Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

Abstract

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

1. Introduction

The integro-differential equations stem from the mathematical modeling of many complex real-life problems. Many scientific phenomena have been formulated using integro-differential equations [1, 2]. Solving nonlinear integro-differential equation is much more difficult than linear one analytically. So different types of numerical methods have been used to obtain an efficient approximation solution [3, 4]. In 1992 Liao [5] proposed the homotopy analysis method (HAM) concept in topology for solving nonlinear differential equations. Liao [6, 7] found that the convergence of series solutions of nonlinear equations cannot be guaranteed by the early HAM. Further, Liao [6] introduced a nonzero auxiliary parameter to solve this limitation. Unlike the special cases of HAM such as Lyapunove’s artificial small parameter method [8], Adomian decomposition method [9–12], and the 𝛿-expansion method [13], this method need not a small perturbation parameter. In the HAM the perturbation techniques [14] need not be converted a nonlinear problem to infinite number of linear problems. The homotopy analysis method is applicable for solving problems having strong nonlinearity [15], even if they do not have any small or large parameters, so it is more powerful than traditional perturbation methods.

The convergence region and the rate of approximation in series can been adjusted by this method. Also it can give us freedom to use different base function to approximate a non linear problem. The convergence of HAM for solving Volterra-Fredholm integro-differential equations is presented in [16].

In 2010, Motsa et al. [17] suggested the so-called spectral homotopy analysis method (SHAM) using the Chebyshev pseudospectral method to solve the linear high-order deformation equations. Since the SHAM combines the HAM with the numerical techniques, it provides us larger freedom to choose auxiliary linear operators. Thus, one can choose more complicated auxiliary linear operators in the frame of the SHAM. In theory, any continuous function in a bounded interval can be best approximated using Chebyshev polynomial. So, the SHAM provides larger freedom to choose the auxiliary linear operator 𝐿 and initial guess. Further, it is easy to employ the optimal convergence-control parameter in the frame of the SHAM. Thus, the SHAM has great potential to solve more complicated nonlinear problems in science and engineering, although further modifications in theory and more applications are needed. Chebyshev polynomial is considered a kind of special function. There are many other special functions such as Hermite polynomial, Legendre polynomial, Airy function, Bessel function, Riemann zeta function, and hypergeometric functions. Since the HAM provides us extremely large freedom to choose the auxiliary linear operator 𝐿 and the initial guess, it should be possible to develop a generalized spectral HAM which can use a proper special function for a given nonlinear problem. The spectral homotopy analysis method has been used for solving partial and ordinary differential equations [18–20]. Spectral homotopy analysis method and its convergency for solving a class of optimal control problems are presented in [21]. Motsa et al. [17–19] found that the spectral homotopy analysis method is more efficient than the homotopy analysis method as it does not depend on the rule of solution expression and the rule of ergodicity. This method is more flexible than homotopy analysis method, since it allows for a wider range of linear and nonlinear operators, and one is not restricted to use the method of higher-order differential mapping for solving boundary value problems in bounded domains, unlike the homotopy analysis method. The range of admissible ℎ values is much wider in spectral homotopy analysis method than in homotopy analysis method. The main restriction of HAM in solving integral equations is to choose the best initial guess, as the series solution is convergent. In SHAM the initial approximation is taken to be the solution of the nonhomogeneous linear part of the given equation. In 2012, Pashazadeh Atabakan et al. solved linear Volterra and Fredholm integro-differential equations using spectral homotopy analysis method; see [22].

In this paper, we apply spectral homotopy analysis method (SHAM) to solve higher-order nonlinear Fredholm type of integro-differential equations. Fredholm integro-differential equation is given by ²_𝑗=0𝑎_𝑗(𝑥)𝑦^(𝑗)(𝑥)=𝑓(𝑥)+𝜇¹₋₁𝑘(𝑥,𝑡)𝑦(𝑡)^𝑟𝑦()(1)𝑑𝑡,−1=𝑦=0, (1) where 𝜇 is constant value, 𝑓(𝑥), 𝑘(𝑥,𝑡), [𝑦(𝑡)]^𝑟, and 𝑎_𝑗(𝑥)𝑟≥1 are functions that have suitable derivatives on interval −1≤𝑡≤𝑥≤1 and 𝑎₂(𝑥)≠0.

The paper is organized in the following way. Section 2 includes a brief introduction in homotopy analysis method. Spectral homotopy analysis method for solving nonlinear Fredholm integral equations is presented in Section 3. The existence and uniqueness of the solution and convergence of the proposed method are proved in Section 4. In Section 5, numerical examples are presented. In Section 6, concluding remarks are given.

