Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudospectral Method

Abstract

A numerical method for solving fourth-order integro-differential equations is presented. This method is based on replacement of the unknown function by a truncated series of well-known shifted Chebyshev expansion of functions. An approximate formula of the integer derivative is introduced. The introduced method converts the proposed equation by means of collocation points to system of algebraic equations with shifted Chebyshev coefficients. Thus, by solving this system of equations, the shifted Chebyshev coefficients are obtained. Special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula. Numerical results are performed in order to illustrate the usefulness and show the efficiency and the accuracy of the present work.

1. Introduction

The integro-differential equation (IDE) is an equation that involves both integrals and derivatives of an unknown function. Mathematical modeling of real-life problems usually results in functional equations, like ordinary or partial differential equations, and integral and integro-differential equations, stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential equations; these equations arise in many fields like physics, astronomy, potential theory, fluid dynamics, biological models, and chemical kinetics. Integro-differential equations; are usually difficult to solve analytically; so, it is required to obtain an efficient approximate solution [1–5]. Recently, several numerical methods to solve IDEs have been given such as variational iteration method [6, 7], homotopy perturbation method [8, 9], spline functions expansion [10, 11], and collocation method [12–15].

Chebyshev polynomials are well-known family of orthogonal polynomials on the interval [−1,1] that have many applications [4, 6, 8, 13]. They are widely used because of their good properties in the approximation of functions. However, with our best knowledge, very little work was done to adapt these polynomials to the solution of integro-differential equations. Orthogonal polynomials have a great variety and wealth of properties. Some of these properties take a very concise form in the case of the Chebyshev polynomials, making Chebyshev polynomials of leading importance among orthogonal polynomials. The Chebyshev polynomials belong to an exclusive band of orthogonal polynomials, known as Jacobi polynomials, which correspond to weight functions of the form (1−𝑥)^𝛼(1+𝑥)^𝛽 and which are solutions of Sturm-Liouville equations [16].

In this work, we derive an approximate formula of the integral derivative 𝑦^(𝑛)(𝑥) and derive an upper bound of the error of this formula, and then we use this formula to solve a class of two-point boundary value problems (BVPs) for the fourth-order integro-differential equations as 𝑦^(𝑖𝑣)(𝑥)=𝑓(𝑥)+𝛾𝑦(𝑥)+^𝑥₀𝑝(𝑡)𝑦(𝑡)+𝑞(𝑡)Θ𝑦(𝑡)𝑑𝑡,0≤𝑥,𝑡≤1, (1) under the boundary and initial conditions 𝑦(0)=𝛼₀,𝑦^′′(0)=𝛼₁,𝑦(1)=𝛽₀,𝑦^′′(1)=𝛽₁, (2) where 𝑓(𝑥),𝑝(𝑥),and𝑞(𝑥) are known functions and 𝛾,𝛼₀,𝛼₁,𝛽₀, and 𝛽₁ are suitable constants. Several numerical methods to solve the fourth-order integro-differential equations have been given such as Chebyshev cardinal functions [17], variational iteration method [7], and others.

