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In this paper, we develop a new numerical method which is based on an exponential spline and Shishkin mesh discretization to solve singularly perturbed boundary value problems, which contain a small uncertain perturbation parameter. The proposed method uses interval analysis principle to deal with the uncertain parameter and the Monte Carlo Simulations (MCS) are used to validate the solution and the accuracy of the proposed method. Furthermore, sensitivity analysis has been conducted using different methods to assess how much the solution is sensitive to the changes of the perturbation parameter. Numerical results are provided to show the applicability and efficiency of the proposed method, which is ε -uniform convergence of almost second order.

In this paper, we consider the following singularly perturbed problem [

− E y ″ + μ f ( x ) y ′ + g ( x ) y = r ( x ) , x ∈ [ a , b ] (1)

subject to the following boundary conditions,

y ( a ) − α = y ( b ) − β = 0 (2)

where 0 < E ≪ 1 and 0 < μ ≪ 1 are two small perturbation parameters; f ( x ) , g ( x ) and r ( x ) are sufficiently smooth functions for x ∈ [ a , b ] ; a, b, α, and β are real constants. In general, the solution y ( x ) may exhibit two boundary layers of exponential type at both end points x = a , b .

Different applications in science and engineering consider these kinds of problems that describe complicated physical and chemical models such as heat transfer problems, Navier-Stokes flow with large Reynolds numbers, chemical reactor theory, convection-diffusion processes, geophysics, aerodynamics, reaction-diffusion processes, quantum mechanics and optimal control, etc. Its solution exhibits two layers at the two endpoints of the domain. The nature of the two-parameter problem was asymptotically examined by [

Many numerical methods have been developed for the solution of two layer boundary value problems, such as described in [

μ = E p + 1 2 with p > 0. Dag and Sahin presented a numerical solution of singularly

perturbed boundary value problems, using finite element method [

For a more reasonable analysis of the system response, uncertainty should be involved. When the exact value of a quantity is unknown, its approximation and corresponding degree of uncertainty can be conveyed via an interval, which estimates a range of possible values expected to include, firstly suggested by several mathematicians for bounding round-off errors, the interval analysis is fully developed by Moore [

The main contribution of this paper is to develop a new spline method based on a Shishkin mesh discretization for obtaining an approximation for the solution of two-layer boundary value problems. As the perturbation parameter is not deterministic, therefore the interval analysis is considered to estimate the solution range. The validation of the developed solver will be done by comparing the exact and the approximate solutions of the proposed method and the MCS results. The convergence analysis also is presented numerically and shows that the presented method is almost second-order.

The paper is organized as follows: In Section 2, we derive our spline scheme. Mesh strategy based on a Shishkin mesh is presented in Section 3. In Section 4, we present interval analysis and sensitivity measures. Numerical results are discussed in Section 5. Section 6 is devoted to the final conclusions, while future work is provided in section 7. Finally, Section 8 is dedicated to references.

We discretize the solution region Ω = [ a , b ] such that a = x 0 < x 1 < x 2 < ⋯ < x N − 1 < x N = b . Where N is the number of mesh points. Let h j = x j + 1 − x j , j = 1 , 2 , 3 , ⋯ , N be the mesh size and the mesh ratio

σ j = h j + 1 h j > 0 , j = 1 , 2 , 3 , ⋯ , N − 1 . When σ = 1 the mesh reduces to a uniform mesh, h j + 1 = h j = h . The interpolating exponential spline approximation function can be defined as [

S ( x ) = a j + b j ( x − x j ) + c j ξ j ( x − x j h j ) + d j ζ j ( x − x j h j ) , j = 0 , 1 , ⋯ , N − 1 (3)

where ξ j ( x ) = 2 / τ 2 [ cosh ( τ x ) − 1 ] , ζ j = 6 / τ 3 [ sinh ( τ x ) − τ x ] , a j , b j , c j , d j are constants and τ is a free parameter such that the non-polynomial spline (3) reduces to usual cubic spline when τ approaches to zero [

1) S ( x j ) ∈ C 2 [ a , b ] ,

2) S ( x j ) = y ( x j ) , S ″ ( x j ) = M j . (4)

The algebraic manipulations of Equations (3) and (4) yield the following expressions:

1) a j = y j ,

2) b j = 1 h j ( y j + 1 − y j ) + ( h j τ sinh τ − h j τ 2 ) M j + 1 + ( h j τ 2 − h j τ coth τ ) M j ,

