Int. J. Anal. Appl. (2022), 20:35 On the Solutions of the Second Kind Nonlinear Volterra-Fredholm Integral Equations via Homotopy Analysis Method A. S. Rahby1,∗, M. A. Abdou2, G. A. Mosa1 1Department of Mathematics, Faculty of Science, Benha University, 13518, Egypt 2Department of Mathematics, Faculty of Education, Alexandria University, 21544, Egypt ∗Corresponding author: a.s.rahby@fsc.bu.edu.eg Abstract. In this paper, we discuss the existence and uniqueness of the solution of the second kind nonlinear Volterra-Fredholm integral equations (NV-FIEs) which appear in mathematical modeling of many phenomena, using Picard’s method. In addition, we use Banach fixed point theorem to show the solvability of the first kind NV-FIEs. Moreover, we utilize the homotopy analysis method (HAM) to approximate the solution and the convergence of the method is investigated. Finally, some examples are presented and the numerical results are discussed to show the validity of the theoretical results. 1. Introduction In the application of physical mathematics and engineering, the second kind of NV-FIEs are often arisen [1–8]. Therefore, there exist great efforts to approximate the solution of this kind of NV-FIEs. Yousefi and Razzaghi [9] presented a numerical method based upon Legendre wavelet approximations for solving the NV-FIEs. Cui and Du [10] obtained the representation of the exact solution for the NV-FIEs in the reproducing kernel space and the exact solution was given by the form of series. The approximate solutions of the NV-FIEs were pesented using modified decomposition method by Bildik and Inc [11]. Ghasemi et al. [12] presented homotopy perturbation method for solving NV-FIEs. In ad- dition, rationalized Haar functions are developed to approximate the solution of the NV-F-Hammerstein IEs by Ordokhani and Razzaghi [13]. He’s variational iteration method was used by Yousefi [14] to approximate the solution of a type of NV-FIEs. Hashemizadeh et al. [15] introduced an approximation Received: Oct. 30, 2021. 2010 Mathematics Subject Classification. 45G10, 46B07, 65R20. Key words and phrases. nonlinear Fredholm-Volterra integral equations; Picard’s method; homotopy analysis method and error analysis. https://doi.org/10.28924/2291-8639-20-2022-35 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-35 2 Int. J. Anal. Appl. (2022), 20:35 method based on hybrid Legendre and Block-Pulse functions for solving the NV-FIEs. A computa- tional technique based on the composite collocation method was presented by Marzban et al. [16] for the solution of the NV-F-Hammerstein IEs. Moreover, Maleknejad et al. [17] utilized a method to solve NV-F-Hammerstein IEs in terms of Bernstein polynomials. Parand and Rad [18] proposed the collocation method based on radial basis functions to approximate the solution of NV-F-Hammerstein IEs. A numerical method based on hybrid of block-pulse functions and Taylor series is proposed by Mirzaee and Hoseini [19] to approximate the solution of NV-FIEs. Chen and Jiang [20] developed a simple and effective method for solving NV-FIEs based on Lagrange interpolation