2. Some Basic Properties and Derivation of an Approximate Formula of the Derivative for Chebyshev Polynomials Expansion

The Chebyshev polynomial of the first kind is a polynomial in 𝑧 of degree 𝑛, defined by the relation 𝑇_𝑛(𝑧)=cos𝑛𝜃,when𝑧=cos𝜃. (3) The Chebyshev polynomials of degree 𝑛>0 of the first kind have precisely 𝑛 zeros and 𝑛+1 local extrema in the interval [−1,1]. The zeros of 𝑇_𝑛(𝑧) are denoted by 𝑧_𝑘()𝜋=cos𝑘−1/2𝑛,𝑘=1,2,…,𝑛. (4) The Chebyshev polynomials can be determined with the aid of the following recurrence formula [18]: 𝑇_𝑛+1(𝑧)=2𝑧𝑇_𝑛(𝑧)−𝑇_𝑛−1(𝑧),𝑇₀(𝑧)=1,𝑇₁(𝑧)=𝑧,𝑛=1,2,…. (5) The analytic form of the Chebyshev polynomials 𝑇_𝑛(𝑧) of degree 𝑛 is given by 𝑇_𝑛(𝑧)=𝑛^[𝑛/2]_𝑖=0(−1)^𝑖2^{𝑛−2𝑖−1}(𝑛−𝑖−1)!(𝑖)!()!𝑧𝑛−2𝑖^𝑛−2𝑖, (6) where [𝑛/2] denotes the integer part of 𝑛/2. The orthogonality condition is ¹₋₁𝑇_𝑖(𝑧)𝑇_𝑗(𝑧)√1−𝑧²𝜋𝑑𝑥=𝜋,for𝑖=𝑗=0;2,for𝑖=𝑗≠0;0,for𝑖≠𝑗. (7) In order to use these polynomials on the interval [0,1], we define the so called shifted Chebyshev polynomials by introducing the change of variable 𝑧=2𝑥−1. The shifted Chebyshev polynomials are denoted by 𝑇^∗_𝑛(𝑥) and defined as 𝑇^∗_𝑛(𝑥)=𝑇_𝑛(2𝑥−1)=𝑇_2𝑛(√𝑥).

The function 𝑦(𝑥), which belongs to the space of square integrable in [0,1], may be expressed in terms of shifted Chebyshev polynomials as 𝑦(𝑥)=^∞_𝑖=0𝑐_𝑖𝑇^∗_𝑖(𝑥), (8) where the coefficients 𝑐_𝑖 are given by 𝑐₀=1𝜋¹₀𝑦(𝑥)𝑇^∗₀(𝑥)√𝑥−𝑥²𝑑𝑥,𝑐_𝑖=2𝜋¹₀𝑦(𝑥)𝑇^∗_𝑖(𝑥)√𝑥−𝑥²𝑑𝑥,𝑖=1,2,…. (9)

In practice, only the first (𝑚+1) terms of shifted Chebyshev polynomials are considered. Then, we have that 𝑦_𝑚(𝑥)=^𝑚_𝑖=0𝑐_𝑖𝑇^∗_𝑖(𝑥). (10)

Lemma 1. The analytic form of the shifted Chebyshev polynomials 𝑇^∗_𝑛(𝑥) of degree 𝑛 is given by 𝑇^∗_𝑛(𝑥)=𝑛^𝑛_𝑘=0(−1)^𝑛−𝑘2^2𝑘(𝑛+𝑘−1)!()!()!𝑥2𝑘𝑛−𝑘^𝑘,𝑛=1,2,…. (11)

Proof. Since we have 𝑇^∗_𝑛(𝑥)=𝑇_2𝑛(√𝑥), then by substituting in (6), we can obtain that 𝑇^∗_𝑛(𝑥)=2𝑛^𝑛_𝑖=0(−1)^𝑖2^{2𝑛−2𝑖−1}(2𝑛−𝑖−1)!(𝑖)!()!𝑥2𝑛−2𝑖^𝑛−𝑖,𝑛=1,2,…. (12) Now, we put 𝑘=𝑛−𝑖 in (12) we obtain the desired result (11).

The main approximate formula of the derivative of 𝑦_𝑚(𝑥), and is given in the following theorem.

Theorem 2. Let 𝑦(𝑥) be approximated by shifted Chebyshev polynomials as (10), and also suppose that 𝑟 is integer; then, 𝐷^𝑟𝑦_𝑚(𝑥)=^𝑚𝑖_{𝑖=𝑟𝑘=𝑟}𝑐_𝑖𝜆_{𝑖,𝑘,𝑟}𝑥^𝑘−𝑟, (13) where 𝜆_{𝑖,𝑘,𝑟} is given by 𝜆_{𝑖,𝑘,𝑟}=(−1)^𝑖−𝑘2^2𝑘𝑖(𝑖+𝑘−1)!𝑘!()!()!()!.𝑖−𝑘2𝑘𝑘−𝑟 (14)

