MODIFIED CHEBYSHEV (VIETA–LUCAS POLYNOMIAL) COLLOCATION 
METHOD FOR SOLVING DIFFERENTIAL EQUATIONS 

1M. Ziaul Arif, 2Ahmad Kamsyakawuni, 3Ikhsanul Halikin 
1 Department of Mathematics, Faculty of Sciences, University of Jember 

Email: ziaul.fmipa@unej.ac.id 

 

Abstract 
 This paper presents derivation of alternative numerical scheme for solving differential equations, 

which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The 
Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both 
Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases.  Finally, 
the performance of the proposed method is compared with finite difference method and the exact 
solution of the example. It is shown that modified Chebyshev collocation method more effective 
and accurate than FDM for some example given. 

 
 Keywords: Vieta-Lucas Polynomial, collocation method, Ordinary Differential Equations, Partial 

Differential Equations, Finite difference methods 

  
 

INTRODUCTION 

Chebyshev polynomials have become 

alternatively numerical solution for differential 

problems, which has developed after digital 

computerizing growth rapidly. It was discovered 

by Chebyshev and developed by some 

mathematicians; Clenshaw [1], Nidekker, Fox 

and Parker [2], etc. after computer was 

discovered. 

It is known that there are four kinds of 

Chebyshev polynomial as developed that 

commonly applied in numerical analysis [3]. For 

the first through four kinds, it is symbolized by 

𝑇𝑁(𝑥),𝑈𝑁(𝑥),𝑉𝑁(𝑥), 𝑎𝑛𝑑 𝑊𝑁(𝑥) respectively. 

First and second kind of Chebyshev polynomial 

had practiced to solving linear differential 

equation [4, 5]. Furthermore, the other kinds 

were used to find solution of some problems 

numerically [6-9]. However, for some reasons of 

the advantages, modified Chebyshev and its 

statistical properties was discovered by Witula 

and Damian and symbolized by  Ω𝑁(𝑥) [10]. 

The main objective of the present article is 

to develop algorithms based on modified 

Chebyshev polynomial (Vieta-Lucas Polynomial) 

[10] for solving differential equations. And In 

this paper, it is practiced for some DEs example 

and compared with finite difference method. 

The contents of this article are 

organized as follows. In Section 2, we discuss 

about a literature review of methods, FDM and 

modified Chebyshev collocations method. In 

addition, we discuss discovering of 

Differentiation matrices of Modified Chebyshev 

(Vieta-Lucas Polynomial) collocation and the 

result. Finally, some concluding remarks are 

presented in Section 3. 

A LITERATURE REVIEW 

Finite Difference Method 

Finite difference method is a superior 

method to approach the derivative of a function 

based on the Taylor series. The accuracy of 

finite difference method can be adapted to 

taking into first part account of the Taylor 

series. In this article, the first and second 

derivatives are approximated using central 

difference (centered difference approximation) 

second order as follows [11]; 

 

𝑓′(𝑥) ≈
𝑓(𝑥+ℎ)−𝑓(𝑥−ℎ)

2ℎ
+ 𝛰(ℎ2)                             (1)    

𝑓′′(𝑥) ≈
𝑓(𝑥+ℎ)−2𝑓(𝑥)+𝑓(𝑥−ℎ)

ℎ2
+𝛰(ℎ2)                   (2) 

𝑓(3)(𝑥) ≈
𝑓(𝑥+2ℎ)−2𝑓(𝑥+ℎ)+2𝑓(𝑥−ℎ)−𝑓(𝑥−2ℎ)

2ℎ3
+ 𝛰(ℎ2) (3) 

 

mailto:ziaul.fmipa@unej.ac.id


Modified Chebyshev Collocation Method for Solving Differential Equations 

 

CAUCHY – ISSN: 2086-0382 29 
 

By this scheme, some of differential equations 

examples shall be solve and compared with the 

exact solution. 