2. Homotopy Analysis Solution

In this section, we give a brief introduction to HAM. We consider the following differential equation in a general form as follows: 𝑁𝑦𝜂=0, (2) where 𝑁 is nonlinear operator, 𝜂 denotes independent variables, and 𝑦(𝜂) is an unknown function, respectively. For simplicity we disregard all initial and all boundary conditions which can be dealt in similar way. The so-called zero-order deformation equation was constructed by Liao as follows: 1−𝑝𝐿𝜓𝜂;𝑝−𝑦₀𝜂=𝑝ℎ𝐻𝜂𝑁𝜓𝜂;𝑝, (3) where 𝑝∈[0,1] is the embedding parameter, ℎ is a nonzero convergence-parameter, 𝐻(𝜂) is an auxiliary function, 𝑦₀(𝜂) is called an initial guess of 𝑦(𝜂), and 𝜓(𝜂;𝑝) is an unknown function. In addition, 𝐿 is an auxiliary linear operator, and 𝑁 is nonlinear operator as follows: 𝐿𝜓𝑥;𝑝=𝑎_𝑘(𝑥)𝜕²𝜓𝑥;𝑝𝜕𝑥² (4) with the property 𝐿(∑²_𝑗=0𝑐_𝑗𝑡^𝑗)=0, where 𝑐_𝑗, are constants and 𝑁𝜓𝑥;𝑝=²_𝑗=0𝑎_𝑗(𝑥)𝜕^𝑗𝜓𝑥;𝑝𝜕𝑥^𝑗−𝑓(𝑥)−𝜇¹₋₁𝑘(𝑥,𝑡)𝜓^𝑟(𝑡)𝑑𝑡 (5) is a nonlinear operator. Obviously, when 𝑝=0 and 𝑝=1, it holds 𝜓(𝜂;0)=𝑦₀(𝜂) and 𝜓(𝜂;1)=𝑦(𝜂). In this way, as 𝑝 increase from 0 to 1, 𝜓(𝜂;𝑝) alter from initial guess 𝑦₀(𝜂) to the solution 𝑦(𝜂), and 𝜓(𝜂;𝑝) is expanded in Taylor series with respect to 𝑝 as follows: 𝜓𝜂;𝑝=𝑦₀𝜂+^+∞_𝑚=1𝑦_𝑚𝜂𝑝^𝑚, (6) where 𝑦_𝑚𝜂=𝐷_𝑚𝜓,𝐷𝜂;𝑝_𝑚1𝜓=𝜕𝑚!^𝑚𝜓𝜕𝑝^𝑚_𝑝=0. (7) The series (6) converges at 𝑝=1 if the auxiliary linear operator, the initial guess, the convergence-parameter, and the auxiliary function are properly selected as follows: 𝜓𝜂=𝑦₀𝜂+^+∞_𝑚=1𝑦_𝑚𝜂. (8) The admissible and valid values of the convergence-parameter ℎ are found from the horizontal portion of the ℎ-curves. Liao proved that 𝑦(𝜂) is one of the solutions of original nonlinear equation. As 𝐻(𝜂)=1, so (3) becomes 1−𝑝𝐿𝜓𝜂;𝑝−𝑦₀𝜂=𝑝ℎ𝑁𝜓𝜂;𝑝. (9) Define the vector 𝑦_𝑚={𝑦₀(𝜂),𝑦₁(𝜂),…,𝑦_𝑚(𝜂)}. Operating on both side of (9) with 𝐷_𝑚, we have the so called 𝑚th-order deformation equation as follows: 𝐿𝑦_𝑚𝜂−𝜒_𝑚𝑦_𝑚−1𝜂=ℎ𝐻𝜂𝑅_𝑚𝑦_𝑚−1𝜂, (10) where 𝑅_𝑚𝑦_𝑚−1=1()!𝜕𝑚−1^𝑚−1𝑁𝜓𝜂;𝑝𝜕𝑝^𝑚−1_𝑝=0,𝜒_𝑚=0,𝑚≤11,otherwise, (11)𝑦_𝑚(𝜂) for 𝑚≥0 that is governed by the linear equation (10) can be solved by symbolic computation software such as MAPLE, MATLAB, and similar CAS.