3) c j = h j 2 2 M j ,

4) d j = τ h j 2 6 sinh τ ( M j + 1 − M j cosh τ ) . (5)

From the aspect of the first derivative continuity at the mesh points yields the expression for the determination of S ″ ( x j ) where i = 0 , ⋯ , N . We can get the following exponential spline identity relation:

y j + 1 − ( 1 + σ j ) y j + σ j y j − 1 = h j h j − 1 [ α j M j + 1 + ( 1 + σ j ) β j M j + γ j M j − 1 ] (6)

where

α j = 1 τ j 2 − 1 τ j sinh τ j , β j = coth τ j τ j − 1 τ j , γ j = 1 τ j 2 + 1 τ j ( sinh τ j − coth τ j cosh τ j ) , j = 1 , 2 , ⋯ (7)

Note that, the exponential spline relation (6) is consistent with the standard variable-mesh cubic spline if τ → 0 , hence α = γ = 1 6 , β = 1 3 [

We consider the simplest possible non-uniform mesh, namely a piecewise-uniform mesh proposed by Shishkin [

Ω l = [ a , a + ω 1 ] , Ω c = [ a + ω 1 , b − ω 2 ] , Ω r = [ b − ω 2 , b ]

where the transition parameters are given by:

ω 1 = min ( 1 4 , 2 φ 1 ln N ) , ω 2 = min ( 1 4 , 2 φ 2 ln N ) and φ 1 = − max x ∈ [ a , b ] λ 1 ( x ) , φ 2 = min x ∈ [ a , b ] λ 2 ( x ) (8)

where λ 1 ( x ) and λ 2 ( x ) are two solutions of the characteristic equation:

− E λ 2 ( x ) + μ f ( x ) λ ( x ) + g ( x ) = 0 (9)

The quantity λ 1 < 0 describes the boundary layer at x = a , while λ 2 > 0 characterizes the layer at x = b , and

λ 1 = μ B − μ 2 B 2 + 4 E C 2 E , λ 2 = μ B + μ 2 B 2 + 4 E C 2 E , where B = max x ∈ [ a , b ] f ( x ) (10)

We take N/4, N/2 and N/4 mesh points, respectively in Ω l , Ω c and Ω r . Denote the step sizes in each subinterval by h 1 = 4 ω 1 / N , h 2 = 2 ( b − a − ω 1 − ω 2 ) / N and h 3 = 4 ω 2 / N , respectively. Accordingly, the resulting piecewise-uniform Shishkin mesh is represented by:

h ∨ = { h 1 = 4 ω 1 N x j = x j − 1 + h 1 ; for j = 1 , 2 , ⋯ , N / 4 , h 2 = 2 ( b − a − ω 1 − ω 2 ) N x j = x j − 1 + h 1 ; for j = N 4 + 1 , ⋯ , 3 N / 4 , h 3 = 4 ω 2 N x j = x j − 1 + h 1 ; for j = 3 N 4 , ⋯ , N . (11)

Since uncertainty will be considered, the interval analysis can be used as descriptive measures of uncertainty in quantitative values. Hence, the perturbation parameter is not deterministic, the solution has to be defined as a range based on the interval of the parameter. Therefore, the upper and lower bounds of the perturbation parameter can be written as:

E ¯ = E c + Δ E , E _ = E c − Δ E (12)

where E ¯ is the upper value, E _ is the lower value and E c is the central value. Then the fluctuation range of solution could be estimated.

Sensitivity measures can be conducted using different techniques for example One-at-a-Time Sensitivity Measures (±SD), the Sensitivity Index (SI), the Importance Index (II), Differential Sensitivity Analysis (PD), etc. We estimated the sensitivity measures using the following methods One-at-a-Time Sensitivity Measures (±SD), the Sensitivity Index, and the Differential Sensitivity Analysis. One-at-a-Time Sensitivity Measures (±SD) is considered the simplest―but powerful―method for conducting sensitivity analysis, which estimates the variation of the solution as the perturbation parameter is increased by a factor of its standard deviation or in other words a percentage of its mean value [