Chebyshev Polynomial Collocation Method 

Chebyshev collocation is one of spectral 

methods based on trial functions Chebyshev 

polynomial. Many practical problems are 

performed by mathematician using Chebyshev 

collocation [1, 4, 5, 7-9].  This is because some 

advantages such as accurate and memory 

minimizing. There are several kinds of 

Chebyshev polynomial describe below, 

First kind, 

𝑇𝑁(𝑥) = cos (𝑁𝜃).                        (4) 

Second kind, 
  

𝑈𝑁(𝑥) =
𝑠𝑖𝑛((𝑁+1)𝜃)

sin (𝜃)
.                      (5) 

Third kind, 

𝑉𝑁(𝑥) =
cos((𝑁+

1
2
)𝜃)

cos (
1
2
𝜃)

.                      (6) 

Fourth kinds, 

𝑊𝑁(𝑥) =
sin((𝑁+

1
2
)𝜃)

sin (
1
2
𝜃)

.                      (7) 

When 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑥) [3] and 𝑥 is collocation 

points. 

Let we have boundary value problem on 

range [-1,1]: 

𝑑2

𝑑𝑥2
𝑢(𝑥) = 𝑓(𝑥),       𝑢(−1) = 𝑎, 𝑢(+1) = 𝑏,     

(8) 

where 𝑓(𝑥) is a function and boundary value is 

given above. 

Suppose that we approximate 𝑢(𝑥) by, 

     𝑢𝑁(𝑥) = ∑ 𝑐𝑘
𝑁
𝑘=0 𝜑𝑘(𝑥)                (9) 

It is involving 𝑛 +1 unknown 

coefficients {𝑐𝑘}, and 𝜑𝑘(𝑥) is a basis function; 

in this case, we perform Chebyshev polynomial 

as a basis function. Then we may select 𝑛 − 1 

points {𝑥1,…,𝑥𝑛−1} in the range of integration 

called collocation points and require 𝑢𝑁(𝑥) 

satisfying the differential equation (8). This 

requires us to solve just the system of 𝑛+ 1  

linear equations by differentiation matrix of the 

method. It is well-known as Chebyshev 

collocation method.  

Modified Chebyshev Collocation Method 

Furthermore, in present paper, we perform first 

kind of modified Chebyshev polynomial as a 

basis function, 

    Ω𝑁(𝑥) = 2𝑇𝑁(
𝑥

2
)                         (10) 

where 𝑇𝑁(𝑥) is the first Chebyshev Polynomial. 

Some properties of Modified Chebyshev 

polynomial presented by Witula, et all [3]. It is 

also called n-th Vieta-Lucas Polynomial.  

 

RESULTS AND DISCUSSION 

Finite Difference Scheme 

To approximate first, second and third 

derivative, we employ central difference second 

order as seen in equation (1)-(3). In this part, 

we can perform its differentiation matrices 

scheme of FDM equations (1)-(3) respectively 

as follows,  

     𝐷𝑁,1 =
1

2ℎ

(

 
 

0 1 0 … 0
−1
0
⋮

0
−1

1 … 0
⋱ ⋮

⋱ 1
0 … −1 0)

 
 

    (11.a) 

  𝐷𝑁,2 =
1

ℎ2

(

 
 

−2 1 0 … 0
1
0
⋮

−2
1

1 … 0
⋱ ⋮

⋱ 1
0 … 1 −2)

 
 

    (11.b) 

         𝐷𝑁,3 =
1

2ℎ3

(

 
 

0 −2 −1 … 0
2
1
⋮

0
2

−2 … ⋮
⋱ ⋱ −1
⋱ ⋱ −2

0 … 1 2 0 )

 
 

     (11.c) 

However, for solving PDEs, the system 

of ODE, which is generated by FDM, can be 

advanced in time by forward Euler scheme or a 

third order Runge-Kutta scheme. 

 



 
M. Ziaul Arif,  Ahmad Kamsyakawuni,  Ikhsanul Halikin 

 

30 Volume 3 No. 4 Mei 2015 

 

Modified Chebyshev (Vieta-Lucas 

Polynomial) Differentiation Matrices 

Basically, solving DEs cannot be denied 

involving system of linear equations.  Hence, we 

need differentiation matrices derived by 

collocation method. In this section, we shall 

explain how to derive differentiation matrices 

of modified Chebyshev (Vieta-Lucas 

Polynomial) collocation method.  