Proof. Since the differential operator 𝐷^𝑟 is linear, we can obtain that 𝐷^𝑟𝑦_𝑚(𝑥)=^𝑚_𝑖=0𝑐_𝑖𝐷^𝑟𝑇^∗_𝑖(𝑥). (15) Since 𝐷^𝑟𝑐=0,𝑐 is a constant, and 𝐷^𝑟𝑥^𝑛=0,for𝑛∈𝑁,𝑛<𝑟,𝑛!()!𝑥𝑛−𝑟^𝑛−𝑟,for𝑛∈𝑁,𝑛≥𝑟. (16) Then, we have that 𝐷^𝑟𝑇^∗_𝑖(𝑥)=0,𝑖=0,1,…,𝑟−1, (17) and for 𝑖=𝑟,𝑟+1,…,𝑚, and by using (16), we get that 𝐷^𝑟𝑇^∗_𝑖(𝑥)=𝑖^𝑖_𝑘=𝑟(−1)^𝑖−𝑘2^2𝑘(𝑖+𝑘−1)!()!()!𝐷𝑖−𝑘2𝑘^𝑟𝑥^𝑘=𝑖^𝑖_𝑘=𝑟(−1)^𝑖−𝑘2^2𝑘(𝑖+𝑘−1)!𝑘!()!()!()!𝑥𝑖−𝑘2𝑘𝑘−𝑟^𝑘−𝑟. (18) A combination of (17), (18), and (14) leads to the desired result and completes the proof of the theorem.

3. Error Analysis

In this section, special attention is given to study the convergence analysis and evaluate the upper bound of the error of the proposed formula.

Theorem 3 (Chebyshev truncation theorem; see [18]). The error in approximating 𝑦(𝑥) by the sum of its first 𝑚 terms is bounded by the sum of the absolute values of all the neglected coefficients. If 𝑦_𝑚(𝑥)=^𝑚_𝑘=0𝑐_𝑘𝑇_𝑘(𝑥), (19) then 𝐸_𝑇(𝑚)≡𝑦(𝑥)−𝑦_𝑚(𝑥)≤^∞_𝑘=𝑚+1𝑐_𝑘, (20) for all 𝑦(𝑥), all 𝑚, and all 𝑥∈[−1,1].

Theorem 4. The derivative of order 𝑟 for the shifted Chebyshev polynomials can be expressed in terms of the shifted Chebyshev polynomials themselves in the following form: 𝐷^𝑟𝑇^∗_𝑖(𝑥)=^𝑖_𝑘=𝑟^𝑘−𝑟_𝑗=0Θ_{𝑖,𝑗,𝑘}𝑇^∗_𝑗(𝑥), (21) where Θ_{𝑖,𝑗,𝑘}=(−1)^𝑖−𝑘2𝑖(𝑖+𝑘−1)!Γ(𝑘−𝑟+1/2)ℎ_𝑗Γ()!!!,ℎ𝑘+1/2)(𝑖−𝑘𝑘−𝑟−𝑗𝑘+𝑗−𝑟₀=2,ℎ_𝑗=1,𝑗=0,1,…. (22)

