^{1}

^{*}

^{1}

^{1}

Chebyshev collocation method is used to approximate solutions of two-point BVP arising in modelling viscoelastic flow. The scheme is tested on four nonlinear problems. The comparison with other methods is made. The results demonstrate the reliability and efficiency of the algorithm developed.

In modelling viscoelastic flows, differential equations of elliptic-hyperbolic operator types arise. Simulations of such flows have been studied extensively lately. The main characteristics of such elliptic-hyperbolic operators can be captured in a nonlinear fifth order two-point boundary value problem in one dimension [

Many researchers have discussed solutions of viscoelastic flow model, like the Galerkin method discussed in [

In this paper, we are going to introduce Chebyshev-collocation method for the numerical solution of the fifth-order non-linear two-point boundary value problem in modelling viscoelastic flow:

[ 1 + ε d y d x d d x ] d 4 y d x 4 = f ( x ) , − 1 2 ≤ x ≤ 1 2 (1)

concerned to the posterior boundary conditions

y ( ± 1 2 ) = d y d x ( ± 1 2 ) = 0 d 2 y d x 2 ( − 1 2 ) = c at ε > 0. (2)

with regard to the given positive constants ε and c which act as elasticity parameter and a boundary stress, respectively. Moreover, c is equal unity in this paper.

In recent years, a lot of attention has been devoted to the study of Chebyshev methods to investigate various scientific models. Using these methods made it possible to solve Troeschs problem [

Chebyshev methods for ordinary differential equations have many salient features due to the properties of the basis functions and the manner in which the problem is discretized. The approximating discrete system depends only on parameters of the differential equation. There are many advantages of using Chebyshev polynomials as expansion function presented in that are good representations of smooth functions. What we do here is to seek a special Chebyshev solution which also satisfies the given boundary conditions.

We organize our paper as follows. In Section 2, we present the preliminaries which we used in this paper. Method of the solution is given in Section 3. Some numerical results are presented in Section 4 to show the efficiency of the proposed method. Finally, we draw some conclusions and closing remarks.

Chebyshev polynomial formula of the first kind of degree m is chiefly defined and bounded in interval [ − 1,1 ] see [

T m ( x ) = cos ( m cos − 1 ( x ) ) , x = cos ϕ , ϕ ∈ [ 0 , π ]

or, in didactic organism,

T m ( cos ϕ ) = cos ( m ϕ ) , m = 0 , 1 , ⋯ , x ∈ [ − 1 , 1 ]

As for the shifted Chebyshev polynomial T ∗ ( x ) of the first kind on interval [ a , b ] .

T m ∗ ( x ) = T m ( q ) , q = 2 b − a ( x − a + b 2 ) .

With leading coefficient is equal to 2 m − 1 ( 2 b − a ) m , by Compensation in the previous equation

T m ∗ ( x ) = T m ( 2 b − a ( x − a + b 2 ) ) .

In addition, the definition of the collocation points is worded as follow

x i = b − a 2 [ ( a + b b − a ) + cos ( i π N ) ] , i = 0 , 1 , ⋯ , N . (3)

Moreover, the relation between Chebyshev coefficient matrix A and A ∗ ( k ) in the interval [ a , b ] is

A ∗ ( k ) = ( 4 b − a ) k M k A ∗

where

A ∗ = [ a 0 ∗ 2 , a 1 ∗ , ⋯ , a N ∗ ] τ

T = T ∗ result from the characteristics of Chebyshev polynomial.

All in all, the use of half interval − 1 2 ≤ x ≤ 1 2 is more favored in modelling viscoelastic flows. The shifted Chebyshev polynomials can also be worded as follows

T m ∗ ( x ) = T m ( 2 x ) = cos ( m cos − 1 ( 2 x ) ) .

This is deduced from definition of the collocation points:

x i = 1 2 [ cos ( i π N ) ] , i = 0 , 1 , ⋯ , N . (4)

Similarly, the relation between Chebyshev coefficient matrix A and A ∗ ( k ) in the interval [ − 1 2 , 1 2 ] is

A ∗ ( k ) = 4 k M k A ∗ , k = 0 , 1 , ⋯ , 5.

