Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

Abstract

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method.

1. Introduction

In recent years, there has been a growing interest in the system of integrodifferential equations (IDE), which arise frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, and so forth [1–8]. The systems of integrodifferential equations are generally difficult to solve analytically, thus finding efficient computational algorithms for obtaining numerical solution is required.

There are various techniques for solving systems of IDE, for example, operational Tau method [9, 10], Adomian decomposition method [11], Galerkin method [12], rationalized Haar functions method [13], He’s homotopy perturbation method (HPM) [14, 15], and Ghebyshev polynomial [16].

A number of problems arising in science and engineering are set in semi-infinite domains. One can apply different spectral methods that are used to solve problems in semi-infinite domains. The first approach is using Laguerre polynomials [17–20]. The second approach is replacing semi-infinite domain with [0,𝐿] interval by choosing 𝐿, sufficiently large. This method is named domain truncation [21]. The third approach is reformulating original problem in semi-infinite domain to singular problem in bounded domain by variable transformation and then using the Jacobi polynomials to approximate the resulting singular problem [22]. The fourth approach of spectral method is based on rational orthogonal functions.

Boyd [23] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Guo et al. [24] introduced a new set of rational Legendre functions which are mutually orthogonal in 𝐿²(0,+∞). They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. [25] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre, and mapped Fourier sine.

The authors of [26–29] applied spectral method to solve nonlinear ordinary differential equations on semi-infinite intervals. Their approach was based on a rational Tau method. They obtained the operational matrices of derivative and product of rational Chebyshev and Legendre functions and then they applied these matrices together with the Tau method to reduce the solution of these problems to the solution of system of algebraic equations.

Zarebnia and Ali Abadi [30] used Sinc-Collocation method for solving system of nonlinear second-order integrodifferential equations of the Fredholm type. Rational second (third) kind Chebyshev (RSC) functions, for the first time, were proposed by Tavassoli Kajani and Ghasemi Tabatabaei [31] to find the numerical solution of Lane-Emden equation.

This paper outlines the application of rational second kind Chebyshev functions and Galerkin method to the following system of linear high-order integrodifferential equations on the interval [0,∞). Two problems of such equations are solved to make clear the application of the proposed method. One has ^𝑙_𝑖=1^𝑚_𝑗=0𝜈_𝑝𝑖𝑗(𝑥)𝑦_𝑖^(𝑗)(𝑥)+𝜆_𝑖𝑝^𝑏_𝑎𝑘_𝑖𝑝(𝑥,𝑡)𝑦_𝑖(𝑡)𝑑𝑡=𝑓_𝑝(𝑥)𝑦,𝑝=1,2,…,𝑙,_𝑖(𝑗)(0)=𝑦_𝑖𝑗[),,𝑖=1,2,…,𝑙,𝑗=0,1,…,𝑚−1,𝑥∈0,∞ (1.1) where 0≤𝑎<𝑏<∞.

2. Properties of RSC Functions

In this section, we present some properties of RSC functions.

2.1. RSC Functions

The second kind Chebyshev polynomials 𝑈_𝑛(𝑥),𝑛≥0, are orthogonal in the interval [−1,1] with respect to the weight function √1−𝑥² and we find that 𝑈_𝑛(𝑥) satisfies the recurrence relation [32] 𝑈₀(𝑥)=1,𝑈₁(𝑥)𝑈=2𝑥,_𝑛(𝑥)=2𝑥𝑈_𝑛−1(𝑥)−𝑈_𝑛−2(𝑥),𝑛≥2. (2.1) The RSC functions are defined by [31, 33] 𝑅_𝑛(𝑥)=𝑈_𝑛𝑥−1,𝑥+1 (2.2) thus RSC functions satisfy 𝑅₀(𝑥)=1,𝑅₁(𝑥)=2𝑥−1,𝑅𝑥+1_𝑛(𝑥)=2𝑥−1𝑅𝑥+1^𝑛−1(𝑥)−𝑅_𝑛−2(𝑥),𝑛≥2. (2.3)

