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In this Paper, we have proposed a new weighted residual method known as orthogonal collocation-based on mixed interpolation (OCMI). Mixed interpolation uses the classical polynomial approximation with two correction terms given in the form of sine and cosine function. By these correction terms, we can control the error in the solution. We have applied this approach to a non-linear boundary value problem (BVP) in ODE which governs the electrohydrodynamic flow in a cylindrical conduit. The solution profiles shown in the figures are in good agreement with the work of Paullet (1999) and Ghasemi
et al
. (2014). Our solution is monotonic decreasing and satisfies
, where, α governs the strength of non-linearity and for large values of α solutions are
. The residual errors are given in
**Table 1**
and
**Table 2**
which are significantly small. Comparison of residual errors between our proposed method, Least square method and Homotopy analysis method is also given and shown via the
**Table 3**
where as the profiles of the residual error are depicted in
**Fig****ures**** ****4****-****8**
. Table and graphs show that efficiency of the proposed method. The error bound and its L_{2}-norm with relevant theorems for mixed interpolation are also given.

The electrohydrodynamic flow (EHD flow) of a fluid in a “ion drag” configuration in a circular conduit (see

d 2 w d r 2 + 1 r d w d r + H a 2 ( 1 − w 1 − α w ) = 0 , 0 < r < 1 , (1)

subject to the boundary conditions

w ′ ( 0 ) = 0 , w ( 1 ) = 0. (2)

where w ( r ) is the fluid velocity, r is the radial distance from the center of the cylindrical conduit, H a is the Hartmann electric number and the parameter α measures the strength of the non-linearity. It has been noted that the nonlinearity occurred in this problem is in the form of a rational function and thus, creates a significant challenge in regard to obtain analytical solutions.

Though, some analytic solutions are introduced by several researchers which are mentioned here. In 1997, Mckee et al. [

For α ≪ 1 , Mckee et al. [

w ( r , α ) = 1 − I 0 ( H r ) I 0 ( H ) .

Similarly, for α ≫ 1 , the authors in [

For any α > 0 and any H a 2 ≠ 0 $ a solution of BVP (1 - 2). Further this

solution is monotonically decreasing and satisfies 0 < w ( r ) < 1 α + 1 ∀ r ∈ ( 0 , 1 ) .

Remark: Clearly the solution w ( r ) of BVP (1 - 2) satisfies w ( r ) < 1 α and w ( r ) never equals 1 α , otherwise the term w ( r ) 1 − α w ( r ) creates the singularity.

Paullet [

α ≪ 1 , our solution obtained by orthogonal collocation method based on mixed interpolation are in complete agreement with those of [

The aim of the present article is to introduce a new weighted residual method based on collocation and mixed interpolation. There are several known weighted residual methods like collocation, Galerkin, Least square method etc. There are several important research contributions to the development of numerical techniques for solving ODE and PDE by different method based on the weighted residual method [

Collocation method is widely used to solve various problems in science and engineering. Its usefulness is due to its simplicity and easy computations. Collocation method has found as one of the important methods to solve various integral and differential equations in science and engineering [

The idea of mixed interpolation was introduced by Mayer et al. (1990) [

a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i .

The present approach of mixed interpolation is inspired by the work of Meyer et al. (1990). Meyer et al. approximates a function f ( x ) by a function f n ( x ) of the form a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i such that f ( j , h ) = f n ( j , h ) for ( n + 1 ) equidistant points jh, j = 0,1,2, ⋯ , n , h is stepsize. Several authors have formulated new quadrature rules and multi-step methods for ordinary differential equations on the basis of mixed interpolation [

In this section, we propose a new type of weighted residual method called orthogonal collocation mixed interpolation method (OCMIM). It is an advancement over existing collocation method in a sense that we interpolate the unknown solution by means of a mixed interpolating function which is actually the mixed version of classical Lagrange polynomial and trigonometric functions. This advancement improves the accuracy of the method. Here, we are using one cosine factor cos ( k x ) and one sine factor sin ( k x ) in interpolating function. These functions can be taken as correction terms of the solution [

The principle of orthogonal collocation method is to minimize the residual function (defect) R ( x , c i ) and set equal to zero at preassigned collocation points (Zeros of some orthogonal polynomial). In this paper, we have considered the zeros of shifted Legendre polynomial as collocation points. The approximate solution is produced by means of the values it assumes in some locations, called collocation points, where the governing differential equation is satisfied. Such approach is called the collocation method.