3. Spectral-Homotopy Analysis Solution

Consider the non linear Fredholm integro-differential equation: ²_𝑗=0𝑎_𝑗(𝑥)𝑦^(𝑗)(𝑥)=𝑓(𝑥)+𝜇¹₋₁𝑘(𝑥,𝑡)𝑦(𝑡)^𝑟𝑦()(1)𝑑𝑡,−1=𝑦=0. (12) We begin by defining the following linear operator: 𝐿𝜓𝑥;𝑝=²_𝑗=0𝑎_𝑗(𝑥)𝜕^𝑗𝜓𝑥;𝑝𝜕𝑥^𝑗, (13) where 𝑝∈[0,1] is the embedding parameter and 𝜓(𝑥;𝑝) is an unknown function. The zeroth-order deformation equation is given by 1−𝑝𝐿𝜓𝜂;𝑝−𝑦₀𝜂=𝑝ℎ𝑁𝜓𝜂;𝑝−𝑓𝜂, (14) where ℎ is the nonzero convergence controlling auxiliary parameter and 𝑁 is a nonlinear operator given by 𝑁𝜓𝜂;𝑝=²_𝑗=0𝑎_𝑗𝜂𝜕^𝑗𝜓𝜂;𝑝𝜕𝜂^𝑗−𝑓𝜂−𝜇¹₋₁𝑘𝜂,𝑡𝜓^𝑟(𝑡)𝑑𝑡. (15) Differentiating (14) 𝑚 times with respect to the embedding parameter 𝑝, setting 𝑝=0, and finally dividing them by 𝑚!, we have the so called 𝑚th-order deformation equation 𝐿𝑦_𝑚𝜂−𝜒_𝑚𝑦_𝑚−1𝜂=ℎ𝑅_𝑚, (16) subject to boundary conditions 𝑦_𝑚(−1)=𝑦_𝑚(1)=0, (17) where 𝑅_𝑚𝜂=²_𝑗=0𝑎_𝑗𝜂𝜕^𝑗𝜓𝜂;𝑝𝜕𝜂^𝑗−𝑓𝜂1−𝜒_𝑚−𝜇¹₋₁𝑘𝜂,𝑡𝜓^𝑟(𝑡)𝑑𝑡. (18) The initial approximation that is used in the higher-order equation (18) is obtained on solving the following equation: ²_𝑗=0𝑎_𝑗(𝑥)𝑦₀^(𝑗)(𝑥)=𝑓(𝑥) (19) subject to boundary conditions 𝑦₀(−1)=𝑦₀(1)=0, (20) where we use the Chebyshev pseudospectral method to solve (19)-(20).

We first approximate 𝑦₀(𝜂) by a truncated series of Chebyshev polynomial of the following form: 𝑦₀𝜂≈𝑦^𝑁₀𝜂_𝑗=^𝑁_𝑘=0𝑦_𝑘𝑇_𝑘𝜂_𝑗,𝑗=0,…,𝑁, (21) where 𝑇_𝑘 is the 𝑘th Chebyshev polynomials, 𝑦_𝑘 are coefficients and Gauss-Lobatto collocation points 𝜂₀,𝜂₁,…,𝜂_𝑁 which are the extrema of the 𝑁th-order Chebyshev polynomial defined by 𝜂_𝑗=cos𝜋𝑗𝑁. (22)

Derivatives of the functions 𝑦₀(𝜂) at the collocation points are represented as 𝑑^𝑠𝑦₀𝜂_𝑘𝑑𝜂^𝑠=^𝑁_𝑗=0𝐷^𝑠_𝑘𝑗𝑦₀𝜂_𝑗,𝑘=0,…,𝑁, (23) where 𝑠 is the order of differentiation and 𝐷 is the Chebyshev spectral differentiation matrix. Following [23], we express the entries of the differentiation matrix 𝐷 as 𝐷_𝑘𝑗=−12𝑐_𝑘𝑐_𝑗×(−1)^𝑘+𝑗𝜋𝜋𝐷sin𝑗+𝑘/2𝑁sin𝑗−𝑘/2𝑁,𝑗≠𝑘,_𝑘𝑗=−12𝑥_𝑘sin²()𝐷𝜋𝑘/𝑁,𝑘≠0,𝑁,𝑘=𝑗,₀₀=−𝐷_𝑁𝑁=2𝑁²+16. (24)