2.2. Function Approximation

Let 𝑤(𝑥)=4√𝑥/(𝑥+1)³ denotes a nonnegative, integrable, real-valued function over the interval 𝐼=[0,+∞). We define 𝐿²_𝑤(𝐼)=𝑦∶𝐼→ℝ∣𝑦ismeasurableand𝑦_𝑤<∞, (2.4) where 𝑦_𝑤=^∞₀𝑦(𝑥)²𝑤(𝑥)𝑑𝑥^1/2 (2.5) is the norm induced by the scalar product 𝑦,𝑧_𝑤=^∞₀𝑦(𝑥)𝑧(𝑥)𝑤(𝑥)𝑑𝑥. (2.6) Thus {𝑅_𝑛(𝑥)}_𝑛≥0 denote a system which is mutually orthogonal under (2.6), that is, ^∞₀𝑅_𝑛(𝑥)𝑅_𝑚(𝑥)𝑤(𝑥)𝜋𝑑𝑥=2𝛿_𝑛𝑚, (2.7) where 𝛿_𝑛𝑚 is the Kronecker delta function. This system is complete in 𝐿²_𝑤(𝐼); as a result, any function 𝑦∈𝐿²_𝑤(𝐼) can be expanded as follows: 𝑦(𝑥)=^∞_𝑘=0𝑦_𝑘𝑅_𝑘(𝑥), (2.8) with 𝑦_𝑘=2𝜋𝑦,𝑅_𝑘𝑤. (2.9) The 𝑦_𝑘’s are the expansion coefficients associated with the family {𝑅_𝑘(𝑥)}. If the infinite series in (2.8) is truncated, then it can be written as 𝑦(𝑥)≃𝑦_𝑁(𝑥)=^𝑁_𝑘=0𝑦_𝑘𝑅_𝑘(𝑥)=𝑌^𝑇𝑅(𝑥), (2.10) where 𝑌=[𝑦₀,𝑦₁,…,𝑦_𝑁]^𝑇 and 𝑅(𝑥)=[𝑅₀(𝑥),𝑅₁(𝑥),…,𝑅_𝑁(𝑥)]^𝑇.

We can also approximate the function 𝑘(𝑥,𝑡) in 𝐿²_𝑤(𝐼×𝐼) as follows: 𝑘(𝑥,𝑡)≃𝑘_𝑁(𝑥,𝑡)=𝑅^𝑇(𝑥)𝐾𝑅(𝑡), (2.11) where 𝐾 is an (𝑁+1)×(𝑁+1) matrix that 𝐾_𝑖𝑗=2𝜋²𝑅_𝑖(𝑥),𝑘(𝑥,𝑡),𝑅_𝑗(𝑡),𝑖,𝑗=0,1,…,𝑁. (2.12) Moreover, from recurrence relation in (2.3) we have 𝑅(0)=1,−2,3,−4,…,(−1)^𝑁(𝑁+1)^𝑇=𝐞₁. (2.13)

2.3. Product Integration of the RSC Functions

We also use the matrix 𝑃_𝑎𝑏 as follows: 𝑃_𝑎𝑏=^𝑏_𝑎𝑅(𝑡)𝑅^𝑇(𝑡)𝑑𝑡. (2.14) To illustrate the calculation 𝑃_𝑎𝑏 we choose 𝑎=0 and 𝑏=1, then we obtain 𝑃₀₁=12−4ln211−16ln228−40ln21673−80ln2⋯2−4ln212−16ln230−44ln22003−96ln23743−180ln2⋯11−16ln230−44ln22033−96ln23803−184ln2332915−320ln2⋯28−40ln22003−96ln23803−184ln2334415−320ln2539615−520ln2⋯1673−80ln23743−180ln2332915−320ln2539615−520ln21945135−800ln2⋯⋮⋮⋮⋮⋮⋱. (2.15)

2.4. Operational Matrix of Derivative

The derivative of the vector 𝑅(𝑥) defined in (2.10) can be approximated by 𝑅^′(𝑥)≃𝐷𝑅(𝑥), (2.16) where 𝐷 is named the 𝑛×𝑛 operational matrix of derivative. Differentiating (2.3) we get 𝑅^′₀(𝑥)=0,𝑅^′₁(𝑥)=54𝑅₀(𝑥)−𝑅₁(𝑥)+14𝑅₂(𝑥),𝑅^′_𝑛(𝑥)=𝑅₁(𝑥)𝑅_𝑛−1(𝑥)^′−𝑅^′_𝑛−2(𝑥),𝑛≥2. (2.17) By using (2.17) the matrix 𝐷 can be calculated. The matrix 𝐷 is a lower Hessenberg matrix and can be expressed as 𝐷=𝐷₁+𝐷₂, where 𝐷₁ is a tridiagonal matrix which is obtained from 𝐷₁=diag−2+7𝑖4𝑖,−𝑖,4,𝑖=0,…,𝑛−1, (2.18) and the 𝑑_𝑖𝑗 elements of matrix 𝐷₂ are obtained from 𝑑_𝑖𝑗=()0,𝑖≤𝑗+1,−1^𝑖+𝑗+12𝑗,𝑖>𝑗+1. (2.19)