Proof. We use the properties of the shifted Chebyshev polynomials [18] and expand 𝑥^𝑘−𝑟 in (18) in the following form: 𝑥^𝑘−𝑟=^𝑘−𝑟_𝑗=0𝑐_𝑘𝑗𝑇^∗_𝑗(𝑥), (23) where 𝑐_𝑘𝑗 can be obtained using (9), and𝑦(𝑥)=𝑥^𝑘−𝑟; then, 𝑐_𝑘𝑗=2ℎ_𝑗𝜋¹₀𝑥^𝑘−𝑟𝑇^∗_𝑗(𝑥)√𝑥−𝑥²𝑑𝑥,ℎ₀=2,ℎ_𝑗=1,𝑗=1,2,…. (24)
At 𝑗=0,  we  find that 𝑐_𝑘0 = (1/𝜋)∫¹₀(𝑥^𝑘−𝑟𝑇^∗₀(𝑥)/√𝑥−𝑥²)𝑑𝑥 = (1/√𝜋)(Γ(𝑘−𝑟+1/2)/(𝑘−𝑟)!); also, at any 𝑗 and using the formula (10), we can find that 𝑐_𝑘𝑗=𝑗√𝜋^𝑗_𝑙=0(−1)^𝑗−𝑙𝑗+𝑙−1!2^2𝑙+1Γ(𝑘+𝑙−𝑟+1/2)!()!()!,𝑗−𝑙2𝑙𝑘+𝑙−𝑟𝑗=1,2,3,…, (25) employing (18) and (23) gives 𝐷^𝑟𝑇^∗_𝑖(𝑥)=^𝑖_𝑘=𝑟^𝑘−𝑟_𝑗=0Θ_{𝑖,𝑗,𝑘}𝑇^∗_𝑗(𝑥),𝑖=𝑟,𝑟+1,…, (26) where Θ_{𝑖,𝑗,𝑘}=𝑖(−1)^𝑖−𝑘(𝑖+𝑘−1)!2^2𝑘𝑘!Γ(𝑘−𝑟+1/2)(𝑖−𝑘)!(2𝑘)!√𝜋(Γ(𝑘+1−𝑟))²(),𝑗=0;−1^𝑖−𝑘𝑖𝑗(𝑖+𝑘−1)!2^2𝑘+1𝑘!√𝜋()!()!()!×𝑘−𝑟𝑖−𝑘2𝑘_𝑗𝑙∑₌₀(−1)^𝑗−𝑙𝑗+𝑙−1!2^2𝑙Γ(𝑘+𝑙−𝑟+1/2)𝑗−𝑙!(2𝑙)!(𝑘+𝑙−𝑟)!,𝑗=1,2,3,…. (27)After some lengthy manipulation, Θ_{𝑖,𝑗,𝑘} can be put in the following form: Θ_{𝑖,𝑗,𝑘}=(−1)^𝑖−𝑘2𝑖(𝑖+𝑘−1)!Γ(𝑘−𝑟+1/2)ℎ_𝑗Γ()!!!,𝑘+1/2)(𝑖−𝑘𝑘−𝑟−𝑗𝑘+𝑗−𝑟𝑗=0,1,…, (28) and this completes the proof of the theorem.

Theorem 5. The error |𝐸_𝑇(𝑚)|=|𝐷^𝑟𝑦(𝑥)−𝐷^𝑟𝑦_𝑚(𝑥)| in approximating 𝐷^𝑟𝑦(𝑥) by 𝐷^𝑟𝑦_𝑚(𝑥) is bounded by 𝐸_𝑇(𝑚)≤^∞_𝑖=𝑚+1𝑐_𝑖^𝑖_𝑘=𝑟^𝑘−𝑟_𝑗=0Θ_{𝑖,𝑗,𝑘}. (29)

Proof. A combination of (8), (10), and (21) leads to 𝐸_𝑇(𝑚)=𝐷^𝑟𝑦(𝑥)−𝐷^𝑟𝑦_𝑚(𝑥=_{∞𝑖=𝑚+1})𝑐_𝑖^𝑖_𝑘=𝑟^𝑘−𝑟_𝑗=0Θ_{𝑖,𝑗,𝑘}𝑇^∗_𝑗(𝑥), (30) but |𝑇^∗_𝑗(𝑥)|≤1; so, we can obtain that 𝐸_𝑇(𝑚)≤^∞_𝑖=𝑚+1𝑐_𝑖^𝑖_𝑘=𝑟^𝑘−𝑟_𝑗=0Θ_{𝑖,𝑗,𝑘}, (31) and subtracting the truncated series from the infinite series, bounding each term in the difference, and summing the bounds complete the proof of the theorem.

