact ADM HAM uADM ErrorADM uHAM ErrorHAM 0.0 0.0 0 0 0 3× 10−8 3× 10−8 0.1 0.1 0.1374382105 0.1374382105 0 0.1374382405 3.00× 10−8 0.2 0.2 0.3157500926 0.3157500925 1× 10−10 0.3157501225 2.99× 10−8 0.3 0.3 0.5136330933 0.5136330933 0 0.5136331233 3.00× 10−8 0.4 0.4 0.725403224 0.7254032241 1× 10−10 0.7254032541 3.01× 10−8 0.5 0.5 0.948137878 0.9481378781 1× 10−10 0.9481379081 3.01× 10−8 0.6 0.6 1.180018979 1.180018979 0 1.180019009 3.0× 10−8 0.7 0.7 1.419793357 1.419793357 0 1.419793387 3.0× 10−8 0.8 0.8 1.666538980 1.666538980 0 1.666539010 3.0× 10−8 0.9 0.9 1.919545908 1.919545908 0 1.919545938 3.0× 10−8 1.0 1.0 2.178248842 2.178248842 0 2.178248872 3.0× 10−8 Figure 5: Exact Solution of (ADM) for α = 0.8 in Application 2 Figure 6: Approximate Solution of (ADM) for α = 0.8 in Application 2 A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 10 of 14 Figure 7: Exact Solution of (HAM) for α = 0.8 in Application 2 Figure 8: Approximate Solution of (HAM) for α = 0.8 in Application 2 Application 3. Consider the following Caputo fractional nonlinear Fredholm integro- differential equation D0.5u(s, t) = 2.256758334 s √ t− 1.235153476 + ∫ 1 0 ∫ 1 0 √ τξ (D0.5u(τ, ξ))2 dτdξ (29) The exact solution is u(s, t) = 2.256758334 s √ t. s t uExact ADM HAM uADM ErrorADM uHAM ErrorHAM 0.0 0.0 0 0 0 −2× 10−8 2× 10−8 0.1 0.1 0.07136496464 0.07136496464 0 0.07136494464 2.000× 10−8 0.2 0.2 0.2018506018 0.2018506017 1× 10−10 0.2018505817 2.01× 10−8 0.3 0.3 0.3708232338 0.3708232338 0 0.3708232138 2.00× 10−8 0.4 0.4 0.5709197172 0.5709197171 1× 10−10 0.5709196971 2.01× 10−8 0.5 0.5 0.797884561 0.7978845608 2× 10−10 0.7978845408 2.02× 10−8 0.6 0.6 1.048846493 1.048846493 0 1.048846473 2.0× 10−8 0.7 0.7 1.321697642 1.321697641 1× 10−9 1.321697621 2.1× 10−8 0.8 0.8 1.614804814 1.614804814 0 1.614804794 2.0× 10−8 0.9 0.9 1.926854045 1.926854045 0 1.926854025 2.0× 10−8 1.0 1.0 2.256758334 2.256758334 0 2.256758314 2.0× 10−8 Table 3: Numerical results and absolute error values using (ADM) and (HAM), N = 10, at fractional order α = 0.5 in Application 3. A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 11 of 14 Figure 9: Exact Solution of (ADM) for α = 0.5 in Application 3 Figure 10: Approximate Solution of (ADM) for α = 0.5 in Application 3 Figure 11: Exact Solution of (HAM) for α = 0.5 in Application 3 Figure 12: Approximate Solution of (HAM) for α = 0.5 in Application 3 Application 4. Consider the following Caputo fractional nonlinear Fredholm integro- differential equation D0.8u(s, t) = 2.178248842 st1/5 − 1.631364025 + ∫ 1 0 ∫ 1 0 √ τξ (D0.8u(τ, ξ))2 dτdξ (30) The exact solution is u(s, t) = 2.178248842 s t1/5. A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 12 of 14 Table 4: Numerical results and absolute error values using (ADM) and (HAM), N = 10, at fractional order α = 0.8 in Application 4. s t uExact ADM HAM uADM ErrorADM uHAM ErrorHAM 0.0 0.0 0 0 0 −5.5× 10−8 5.5× 10−8 0.1 0.1 0.1374382105 0.1374382105 0 0.1374381555 5.50× 10−8 0.2 0.2 0.3157500926 0.3157500925 1× 10−10 0.3157500375 5.51× 10−8 0.3 0.3 0.5136330933 0.5136330933 0 0.5136330383 5.50× 10−8 0.4 0.4 0.725403224 0.7254032241 1× 10−10 0.7254031691 5.49× 10−8 0.5 0.5 0.948137878 0.9481378781 1× 10−10 0.9481378231 5.49× 10−8 0.6 0.6 1.180018979 1.180018979 0 1.180018924 5.5× 10−8 0.7 0.7 1.419793357 1.419793357 0 1.419793302 5.5× 10−8 0.8 0.8 1.66653898 1.66653898 0 1.666538925 5.5× 10−8 0.9 0.9 1.919545908 1.919545908 0 1.919545853 5.5× 10−8 1.0 1.0 2.178248842 2.178248842 0 2.178248787 5.5× 10−8 Figure 13: Exact Solution of (ADM) for α = 0.8 in Application 4 Figure 14: Approximate Solution of (ADM) for α = 0.8 in Application 4 A. M. Al-Bugami, N. A. Alharbi, A. M. S. Mahdy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5924 13 of 14 Figure 15: Exact Solution of (HAM) for α = 0.8 in Application 4 Figure 16: Approximate Solution of (HAM) for α = 0.8 in Application 4 6. Conclusion From the previous study, we concluded the following: as s, t are increasing in [0, 1]×[0, 1] the error for (ADM) is increasing and for (HAM) is remain constant at 108. In nonlinear case the using 0.8 is a smaller than by 0.5. The error using 0.8 is it smaller than in (ADM) and (HAM),(ADM) using 0.8 is more accurate than using 0.5. At N = 10 the error is decreasing in linear and nonlinear case.When using the same fractional order (ADM) and (HAM), exactly the same error, but using fractional order 0.8, (ADM) convergences faster than (HAM). The error decreases when the fractional order of (ADM) increases and the error for (HAM) remains the same. 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