2.5. The Product Operational Matrix

The following property of the product of two rational Chebyshev vectors will also be used: 𝑅(𝑥)𝑅^𝑇(𝑥)𝑌=𝑌𝑅(𝑥), (2.20) where 𝑌 is called (𝑁+1)×(𝑁+1) product operational matrix for the vector 𝑌. Using (2.20) and the orthogonal property, the elements 𝑌_𝑖𝑗,𝑖=0,…,𝑁,𝑗=0,…,𝑁 of the matrix 𝑌 can be calculated from 𝑌_𝑖𝑗=2𝜋^𝑁_𝑘=0𝑐_𝑘𝑔_𝑖𝑗𝑘, (2.21) where 𝑔_𝑖𝑗𝑘 is given by 𝑔_𝑖𝑗𝑘=^∞₀𝑅_𝑖(𝑥)𝑅_𝑗(𝑥)𝑅_𝑘(𝑥)𝑤(𝑥)𝑑𝑥. (2.22)

3. Solving System of Integrodifferential Equations over Semi-Infinite Interval

Consider the following system of integrodifferential equations: ^𝑙_𝑖=1^𝑚_𝑗=0𝜈_𝑝𝑖𝑗(𝑥)𝑦_𝑖^(𝑗)(𝑥)+𝜆_𝑖𝑝^𝑏_𝑎𝑘_𝑖𝑝(𝑥,𝑡)𝑦_𝑖(𝑡)𝑑𝑡=𝑓_𝑝(𝑥)𝑦,𝑝=1,2,…,𝑙,_𝑖(𝑗)(0)=𝑦_𝑖𝑗[).,𝑖=1,2,…,𝑙,𝑗=0,1,…,𝑚−1,𝑥∈0,∞ (3.1) Using (2.10) and (2.11) to approximate 𝑦_𝑖, 𝑓_𝑝, 𝑘_𝑖𝑝, and 𝜈_𝑝𝑖𝑗 when 𝑖,𝑝=1,2,…,𝑙 and 𝑗=0,…,𝑚, we have 𝑦_𝑖(𝑥)≃𝑌^𝑇_𝑖𝑅(𝑥),𝑓_𝑝(𝑥)≃𝐹^𝑇_𝑝𝑅(𝑥),𝑘_𝑖𝑝(𝑥,𝑡)≃𝑅^𝑇(𝑡)𝐾_𝑖𝑝𝑅(𝑥),𝜈_𝑝𝑖𝑗(𝑥)≃𝑉^𝑇_𝑝𝑖𝑗𝑅(𝑥). (3.2) According to the operational matrix of derivative we can approximate 𝑦_𝑖^(𝑗) as 𝑦_𝑖^(𝑗)(𝑥)≃𝑌^𝑇_𝑖𝑅^(𝑗)(𝑥)≃𝑌^𝑇_𝑖𝐷^𝑗𝑅(𝑥),𝑦_𝑖^(𝑗)(0)≃𝑌^𝑇_𝑖𝐷^𝑗𝑅(0)=𝑌^𝑇_𝑖𝐷^𝑗𝐞₁. (3.3) With substituting these approximations in (3.1) we have ^𝑙_𝑖=1^𝑚_𝑗=0𝑌^𝑇_𝑖𝐷^𝑗𝑅(𝑥)𝑅^𝑇(𝑥)𝑉_𝑝𝑖𝑗+𝜆_𝑖𝑝^𝑏_𝑎𝑌^𝑇_𝑖𝑅(𝑡)𝑅^𝑇(𝑡)𝐾_𝑖𝑝𝑅(𝑥)𝑑𝑡=𝐹^𝑇_𝑝𝑅(𝑥),𝑌^𝑇_𝑖𝐷^𝑗𝐞₁=𝑦_𝑖𝑗,𝑖=1,2,…,𝑙,𝑗=0,1,…,𝑚−1,𝑝=1,2,…,𝑙. (3.4) Then using (2.20) we obtain ^𝑙_𝑖=1^𝑚_𝑗=0𝑌^𝑇_𝑖𝐷^𝑗𝑉_𝑝𝑖𝑗𝑅(𝑥)+𝜆_𝑖𝑝𝑌^𝑇_𝑖^𝑏_𝑎𝑅(𝑡)𝑅^𝑇(𝑡)𝑑𝑡𝐾_𝑖𝑝𝑅(𝑥)=𝐹^𝑇_𝑝𝑅(𝑥),𝑌^𝑇_𝑖𝐷^𝑗𝐞₁=𝑦_𝑖𝑗,𝑖=1,2,…,𝑙,𝑗=0,1,…,𝑚−1,𝑝=1,2,…,𝑙, (3.5) which can be simplified using (2.14) ^𝑙_𝑖=1^𝑚_𝑗=0𝑌^𝑇_𝑖𝐷^𝑗𝑉_𝑝𝑖𝑗+𝜆_𝑖𝑝𝑌^𝑇_𝑖𝑃_𝑎𝑏𝐾_𝑖𝑝=𝐹^𝑇_𝑝𝑌,𝑝=1,2,…,𝑙,_𝑇𝑖𝐷^𝑗𝐞₁=𝑦_𝑖𝑗,𝑖=1,2,…,𝑙,𝑗=0,1,…,𝑛−1. (3.6) By solving this linear system of algebraic equations we can find vectors 𝑌_𝑖, 𝑖=1,2,…,𝑛, and then approximate the solutions 𝑦_𝑖(𝑥) as 𝑦_𝑖(𝑥)≃𝑌^𝑇_𝑖𝑅(𝑥). (3.7)

