# Perturbed Galerkin Method for Solving Integro-Differential Equations

Authors: K. Issa; J. Biazar; T. O. Agboola; T. Aliu
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

In this paper, perturbed Galerkin method is proposed to find numerical solution of an integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions which transform the integro-differential equation into a system of linear equations. The systems of linear equations are then solved to obtain the approximate solution. Examples to justify the effectiveness and accuracy of the method are presented and their numerical results are compared with Galerkin’s method, Taylor’s series method, and Tau’s method which provide validation for the proposed approach. The errors obtained justify the effectiveness and accuracy of the method.


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## Body

## 1. Introduction

Many researchers in the recent times have been concentrating their attention on the solution of integro-differential equation due to its application in many branches of Mathematical sciences such as Mathematical modeling (in biological application, engineering, and physical problems), Financial mathematics, control theory, and many of the problems are difficult to solve analytically. Different methods have been employed by many researchers to find the approximate solution of integro-differential equations such as Biazar and Salehi [1, 2] who applied the Chebyshev Galerkin method to study integro-differential equations of second kind, Jangveladze et al. [3] proposed finite element method for nonlinear integro-differential model, Fathy [4] proposed the Legendre-Galerkin method for solving linear the Fredholm integro-differential equations, Hosseini and Shahmorad [5–7] implemented the Tau method to obtain approximate solution of Fredholm-Volterra integro-differential equations with arbitrary polynomial bases and its corresponding error estimation. Homotopy perturbation methods were used in [8–10] to solve integro-differential equations, Chen et al. [11] studied the convergence and stability of Galerkin’s method in their paper entitled “A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation”. The PellLucas matrix-collocation method was proposed by Demir et al. [12] to solve a class of Fredholm-type delay integro-differential equations with variable delays, convergence of a block-by-block method was proposed by Makroglou [13] for the solution of nonlinear Volterra integro-differential equations. Khater et al. [14] implemented Legendre polynomials to solve integral and integro-differential equations, Chang [15] implemented the extrapolation methods to obtain numerical solution of integro-differential equations, variational iteration method was used by Bildik [16] to obtain the approximate solution of general linear Fredholm integro-differential equations, Shahsavara [17] employed the Haar wavelets to solve the linear Volterra and Fredholm integro-differential equations, Yalcinbas [18–20] used the Taylor polynomials to obtain approximate solution of high-order linear Volterra-Fredholm integro-differential equations, Golbabai and Seifollahi [21] implemented radial basis function networks to obtain numerical solution of linear integro-differential equations, Gümgüm [22] employed the Lucas polynomial together with standard and the Chebyshev-Lobatto collocation points to solve functional integro-differential equations involving variable delays, Šmarda and Khan [23] solved analytical properties and asymptotic behaviour of solutions for system of integro-differential equations and also in another paper they used an approach which combines topological method of Wazewski and Schauder’s fixed point theorem to solve singular initial value problem for a system of integro-differential equations unsolved with respect to the derivative (see [24] for details).The main motivation in this present work is proposed perturbed Galerkin approach to find approximate solution for integro-differential equations. The approximate solutions obtain by the proposed approach are more accurate and effective compared to the numerical solutions obtained by the Tau method and Galerkin method (GM).