The Sensitivity Index (SI), another simple method of estimating the sensitivity measure is to calculate the relative solution difference when varying one input parameter from its minimum value to its maximum value, which provides a good indication of parameter and model variability. The SI is calculated using, SI = ( Ymax − Ymin ) / Ymax , where Ymin and Ymax represent the minimum and maximum solution values, respectively, resulting from varying the perturbation parameter over its entire range [

Differential Sensitivity Analysis (PD) method considers all random parameters equal to their mean values and partial differentiation of the system with respect to the random parameters should be done. The sensitivity coefficient for a specific parameter ?perturbation parameter can be measured from the partial derivative relation ∂ y / ∂ E . The results are normalized by multiplying the derivatives by the ratio of the parameter value to the solution for the mean value, sensitivity coefficient = ∂ y / ∂ E ∗ ( E / y ) [

We consider the following reaction-diffusion problem; see [

− E y ″ + y = cos π x , x ∈ [ 0 , 1 ] , y ( 0 ) = y ( 1 ) = 0 (13)

whose exact solution is given by:

Y = c 1 cos π x + c 2 e λ 1 x + c 3 e − λ 2 ( 1 − x ) c 1 = 1 E π 2 + 1 , c 2 = − c 1 1 + e − λ 2 1 − e λ 1 − λ 2 , c 3 = c 1 1 + e λ 2 1 − e λ 1 − λ 2 , λ 1 , 2 = ∓ 1 E (14)

The estimated maximum error E N and the rate of convergence r N are computed by the formulas:

E N = max 0 < j < N | Y j − y j | , r N = log 2 ‖ E N ‖ − log 2 ‖ E 2 N ‖ (15)

Figures 2-4 show the comparison of the solution between the proposed method and the MCS (50000 samples), as the perturbation parameter changes by 20%.

N | k = 8 | k = 10 | k = 12 | k = 14 | ||||
---|---|---|---|---|---|---|---|---|

E_{N} | r_{N} | E_{N} | r_{N} | E_{N} | r_{N} | E_{N} | r_{N} | |

2^{7} | 9.51E−04 | 1.64 | 8.66E−04 | 1.80 | 8.57E−04 | 1.82 | 8.56E−04 | 1.82 |

2^{8} | 3.05E−04 | 1.57 | 2.49E−04 | 1.85 | 2.43E−04 | 1.89 | 2.43E−04 | 1.89 |

2^{9} | 1.03E−04 | 1.44 | 6.91E−05 | 1.84 | 6.56E−05 | 1.93 | 6.52E−05 | 1.94 |

2^{10} | 3.78E−05 | 1.32 | 1.93E−05 | 1.77 | 1.72E−05 | 1.94 | 1.70E−05 | 1.96 |

2^{11} | 1.51E−05 | 1.26 | 5.64E−06 | 1.64 | 4.48E−06 | 1.93 | 4.36E−06 | 1.98 |

The upper, centre and lower solutions of the two methods are very close which shows the accuracy and validation of the proposed method. Further sensitivity measures have been conducted such as the SI method gives an index of value 0.0039%, which is very small. The differential method indicates that the sensitivity coefficient is also very small by value 1.34E−15 at the mid-point of the scheme. Therefore, the solution is not sensitive to the changes in the perturbation parameter.

A numerical method based on exponential spline with Shishkin mesh discretization is combined with interval analysis perspective to evaluate the range of the solution for the singularly perturbed two-point boundary value problems with uncertain parameter. The numerical results show that the present method approximates the solution very well compared with the exact solution. Therefore, the proposed method is almost second-order uniformly convergent with respect to the perturbation parameter. MCS are used to prove the validation and the accuracy of the proposed method. Sensitivity analysis has been conducted using different methods and it is found that the solution is not sensitive to the perturbation parameter.

This paper proposed a new numerical spline method combined with interval analysis to solve singularly perturbed boundary value problems. We used interval analysis to estimate the solution range as the perturbation parameter is not deterministic and compared the results with the MCS method to validate the proposed method. In the future, we shall use another stochastic method to deal with the uncertainty, which appears in the perturbation parameter for example polynomial chaos expansion.

Zahra, W.K., El-Beltagy, M.A., El Mhlawy, A.M. and Elkhadrawy, R.R. (2018) Exponential Spline Solution for Singularly Perturbed Boundary Value Problems with an Uncertain―But―Bounded Parameter. Journal of Applied Mathematics and Physics, 6, 854-863. https://doi.org/10.4236/jamp.2018.64073