 Firstly, we define Vieta-Lucas 

Polynomial as a basis function of collocation 

approximations. We have first kind of modified 

Chebyshev (Vieta-Lucas Polynomial) 

Collocation Method defined by Witula [3] as 

equation (10), consequently,  

Ω𝑁(𝑥) = 2.𝑐𝑜𝑠(𝑁.𝑎𝑟𝑐𝑐𝑜𝑠(
𝑥

2
))        [-2,2]  (12) 

Furthermore, discover zero and 

extrema points of first kind of modified 

Chebyshev (Vieta-Lucas Polynomial) used to 

construct differentiation matrices. We show that 

zero points of modified Chebyshev (Vieta-Lucas 

Polynomial) is  

 𝑥𝑘 = 2.𝑐𝑜𝑠(
2𝑘−1

2𝑁
)𝜋             𝑘 ∈ ℤ        (13) 

By first derivative, 
𝑑Ω𝑁(𝑥)

𝑑𝑥
= 0 , we discover the 

extrema points of modified Chebyshev as 

follows, 

𝑥𝑗 = 2.𝑐𝑜𝑠(
𝜋𝑗

𝑁
),       0 ≤ 𝑗 ≤ 𝑁         (14) 

It is well-known as first kind modified 

Chebyshev (Vieta-Lucas Polynomial) Collocation 

points. Considering the Chebyshev points above, 

we construct differentiation matrices of 

Modified Chebyshev Collocation method as 

follows. First, we need the following results: 

Ω𝑁(𝑥𝑗) = 2.𝑐𝑜𝑠(𝑁.𝑎𝑟𝑐𝑐𝑜𝑠(
𝑥𝑗

2
))   [-2,2] 

Ω′𝑁(𝑥𝑗) =
𝑁.sin (𝑁𝜃)

sin (𝜃)
= 0,     0 ≤ 𝑗 ≤ 𝑁 − 1   (15.a) 

 Ω′′𝑁(𝑥𝑗) =
(−1)𝑗+1.2.𝑁2

(4−𝑥𝑗
2)
,      0 ≤ 𝑗 ≤ 𝑁 −1     (15.b) 

Ω′′′𝑁(𝑥𝑗) =
(−1)𝑗+1.6.𝑁2𝑥𝑗

(4−𝑥𝑗
2)2

,    0 ≤ 𝑗 ≤ 𝑁 −1   (15.c) 

Ω′𝑁(±2) = (±1)
𝑁+1𝑁2   and 

  Ω′′𝑁(±2) =
(±1)𝑁𝑁2(𝑁2−1)

6   
(15.d) 

Since modified Chebyshev  Ω′𝑁(𝑥) is polynomial 

and we know all zeros points of it (called as 

Vieta-Lucas Collocation points) we may write, 

   Ω′𝑁(𝑥) = 𝛼Ω𝑁 ∏ (𝑥 − 𝑥𝑗)
𝑁−1
𝑗=1                 (16) 

For 𝑥0=−2 and  𝑥𝑁=2, we can write, 

     𝛾𝑁(𝑥) = ∏ (𝑥 −𝑥𝑗) = 𝛽𝑁(𝑥
2 −4)Ω′𝑁(𝑥)

𝑁
𝑗=0   (17) 

 
Where 𝛽𝑁 is a positive a constant such that 

coefficient  𝑥𝑁+1 equals to 1. 

From equation (15.a) and (15.d) and if  𝑥 = 𝑥𝑗 

and  0 ≤ 𝑗 ≤ 𝑁, it follows that 

𝛾′𝑁(𝑥) = 𝛽𝑁2𝑥Ω′𝑁(𝑥)+𝛽𝑁(𝑥
2 −4)Ω′′𝑁(𝑥)  (18) 

However, for 0 ≤ 𝑗 ≤ 𝑁 −1, Ω′𝑁(𝑥𝑗) = 0 and 

when 𝑗 = 0 and 𝑗 = 𝑁 as seen in equation (15.d), 

In consequence, we have  

    𝛾′𝑁(𝑥𝑗) = (−1)
𝑗𝑐�̃�𝑁

2𝛽𝑁                  (19) 

where 𝑐0̃ = �̃�𝑁 = 4 and 𝑐�̃� = 2  for 1 ≤ 𝑗 ≤ 𝑁 − 1. 