Substituting (21)–(23) into (19) will result in 𝐀𝐘₀=𝐅 (25) subject to the boundary conditions 𝑦₀𝜂₀=𝑦₀𝜂_𝑁=0, (26) where 𝐀=²_𝑗=0𝐚_𝑗𝐃^𝑗,𝐘₀=𝑦₀𝜂₀,𝑦₀𝜂₁,…,𝑦₀𝜂_𝑁^𝑇,𝐅=𝑓𝜂₀,𝑓𝜂₁,…,𝑓𝜂_𝑁^𝑇,𝐚_𝑟=diag𝑎_𝑟𝜂₀,𝑎_𝑟𝜂₁,…,𝑎_𝑟𝜂_𝑁. (27) The values of 𝑦₀(𝜂_𝑖), 𝑖=0,…,𝑁 are determined from the equation 𝐘₀=𝐀⁻¹𝐅, (28) which is the initial approximation for the SHAM solution of the governing equation (12). Apply the Chebyshev pseudospectral transformation on (16)–(18) to obtain the following result: 𝐀𝐘_𝑚=𝜒_𝑚+ℎ𝐀𝐘_𝑚−1−ℎ𝐒_𝑚−1−1−𝜒_𝑚𝐅, (29) subject to the boundary conditions 𝑦_𝑚𝜂₀=𝑦_𝑚𝜂_𝑁=0, (30) where 𝐀 and 𝐅 were defined in and 𝐘_𝑚=𝑦_𝑚𝜂₀,𝑦_𝑚𝜂₁,…,𝑦_𝑚𝜂_𝑁^𝑇,𝐬_𝑚=¹₋₁𝑘(𝜏,𝑡)𝐘_𝑚^𝑟𝑑𝑡. (31) To implement the boundary condition (30) we delete the first and the last rows of 𝐒_𝑚−1, 𝐅 and the first and the last rows and columns of 𝐀. Finally this recursive formula can be written as follows: 𝐘_𝑚=𝜒_𝑚+ℎ𝐘_𝑚−1−ℎ𝐀⁻¹𝐒_𝑚−1−1−𝜒_𝑚𝐅_𝑚−1, (32) with starting from the initial approximation we can obtain higher-order approximation 𝐘_𝑚 for 𝑚≥1 recursively. To compute the integral in (32) we use the Clenshaw-Curtis quadrature formula as follows: 𝐒_𝑚𝜂=¹₋₁𝑘𝜂,𝑡,𝐘_𝑚𝑑𝑡=^𝑁_𝑗=0𝑤_𝑗𝑘𝜂,𝜂_𝑗,𝐘_𝑚, (33) where the nodes 𝜂_𝑗 are given by (22) and the weights 𝑤_𝑗 are given by 𝑤₀=𝑤_𝑁=1𝑁²1,𝑁odd,𝑁²𝑤−1,𝑁even,_𝑙=2𝑁𝛾_𝑙1−^⌊𝑁/2⌋_𝑘=12𝛾_2𝑘4𝑘²−1cos2𝑘𝑙𝜋𝑁,𝑙=1,…,𝑁−1, (34)(35) where 𝛾₀=𝛾_𝑁=2 and 𝛾_𝑙=1, for 𝑙=1,…,𝑁−1. 𝐘 is a column vector of the elements of the vector 𝐘 that is computed as follows: 𝐘_𝑚=^𝑚_𝑛1=0𝑦_𝑚−𝑛1^𝑛1_𝑛2=0𝑦_{𝑛1−𝑛2}⋯^𝑛𝑟−2_{𝑛𝑟−1=0}𝑦_{𝑛𝑟−2−𝑛𝑟−1}𝑦_𝑛𝑟−1, (36) where 𝑚, 𝑟≥0 are positive integers [24].

Regarding to accuracy, the stability, and the error of previous quadrature formula at the Gauss-Lobatto points we refer the reader to [25].

4. Convergence Analysis

Following the authors in [7, 16, 26], we present the convergence of spectral homotopy analysis method for solving Fredholm integro-differential equations.

In view of (13) and (27), (12) can be written as follows: 𝐀𝐘=𝐅+𝜇¹₋₁𝑘(𝑥,𝑡)𝐆(𝐘)𝑑𝑡, (37) where 𝐘, 𝐅, and 𝐆(𝐘) are vector functions.

We obtain 𝐘=𝐀⁻¹𝐅+𝜇¹₋₁𝑘(𝑥,𝑡)𝐀⁻¹𝐆(𝐘)𝑑𝑡. (38) By substituting 𝐅=𝐀⁻¹𝐅 and 𝐆(𝐘)=𝐀⁻¹𝐆(𝐘) in (38), we obtain 𝐘=𝐅+𝜇¹₋₁𝑘(𝑥,𝑡)𝐆(𝐘)𝑑𝑡. (39) In (39), we assume that 𝐅 is bounded for all 𝑡 in 𝐶=[−1,1] and |𝑘(𝑥,𝑡)|≤𝑀. (40) Also, we suppose that the non linear term 𝐆(𝐘) is Lipschitz continuous with 𝐆(𝐘)−𝐆𝐘^∗≤𝐿𝐘−𝐘^∗. (41) If we set 𝛼=2𝜇𝐿𝑀, then the following can be proved by using the previous assumptions.

