Numerical techniques for behavior of incompressible flow in steady two-dimensional motion due to a linearly stretching of porous sheet based on radial basis functions

1.1 Introduction of the problem

In recent researches the flows of micro-polar fluids has been increasingly favored due to the occurrence of these fluids in industrial processes, such as solidification of the liquid crystals, cooling of a metallic plate in a bath, exotic lubricants extrusion of metals and polymers drawing of plastic films, production of glass and paper sheets and solution of colloidal suspension. The theory of micro-polar fluids introduced by Erington [1, 2] can display the effects of local rotary inertia and couple stresses, also it is capable of explaining the flow behavior in which the classical Newtonian fluids theory is inadequate. Extensive reviews of the theory and applications were given by Łukaszewicz [3]. In this book the mathematical aspects of micro-polar fluid flow are presented. Also Ariman et al. [4] have given an excellent review of micro-polar fluids and their applications. As the problem of stretching sheet has been of great use in engineering studies, Crane [5] first studied the flow caused by an elastic sheet whose velocity varies linearly with the distance from a fixed point on the surface. Since then many researchers have reported results on stretching sheet. The micro-polar flow on a moving flat plate was studied by Ishak et al. [6, 7, 8]. Hassanien and Gorla [9] by considering suction and injection obtained numerical solution. The same problem, using a method of successive approximations, was analyzed by Hady [10]. The Newtonian fluid counterpart was studied by Gupta and Gupta [11]. Recently, some new aspects of micro-polar fluids and their applications have been investigated by Rahman et al. [12, 13, 14, 15, 16, 17]. More recently, limiting behavior of micro-polar flow due to a linearly stretching porous sheet have been investigated by [18, 19]. In present investigation we deal with the boundary layer flow of a micro-polar fluid due to linearly stretching sheet.

1.2 Introduction of radial basis functions

Radial basis functions (RBFs) interpolations are techniques for representing a function starting with data on scattered nodes. This technique first appears in the literature as a method for scattered data interpolation, and the method was highly favored after being reviewed by Franke [20], who found it to be the most impressive of the many methods he tested. Later, Kansa [21, 22] in 1990 proposed an approximate solution of linear and nonlinear differential equations (DEs) using RBFs. For the last years, the radial basis functions (RBFs) method was known as a powerful tool for the scattered data interpolation problem. The main advantage of numerical methods which use radial basis functions is the meshless characteristic of these methods. The use of radial basis functions as a meshless method for the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs) is based on the collocation method. Kansa’s method has recently received a great deal of attention from researchers [23, 24, 25, 26, 27, 28, 29, 30, 31, 32].

Recently, Kansa’s method was extended to solve various ordinary and partial differential equations including the nonlinear Klein-Gordon equation [27], regularized long wave (RLW) equation [33], high order ordinary differential equations [34], the case of heat transfer equations [35], Hirota-Satsuma coupled KdV equations [36], second-order parabolic equation with nonlocal boundary conditions [37], Second-order hyperbolic telegraph equation [38], incompressible Eyring-Powell fluid over a linear stretching sheet [66], pricing of options [67, 68], inverse Cauchy-Stefan problem [69, 70], non-Fickian flows with mixing length growth in porous media [71], unsteady flow of gas in a semi-infinite porous medium [72], and so on [73]. All of the radial basis functions have global support, and in fact many of them, such as multiquadrics (MQ), do not even have isolated zeros [27, 33, 39]. The RBFs can be compactly and globally supported, infinitely differentiable, and contain a free parameter c, called the shape parameter [33, 39, 40]. For more basic details about RBFs the interested readers can refer to the recent books and paper by Buhmann [39, 41] and Wendland [42], compactly and globally supported; and convergence rate of the radial basis functions.