For an arbitrary function 𝑓(𝑥) and nodes 𝑥 = 𝑥𝑗, 

we can estimate the function using Lagrange’s 

interpolation polynomial below, 

𝑝(𝑥) = ∑ 𝐹𝑗(𝑥).𝑓(𝑥)
𝑁
𝑗=0                    (20)  

where  

           𝐹𝑗(𝑥) =
𝛾𝑁(𝑥)

𝛾′𝑁(𝑥𝑗)(𝑥−𝑥𝑗)
  ,0 ≤ 𝑗 ≤ 𝑁         (21) 

since  𝛾𝑁(𝑥) and 𝛾
′
𝑁
(𝑥𝑗) 

are described by (17) 

and (19) respectively. By definition of 

differentiation matrix, hence we obtain 

differentiation matrix below, 

𝐷𝑘𝑗
1 = 𝐹′𝑗(𝑥𝑘) 

The element of the matrices can be evolved from 

first derivative of 𝐹(𝑥𝑗) with some point’s 

defined equation (14). Scheme below is 

determining the element of differentiation 

matrices; 

1. For 1 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁 −1  for equation (21), 

together with (15.a) and (15.d), leads to 

𝐹′𝑗(𝑥𝑘) =
(−1)𝑗+𝑘

𝑐�̃�(𝑥𝑘−𝑥𝑗)
                  (22.a) 



Modified Chebyshev Collocation Method for Solving Differential Equations 

 

CAUCHY – ISSN: 2086-0382 31 
 

2. For 𝑗 ≠ 𝑘 and 𝑘 = 0, 1 ≤ 𝑗 ≤ 𝑁 −1 

           𝐹′𝑗(𝑥0) =
(−1)𝑗.4

𝑐̃𝑗(𝑥0−𝑥𝑗)
                  (22.b) 

 

3.  For 𝑗 ≠ 𝑘
 
and 𝑗 = 𝑁, 1 ≤ 𝑘 ≤ 𝑁 −1 

       𝐹′𝑁(𝑥𝑘) =
(−1)𝑁+𝑘.2

𝑐�̃�(𝑥𝑘−𝑥𝑁)
               (22.c) 

4. For 𝑗 ≠ 𝑘
 
and 𝑘 = 𝑁, 1 ≤ 𝑗 ≤ 𝑁 −1  

   𝐹′𝑗(𝑥𝑁) =
(−1)𝑗+𝑁.4

𝑐̃𝑗(𝑥𝑁−𝑥𝑗)
                 (22.d) 

5. For 𝑗 ≠ 𝑘
 
and 𝑗 = 0, 1 ≤ 𝑘 ≤ 𝑁 −1  

   𝐹′0(𝑥𝑘) =
(−1)𝑘.2

𝑐0̃(𝑥𝑘−𝑥0)
                    (22.e) 

6.  For 𝑗 ≠ 𝑘,
 
and 𝑘 = 0,𝑗 = 𝑁,  

         𝐹′𝑁(𝑥0) =
1

𝑐�̃�
                                 (22.f) 

7. For 𝑗 ≠ 𝑘,
 
and 𝑗 = 0,𝑘 = 𝑁,  

   𝐹′0(𝑥𝑁) = −
(−1)𝑁

𝑐0̃
                        (22.g) 

From equation (22.a) to (22.g), in general, for 

0 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁, we have 

    𝐹′𝑗(𝑥𝑘) =
(−1)𝑗+𝑘�̃�𝑘

𝑐�̃�(𝑥𝑘−𝑥𝑗)
                           (23) 

where 𝑐0̃ = 𝑐�̃� = 4 and 𝑐�̃� = 2. 

Furthermore, by applying L’Hospital for 𝑗 = 𝑘 =

0 we have, 

         𝐹′0(𝑥0) = (
1+2𝑁2

6
)                      (24)        

and for 𝑗 = 𝑘 = 𝑁 

             𝐹′𝑁(𝑥𝑁) = −(
1+2𝑁2

6
)                  (25)              

Again using L’Hospital’s rule for  1 ≤ 𝑗 = 𝑘 ≤ 𝑁 −1
 
 

shows that 

             𝐹′𝑗(𝑥𝑗) =
−𝑥𝑗

𝑐̃𝑗(4−𝑥𝑗
2)

                      (26) 

Vieta-Lucas Differentiation Matrices 

For each 𝑁 ≥ 1, modified Chebyshev (Vieta-

Lucas Polynomial) collocation differentiation 

matrix is defined below 

 (𝐷𝑁
1)00 = (

1+2𝑁2

6
)  (27.a) 