4. Procedure Solution for the Fourth-Order Integro-Differential Equation

In this section, we will present the proposed method to solve numerically the fourth-order integro-differential equation of the form in (1). The unknown function 𝑦(𝑥) may be expanded by finite series of shifted Chebyshev polynomials as in the following approximation: 𝑦_𝑚(𝑥)=^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑥), (32) and approximated formula of its derivatives can be defined in Theorem 2. From (1), (32), and Theorem 2, we have that ^𝑚𝑖_{𝑖=𝑟𝑘=𝑟}𝑐_𝑖𝜆_{𝑖,𝑘,𝑟}𝑥^𝑘−𝑟=𝑓(𝑥)+𝛾^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑥)+^𝑥₀𝑝(𝑡)^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)+𝑞(𝑡)Θ^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)𝑑𝑡. (33) We now collocate (33) at (𝑚−1+𝑟) points 𝑥_𝑠,𝑠=0,1,…,𝑚−𝑟 as ^𝑚𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_𝑖,𝑘,4𝑥_𝑠^𝑘−4=𝑓𝑥_𝑠+𝛾^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛𝑥_𝑠+^𝑥𝑠₀𝑝(𝑡)^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)+𝑞(𝑡)Θ^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)𝑑𝑡. (34) For suitable collocation points, we use roots of shifted Chebyshev polynomial 𝑇^∗_{𝑚+1−𝑟}(𝑥). The integral terms in (34) can be found using composite trapezoidal integration technique as ^𝑥𝑠₀𝑝(𝑡)^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)+𝑞(𝑡)Θ^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡≅ℎ)𝑑𝑡_𝑠2Ω𝑡₀+Ω𝑡_𝐿+2^𝐿−1_𝑘=1Ω𝑡_𝑘, (35) where Ω(𝑡)=𝑝(𝑡)∑^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)+𝑞(𝑡)Θ(∑^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)),ℎ_𝑠=𝑥_𝑠/𝐿, for an arbitrary integer 𝐿,𝑡_𝑗+1=𝑡_𝑗+ℎ_𝑠,𝑠=0,1,…,𝑚−𝑟,and𝑗=0,1,…,𝐿. So, by using (34) and (35), we obtain ^𝑚𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_{𝑖,𝑘,𝑟}𝑥_𝑠^𝑘−𝑟=𝑓𝑥_𝑠+𝛾^𝑚_𝑛=0𝑐_𝑛𝑇^∗_𝑛𝑥_𝑠+ℎ_𝑠2Ω𝑡₀+Ω𝑡_𝐿+2^𝐿−1_𝑘=1Ω𝑡_𝑘. (36) Also, by substituting (32) in the boundary conditions (2), we can obtain 𝑟 equations as follows: ^𝑚_𝑖=0(−1)^𝑖𝑐_𝑖=𝛼₀,^𝑚_𝑖=0𝑐_𝑖=𝛽₀,_𝑚𝑖=2𝑐_𝑖𝑇^∗_𝑖^′′(0)=𝛼₁,^𝑚_𝑖=2𝑐_𝑖𝑇^∗_𝑖^′′(1)=𝛽₁. (37) Equation (36), together with 𝑟 equations of the boundary conditions (37), give (𝑚+1) of system of algebraic equations which can be solved, for the unknowns 𝑐_𝑛,𝑛=0,1,…,𝑚, using conjugate gradient method or Newton iteration method.

5. Numerical Results

In this section, to verify the validity and the accuracy and support our theoretical discussion of the proposed method, we give some computations results of numerical examples.