Despite many studies done to find algorithms for selecting the optimum values of c [43, 44, 45], the optimal choice of shape parameter is an open problem which is still under intensive investigation. For example, Carlson and Foley [44] found that the shape parameter is problem dependent. They [44] observed that for rapidly varying functions, a small value of c should be used, but a large value should be used if the function has a large curvature. Tarwater [45] found that by increasing c, the Root-Mean-Square (RMS) of error dropped to a minimum and then increased sharply afterwards. In general, as c increases, the system of equations to be solved becomes ill-conditioned. Cheng et al. [43] showed that when c is very large then the RBFs system error is of exponential convergence. But there is a certain limit for the value c after which the solution breaks down. In general, as the value of the shape parameter c increases, the matrix of the system to be solved becomes highly ill-conditioned and hence the condition number can be used in determining the critical value of the shape parameter for an accurate solution 43]. For some new work on optimal choice of shape parameter, we refer the interested reader to the recent work of [46].

There are two basic approaches for obtaining basis functions from RBFs, namely direct approach (DRBF) based on a differential process [22] and indirect approach (IRBF) based on an integration process [25, 34, 47]. Both approaches were tested on the solution of second order DEs and the indirect approach was found to be superior to the direct approach [25].

In contrast, the integration process is much less sensitive to noise [34, 48]. Based on this observation, it is expected that through the integration process, the approximating functions will be much smoother and therefore have higher approximation power [34, 48].

To numerically explore the IRBF methods with shape parameters for which the interpolation matrix is too poorly conditioned to use standard methods mentioned by the authors as ref [49]. This is perhaps the major advantage of the IRBFs as RBFs methods are typically not employed in applications using the optimal shape parameters, but using some value of the parameter safely away from the region of ill-conditioning [50].

Some of the infinitely smooth RBFs choices are listed in Table 1. The RBFs can be of various types, for example: polynomials of any chosen degree such as linear, cubic, etc., thin plate spline (TPS), multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian forms (GA), hyperbolic secant (sech) form etc. Regarding the inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA), the coefficient matrix of RBFs interpolating is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [51].

In this paper, we use the multiquadric (MQ) to direct approach and inverse multiquadrics (IMQ) to indirect approach radial basis functions for finding the solution of main problem. The MQ was first developed by Hardy [52] in 1971 as a multidimensional scattered interpolation method in modeling of the earth gravitational field. It was not recognized by most of the academic researchers until Franke [20] published a review paper on the evaluation of two-dimensional interpolation methods, whereas MQ was ranked as the best, based on its accuracy, visual aspect, sensitivity to parameters, execution time, storage requirements, and ease of implementation.

For convenience the solution we use RBFs with collocation nodes {xj}j=1N which are the zeros of the shifted Legendre polynomial L_N(x), 0 ≤ x ≤ t_f. The shifted Legendre polynomials L_i(x) are defined on the interval [0, t_f] and satisfy the following formulae [53]:

This paper is arranged as follows: in Section 2 we present a brief formulation of the problem. In Section 3, we describe the properties of radial basis functions. In Section 4 we implement the problem with direct and indirect radial basis functions methods and in Section 5 we report our numerical finding and demonstrate the accuracy of the proposed methods. The conclusions are discussed in the final Section.

3.1 Definition of radial basis functions

Let ℝ⁺ = {x ∈ ℝ, x ≥ 0} be the non-negative half-line and let ϕ : ℝ⁺ → ℝ be a continuous function with ϕ(0) ≥ 0. A radial basis functions on ℝ^d is a function of the form

where X, X_i ∈ ℝ^d and ‖.‖ denotes the Euclidean distance between X and X_is. If one chooses N points {Xi}i=1N in ℝ^d then by custom

is called a radial basis functions as well [54].

The standard radial basis functions are categorized into two major classes [36]:

Class 1. Infinitely smooth RBFs [36, 55]:

These basis functions are infinitely differentiable and heavily depend on the shape parameter c e.g. Hardy multiquadric (MQ), Gaussian(GA), inverse multiquadric (IMQ), and inverse quadric(IQ) (See Table 1).

Class 2. Infinitely smooth (except at centers) RBFs [36, 55]: The basis functions of this category are not infinitely differentiable. These basis functions are shape parameter free and have comparatively less accuracy than the basis functions discussed in the Class 1. For example, thin plate spline, etc [36].