functions. The approximate solution of the NV-F-Hammerstein IEs is obtained by Gouyandeh et al. [21] using the Tau-Collocation method. The present paper shall utilize HAM for solving the NFVIEs of the first and second kind. Foremost, in Section 2, we discuss the solvability of the second kind NF-VIEs using Picard’s method. Moreover, in Section 3, Banach fixed point theorem is used to discuss the existence and uniqueness of the solution of the first kind NF-VIEs. In addition, the basic idea of HAM and how to utilize HAM for the NF-VIEs of the second and first kind are presented in Section 4. Finally, we present the numerical results in Section 5. 2. Existence and uniqueness of the second kind NV-FIEs Consider the following second kind NV-FIEs of the form µu(t) = f (t) + λ ∫ b a K1(t,s)N1(u(s))ds + λ ∫ t 0 K2(t,s)N2(u(s))ds. (2.1) Now, we shall discuss the solvability of Eq.(2.1) under the following assumptions (1) The function f (t) is continuous in the space C[0,T ], such that ‖f (t)‖C[0,T] = max t∈[0,T] |f (t)| ≤ P1 and µ ∈R−{0}. (2) The Kernels K1(t,s) and K2(t,s) are continuous in C[0,T ] and satisfy |K1(t,s)| ≤ P2 and |K2(t,τ)| ≤ P3, ∀t,τ ∈ [0,T ], and 0 ≤ τ ≤ t ≤ T < 1. (3) The nonlinear functions Ni (u(t)), i = 1, 2 satisfy i. the Lipschitz condition |Ni (u2(t)) −Ni (u1(t))| ≤Li |u2(t) −u1(t)|, ii. the following inequality ‖Ni (u(t))‖≤ σi ‖u(t)‖, where P1 : P3,L1,L2,σ1 and σ2 are positive constants. Theorem 2.1. If assumptions (1), (2) and (3.i) are satisfied and |λ| < |µ| P2L1 + P3L2T , (2.2) then Eq.(2.1) has a unique solution u(t) in the space C[0,T ]. Int. J. Anal. Appl. (2022), 20:35 3 Proof. Foremost, using Picard’s method, the solution of Eq.(2.1) can be expressed as a sequence of functions {un(t)} as n →∞ based on um(t) = f (t) + λ ∫ b a K1(t,s)N1(um−1(s))ds + λ ∫ t 0 K2(t,s)N2(um−1(s))ds, (2.3) with u0(t) = f (t). Let vm(t) = um(t) −um−1(t), and v0(t) = f (t). (2.4) with un(t) = n∑ i=0 vi (t), n = 1, 2, 3, ... , (2.5) where vm(t),m = 1, 2, ..., are continuous functions. Now we shall prove that the series ∞∑ i=0 vi (t) is uniformly convergent. Using Eqs.(2.3), (2.4) and norm properties yield |µ|‖um(t) −um−1(t)‖≤|λ| ∥∥∥∥ ∫ b a K1(t,s) [N1(um−1(s)) −N1(um−2(s))] ds ∥∥∥∥ + |λ| ∥∥∥∥ ∫ t 0 K2(t,s) [N2(um−1(s)) −N2(um−2(s))] ds ∥∥∥∥ , (2.6) at n = 1, from (2.6) and using the given assumptions, we get |µ|‖v1(t)‖≤ |λ| ∥∥∥∥ ∫ b a |K1(t,s)|L1|v0|ds ∥∥∥∥ + |λ| ∥∥∥∥ ∫ t 0 |K2(t,s)|L2|v0|ds ∥∥∥∥ . (2.7) Hence, we obtain ‖v1(t)‖≤ |λ| |µ| (P2L1 + P3L2T )P1, (2.8) where max t∈[0,T] |t| = T. In addition, at n = 2, we have |µ|‖v2(t)‖≤ |λ|‖ ∫ b a |K1(t,s)|L1|v1|ds + |λ|‖ ∫ t 0 |K2(t,s)|L2|v1|ds‖, (2.9) which leads to ‖v2(t)‖≤ ( |λ| |µ| (P2L1 + P3L2T ) )2 P1. (2.10) Subsequently, the mathematical induction is applied to obtain ‖vm(t)‖≤ γm1 P1, γ1 = (P2L1 + P3L2T )|λ| |µ| . (2.11) Note that, the series ∞∑ i=0 vi (t) is uniformly convergent if and only if the series ∞∑ m=0 γm1 P3 is convergent. Therefore, since |λ| < |µ| P2L1+P3L2T , we get γ1 < 1 and this implies that the series ∞∑ m=0 γm1 P3 is convergent. Thus, for n →∞, we get u(t) = ∞∑ i=0 vi (t) (2.12) 4 Int. J. Anal. Appl. (2022), 20:35 represents a solution of Eq (2.1). Now, to show the solution is unique, we assume that there exists another continuous solution ũ(t) of Eq.