## 2. Review of Some Orthogonal Polynomials

A sequence of real functionsψrz,r=0,1,⋯ is said to be orthogonal with respect to the weight function ωz over a,b, if
(1)ψmz,ψnz=∫abψmzψnzωzdz=0,m≠nλn,m=n,where
(2)ψnz=∑i=0naizi.Some commonly used orthogonal polynomialsψnz are as follows:Legendre polynomialsPmz: An orthogonal polynomial with recurrence relation and weight function:
(3)Pm+1z=2m+1m+1zPmz−mm+1Pm−1z,m≥1,andωz=1,z∈−1,1,respectively, where P0z=1,P1z=z.Shifted Gegenbauer polynomialsCmαz: An orthogonal polynomial in the interval a,b having recurrence relation:
(4)Cmαz=1m2m−1+α2z−a+bb−aCm−1αz−m−2+2αCm−2αz,m≥2,whereCmα=1,C1α=2α2z−a+bb−a,with respect to the weight function ωz=1−2z−a+b/b−a2α−1/2 (see [25–27] for more details).Jacobi polynomials: The well-established Jacobi polynomials Pmα,γz,m=0,1,⋯, with parameters α,γ>−1 reference to the weight function ωz=1−zα1+zγ. The explicit forms of Jacobi polynomials are used in [28] as
(5)Pmα,γz=Γα+m+1m!Γα+γ+m+1∑n=0mm!Γα+γ+n+m+1m−n!n!Γα+n+1z−12n.Equation (5) becomes the Legendre polynomial whenever α=γ=0, it reduces to the Chebyshev polynomial of the second kind when α=γ=1/2, while it gives the Gegenbauer polynomial of the form Cm3/2z whenever α=γ=1.Chebyshev polynomials ofνkinds: Chebyshev polynomials are of different kinds ν with respect to their respective weight functions ωz defined here in the interval −1,1 are
(6)ψν,m=Tm=cosmϕ,ν=1,Um=sinm+1ϕsinϕ,ν=2,Vm=cosm+1/2ϕcosϕ/2,ν=3,Wm=sinm+1/2ϕsinϕ/2,ν=4,and their corresponding weight functions ωz are
(7)ωzν=1−z2−12,ν=1,1−z212,ν=2,1+z1−z12,ν=3,1−z1+z12,ν=4.The Chebyshev polynomials of all kinds are said to be orthogonal reference to their respective kindν in the interval −1,1, if
(8)ψν,mz,ψν,nz=ψν,mz,ψν,nz=∫−11ωzνψν,mzψν,nzdz=0,m≠nπ2,m=n,ν=1,2,Vmz,Vnz=∫−11ωzνVmzVnzdz=0,m≠nπ,m=n,ν=3,Wmz,Wnz=∫−11ωzWmzWnzdz=0,m≠nπ,m=n,ν=4,see details in [1, 29].


## 3. Description of Perturbed Galerkin Method

In this work, we consider integro-differential equations of the form:(9)∑i=0nsizdidziφz=gz+β1∫abρ1z,tφtdt+β2∫azρ2z,tφtdt,subject to the initial condition:
(10)φia=τi,i=0,1,⋯,n−1,where φz is an unknown function, siz,gz,ρ1z,t, and ρ2z,t are known functions, n is the order of Equation (9), β1,β2 are real numbers and z is the independent variable of the functions which appear throughout this paper, defined in the interval a,b except stated otherwise.Supposeφmz is the approximate solution of degree m to the function φz, then we write
(11)φmz=∑j=0mαjWj2z−a+bb−a,where αj are unknowns to be determined and Wj,j=0,1,⋯,m are fourth kind shifted Chebyshev polynomials.Adding perturbation termsHmz to the right-hand side of approximate form of Equation (9), we obtain
(12)∑i=0nsizdidziφmz−Hmz−β1∫abρ1z,tφmtdt−β2∫azρ2z,tφmtdt=gz,then multiply both sides of Equation (12) by Wk2z−a+b/b−a,k=n,n+1,n+m+1 and integrate the resulting equation with respect to the independent variable, over a,b, we obtain
(13)∫ab∑i=0nsiz∑j=0m2b−aiαjdidziWj2z−a+bb−a−β1∫abρ1z,t∑j=0mαjWj2t−a+bb−adt−β2∫azρ2z,t∑j=0mαjWj2t−a+bb−adt−∑r=0nηr+1Wm−n+r+12z−a+bb−aWk2z−a+bb−adz=∫abgzWk2z−a+bb−adz,k=n,n+1,⋯,n+m+1,where
(14)Hmz=∑r=0nηr+1Wm−n+r+12z−a+bb−a,m is the degree of approximation.Putting Equation (13) in matrix form, we have
(15)χα=G,where χ is a matrix of m+2×m+n+2, α and G are column matrices of m+2×1. We derived the remaining equations from the initial conditions (10), that is
(16)didziφma=∑i=0m2b−aiαididziWi2z−a+bb−az=a=τi,i=0,1,n−1.Solving Eqs. (13) and (16) simultaneously, we obtain the values of the unknowns αj,j=0,1,⋯,m, we then substitute in Equation (11) to obtain the approximate solution of degree m.