Let is assume the approximate solution y ( x ) of the main problem (1) and its derivatives is

y ( x ) = ∑ r = 0 N a r ∗ T r ∗ ( x ) , − 0.5 ≤ x ≤ 0.5 (5)

y ( k ) ( x ) = ∑ r = 0 N ( a r ∗ ) ( k ) T r ∗ ( x ) , k = 0 , 1 , 2 , ⋯ , 5. (6)

where N is chosen as any positive integer such that 0 ≤ r ≤ N . Besides, the anonymous Chebychev coefficients of y ( x ) and its derivatives are a r and a r ( k ) , respectively. The approximate solution and its derivatives in the matrix format are

[ y ( x ) ] = T ∗ ( x ) A ∗ [ y ( k ) ( x ) ] = T ∗ ( x ) ( A ∗ ) k = 4 k T ∗ ( x ) M k A ∗ . (7)

whereas the definitions of ultimate matrices are:

Y = [ y ( x 0 ) y ( x 1 ) ⋮ y ( x N ) ] , Y ( k ) = [ y ( k ) ( x 0 ) y ( k ) ( x 1 ) ⋮ y ( k ) ( x N ) ] , T = [ T ( x 0 ) T ( x 1 ) ⋮ T ( x N ) ]

and

M = [ 0 1 2 0 3 2 ⋯ 0 N / 2 0 0 2 0 ⋯ N − 1 0 0 0 0 3 ⋯ 0 N ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 0 0 ⋯ N 0 0 0 0 0 ⋯ 0 ] ( N + 1 ) × ( N + 1 ) (8)

for odd N.

M = [ 0 1 2 0 3 2 ⋯ N − 1 2 0 0 0 2 0 ⋯ 0 N 0 0 0 3 ⋯ N − 1 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 0 0 0 ⋯ 0 N 0 0 0 0 ⋯ 0 0 ] ( N + 1 ) × ( N + 1 ) (9)

for even N. We need the following lemma where l and K are both positive integer.

Lemma 1. [

[ y ( k ) ( x 0 ) y ( l ) ( x 0 ) y ( k ) ( x 1 ) y ( l ) ( x 1 ) ⋮ y ( k ) ( x N ) y ( l ) ( x N ) ] = [ y ( k ) ( x 0 ) 0 ⋯ 0 0 y ( k ) ( x 1 ) ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ y ( k ) ( x N ) ] [ y ( l ) ( x 0 ) y ( l ) ( x 1 ) ⋮ y ( l ) ( x N ) ] = Y ¯ ( k ) Y ( l ) = 4 k + l ( T * ¯ M ¯ k A * ¯ ) T * M l A * (10)

where

T * ¯ = [ T ( x 0 ) 0 ⋯ 0 0 T ( x 1 ) ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ T ( x N ) ] ,

A * ¯ = [ A * 0 ⋯ 0 0 A * ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A * ] , and M ¯ = [ M 0 ⋯ 0 0 M ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ M ] .

We need the following the theorem:

Theorem 2. If the considered approximate solution of the problem (1) is (7), so that the discrete Chebyshev system is availed by

W A ∗ = F , (11)

where

W = 4 4 T ∗ M 4 + ε 4 6 ( T * ¯ M ¯ A * ¯ ) T ∗ M 5

Proof. Replacing each term of (1) with the approximation defined in (7) and (10), and applying the collocation points (4) to it.

The matrices for the boundary conditions subjected to Equation (2) are

T ∗ ( 1 2 ) = 4 T ∗ ( 1 2 ) M = 0 , T ∗ ( − 1 2 ) = 4 T ∗ ( − 1 2 ) M = 0 , 4 2 T ∗ ( − 1 2 ) M 2 = c (12)

Thus, in the matrix [ W ; F ] we will replace 5_{th} rows by the Equation (12), we have the augmented matrix [ W ˜ ; F ˜ ]

W ˜ A ∗ = F ˜ . (13)

Now we will solve a linear system (13) of N + 1 equations of the N + 1 unknown coefficients a r , r = 0 , 1 , ⋯ , N . So as to gain the coefficients of the approximate solution A ∗ by the Q-R method.

Algorithm

· Input (integer) N.

· Input (double) tol.

· Input (array) A o l d = A 0 (Initial approximation, A 0 with N + 1 dimension, can be chosen so that the boundary conditions are satisfied).

