Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method

Abstract

Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

1. Introduction

Singularly perturbed second-order two-point boundary value problems, which received a significant amount of attention in past and recent years, arise very frequently in fluid mechanics, quantum mechanics, optimal control, chemical-reactor theory, aerodynamics, reaction-diffusion process, geophysics, and so forth. In these problems a small parameter multiplies to a highest derivative. A well-known fact is that the solution of such problems display sharp boundary or interior layers when the singular perturbation parameter 𝜀 is very small. Numerically, the presence of the perturbation parameter leads to difficulties when classical numerical techniques are used to solve such problems, and convergence will not be uniform. The solution varies rapidly in some parts and varies slowly in some other parts. There are thin transition boundary or interior layers where the solutions can change rapidly, while away from the layers the solution behaves regularly and varies slowly. There are a wide variety of techniques for solving singular perturbation problems (see [1–7]). Furthermore different numerical methods have been proposed by various authors for singularly perturbed two-point boundary value problems, such as non-uniform mesh tension spline methods [8], non-uniform mesh compression spline numerical method [9], and the least squares methods based on the Bézier control points [10].

The aim of our study is to introduce the differential transform method [11] as an alternative to existing methods in solving singularly perturbed two-point boundary value problems and the method is implemented to four numerical examples. The present method is the first time applied by the authors to singularly perturbed two-point boundary value problems.

The rest of the paper is organized as follows. In Section 2, we give a brief description of the method. In Section 3, we have solved four numerical examples to demonstrate the applicability of the present method. The discussion on our results is given in Section 4.

2. Fundamental of Differential Transform Method

In this section, the concept of the differential transformation method (DTM) is briefly introduced. The concept of differential transform was first introduced by Pukhov [11], who solved linear and nonlinear initial value problems in electric circuit analysis. This method constructs, for differential equations, an analytical solution in the form of a polynomial. It is a seminumerical and semianalytic technique that formulizes the Taylor series in a totally different manner. The Taylor series method is computationally taken long time for large orders. With this technique, the given differential equation and its related boundary conditions are transformed into a recurrence equation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. This method is useful to obtain exact and approximate solutions of linear and nonlinear differential equations. No need to linearization or discretization, large computational work and round-off errors are avoided. It has been used to solve effectively, easily, and accurately a large class of linear and nonlinear problems with approximations. The method is well addressed in [12–19]. The basic principles of the differential transformation method can be described as follows.

The differential transform of the 𝑘th derivative of a function 𝑓(𝑥) is defined as follows.𝐹(𝑘)=1𝑑𝑘!^𝑘𝑓(𝑥)𝑑𝑥^𝑘_𝑥=𝑥0, (2.1) and the differential inverse transform of 𝐹(𝑘) is defined as follows:𝑓(𝑥)=^∞_𝑘=0𝐹(𝑘)𝑥−𝑥₀^𝑘. (2.2) In real applications, function 𝑓(𝑥) is expressed by a finite series and (2.2) can be written as𝑓(𝑥)=^𝑁_𝑘=0𝐹(𝑘)𝑥−𝑥₀^𝑘. (2.3) The following theorems that can be deduced from (2.1) and (2.2) are given [20].

Theorem 2.1. If 𝑓(𝑥)=𝑔(𝑥)±ℎ(𝑥), then 𝐹(𝑘)=𝐺(𝑘)±𝐻(𝑘).

Theorem 2.2. If 𝑓(𝑥)=𝑎𝑔(𝑥), then 𝐹(𝑘)=𝑎𝐺(𝑘), where 𝑎 is constant.

Theorem 2.3. If 𝑓(𝑥)=(𝑑^𝑚𝑔(𝑥)/𝑑𝑥^𝑚), then 𝐹(𝑘)=((𝑚+𝑘)!/𝑘!)𝐺(𝑘+𝑚).

Theorem 2.4. If 𝑓(𝑥)=𝑔(𝑥)ℎ(𝑥), then 𝐹(𝑘)=∑^𝑘_𝑘1=0𝐺(𝑘₁)𝐻(𝑘−𝑘₁).