(2.1). So, we get ‖u(t) − ũ(t)‖≤ ∥∥∥∥λ ∫ b a K1(t,s)(N1(u(s) −N1(ũ(s)))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t a K2(t,s)(N2(u(s) −N1(ũ(s)))ds ∥∥∥∥ . (2.13) Note that under the given conditions, inequality (2.13) yields ‖u(t) − ũ(t)‖≤ γ1‖u(t) − ũ(t)‖. (2.14) If ‖u(t)− ũ(t)‖ 6= 0, then (2.14) yields γ1 ≥ 1 which is a contradiction. Therefore, ‖u(t)− ũ(t)‖ = 0 and it is implied that u(t) = ũ(t) which means the solution is unique. � 3. Existence and uniqueness of the first kind NV-FIEs If we have µ = 0 in Eq (2.1), we get the first kind NV-FIEs f (t) + λ ∫ b a K1(t,s)N1(u(s))ds + λ ∫ t 0 K2(t,s)N2(u(s))ds = 0. (3.1) Now, we shall use Banach fixed point theorem which is used in case of failure of Picard’s method at µ = 0. So, Eq.(3.1) will be first expressed in its integral operator form Uu = f + Uu, (3.2) where Uu = U1u + U2u, U1u = λ ∫ b a K1(t,s)N1(u(s))ds, and U2u = λ ∫ t 0 K2(t,s)N2(u(s))ds. For the normality of the integral operator, we use (3.2) with the help of the given assumptions and norm properties to obtain ‖Uu‖≤ ∥∥∥∥λ ∫ b a K1(t,s)N1(u(s))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t 0 K2(t,s)N2(u(s))ds ∥∥∥∥ ≤ |λ|P2σ1‖u(s)‖ + |λ|P3σ2T‖u(s)‖ ≤ γ2‖u(s)‖, γ2 = |λ|(P2σ1 + P3σ2T ). (3.3) If |λ| < 1 P2σ1‖+P3σ2T , we get γ2 < 1 which means U is a contraction operator and this implies that the integral operator U has a normality which leads directly after using the condition (1) to the normality of the operator U. Int. J. Anal. Appl. (2022), 20:35 5 For the continuity of the integral operator, if we assume that the two functions u1(t) and u2(t) ∈ C[0,T ] with the help of the norm properties under the given conditions, then we get ‖Uu1 −Uu2‖ = ‖Uu1 −Uu2‖ ≤ ∥∥∥∥λ ∫ b a K1(t,s)(N1(u1(s)) −N1(u2(s)))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t 0 K2(t,s)(N2(u1(s)) −N2(u2(s)))ds ∥∥∥∥ ≤ |λ|(P2L1 + P3L2T )‖u1(s) −u2(s)‖ ≤ γ3‖u1(s) −u2(s)‖, γ3 = (P2L1 + P3L2T )|λ|. (3.4) If |λ| < 1 P2L1+P3L2T , we get γ2 < 1 which means U is a contraction operator and leads to the continuity of the integral operator U in the space C[0,T ]. Using Banach fixed point theorem, U has a unique fixed point that means the NV-FIEs (3.1) of the first kind has a unique solution. 4. Homotopy analysis method for NV-FIEs We shall introduce the basic idea of HAM [22,23] for solving the operator equation N(u(t)) = 0, t ∈ [0,T ] (4.1) where N denotes the nonlinear operator, and u(t) is an unknown function. Foremost, we define the homotopy operator H, H(Φ,p) = (1 −p)(Φ(t; p) −u0(t)) −phN(Φ(t; p)), (4.2) where p ∈ [0, 1] is the embedding parameter, h 6= 0 denotes the convergence control parameter, u0(t) describes the initial approximation of the solution of (4.1). Considering H(Φ,p) = 0, we get the so-called zero-order deformation equation (1 −p)(Φ(t; p) −u0(t)) = phN(Φ(t; p)). (4.3) For p = 0, we have Φ(t; 0) −u0(t) = 0 which implies that Φ(t; 0) = u0(t), whereas for p = 1, we have N(Φ(t; 1)) = 0 that