· W ˜ ( A o l d ) A n e w = F ˜ is a linear algebraic equation system. Then solve this system to find A n e w .

· If | A o l d − A n e w | < t o l then A n e w = A , break (the program is finished).

· Else then A o l d ← A n e w .

In this section we give an illustrative example to authenticate the obtained results on Equation (1). The performance of Chebyshev method is measured by the root mean square errors E chebyshev which is defined as

‖ E Chebyshev ‖ = ∑ i = 0 N | y Exact ( x i ) − y Chebyshev ( x i ) | 2

Example 1: [

[ 1 + ε d y d x d d x ] d 4 y d x 4 = 12 , − 1 2 ≤ x ≤ 1 2 ,

whose exact solution is

y ( x ) = 1 2 ( x 2 − 1 4 ) 2 .

Example 2: [

[ 1 + ε d y d x d d x ] d 4 y d x 4 = − 120 x + 600 ε ( x 2 − 1 4 ) ( x 2 − 1 20 ) , − 1 2 ≤ x ≤ 1 2 ,

whose exact solution is

y ( x ) = − x ( x 2 − 1 4 ) 2 .

Method | The root mean square errors |
---|---|

Chebyshev, N = 5 | 1.743E-18 |

Chebyfun | 8.664E-18 |

Runge-Kutta methods, N = 100 [ | 6.899E-03 |

Beam function using Galerkin, N = 4 [ | 0.975E-04 |

Beam function using Collocation, N = 4 [ | 0.855E-03 |

ε | ‖ E Chebyshev ‖ , N = 5 | ‖ E Runge-Kutta ‖ , N = 100 [ | ‖ E Galerkin ‖ , N = 5 [ | ‖ E Collocation ‖ , N = 5 [ |
---|---|---|---|---|

10^{−1} | 4.7300E-13 | 3.98537E-03 | 0.25E-03 | 0.24E-02 |

10^{−2} | 1.7820E-13 | 3.98585E-03 | 0.16E-03 | 0.22E-02 |

10^{−3} | 1.2704E-16 | 3.98599E-03 | 0.16E-03 | 0.22E-02 |

10^{−4} | 2.0230E-16 | 3.98586E-03 | ---- | ---- |

Method | The root mean square errors |
---|---|

Chebyshev | 1.2704E-16 |

Chebyfun | 2.7800E-16 |

Example 3: [

[ 1 + ε d y d x d d x ] d 4 y d x 4 = f ( x ) , − 1 2 ≤ x ≤ 1 2 ,

whose exact solution is

y ( x ) = 1 2 π ( x 2 − 1 4 ) sin π ( x − 1 2 ) .

and

f ( x ) = π 2 [ π 2 ( x 2 − 1 4 ) − 12 ] sin π ( x − 1 2 ) − 4 π 2 x cos π ( x − 1 2 ) + 1 8 ε π 2 [ π 2 ( x 2 − 1 4 ) 2 − 40 ( x 2 − 1 8 ) ] cos 2 π ( x − 1 2 ) + 1 8 ε π x [ 12 π 2 ( x 2 − 1 4 ) − 40 ] sin 2 π ( x − 1 2 ) + 1 8 ε π 2 [ π 2 ( x 2 − 1 4 ) 2 + 5 ] .

Tables 4-7 illustrate the comparison between result of Chebyshev polynomial method and result of methods in [^{3} with N from 6 to 16. Besides, ^{−1}, 10^{−2} and 10^{−3} that has an impact on the result of Chebyshev polynomial method at N = 18 , Chebfun matlab program and the Runge-Kutta in [