The important step in collocation technique is the choice of collocation points. It is the most important part of collocation technique as the wrong choice of collocation points may lead to divergent results. Preferably the zeros of the orthogonal polynomial are used as collocation points to keep the error minimum.

Jacobi polynomial of degree n, denoted as P n ( α , β ) , forms a basis for the vector space of polynomials of degree at most n. Jacobi polynomial is defined on the interval [−1, 1] and can be determined with the aid of the following recurrence formulae:

2 ( n + 1 ) ( α + β + n + 1 ) ( α + β + 2 n ) P n + 1 ( α + β ) ( x ) = ( α + β + 2 n + 1 ) [ α 2 − β 2 + x ( α + β + 2 n + 2 ) ( α + β + 2 n ) ] P n α , β ( x ) − 2 ( α + n ) ( β + n ) ( α + β + 2 n + 2 ) P n − 1 ( α , β ) ( x ) , ∀ x ∈ [ − 1,1 ]

P 0 ( α , β ) ( x ) = 1, (3)

P 1 ( α , β ) ( x ) = ( α + 1 ) ( β + 1 ) ( x + 1 ) 2 + x − 2 2 .

The interpolation points (‘n’ in number) are chosen to be the extreme values of an n^{th} order shifted Jacobi polynomial. For the interval [0, 1], the collocation points are obtained by mapping the computational domain of the interval [−1, 1] to [0, 1] with the help of the following relationship:

ξ j = x j + 1 2 ; j = 2 , 3 , ⋯ , n − 1 ,

where, x j is the jth zero of P n − 2 ( α , β ) ( x ) in the interval [−1, 1] with ξ 1 = 0 and ξ n = 1 .

For α = β = 0 , Jacobi becomes Legendre polynomial which is defined by

P n ( x ) = 1 2 n n ! d n d x n ( x 2 − 1 ) n .

The first five zeros of shifted Legendre polynomial in the interval [0, 1] are given by

x 1 = 0.046910077030668 , x 2 = 0.23076534494715875 , x 3 = 0.5 ,

x 4 = 0.7692346550528495 , and x 5 = 0.9530899229693268 .

Suppose a diffrential operator A is acted on a function u to produce a function f. i.e.

A ( u ( x ) ) = f ( x ) . (4)

It is considered that u is approximated by a function u N , which is a linear combination of basic functions chosen from a linearly dependent set. That is,

u ≅ u N ( x ) = a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i . (5)

Consider a set of collocation (grid) points { u i : i = 1 , 2 , ⋯ , n + 1 } in the domain [0, 1] such that u 1 = 0 and u n + 1 = 1 and { u 2 , u 3 , ⋯ , u n } ∈ ( 0,1 ) , such that u 1 = 0 < u 2 < u 3 < ⋯ < u n + 1 = 1 . In the present article, we have taken { u i : i = 2 , 3 , 4 , ⋯ , n } the interior collocation points as zero of shifted Legendre polynomial of order ( n − 1 ) . Let { u ( u i ) = u i : i = 1 , 2 , ⋯ , n + 1 } represent the values of dependent variable at these collocation points. Suppose the approximate solution u N ( u i ) given in (4) agrees with the exact one i.e. with u ( u i ) i.e.

u ( u i ) = u i = u N ( u i ) ; i = 1 , 2 , ⋯ , n + 1. (6)

So, we get a set of ( n + 1 ) equations in unknown coefficients c i .

Solving Equation (5) with the combination of Equation (4), we get c i in terms of unknown numerical solutions { u i : i = 1 , 2 , 3 , ⋯ , n + 1 } .