Theorem 1. The nonlinear Fredholm integro-differential equation in (32) has a unique solution whenever 0<𝛼<1.

Proof. Let 𝐘 and 𝐘^∗ be two different solutions of (39), then 𝐘−𝐘^∗=𝜇¹₋₁𝑘(𝑥,𝑡)𝐆(𝐘)−𝐆𝐘^∗𝑑𝑡≤𝜇¹₋₁|𝑘(𝑥,𝑡)|𝐆(𝐘)−𝐆𝐘^∗𝑑𝑡≤2𝜇𝐿𝑀𝐘−𝐘^∗. (42) So we get (1−𝛼)‖𝐘−𝐘^∗‖≤0. Since 0<𝛼<1, so ‖𝐘−𝐘^∗‖=0, therefore 𝐘=𝐘^∗, and this completes the proof.

Theorem 2. If the series solution 𝐘=∑^∞_𝑚=0𝐘_𝑚 obtained from (32) is convergent, then it converges to the exact solution of the problem (39).

Proof. We assume 𝐘=^∞_𝑚=0𝐘_𝑚,𝐕(𝐘)=^∞_𝑚=0𝐆𝐘_𝑚, (43) where lim_𝑚→∞𝐘_𝑚=0. We can write ^𝑛_𝑚=1𝐘_𝑚−𝜒_𝑚𝐘_𝑚−1=𝐘₁+𝐘₂−𝐘₁+⋯+𝐘_𝑛−𝐘_𝑛−1=𝐘_𝑛. (44) Hence, from (44), ^∞_𝑚=1𝐘_𝑚−𝜒_𝑚𝐘_𝑚−1=0, (45) so using (45) and the definition of the linear operator 𝐿, we have ^∞_𝑚=1𝐿𝐘_𝑚−𝜒_𝑚𝐘_𝑚−1=𝐿^∞_𝑚=1𝐘_𝑚−𝜒_𝑚𝐘_𝑚−1=0. (46) Therefore, from (16), we can obtain that ^∞_𝑚=1𝐿𝐘_𝑚−𝜒_𝑚𝐘_𝑚−1=ℎ^∞_𝑚=1𝑅_𝑚𝐘_𝑚−1=0. (47) Since ℎ≠0, we have ^∞_𝑚=1𝑅_𝑚𝐘_𝑚−1=0. (48) By applying (39) and (43), ^∞_𝑚=1𝑅_𝑚𝐘_𝑚−1=^∞_𝑚=1𝐘_𝑚−1−1−𝜒_𝑚−1𝐅−𝜇¹₋₁𝑘(𝑥,𝑡)𝐆𝐘_𝑚−1𝑑𝑡=𝐘−𝐅−𝜇¹₋₁𝑘(𝑥,𝑡)𝐕(𝐘)𝑑𝑡. (49) Therefore, 𝐘 must be the exact solution of (39).

5. Numerical Examples

In this section we apply the technique described in Section 3 to some illustrative examples of higher-order nonlinear Fredholm integro-differential equations.

Example 1. Consider the second-order Fredholm integro-differential equation 𝑦^′′(𝑥)=6𝑥+¹₋₁𝑥𝑡𝑦^′(𝑡)²𝑦(𝑡)²𝑑𝑡 (50) subject to 𝑦(−1)=𝑦(1)=0 with the exact solution 𝑦(𝑥)=𝑥³−𝑥. We employ SHAM and HAM to solve this example. From the ℎ-curves (Figure 1), it is found that when −1.5≤ℎ≤1.5 and −1≤ℎ≤0, the SHAM solution and HAM solution converge to the exact solution, respectively. A numerical results of Example 1 against different order of SHAM approximate solutions is shown in Table 1.

Table 1 

The numerical results of Example 1 against different order of SHAM approximate solutions with ℎ=−0.01.

(a)

(b)

Figure 1 

The ℎ-curve 𝑦′′(0) and 𝑦′′′(0) for 10th-order (a) SHAM, (b) HAM.