means Φ(t; 1) = u(t), where u(t) is the solution of (4.1). In this way, the variation of parameter p : 0 → 1 corresponds with the change of problem from the trivial problem to the original one (and with the change of solution from u0(t) → u(t)). Expanding Φ(x; p) into the Maclaurin series with respect to p, we get Φ(t; p) = Φ(t; 0) + ∞∑ m=1 1 m! ∂mΦ(t; p) ∂pm ∣∣∣∣∣ p=0 pm, (4.4) By distinguishing vm(t) = ∞∑ m=1 1 m! ∂mΦ(t; p) ∂pm ∣∣∣∣∣ p=0 , m = 1, 2, 3, ... . (4.5) 6 Int. J. Anal. Appl. (2022), 20:35 Eq.(4.4) becomes Φ(t; p) = v0(t) + ∞∑ m=1 vm(t)p m. (4.6) If the above series is convergent at p = 1, we obtain u(t) = ∞∑ m=0 vm(t). (4.7) To determine function vm(t), we differentiate the both sides of Eq.(4.3) m times with respect to p, next we divide the received result by m! and we substitute p = 0. Herein, we get the so-called m th-order deformation equation (m > 0) vm(t) −χmvm−1(t) = hRm (v̄m−1,t) (4.8) where v̄m−1 = {v0(t),v1(t), . . . ,vm−1(t)}, χm = { 0 m ≤ 1 1 m > 1 (4.9) and Rm (v̄m−1,t) = 1 (m− 1)! ( ∂m−1 ∂pm−1 N ( ∞∑ i=0 vi (t)p i ))∣∣∣∣∣ p=0 . (4.10) Since, we can not determine the sum of series in (4.7), we shall accept the partial sum of this series u(t) ≈ un(t) = n∑ m=0 vm(t) (4.11) as the approximate solution of considered equation. Secondly, we introduce HAM for NF-VIE (2.1) and operator N can be defined as N(v(t)) =µv(t) − f (t) −λ ∫ b a K1(t,s)N1(v(s))ds −λ ∫ t 0 K2(t,s)N2(v(s))ds. (4.12) Applying the HAM, we get the following formula for functions vm(t) vm(t) = χmvm−1(t) + hRm (v̄m−1,t) , (4.13) where χm and Rm are defined by (4.9) and (4.10), respectively. Using definitions of the respective operators, we obtain v1(t) = hR1 (v̄0,t) = h ( µv0(t) − f (t) −λ ∫ b a K1(t,s)N1(v0(s))ds −λ ∫ t 0 K2(t,s)N2(v0(s))ds ) (4.14) Int. J. Anal. Appl. (2022), 20:35 7 and for m ≥ 2, we get vm(t) =(1 + hµ)vm−1(t) − λ h (m− 1)! ∫ b a K1(t,s) [ ∂m−1 ∂pm−1 N2 ( ∞∑ i=0 vi (s)p i )] p=0 ds − λ h (m− 1)! ∫ t a K2(t,s) [ ∂m−1 ∂pm−1 N2 ( ∞∑ i=0 vi (s)p i )] p=0 ds. (4.15) In case of µ = 0, Eq.(4.14) becomes v1(t) = hR1 (v̄0,t) = h ( −f (t) −λ ∫ b a K1(t,s)N1(v0(s))ds −λ ∫ t 0 K2(t,s)N2(v0(s))ds ) (4.16) and Eq.(4.15) gives vm(t) =vm−1(t) − λ h (m− 1)! ∫ b a K1(t,s) [ ∂m−1 ∂pm−1 N2 ( ∞∑ i=0 vi (s)p i )] p=0 ds − λ h (m− 1)! ∫ t a K2(t,s) [ ∂m−1 ∂pm−1 N2 ( ∞∑ i=0 vi (s)p i )] p=0 ds. (4.17) Theorem 4.1. Suppose that the nonlinear operators N1 and N2 satisfies Lipschitz condition (3.i). If the series ∑+∞ m=0vm(t) converges to u(t), where vm(t) is governed by Eq.(4.8) under the definitions (4.9) and (4.10), then u(t) will be the exact solution of the NF-VIE (2.1). Proof. Firstly, we define Him (t) = 1 m!   ∂m ∂pm Ni   ∞∑ j=0 vj(t)p j     ∣∣∣∣∣∣ p=0 , i = 1, 2. (4.18) From (4.9), we have n∑ m=1 [vm(t) −χmvm−1(t)] =v1(t) + [v2(t) −v1(t)] + [v3(t) −v2(t)] + · · · + [vn(t) −vn−1(t)] = vn(t). (4.19) From the convergence of ∑+∞ m=0vm(t), lim m→∞ vm(t) = 0 , t ∈ [0,T ]. (4.20) Using Eq.(4.20), Eq.(4.19) becomes ∞∑ m=1 [vm(t) −χmvm−1(t)] = lim n→∞ vn(t) = 0. (4.21) 8 Int. J. Anal. Appl. (2022), 20:35 Eq.(4.21) and Eq.(4.13) yield h ∞∑ m=1 Rm (v̄m−1,t) = ∞∑ m=1 vm(t) −χmvm−1(t) = 0. (4.22) Since h 6= 0, Eq.