## 4. Numerical Experiments

In this section, we illustrate the method discussed (present method (PM)) in Section3 to solve some selected examples from the literature [1, 2, 7, 21]. To achieve the desired results, we used Maple for all the computation and MATLAB for graphical interpretation of the results. The results are compared with the results obtained using radial basis function (RBF) [21], the Galerkin method (GM) [2], the Chebyshev-Galerkin method (CGM) of first kind [1], and Tau method [7]. We compute the absolute maximum error ξm and compare with the existing results from the literature, written as below:
(17)ξm=max0≤i≤100φzi−φmzi,zi=a+ih.Example 1.
Considering the Fredholm integro-differential equation in [1, 2, 21],
(18)φ′z=φz−12z+11+z−log1+z+1log22∫01z1+tφtdt,0≤z≤1subject to initial condition φ0=0.
The exact solutionφz=log1+z.
Replacingφz in Equation (18) with the corresponding approximate solution defined in Equation (11) and add the perturbation terms defined in Equation (14), we obtain
(19)φ′mz−φmz−1log22∫01z1+tφmtdt−∑r=0nηr+1Wm−n+r+12z−1=11+z−12z−log1+z,0≤z≤1.
Seeking an approximate solutions of degrees 4,8, and12 (that is m=4,8, and 12), we multiply both sides of Equation (19) by Wk2z−1,k=1,2,⋯,m+2 (since k=n,n+1,⋯,n+m+1, as explained in Equation (13)) and then integrate the resulting equation in the interval 0,1, we obtain
(20)∫01∑j=0m2jαjddzWj2z−1−∑j=0mαjWj2z−1−1log22∫01z1+t∑j=0mαjWj2t−1dt−∑r=01ηr+1Wr+42z−1Wk2z−1dz=∫0111+z−12z−log1+zWk2z−1dz,k=1,2,⋯,m+2.
From Equation (20) we obtain m+2 equations with m+3 unknowns, to obtain a unique solutions, we use initial condition φ0=0≅φm0, that is
(21)α0−α1+⋯+−1mαm=0.We then solve Equations (20) and (21) to obtain the approximate solutions.Table1 shows the maximum absolute errors at different degrees of approximation. Figure 1 shows the exact and its corresponding approximate solutions at different degrees of approximation m.Table 1
Maximum absolute errors for Example1 at different degrees of approximation m.
mPMGM [2]CGM [1]48.08×10−44.63×10−31.91×10−485.79×10−75.61×10−61.57×10−7122.93×10−105.93×10−91.88×10−10Figure 1
Approximate solutions and its corresponding absolute errors at different degrees of approximationm.Example 2.
Consider the Volterra integro-differential equation in [1, 2, 30]
(22)φ′z−∫0zφtdt=1−2zsinz;0≤z≤1,φ0=0,with analytic solution φz=zcosz.Table2 displays the comparison of the absolute maximum errors at different degrees of approximation relative to the existing ones from the literature. Figure 2 exhibits the exact solution and the corresponding approximate solutions.Table 2
Maximum absolute errors for Example2 at different degrees of approximation m.
mPMGM [2]CGM [1]48.12×10−42.30×10−34.19×10−487.49×10−101.38×10−84.19×10−10129.40×10−168.09×10−151.20×10−16Figure 2
Approximate solution for Example2 and its absolute errors.Example 3.
Consider the following Volterra integro-differential equation [1, 2, 30]
(23)φ′z+∫0ztφtdt=−1+12z2−zexpz;0≤z≤1,φ0=0,the exact solution is φz=1−expz.Absolute maximum errors obtained are tabulated in Table3. Figure 3 presents the exact solution and the corresponding approximate solutions at different degrees of approximation m.Table 3
Maximum absolute errors for Example3 at different degree of approximation m.
mPMGM [2]CGM [1]47.38×10−57.20×10−38.14×10−588.59×10−112.31×10−98.61×10−11128.91×10−179.24×10−161.67×10−17Figure 3
Approximate solutions and its absolute errors.Example 4.
Consider the Fredholm-Volterra integro-differential equation ([1, 2, 7, 18])
(24)φ′′z+zφ′z−zφz=ez−2sinz+∫−11sinzetφtdt,−1≤z≤1,subject to initial conditions φ0=1,φ′0=1 and with the exact solution given as yx=expz.The absolute maximum errors are presented in Table4 together with the results from the literature with Figure 4 showing the exact and the corresponding approximate solutions.Table 4
Absolute maximum errors for Example4 at different degrees of approximation m.
mPMGM [2]CGM [1]Tau method [7]54.66×10−43.19×10−37.12×10−43.19×10−3105.49×10−102.10×10−61.00×10−92.10×10−6151.27×10−177.50×10−111.44×10−177.50×10−11Figure 4
The approximate solution for Example4 and its absolute errors at different values of m.