C = A − 1 U ,

where C = [ a , b , c 0 , c 1 , ⋯ , c n − 2 ] T and U = [ u 1 , u 2 , u 2 , ⋯ , u n + 1 ] T and A is a coefficient matrix whose rows are of the form [ cos ( k u i ) , sin ( k u i ) , 1 , u i , u i 2 , ⋯ , u i n − 2 ] , i = 1 , 2 , 3 , ⋯ , n . Following the procedure of Mayer et al. (1989), we can prove A is non-singular and | A | ≠ 0 as u i are distinct grid points in the domain. Thus, u N ( x ) in (4) can be rewritten as

u N ( x ) = ∑ i = 0 n u i l i ( x ) . (7)

When an approximate solution u N ( x ) given in (5) is substituted into the differential Equation (4), the result of the operations generally not equal to f ( x ) . Hence, an error or residual will exist which is denoted and defined by

R ( x , c i ) ≅ A ( u N ) − f = a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i − f ≠ 0. (8)

Here, the residual R ( x , c i ) is a function of position as well as of the parameters c i .

Combining (7) and (8), we have the residual error as:

R ( x ) ≡ R ( x , u i ) ≅ A ( ∑ i = 0 n u i l i ( x ) ) − f . (9)

To find the u i from (8), we set R ( x , u i ) = R ( x ) equal to zero at interior collocation points { u i : i = 2 , 3 , 4 , ⋯ , n } with combination of boundary conditions u ′ ( 0 ) = 0 and u ( 1 ) = 0 . i.e. we solve the set of ( n + 1 ) equations in u i , i = 0 , 1 , 2 , ⋯ , n which are

∫ 0 1 R ( x , c i ) δ ( x − u i ) d x = 0, i = 2,3,4, ⋯ , n , (10)

and

u ′ ( 0 ) = 1 , u ( 1 ) = 0. (11)

where, Dirac delta function is defined by

δ ( x − u i ) = 0 when x ≠ u i ,

and

∫ − ∞ ∞ R ( x ) δ ( x − u i ) = R ( u i ) .

Solving (10) and (11), we get the desired unknown numerical solutions u i which on substituting in (7) gives us the approximates solution u N ( x ) .

In this section, we consider the convergence analysis and error bound of the orthogonal collocation method introduced in section 2 to compute the approximate solution of the EHD flow equation (Equations ((1), (2))).

We denote,

e n ( x ) = y ( x ) − ϕ n ( x ) .

To compute the error bound, we use the following results.

Theorem (3.1). (Weierstrass Approximation Theorem). Any Continous function defined on the closed and bounded interval [a,b] can be approximated uniformly by polynomials to any degree of accuracy on that interval. If f ∈ C [ a , b ] is approximated by a polynomial p ( x ) of degree n then

| f ( x ) − p ( x ) | ≤ ϵ , ∀ x ∈ [ a , b ] .

Theorem (3.2). If x 1 , ⋯ , x n are distinct n points defined on [0, 1] and f ( x ) ∈ C n + 1 [ 0,1 ] is any function defined on [0, 1] then there exists a unique polynomial L ( x ) of degree atmost n such that

f ( x i ) = L ( x i ) ; i = 1,2, ⋯ , n ,

where,

L ( x ) = ∑ i = 1 n l i ( x ) f i .

Proof. Result is straightforward and proof is followed by use of theorem (3.1).

Theorem (3.3). ( [^{th} degree shifted Legendre polynomial in [ 0,1 ] . If ϕ n ( x ) is the interpolating polynomial to y ( x ) in [ 0,1 ] such that y ( x i ) = ϕ n ( x i ) , i = 0,1,2, ⋯ , n . Then,

y ( x ) − ϕ n ( x ) = y n + 1 ( ξ ) ∏ i = 0 n ( x − x i ) ( n + 1 ) ! , 0 < ξ < 1 ,

and

| y ( x ) − ϕ n ( x ) | ≤ M n 2 2 n + 1 ( n + 1 ) ! (12)

where,

M n = m a x { | y n + 1 ( ξ ) | : ξ ∈ [ 0,1 ] } .

Proof: Let

f ( x ) = y ( x ) − ϕ n ( x ) − L ∏ i = 0 n ( x − x 0 ) ,

where, L is constant such that f ( x ) vanishes at some interior point x ′ in [ x 0 , x n ] , where x 0 = 0 < x 1 < x 2 < ⋯ < x n = 1 .