Theorem 2.5. If 𝑓(𝑥)=𝑥^𝑛, then 𝐹(𝑘)=𝑛𝑘𝑥₀^𝑛−𝑘,𝑘<𝑛1,𝑘=𝑛0,𝑘>𝑛. (2.4) Here 𝑛∈𝑁, 𝑁 is the set of natural numbers, and 𝑊(𝑘) is the differential transform function of 𝑤(𝑥). In the case of 𝑥₀=0, one has the following result: 𝑊(𝑘)=𝛿(𝑘−𝑛)=1,𝑘=𝑛0,𝑘≠𝑛. (2.5)

Theorem 2.6. If 𝑓(𝑥)=𝑔₁(𝑥)𝑔₂(𝑥)⋯𝑔_𝑛−1(𝑥)𝑔_𝑛(𝑥), then 𝐹(𝑘)=^𝑘_{𝑘𝑛−1=0}^𝑘𝑛−1_{𝑘𝑛−2=0}⋯^𝑘3_𝑘2=0^𝑘2_𝑘1=0𝐺₁𝑘₁𝐺₂𝑘₂−𝑘₁⋯𝐺_𝑛−1𝑘_𝑛−1−𝑘_𝑛−2𝐺_𝑛𝑘−𝑘_𝑛−1. (2.6)

3. The Applications of Differential Transformation Method and Numerical Results

In order to evaluate the accuracy of DTM for solving singularly perturbed two-point boundary value problems, we will consider the following examples. These examples have been chosen because they have been widely discussed in the literature and also approximate solutions are available for a concrete comparison.

Example 3.1. We first consider the following problem [21]: 𝜀𝑦^′′+𝑦=0;𝑥∈[0,1], (3.1) with the boundary conditions 𝑦(0)=0,𝑦(1)=1. (3.2) The exact solution for this problem is 𝑦(𝑥)=sin𝑥/√𝜀sin1/√𝜀. (3.3) Taking the differential transform of both sides of (3.1), the following recurrence relation is obtained: 𝑌()𝑌(𝑘)𝑘+2=−𝜀(𝑘+1)(𝑘+2). (3.4) The boundary conditions given in (3.2) can be transformed at 𝑥₀=0 as follows: 𝑌(0)=0,^𝑁_𝑘=0𝑌(𝑘)=1. (3.5) Using (3.4) and (3.5) and by taking 𝑁=5, the following series solution is obtained: 𝑦(𝑥)𝑎=𝑎𝑥−𝑥6𝜀³+𝑎120𝜀²𝑥⁵+𝑂𝑥⁷, (3.6) where, according to (2.1), 𝑎=𝑦^′(0). The constant 𝑎 is evaluated from the second boundary condition given in (3.2) at 𝑥=1 as follows: 𝑎=120𝜀²1−20𝜀+120𝜀². (3.7) Then, by using the inverse transform rule in (2.2), we get the following series solution: 𝑦(𝑥)=120𝜀²1−20𝜀+120𝜀²𝑥−20𝜀1−20𝜀+120𝜀²𝑥³+11−20𝜀+120𝜀²𝑥⁵+𝑂𝑥⁷. (3.8) The evolution results for the exact solution (3.3) and the approximate solution (3.8) obtained by using the differential transform method, for 𝜀=2⁻⁹, are shown in Figure 1.

Figure 1 

The approximate solution (dotted curve) versus the analytic solution (solid curve) for 𝜀=2−9.