(𝐷𝑁
1)𝑁𝑁 = −(

1+2𝑁2

6
)              (27.b) 

(𝐷𝑁
1)𝑘𝑘 =

−𝑥𝑘

2(4−𝑥𝑘
2)
, 1 ≤ 𝑘 ≤ 𝑁 −1

 
(27.c) 

   (𝐷𝑁
1)𝑘𝑗 =

(−1)𝑗+𝑘𝑐̃𝑘

𝑐̃𝑗(𝑥𝑘−𝑥𝑗)
, 0 ≤ 𝑗 ≠ 𝑘 ≤ 𝑁

 
(27.d) 

where 𝑐0̃ = �̃�𝑁 = 4 and 𝑐�̃� = 2. 

Furthermore, for high order differential 

problem, modified Chebyshev (Vieta-Lucas 

Polynomial) differentiation matrices can be 

applied by define high order modified 

Chebyshev (Vieta-Lucas Polynomial) 

differentiation as power of matrix below, 

   𝐷𝑁
𝑚 = (𝐷𝑁

1)𝑚                              (28) 

 

Examples and Solutions 

For comparison the performance of Vieta-Lucas 

Collocation scheme and FDM, some of example 

are given below. 

 

Table 1. Example of Boundary and initial value 

of differential equations problems. 

No Example 

1 
𝑦(2) + 𝑦 = 𝑥2 + 𝑥 

𝑦(−2) = 𝑦(2) = 0,𝑦′(−2) = 2.285 

2 
𝑦(2) −𝑦(−1) = sin (𝑥) 

𝑦(−2) = 𝑦(2) = 0,𝑦′(−2) = 0.67 

3 

𝑢𝑡 + 𝑢𝑥 = 0,       𝑥 ∈ [−1,1] 

𝑢(𝑥,0) = sin (𝜋𝑥) 

𝑢(−1,𝑡) = sin (𝜋(−1− 𝑡)) 

 

NUMERICAL EXPERIMENT 

Numerical result below is comparison belong to 

the Vieta-Lucas collocation, FDM and its exact 

solution. 

Example of ODE no 1 can be solved by Vieta-

Lucas Collocation scheme as equation 𝐷𝑁
2𝑢𝑁 +

𝑢𝑁 = 𝐹. Where 𝐹 = (𝑋0,𝑋1,…,𝑋𝑁), X  is right 

hand side of the example of ODE no 1 and N=64. 

While for FDM, we perform the second order of 

FDM and N=400 to solve the example. The result 

of both scheme is compared with the exact 

solution as table and the picture below, 



 
M. Ziaul Arif,  Ahmad Kamsyakawuni,  Ikhsanul Halikin 

 

32 Volume 3 No. 4 Mei 2015 

 

 
Figure 1. Numerical Solution of example 1 

Table 2. Error of the example 1 

Scheme Error 

Vieta-Lucas 2.3 x 10−6 

FDM 1.5 x 10−2 

 

Furthermore, for ODE example no 2, Vieta-Lucas 

Scheme can be write as  𝐷𝑁
2𝑢𝑁 +𝐷𝑁𝑢𝑁 = 𝐹 

and 

N=64. Whereas FDM is performed with N=400. 

 
Figure 2. Numerical Solution of example 2 

Table 3. Error of the example 2 

Scheme Error 

Vieta-Lucas 3.9 x 10−5 

FDM 3.5 x 10−2 

 

In addition, the boundary and initial problem of 

linear PDE is solved by Vieta-Lucas Collocation 

for spatial part with N=64, then it is evaluate by  

forward Euler scheme for time differencing as 

equation 

 u𝑁(𝑡
𝑖+1) = 𝐮𝑁(𝑡

𝑖)+∆𝑡 𝐷𝑁𝐮𝑁(𝑡
𝑖),    𝐮(𝑡0) = 𝑠𝑖𝑛 (𝜋𝑋𝑁).  

Moreover, second order of FDM with N=400 is 

applied to solve it with time differencing as 

Vieta-Lucas Collocation. 

 

Figure 3. Numerical Solution of example 3 

CONCLUSION 

The problem of solving differential equations 

occurs frequently in mathematical work. In this 

paper, we have introduced modified Chebyshev 

collocation scheme called Vieta-Lucas 

collocation to for solving differential equation. 