Under the assumption of the theorem (3.3), it is clear that f ( x ) vanishes at ( n + 2 ) values x 0 , x 1 , ⋯ , x n and x ′ . So, by repeated use of Rolle’s theorem, one can prove that,

y ( x ) − ϕ n ( x ) = y n + 1 ( ξ ) ∏ i = 0 n ( x − x i ) ( n + 1 ) ! , 0 < ξ < 1.

So,

| y ( x ) − ϕ n ( x ) | ≤ y n + 1 ( ξ ) | ∏ i = 0 n ( x − x 0 ) | ( n + 1 ) ! ,

or,

| y ( x ) − ϕ n ( x ) | ≤ M n 2 2 n + 1 ( n + 1 ) ! .

Theorem (3.4). Suppose the solution of boundary value problem (1 - 2) is ( n + 1 ) times continuously differential on [0, 1] and ϕ n ( x ) be the Lagrange polynomial approximation of y ( x ) . If ψ n ( x ) is the approximate solution of BVP (1 - 2) based on mixed interpolation where ψ n ( x ) = a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i and M n = m a x { | y n + 1 ( ξ ) | : ξ ∈ [ 0,1 ] } then $ two real numbers α n and β n such that

‖ y ( x ) − ψ n ( x ) ‖ 2 ≤ M n 2 2 n + 1 ( n + 1 ) ! + α n ‖ C − C ¯ ‖ 2 + β n ,

where, C = ( C 0 , C 1 , C 2 , ⋯ , C n − 2 ) T and C ¯ = ( C ¯ 0 , C ¯ 1 , C ¯ 2 , ⋯ , C ¯ n − 2 ) T .

Proof: Consider,

‖ e n ‖ = ‖ y ( x ) − ψ n ( x ) ‖ 2 ≤ ‖ y ( x ) − ϕ n ( x ) ‖ 2 + ‖ ϕ n ( x ) − ψ n ( x ) ‖ 2 . (13)

Using theorem (3.3) and Equation (12)

‖ y ( x ) − ϕ n ( x ) ‖ ≤ M n 2 2 n + 1 ( n + 1 ) ! . (14)

Again,

‖ ϕ n ( x ) − ψ n ( x ) ‖ 2 ≤ α n ‖ C − C ¯ ‖ + β ,

where,

α n = ∑ i = 0 n − 2 1 2 i + 1

and

β n = C n − 1 2 2 n − 1 + C n 2 2 n + 1 + a 2 + b 2 4 k + ( b 2 − a 2 ) sin 2 k 4 k .

Remark: If f ( x ) ∈ L 2 [ 0,1 ] be aprroximated by

f n ( x ) = a cos ( k x ) + b sin ( k x ) + ∑ i = 0 n − 2 c i x i . (15)

Then (7) can be rewritten as

f n ( x i ) = ∑ i = 0 n f ( x i ) l i ( x ) , (16)

where, l i ( x ) are called trial function and f ( x i ) = f i are unknown numerical solution at node x i .

Consider the EHD flow equation (Equations ((1), (2))). Using (5) the approximate solution of BVP (1 - 2) is

u N ( r ) = a cos ( k r ) + b sin ( k r ) + c 0 + c 1 r + c 2 r 2 + c 3 r 3 + c 4 r 4 , (17)

such that

u N ( u i ) = u i , i = 0 , 1 , ⋯ , 6 (18)

So, by Equation (7),

u N ( r ) = ∑ i = 0 6 u i l i ( r ) , (19)

where l i ( x ) are base functions. Differentiating Equation (19) two times,

d d r u N ( r ) = ∑ i = 0 6 u i l ′ i ( r ) ,

d 2 u N ( r ) d r 2 = ∑ i = 0 6 u i l ″ i ( r ) .

So, in view of (1), the residual R ( r ) (given in (9)) is:

R ( r ) = ∑ i = 0 6 u i l ″ i ( r ) + ∑ i = 0 6 u i l ′ i ( r ) + H a 2 ( 1 − ∑ i = 0 6 u i l i ( r ) 1 − α ∑ i = 0 6 u i l i ( r ) ) = 0 , 0 < r < 1.