As far as the radial basis functions are concerned, several choices are possible, such as the so-called multiquadrics, inverse multiquadrics, Gaussian RBFs (see, for example, [41]). In particular, the multiquadrics, the inverse multiquadrics and the Gaussian RBFs contain a free shape parameter on which the performances of the RBF approximation strongly depend. Precisely, values of the shape parameters that yield a high spatial resolution (i.e. a high level of accuracy) also lead to severely ill-conditioned linear systems. Therefore, one has to find a value of the shape parameter such that the numerical results are satisfactorily accurate but at the same time the overall approximation do not blow up (due to ill-conditioning problems). Now, in the technical literature, various approaches for selecting the RBF shape parameter have been proposed, see, e.g., [20, 43, 44, 46, 59, 60, 61, 62, 63]. These algorithms, which are often based on rules of thumbs and on semi-analytical relations, can yield satisfactory results in some circumstances. Nevertheless, to the best of our knowledge, a method to choose the RBF shape parameter which is rigorously established and is proven to perform well in the general case is still lacking.

3.2 RBFs interpolation

The one dimensional function y(x) to be interpolated or approximated can be represented by an RBFs as:

where

x is the input and {λi}i=1N are the set of coefficients to be determined. By choosing N interpolate nodes {xi}i=1N, we can approximate the function y(x).

To summarize discussion on coefficient matrix, we define:

where

Note that ϕ_i(x_j) = φ(‖x_i − x_j‖) therefore we have ϕ_i(x_j) = ϕ_j(x_i) consequently A = A^T.

All the infinitely smooth RBFs choices are listed in Table 1 will give coefficient matrices A in Eq. (3.3) which are symmetric and nonsingular [51], i.e. there is a unique interpolant of the form Eq. (3.1) no matter how the distinct data points are scattered in any number of space dimensions. In the cases of inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA) the matrix A is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [51].

3.3 Direct radial basis functions (DRBF)

In this method, we write the goal function y(x) by using a linear combination of N RBFs:

which Φ(x) and Λ are intruduced in the Eq. (3.2). Now, for derivative of any order y(x), we have

where

Let p be the highest order of the derivative under consideration the boundary value ODEs in general form:

where e_i ∈ {a, b} and yⁱ(x) = dⁱy(x)/dxⁱ, F is known function and {αi}i=1p are known constants. By substituting Eq. (3.4) and Eq. (3.5) in (3.6), we obtain the residual functions:

The set of equations for obtaining the coefficients {λi}i=1N come from equalizing Eq. (3.7) to zero at N − p collocation nodes {xj}j=1N−p plus p boundary conditions:

3.4 Indirect radial basis functions (IRBF)

In this approach [34, 56, 57, 58], RBFs are used to represent the highest order derivatives p of a function y(x), e.g.

where

The obtained derivative expression is then integrated to yield expressions for lower order derivatives and finally for the original function itself [34]. Let p be the highest order of the derivative under consideration the boundary value ODEs in general form Eq. (3.6). We define integral operation I as:

then we can define:

⋮

where:

Now to obtain {λi}i=1N, we define the residual function:

The set of equations for obtaining the coefficients {λi}i=1N and {dj}j=0p−1 come from equalizing Eq. (3.14) to zero at N interpolate nodes {xj}j=1N

4.1 Solving the model by DRBF

For obtaining the solution by DRBF approach, the residual functions are constructed by substituting Eq. (4.1) in Eq. (4.4):

By using N − 3 interpolate nodes {xj}j=1N−3 in Res₁(η) and N − 2 interpolate nodes {xj}j=1N−2 in Res₂(η) plus five boundary conditions F(0) = ϕ, F′(0) = 1, F′(η_∞) = 0, H(0) = 0, H(η_∞) = 0, the set of equations can be solved, consequently the coefficients {λi[1]}i=1Nand{λi[2]}i=1N will be obtained:

4.2 Solving the model by IRBF

For obtaining the solution by IRBF approach, the residual functions are constructed by substituting Eq. (4.2) in Eq. (4.5) and using Eq. (3.12):

By using N interpolate nodes {xj}j=1N in Res₁(η) and N interpolate nodes {xj}j=1N in Res₂(η) plus five boundary conditions of Eq. (4.3) the set of equations can be solved, consequently the coefficients matrix Λ̂^[1], Λ̂^[2] and coefficients {di[1]}i=13and{di[2]}i=12 will be obtained.