(4.22) gives ∞∑ m=1 Rm (v̄m−1,t) = 0. (4.23) Now, Eq.(4.12) and definitions (4.5) and (4.10) give 0 = +∞∑ m=1 [Rm (~vm−1,t)] = +∞∑ m=1  µvm−1(t)−(1−χm)f (t)−λ∫ b a K1(t,s) ∂m−1 (m−1)!∂pm−1 N1 [ k=+∞∑ k=0 vk(s)p k ]∣∣∣∣∣ p=0 ds −λ ∫ t a K2(t,s) ∂m−1 (m− 1)!∂pm−1 N2 [ k=+∞∑ k=0 vk(s)p k ]∣∣∣∣∣ p=0 ds   = +∞∑ m=1 [ µvm−1(t)−(1−χm)f (t)−λ ∫ b a K1(t,s)H1m−1(s)ds−λ ∫ t a K2(t,s)H2m−1(s)ds ] = +∞∑ m=1 µvm−1(t) −f (t) −λ ∫ b a K1(t,s) +∞∑ m=1 H1m−1(s)ds −λ ∫ t a K2(t,s) +∞∑ m=1 H2m−1(t)ds (4.24) Since the nonlinear operators N1 and N2 are contraction; therefore, if the series ∑+∞ m=0 vm(t) converges to u(t), then the series ∑+∞ m=0 H1m−1(t) and ∑+∞ m=0 H2m−1(t) will converge to N1(u(t)) and N2(u(t)), respectively [24]. So, Eq.(4.24) becomes µu(t) = f (t) + λ ∫ b a K1(t,s)N1(u(s))ds + λ ∫ t 0 K2(t,s)N2(u(s))ds. (4.25) Hence, u(t) is the exact solution of NF-VIE (2.1). In case of µ = 0, Eq.(4.24) gives f (t) + λ ∫ b a K1(t,s)N1(u(s))ds + λ ∫ t 0 K2(t,s)N2(u(s))ds = 0. (4.26) which indicates that u(t) is the exact solution of NF-VIE (3.1). � Theorem 4.2. (Convergence Theorem) Assume that h is properly chosen for which there exists 0 < α < 1 so that ‖vk+1‖6 α‖vk‖ ,∀k > k0, for some k0 ∈N, then un (t,h) in (4.11) converges as n −→ +∞. Proof. Define the sequence Um as U0 = v0 U1 = v0 + v1 ... Un = v0 + v1 + · · · + vm (4.27) Int. J. Anal. Appl. (2022), 20:35 9 Now, we show that Um is a Cauchy sequence in the space C[0,T ]. Consider ‖Um+1 −Um‖ = ‖vm+1‖6 α‖vm‖6 α2‖vm−1‖6 · · ·6 αm−k0+1‖vk0‖ (4.28) For every l,m ∈N,m > l > k0, we have ‖Um−Ul‖ = ‖(Um −Um−1)+(Um−1 −Um−2) + · · · + (Ul+1 −Ul)‖ 6 ‖(Um −Um−1)‖ + ‖(Um−1 −Um−2)‖ + · · · + ‖(Ul+1 −Ul)‖ 6 αm−k0 ‖vk0‖ + α m−k0−1‖vk0‖ + · · · + α m−k0+1‖vk0‖ = 1 −αm−l 1 −α αl−k0+1‖vk0‖ . (4.29) Since 0 < α < 1, we get lim m,l→∞ ‖Um −Ul‖ = 0. (4.30) Therefore, Um is a Cauchy sequence in the space C[0,T ] and Un = un(t,h) converges as n →∞ and the proof is complete. � Theorem 4.3. If assumptions of Theorem 4.2 are satisfied, n ∈ N and n ≥ k0, then we obtain the estimation of error of the approximate solution defined by ‖u(t) −un(t)‖≤ αn+1−k0 1 −α ‖vk0‖ . (4.31) Proof. . Let n ∈ N and n ≥ k0, we get ‖u(t) −un(t)‖ = sup t∈[0,T] ∣∣∣∣∣u(t) − n∑ m=0 vm(x,t) ∣∣∣∣∣ ≤ sup t∈[0,T] ( ∞∑ m=n+1 |vm(t)| ) ≤ ∞∑ m=n+1 sup t∈[0,T] (|vm(x,t)|) ≤ ∞∑ m=n+1 αm−k0 ‖vk0‖ = αn+1−k0 1 −α ‖vk0‖ . (4.32) � 5. Illustrating examples In this section, some examples will be given to investigate the efficiency and accuracy of the proposed method 10 Int. J. Anal. Appl. (2022), 20:35 Example 5.1. Consider the following strongly NV-FIE u(t) = f (t) + ∫ 1 0 (t2 − s)u2(s)ds + ∫ t 0 ts2u3(s)ds, (5.1) where f (t) = e−tt2− 6(e2−7)t2+109 8e2 + e −3tt 2187 (3t(3t(t(3t(3t(t(3t(3t + 8) + 56) + 112) + 560) + 2240) + 2240) + 4480) + 4480) − 4480t 2187 + 15 8 and (the exact solution u(t) = t2e−t). In this example, Fig.(1) shows the behaviour of error using HAM at n = 10 and h = −1.9, t ∈ [0, 0.8]. Also, Fig.(2) presents the valid region of h at n = 10. 0.2 0.4 0.6 0.8 t 1.×10-16 2.×10-16 3.×10-16 4.