In light of (9) and (10), we have the following set of 7 equations

∫ 0 1 R ( r ) δ ( x − u i ) d x = 0 ; i = 0,1,2, ⋯ ,6. (20)

For α = 0.5 , H a 2 = 0.5 and k = 1.5 , solving the system (20), we have

u 0 = 0.11374565155265949 , u 1 = 0. 1135 0 37385434152 ,

u 2 = 0. 1 0 788 0 9362815 0 8 , u 3 = 0.0 86 0 238 0 525867 0 34 ,

u 4 = 0.0 473571247957 0 691 , u 5 = 0.0 1 0 73621856 0 82 0 429 ,

u 6 = 0 .

Similarly, For other values of parameters α and H a 2 the numerical results are shown in the upcoming

Note: There are several choices for the parameter k but here we have considered particular k = 1.5 . The optimum value of k can be computed and it takes a tedious computation which can be the part of future research in this direction.

α | H a 2 = 1 | H a 2 = 0.5 | H a 2 = 2 |
---|---|---|---|

0.5 | 1.3 × 10^{−5} | 1.35 × 10^{−6} | 1.5 × 10^{−4} |

1 | 2.3 × 10^{−6} | 8 × 10^{−7} | 9 × 10^{−5} |

2 | 1.4 × 10^{−4} | 1 × 10^{−5} | 1.55 × 10^{−3} |

4 | 0.54 | 6 × 10^{−5} | 1.7 |

10 | 0.42 | 0.11 | 0.32 |

H a 2 | α = 0.5 | α = 1 | α = 2 |
---|---|---|---|

0.5 | 1.35 × 10^{−6} | 8 × 10^{−7} | 1 × 10^{−5} |

1 | 1.3 × 10^{−5} | 2.3 × 10^{−6} | 1.4 × 10^{−4} |

2 | 1.5 × 10^{−4} | 9 × 10^{−5} | 1.55 × 10^{−3} |

4 | 1.7 × 10^{−3} | 2.5 × 10^{−3} | 9 × 10^{−3} |

10 | 3 × 10^{−2} | 8 × 10^{−2} | 0.22 |

α | H a 2 = 1 | H a 2 = 2 | ||||
---|---|---|---|---|---|---|

LSM | HAM | OCMI | LSM | HAM | OCMI | |

0.5 | 2 × 10^{−}^{5} | 8 × 10^{−}^{5} | 1.3 × 10^{−}^{5} | 2 × 10^{−}^{4} | 6 × 10^{−}^{4} | 1.5 × 10^{−}^{4} |

1 | 4 × 10^{−}^{5} | 4 × 10^{−}^{5} | 2.3 × 10^{−}^{6} | 5 × 10^{−}^{4} | 1.5 × 10^{−}^{3} | 9 × 10^{−}^{5} |

The solution profiles of numerical solution for several values of α and H a 2 are shown via the

The main aim of the discussion is to recognize the effects of Hartmann number H a 2 and nonlinearity parameter α on conduit velocity profiles. In this paper, the solution profiles for both large and small values of α are considered. For α ≪ 1 , the solution profiles obtained by our proposed method are similar to Mckee et al. [

large H a 2 ) are monotonically decreasing and satisfy 0 < w ( r ) < 1 α + 1 which is quite evident from

errors for varying values of α and H a 2 are tabulated in

In this paper, we have successfully introduced a very simple numerical method named as “orthogonal collocation-based on mixed interpolation” (OCMI). We have solved a non-linear boundary value problem governing the electrohydrodynamic

flow (EHD) in cylindrical conduit where the unknown solution is approximated by means of mixed interpolation. The approximate solutions are in excellent agreement with those of the solution given by previous researchers Mckee et al. (1997), Paullet (1999) and Ghasemi et al. (2014). The comparison of residual errors is made in _{2}-norm has computed and given in section 3. In this approach of mixed interpolation we have used a fixed value of parameter k = 1.5 in correction terms. The computation of optimum value of correction parameter k can be taken as future development in this direction.

The first author acknowledges the financial support in form of fellowship given by Dr. Harisingh Gour Vishwavidyalaya, Sagar (M.P.), India.

Tiwari, S. and Pandey, R.K. (2018) Solution of Electrohydrodynamic Flow Equation Using an Orthogonal Collocation Method Based on Mixed Interpolation. Journal of Applied Mathematics and Physics, 6, 573-587. https://doi.org/10.4236/jamp.2018.63050