Example 2. Consider the second order Fredholm integro-differential equation 𝑥𝑦^′′(𝑥)+𝑥²𝑦^′(𝑥)(𝑥)=+2𝑦−𝜋²𝑥+2sin(𝜋𝑥)+𝜋𝑥²()+cos𝜋𝑥₁₋₁cos(𝜋𝑡)𝑦⁴(𝑡)𝑑𝑡 (51) subject to 𝑦(−1)=𝑦(1)=0 with the exact solution 𝑦(𝑥)=sin(𝜋𝑥). We employ HAM and SHAM to solve this example. The numerical results of Example 2 against different order of SHAM approximate solutions with ℎ=−0.01 is shown in Table 2. In Table 3, there is a comparison of the numerical result against the HAM and SHAM approximation solutions at different orders with ℎ=−0.001. It is worth noting that the SHAM results become very highly accurate only with a few iterations, and fifth-order solutions are very close to the exact solution. Comparison of the numerical solution with the 4th-order SHAM solution for ℎ=−0.01 is made in Figure 2. As it is shown in Figure 3, the rate of convergency in SHAM is faster than HAM. In Figure 4, it is found that when −2.5≤ℎ≤0.5 and −1≤ℎ≤1, the SHAM solution and HAM solution converge to the exact solution, respectively. In HAM we choose 𝑦₀(𝑥)=1−𝑥² as initial guess.

Table 2 

The numerical results of Example 2 against different order of SHAM approximate solutions with ℎ=−0.01.

Table 3 

Numerical result of Example 2 against the HAM and the SHAM solutions with ℎ=−0.001.

Figure 2 

Comparison of the numerical solution of Example 2 with the 4th-order SHAM solution for ℎ=−0.01.

(a)

(b)

Figure 3 

Comparison of the absolute error of third-order (a) SHAM, (b) HAM.

(a)

(b)

Figure 4 

The ℎ-curve 𝑦′′(−1) and 𝑦′′′(1) for 6th-order (a) SHAM, (b) HAM.

Example 3. Consider the first-order Fredholm integro-differential equation [27, 28] 𝑦^′(𝑥)1=−2𝑒^𝑥+2+32𝑒^𝑥+¹₀𝑒^𝑥−𝑡𝑦³(𝑡)𝑑𝑡 (52) subject to the boundary condition 𝑦(0)=1. In order to apply the SHAM for solving the given problem, we should transform using an appropriate change of variables as 𝑥=𝜁+12,𝜁∈[−1,1]. (53) Then, we use the following transformation: 𝑦(𝑥)=𝑌(𝜁)+𝑒^(𝑥+1)/2. (54) We make the governing boundary condition homogeneous. Substituting (54) into the governing equation and boundary condition results in 𝑌^′(𝜁)=14¹₋₁𝑒^{(𝜁−𝑡)/2}𝑌³(𝑡)+3𝑒^𝑡+1𝑌(𝑡)+3𝑒^(𝑡+1)/2𝑌²(𝑡)𝑑𝑡 (55) subject to the boundary condition 𝑌(−1)=0. A comparison between absolute errors in solutions by SHAM, Lagrange interpolation, and Rationalized Haar functions is tabulated in Table 4. It is also worth noting that the SHAM results are very close to exact solutions only with two iterations.

Table 4 

A comparison of absolute errors between SHAM, LIM, and RHFS.

6. Conclusion

In this paper, we presented the application of spectral homotopy analysis method (SHAM) for solving nonlinear Fredholm integro-differential equations. A comparison was made between exact analytical solutions and numerical results obtained by the spectral homotopy analysis method, Rationalized Haar functions, and Lagrange interpolation solutions. In Example 1, the numerical results indicate that the rate of convergency in SHAM is faster than HAM. In this example, we found that the forth-order SHAM approximation sufficiently gives a match with the numerical results up to eight decimal places. In contrast, HAM solutions have a good agreement with the numerical results in 20th order with six decimal places. As we can see in Table 4, the spectral homotopy analysis results are more accurate and efficient than Lagrange interpolation solutions and rationalized Haar functions solutions [27, 28]. As it is shown in Figures 1 and 4 the rang of admissible values of ℎ is much wider in SHAM than HAM.

In this paper, we employed the spectral homotopy analysis method to solve nonlinear Fredholm integro-difflerential equations; however, it remains to be generalized and verified for more complicated integral equations that we consider it as future works.

Acknowledgment

The authors express their sincere thanks to the referees for the careful and details reading of the earlier version of the paper and very helpful suggestions. The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068.