Example 6. Consider the nonlinear fourth-order integro-differential equation as in (1) and (2) with 𝑓(𝑥)=1,𝛾=0,𝑝(𝑡)=0,𝑞(𝑡)=𝑒^−𝑡,andΘ(𝑦)=𝑦²(𝑥); then, the integro-differential equation will be 𝑦^(𝑖𝑣)(𝑥)=1+^𝑥₀𝑒^−𝑡𝑦²(𝑡)𝑑𝑡,0≤𝑥≤1, (38) subject to the boundary conditions 𝑦(0)=𝑦^′′(0)=1,𝑦(1)=𝑦^′′(1)=𝑒. (39) The exact solution of this problem is 𝑦(𝑥)=𝑒^𝑥 [7].
We apply the suggested method with 𝑚=5 and approximate the solution 𝑦(𝑥) as follows: 𝑦₅(𝑥)=⁵_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑥). (40) From (38), (40), and Theorem 2, we have that ^5𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_𝑖,𝑘,4𝑥^𝑘−4=1+^𝑥₀𝑒^−𝑡5_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)²𝑑𝑡. (41) We now collocate (41) at points, 𝑥_𝑠, 𝑠=0,1 as ^5𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_𝑖,𝑘,4𝑥_𝑠^𝑘−4=1+^𝑥𝑠₀𝑒^−𝑡5_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)²𝑑𝑡. (42) For suitable collocation points we use roots of shifted Chebyshev polynomial 𝑇^∗₂(𝑥). The integral terms in (42) can be found using composite trapezoidal integration technique as ^𝑥𝑠₀𝑒^−𝑡5_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡)²=ℎ𝑑𝑡_𝑠2Ω𝑡₀+Ω𝑡_𝐿+2^𝐿−1_𝑘=1Ω𝑡_𝑘, (43) where Ω(𝑡)=𝑒^−𝑡(∑⁵_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡))²,ℎ_𝑠=𝑥_𝑠/𝐿, for an arbitrary integer 𝐿,𝑡_𝑗+1=𝑡_𝑗+ℎ_𝑠,𝑠=0,1,and𝑗=0,1,…,𝐿. So, by using (43) and (42), we obtain ^5𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_𝑖,𝑘,4𝑥_𝑠^𝑘−4ℎ=1+^𝑠2Ω𝑡₀+Ω𝑡_𝐿+2^𝐿−1_𝑘=1Ω𝑡_𝑘. (44) Also, by substituting (40) in the boundary conditions (39), we can obtain four equations as follows: 𝑐₀−𝑐₁+𝑐₂−𝑐₃+𝑐₄−𝑐₅𝑐=1,₀+𝑐₁+𝑐₂+𝑐₃+𝑐₄+𝑐₅𝑙=𝑒,₀𝑐₀+𝑙₁𝑐₁+𝑙₂𝑐₂+𝑙₃𝑐₃+𝑙₄𝑐₄+𝑙₅𝑐₅𝑠=1,₀𝑐₀+𝑠₁𝑐₁+𝑠₂𝑐₂+𝑠₃𝑐₃+𝑠₄𝑐₄+𝑠₅𝑐₅=𝑒, (45) where 𝑙_𝑖=𝑇^∗_𝑖^′′(0) and 𝑠_𝑖=𝑇^∗_𝑖^′′(1).
Equation (44), together with four equations of the boundary conditions (45), represent, a nonlinear system of six algebraic equations in the coefficients 𝑐_𝑛; by solving it using the Newton iteration method, we obtain 𝑐₀=1.75379,𝑐₁=0.85039,𝑐₂𝑐=0.10478,₃=0.00872,c₄=0.00057,𝑐₅=0.00003. (46) The behavior of the approximate solution using the proposed method with 𝑚=5, the approximate solution using variational iteration method (VIM), and the exact solution are presented in Figure 1. Table 1 shows the behavior of the absolute error between exact solution and approximate solution using the presented method at 𝑚=6 and 𝑚=8. From Figure 1 and Table 1, it is clear that the proposed method can be considered as an efficient method to solve the non-linear integro-differential equations. Table 1 indicates that as 𝑚 increases the errors decrease more rapidly; hence, for better results, using number 𝑚 is recommended. Also, we can conclude that the obtained approximated solution is in excellent agreement with the exact solution.

Table 1 

The behavior of the absolute error between the exact solution and approximate solution at 𝑚=6 and 𝑚=8.

Figure 1 

The behavior of the exact solution, the approximate solution using VIM, and the approximate solution using the proposed method at 𝑚=5.