Numerical Experiment shows that Vieta-Lucas 

Collocation scheme s more effective and 

accurate than FDM even with small number of N. 

Meanwhile, the scheme introduced in this paper 

can be used to more class of nonlinear 

differential equations 

REFERENCES 

[1] C. Clenshaw, "The numerical solution of 
linear differential equations in Chebyshev 
series," in Mathematical Proceedings of 
the Cambridge Philosophical Society, 1957, 
pp. 134-149. 

[2] I. Nidekker, "Chebyshev polynomials in 
numerical analysis: L. Fox and IB Parker. 
Oxford University Press, Oxford, London, 
1968, ix+ 205 pp," Zhurnal Vychislitel'noi 
Matematiki i Matematicheskoi Fiziki, vol. 
9, pp. 491-491, 1969. 

[3] J. C. Mason and D. C. Handscomb, 
Chebyshev polynomials: CRC Press, 2010. 

[4] A. Akyüz and M. Sezer, "Chebyshev 
polynomial solutions of systems of high-
order linear differential equations with 
variable coefficients," Applied 
mathematics and computation, vol. 144, 
pp. 237-247, 2003. 

[5] A. Akyüz and M. sezer, "A Chebyshev 
collocation method for the solution of 
linear integro-differential equations," 

1.4 1.5 1.6 1.7 1.8 1.9 2

-0.2

-0.15

-0.1

-0.05

0

0.05

x

y
(x

)
Example of ODE no 1

 

 

FDM

Exact SOlution

Vieta-Lucas Coll

0.6 0.8 1 1.2 1.4 1.6 1.8

-0.15

-0.1

-0.05

0

0.05

x

y
(x

)

Example of ODE no 2

 

 

FDM

Exact Solution

Vieta-Lucas Coll

-0.528 -0.526 -0.524 -0.522 -0.52 -0.518
-1

-0.995

-0.99

x

 u
(
x

,t
)

Numeric Vs Exact 

 

 

0 0.5 1 1.5 2 2.5 3 3.5 4
0

0.05

0.1

t

 E
r
r
o

r

 

 

Absolute Error 

FDM

Exact Solution

Vieta-Lucas

FDM

Vieta-Lucas



Modified Chebyshev Collocation Method for Solving Differential Equations 

 

CAUCHY – ISSN: 2086-0382 33 
 

International journal of computer 
mathematics, vol. 72, pp. 491-507, 1999. 

[6] M. Y. Hussaini, C. L. Streett, and T. A. Zang, 
"Spectral methods for partial differential 
equations," II. s. ARMY RESEARCH OFFICE, 
p. 883, 1984. 

[7] J. Mason, "Chebyshev polynomials of the 
second, third and fourth kinds in 
approximation, indefinite integration, and 
integral transforms," Journal of 
Computational and Applied Mathematics, 
vol. 49, pp. 169-178, 1993. 

[8] W. Abd-Elhameed, E. Doha, and Y. Youssri, 
"New wavelets collocation method for 
solving second-order multipoint 
boundary value problems using 

Chebyshev polynomials of third and 
fourth kinds," in Abstract and Applied 
Analysis, 2013. 

[9] E. Doha, W. Abd-Elhameed, and M. 
Alsuyuti, "On using third and fourth kinds 
Chebyshev polynomials for solving the 
integrated forms of high odd-order linear 
boundary value problems," Journal of the 
Egyptian Mathematical Society, 2014. 

[10] R. Wituła and D. Słota, "On modified 
Chebyshev polynomials," Journal of 
mathematical analysis and applications, 
vol. 324, pp. 321-343, 2006. 

[11] S. C. Chapra and R. P. Canale, Numerical 
methods for engineers: McGraw-Hill 
Higher Education New York,2010. 

 


	INTRODUCTION
	A LITERATURE REVIEW
	Finite Difference Method
	Chebyshev Polynomial Collocation Method
	Modified Chebyshev Collocation Method

	RESULTS AND DISCUSSION
	Finite Difference Scheme
	Modified Chebyshev (Vieta-Lucas Polynomial) Differentiation Matrices
	Vieta-Lucas Differentiation Matrices
	Examples and Solutions

	NUMERICAL EXPERIMENT
	CONCLUSION
	REFERENCES