## 5. Discussion of Results and Conclusion

### 5.1. Discussion of Results

Tables1–4 show the absolute maximum errors obtained for Examples 1–4, as compared with the existing results in the literature. It was observed from the results obtained that the present method gives a better accuracy compared to the results obtained in [2, 7, 21], while it produced the same degree of accuracy when compare with CGM [1]. Figures 1–4 display the exact and its corresponding approximate solutions at different values of m, it is obvious from the figures that, as the degree of approximation increases the accuracy improves, that is, the maximum absolute errors decline.


### 5.2. Conclusion

In this work, we have proposed the perturbed Galerkin method for solving integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions. The Galerkin approach was used to transform the equation into a system of algebraic equations with the resulting equations solved to obtain the coefficients of the approximate solution of degreem. The method was implemented on some selected problems from the literature. The results have been compared with the GM, CGM, and Tau method solutions. The present approach gives a better accuracy compared to the results obtained via GM and Tau method while it is in good agreement with the results obtained via CGM. Sequel to the results, it becomes obvious that the proposed method is effective and accurate. In conclusion, the perturbed Galerkin approach can be considered a nice and alternative method to the existing numerical techniques.


## 5.1. Discussion of Results

Tables1–4 show the absolute maximum errors obtained for Examples 1–4, as compared with the existing results in the literature. It was observed from the results obtained that the present method gives a better accuracy compared to the results obtained in [2, 7, 21], while it produced the same degree of accuracy when compare with CGM [1]. Figures 1–4 display the exact and its corresponding approximate solutions at different values of m, it is obvious from the figures that, as the degree of approximation increases the accuracy improves, that is, the maximum absolute errors decline.


## 5.2. Conclusion

In this work, we have proposed the perturbed Galerkin method for solving integro-differential equations using fourth kind shifted Chebyshev polynomials as basis functions. The Galerkin approach was used to transform the equation into a system of algebraic equations with the resulting equations solved to obtain the coefficients of the approximate solution of degreem. The method was implemented on some selected problems from the literature. The results have been compared with the GM, CGM, and Tau method solutions. The present approach gives a better accuracy compared to the results obtained via GM and Tau method while it is in good agreement with the results obtained via CGM. Sequel to the results, it becomes obvious that the proposed method is effective and accurate. In conclusion, the perturbed Galerkin approach can be considered a nice and alternative method to the existing numerical techniques.



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