N | ‖ E Chebyshev ‖ | ‖ E Collocation ‖ [ | ‖ E Galerkin ‖ [ |
---|---|---|---|

6 | 4.3706E-04 | 0.22E-02 | ---- |

8 | 1.0349E-05 | 0.43E-04 | 0.84E-02 |

10 | 1.1865E-07 | 0.44E-06 | 0.19E-06 |

14 | 3.8885E-12 | 0.12E-10 | 0.20E-09 |

16 | 1.2817E-14 | 0.38E-13 | 0.18E-09 |

N | ‖ E Chebyshev ‖ | ‖ E Collocation ‖ [ | ‖ E Galerkin ‖ [ |
---|---|---|---|

6 | 4.7455E-04 | 0.24E-02 | 0.60E-02 |

8 | 8.7264E-06 | 0.37E-04 | 0.84E-04 |

10 | 8.7156E-08 | 0.32E-06 | 0.29E-06 |

14 | 2.3566E-12 | 0.73E-11 | 0.31E-09 |

16 | 8.2138E-15 | 0.21E-13 | 0.29E-09 |

N | ‖ E Chebyshev ‖ | ‖ E Collocation ‖ [ | ‖ E Galerkin ‖ [ |
---|---|---|---|

6 | 0.19E-02 | 0.95E-02 | ---- |

8 | 1.1786E-04 | 0.11E-02 | 0.27E-04 |

10 | 2.3777E-06 | 0.91E-05 | 0.16E-06 |

14 | 1.9856E-10 | 0.69E-09 | 0.13E-10 |

16 | 1.4004E-12 | 0.41E-11 | 0.13E-10 |

N | ‖ E Chebyshev ‖ | ‖ E Collocation ‖ [ | ‖ E Galerkin ‖ [ |
---|---|---|---|

6 | 7.9662E-04 | 0.40E-02 | ---- |

8 | 1.5049E-05 | 0.20E-03 | 0.14E-03 |

10 | 1.8040E-06 | 0.69E-05 | 0.16E-06 |

14 | 1.1496E-10 | 0.47E-09 | 0.13E-10 |

16 | 4.9300E-13 | 0.20E-10 | 0.13E-10 |

ε | ‖ E Chebyshev ‖ , N = 18 | ‖ E Chebyfun ‖ | ‖ E Runge-Kutta ‖ , N = 100 [ |
---|---|---|---|

10^{−1} | 1.9836E-16 | 2.84E-17 | 2.52017E-02 |

10^{−2} | 4.1631E-16 | 1.01E-16 | 9.238658E-03 |

10^{−3} | 1.0646E-15 | 6.71E-16 | 8.836376E-03 |

Example 4: Now we turn to a singular problem

[ 1 + ε d y d x d d x ] d 4 y d x 4 + 1 x y ′ + 1 x 2 y = f ( x ) , − 1 2 ≤ x ≤ 1 2 ,

and

f ( x ) = ε 1440 x [ 4 x ( x 2 − 1 4 ) 2 + 8 x 3 ( x 2 − 1 4 ) ] + 6 ( x 2 − 1 4 ) 2 + 8 x 2 ( x 2 − 1 4 ) + 720 x 2 − 24,

whose exact solution is

y ( x ) = 2 x 2 ( x 2 − 1 4 ) 2 .

Applying L’Hospital rule to BVP in order to remove singularity. The latter form is

[ 1 + ε d y d x d d x ] d 4 y d x 4 + p ( x ) y ″ + d ( x ) y ′ + b ( x ) y = f (x)

p ( x ) = { 0 , x ≠ 0 ; 1.5 , x = 0. d ( x ) = { 1 / x , x ≠ 0 ; 0 , x = 0. b ( x ) = { 1 / x 2 , x ≠ 0 ; 0 , x = 0.

The computational results are summarized in

In this article, we present a method to approximate the solution of viscoelastic flows. The numerical method is based on the operational matrix of Chebychev polynomials. We present four examples, the first three examples are nonlinear fifth-order differential equation and fourth example is singular nonlinear. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method. The proposed method is a powerful tool for obtaining novel numerical solutions of such equations. It is advisable to use it for other nonlinear differential equations.

ε | ‖ E Chebyshev ‖ |
---|---|

10^{−1} | 8.7184E-12 |

10^{−2} | 2.9893E-14 |

10^{−3} | 9.3948E-15 |

Method | The root mean square errors |
---|---|

Chebyshev | 8.718E-12 |

Chebfun | 11.40E-11 |

The authors declare no conflicts of interest regarding the publication of this paper.

El-Gamel, M., Mohamed, O. and El-Shamy, N. (2020) A Robust and Effective Method for Solving Two-Point BVP in Modelling Viscoelastic Flows. Applied Mathematics, 11, 23-34. https://doi.org/10.4236/am.2020.111003