×10-16 en(t) Figure 1. The error behaviour at n =10,h =−0.1 using HAM un(0.8,h) un ′ (0.8,h) u n ′′(0.8,h) -40 -20 20 40 h -4 -2 2 Figure 2. The h-curves at n =10 Int. J. Anal. Appl. (2022), 20:35 11 Example 5.2. Consider the following strongly NV-FIE [20] u(t) = f (t) + ∫ 1 0 (t − s)u5(s)ds + ∫ t 0 (t + s)u4(s)ds, (5.2) where f (t) = 77107623−70t(792t5+324787) 151200 + 5t−6 25e5 + 1 16 e−4t(8t + 1) + 5(10t−13) 32e4 + 10(17t−26) 27e3 + 5(19t−42) 4e2 + 5(65t−326) e + 4 27 e−3t(9t(2t+1)+2)+ 1 4 e−2t(6t(t(4t+5)+4)+9)+e−t(4t(t(t(2t+7)+18)+30)+97) and (the exact solution u(t) = t + et). In this example, Fig.(3) displays the behaviour of error using HAM at n = 10 and h = −1.9. In addition, Fig.(4) explains the valid region of h at n = 10. Moreover, Table 1 presents the maximum error emax = max i |u(ti ) −un(ti )| ∀ti ∈ [0, 0.9] for every n = 2, 5, 8, 12, 16, 20 and a comparison with results in Ref. [20]. 0.2 0.4 0.6 0.8 t 2.×10-14 4.×10-14 6.×10-14 8.×10-14 1.×10-13 en(t) Figure 3. The error behaviour at n =10,h =−1.9 using HAM un(0.9,h) un ′ (0.9,h) u n ′′(0.9,h) -30 -20 -10 10 20 30 h -4 -2 2 4 Figure 4. The h-curves at n =10 12 Int. J. Anal. Appl. (2022), 20:35 Table 1. The maximum error emax for different values of n with corresponding h for example (5.2) n emax in [20] emax of HAM at h = −0.1 2 2.94 × 10−2 4.3876 × 10−12 5 5.95 × 10−4 9.45667 × 10−13 8 1.27E × 10−4 1.3152 × 10−13 12 3.93 × 10−5 1.65706 × 10−13 16 1.60 × 10−5 1.96266 × 10−13 20 7.66 × 10−6 2.02851 × 10−13 Example 5.3. Consider the following strongly NV-FIE f (t) = ∫ 1 0 s2tu2(s)ds + ∫ t 0 st2u3(s)ds, (5.3) where f (t) = −1 4 3t3 cosh(t)+ 1 12 t3 cosh(3t)+ 3 4 t2 sinh(t)− 1 36 t2 sinh(3t)− t 6 + 3 8 t sinh(2)−1 4 t cosh(2) and (the exact solution u(t) = sinh(t)). In this example, Fig.(5) shows the behaviour of error using HAM at at n = 20, h = 0.1 and t ∈ [0, 0.8]. In addition, Fig. (6) investigates the valid region of h at n = 20. 0.2 0.4 0.6 0.8 t 5.×10-17 1.×10-16 1.5×10-16 2.×10-16 en(t) Figure 5. The error behaviour at n =20,h =0.1 using HAM Int. J. Anal. Appl. (2022), 20:35 13 un(0.8,h) un ′ (0.8,h) u n ′′(0.8,h) -40 -20 20 40 h -0.5 0.5 1.0 1.5 Figure 6. The h-curves at n =20 6. Conclusion In this paper, we used Picard’s method to prove the existence and uniqueness of the solution of the second kind NV-FIEs which has many application in mathematical physics. Moreover, we utilized Banach fixed point theorem to discuss the solvability of the first kind NV-FIEs. In addition, we applied the HAM to approximate the solution and discussed the convergence analysis. Furthermore, we investigated illustrative examples to indicate the validity and accurately of the presented method showing the error behaviour. Based on the results, we observed that that HAM is an effective method for solving the first and second kind NF-VIEs. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] G.A. Mosa, M.A. Abdou, A.S. 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Introduction 2. Existence and uniqueness of the second kind NV-FIEs 3. Existence and uniqueness of the first kind NV-FIEs 4. Homotopy analysis method for NV-FIEs 5. Illustrating examples 6. Conclusion References