4. Numerical Examples

Example 4.1. Consider the following system of linear integrodifferential equations: 1𝑦𝑥+1^1′′(𝑥)+𝑦^′₁(𝑥)+𝑦₂^′′(𝑥)+¹₀𝑦₁(𝑡)+𝑦₂(𝑡)(𝑥+1)²(𝑡+1)²𝑑𝑡=𝑥−154(𝑥+1)³,𝑦^′₁(𝑥)−2𝑦^′₂(𝑥)+𝑦₂^′′(𝑥)+24¹₀𝑦₁(𝑡)(𝑥+1)²(𝑡+1)²+𝑦₂(𝑡)(𝑥+1)³(𝑡+1)³𝑑𝑡=8𝑥−1(𝑥+1)³,𝑦₁(0)=1,𝑦^′₁(0)=0,𝑦₂(0)=−1,𝑦^′₂(0)=2. (4.1)The exact solution of this example is 𝑦₁(𝑥)=1 and 𝑦₂(𝑥)=(𝑥−1)/(𝑥+1).
We solved Example 4.1 using the present method with 𝑁=3, and we obtained 𝑌^𝑇₁=[1,0,0,0] and 𝑌^𝑇₂=[0,0.5,0,0], which imply 𝑦₁(𝑥)=𝑌^𝑇₁𝑅(𝑥)=1,𝑦₂(𝑥)=𝑌^𝑇₂𝑅(𝑥)=𝑥−1𝑥+1 (4.2) that are the exact solutions.

Example 4.2. Next, consider the following system of integrodifferential equations with the exact solution 𝑦₁(𝑥)=𝑒^−𝑥 and 𝑦₂(𝑥)=𝑒^−2𝑥: 𝑦₁^′′(𝑥)+𝑦^′₁(𝑥)+𝑦₂^′′(𝑥)+2𝑦^′₂(𝑥)+¹₀6𝑒^{−𝑡−𝑥}𝑦₁(𝑡)+𝑦₂(𝑡)𝑑𝑡=5−3𝑒⁻²−2𝑒⁻³𝑒^−𝑥,4𝑒^−𝑥𝑦₁^′′(𝑥)−𝑦₂^′′(𝑥)+¹₀12𝑒^{−2(𝑡+𝑥)}𝑦₁(𝑡)+𝑦₂(𝑡)𝑑𝑡=5−4𝑒⁻³−3𝑒⁻⁴𝑒^−2𝑥,𝑦₁(0)=1,𝑦^′₁(0)=−1,𝑦₂(0)=1,𝑦^′₂(0)=−2. (4.3) We solved this example by using the method described in Section 3 for 𝑁=11 and 𝑁=13. Results are shown in Table 1 and Figures 1 and 2. The errors for large values of 𝑥 are shown in Table 2. It is seen that the proposed method provides accurate results even for large values of 𝑥.

Table 1 

Numerical results of Example 4.2.

Table 2 

Absolute error for large values of 𝑥.

Figure 1 

Graph of the exact and numerical solutions of 𝑦1(𝑥) for 𝑁=13; symbols correspond to the numerical solution.

Figure 2 

Graph of the exact and numerical solutions of 𝑦2(𝑥) for 𝑁=13; symbols correspond to the numerical solution.

5. Conclusion

The fundamental goal of this paper has been to construct an approximation to the solution of the integrodifferential equations system in a semi-infinite interval. In the above discussion, the Galerkin method with RSC functions, which have the property of orthogonality, is employed to achieve this goal. Advantages of this method is that we do not reform the problem to a finite domain, and with a small value of 𝑁 accurate results are obtained. There is a good agreement between obtained results and exact values that demonstrates the validity of the present method for this type of problems and gives the method a wider applicability.