Example 7. Consider the linear fourth-order integro-differential equation as in (1) and (2) with 𝑓(𝑥)=𝑥+(𝑥+3)𝑒^𝑥,𝛾=1,𝑝(𝑡)=−1,ℎ(𝑡)=0,andΘ(𝑦)=𝑦(𝑥); then, the integro-differential equation will be 𝑦^(𝑖𝑣)(𝑥)=𝑥+(𝑥+3)𝑒^𝑥(𝑥)−+𝑦_𝑥0𝑦(𝑡)𝑑𝑡,0≤𝑥≤1, (47) subject to the boundary conditions 𝑦(0)(1)𝑦=1,=1+𝑒,_′′(0)=2,𝑦^′′(1)=3𝑒. (48) The exact solution of this problem is 𝑦(𝑥)=1+𝑥𝑒^𝑥 [17].
We apply the suggested method with 𝑚=5 and approximate the solution 𝑦(𝑥) as follows: 𝑦(𝑥)≅⁵_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑥). (49) By the same procedure in the previous example, we have ^5𝑖_𝑖=4𝑘=4𝑐_𝑖𝜆_𝑖,𝑘,4𝑥_𝑠^𝑘−4=𝑓𝑥_𝑠+⁵_𝑛=0𝑐_𝑛𝑇^∗_𝑛𝑥_𝑠+ℎ_𝑠2×Ω𝑡₀+Ω𝑡_𝐿+2^𝐿−1_𝑘=1Ω𝑡_𝑘,𝑠=0,1,2, (50) where Ω(𝑡)=∑⁵_𝑛=0𝑐_𝑛𝑇^∗_𝑛(𝑡), and the nodes 𝑡_𝑗+1=𝑡_𝑗+ℎ_𝑠,𝑗=0,1,…,𝐿, 𝑡₀=0,andℎ_𝑠=𝑥_𝑠/𝐿. We can write the initialboundary conditions in the form 𝑐₀−𝑐₁+𝑐₂−𝑐₃+𝑐₄−𝑐₅+𝑐₆𝑐=1,₀+𝑐₁+𝑐₂+𝑐₃+𝑐₄+𝑐₅+𝑐₆𝑙=1+𝑒,₀𝑐₀+𝑙₁𝑐₁+𝑙₂𝑐₂+𝑙₃𝑐₃+𝑙₄𝑐₄+𝑙₅𝑐₅+𝑙₆𝑐₆𝑠=2,₀𝑐₀+𝑠₁𝑐₁+𝑠₂𝑐₂+𝑠₃𝑐₃+𝑠₄𝑐₄+𝑠₅𝑐₅+𝑠₆𝑐₆=3𝑒. (51) By using (50) and (51), we obtain a linear system of seven algebraic equations in the coefficients 𝑐_𝑛; by solving it using the conjugate gradient method, we obtain 𝑐₀=2.09189,𝑐₁=1.32820,𝑐₂𝑐=0.26461,₃=0.03079,𝑐₄=0.00264,𝑐₅=0.00015. (52) The behavior of the approximate solution using the proposed method with 𝑚=6, the approximate solution using variational iteration method (VIM) and the exact solution are presented in Figure 2. Table 2 shows the behaviour of the absolute error between exact solution and approximate solution using the presented method at 𝑚=7 and 𝑚=9. From this figure, it is clear that the proposed method can be considered as an efficient method to solve the linear integro-differential equations. Also, we can conclude that the obtained approximate solution is in excellent agreement with the exact solution.

Table 2 

The behaviour of the absolute error between the exact solution and approximate solution at 𝑚=7 and 𝑚=9.

Figure 2 

The behavior of the exact solution, the approximate solution using VIM, and the approximate solution using the proposed method at 𝑚=6.

6. Conclusion and Discussion

Integro-differential equations are usually difficult to solve analytically; so, it is required to obtain the approximate solution. In this paper, we proposed the pseudospectral method using shifted Chebyshev method for solving the integro-differential equations. The Chebyshev method is useful for acquiring both the general solution and particular solution as demonstrated in examples. Special attention is given to study the convergence analysis and derive an upper bound of the error of the derived approximate formula. From our obtained results, we can conclude that the proposed method gives solutions in excellent agreement with the exact solution and better than the other methods. An interesting feature of this method is that when an integral system has linearly independent polynomial solution of degree 𝑚 or less than 𝑚, the method can be used for finding the analytical solution. All computations are done using MATLAB 8.