Example 3.2. Secondly, we consider the following problem: 𝜀𝑦^′′+𝑦=−𝑥;𝑥∈[0,1], (3.9) with the boundary conditions 𝑦(0)=0,𝑦(1)=0. (3.10) The exact solution for this boundary value problem is 𝑦(𝑥)√=−𝑥+sin𝑥/𝜀sin1/√𝜀. (3.11) Taking the differential transform of (3.9), we have 𝑌()=()(𝑘)𝑘+2−𝛿𝑘−1−𝑌𝜀(𝑘+1)(𝑘+2). (3.12) Choosing 𝑥₀=0, the boundary conditions given in (3.10) can be transformed to give 𝑌(0)=0,^𝑁_𝑘=0𝑌(𝑘)=0. (3.13) By using (3.12) and (3.13), and, by taking 𝑁=5, we get the following series solution: 𝑦(𝑥)−1=𝑎𝑥+−𝑎6𝜀𝑥6𝜀³+1120𝜀²+𝑎120𝜀²𝑥⁵+−15040𝜀³−𝑎5040𝜀³𝑥⁷+1362880𝜀⁴+𝑎362880𝜀⁴𝑥⁹+−139916800𝜀⁵−𝑎39916800𝜀⁵𝑥¹¹[phone]𝜀⁶+𝑎6227020800𝜀⁶𝑥¹³+−11307674368000𝜀⁷−𝑎1307674368000𝜀⁷𝑥¹⁵+𝑂𝑥⁷, (3.14) where, according to (2.1), 𝑎=𝑦^′(0).
The constant 𝑎 is evaluated from the second boundary condition given in (3.10) at 𝑥=1 as follows: 𝑎=1−210𝜀+32760𝜀²−3603600𝜀³[phone]𝜀⁴−10897286400𝜀⁵[phone]𝜀⁶/−1+210𝜀−32760𝜀²+3603600𝜀−259459200𝜀⁴[phone]𝜀⁵−217945728000𝜀⁶[phone]𝜀⁷. (3.15)
Then, by using the inverse transform rule in (2.2), one can obtain the approximate solution. We do not give it because of long terms in the approximate solution.
In Figure 2, we plot the exact solution (3.11) and the approximate solution for 𝜀=10⁻³.

Figure 2 

The approximate solution (dotted curve) versus the analytic solution (solid curve) for 𝜀=10−3.

Example 3.3. Thirdly, we consider the following problem [22] 𝜀𝑦^′′+𝑦^′=0;𝑥∈[0,1] (3.16) subject to the boundary conditions 𝑦(0)=1,𝑦(1)=𝑒^−1/𝜀. (3.17) The exact solution for this problem is 𝑦(𝑥)=𝑒^−𝑥/𝜀. (3.18) Applying the operations of the differential transform to (3.16), we obtain the following recurrence relation: 𝑌()()𝑌()𝑘+2=−𝑘+1𝑘+1𝜀(𝑘+1)(𝑘+2). (3.19) By using the basic definitions of the differential transform and (3.17), the following transformed boundary conditions at 𝑥₀=0 can be obtained: 𝑌(0)=1,^𝑁_𝑘=0𝑌(𝑘)=𝑒^−1/𝜀. (3.20) By utilizing the recurrence relation in (3.19) and the transformed boundary conditions in (3.20), the following series solution up to 15-term is obtained: 𝑦(𝑥)𝑎=1+𝑎𝑥−𝑥2𝜀²+𝑎6𝜀²𝑥³−𝑎24𝜀³𝑥⁴+𝑎120𝜀⁴𝑥⁵−𝑎5040𝜀⁵𝑥⁶+𝑎5040𝜀⁶𝑥⁷−𝑎40320𝜀⁷𝑥⁸+𝑎362880𝜀⁸𝑥⁹−𝑎3628800𝜀⁹𝑥¹⁰+𝑎39916800𝜀¹⁰𝑥¹¹−𝑎479001600𝜀¹¹𝑥¹²+𝑎6227020800𝜀¹²𝑥¹³−𝑎87178291200𝜀¹³𝑥¹⁴+𝑎1307674368000𝜀¹⁴𝑥¹⁵−𝑂𝑥¹⁶, (3.21) where 𝑎=𝑦^′(0).
By taking 𝑁=15, the following equation can be obtained from (3.20): 𝑎1+𝑎+1307674368000𝜀¹⁴−𝑎87178291200𝜀¹³+𝑎6227020800𝜀¹²−𝑎479001600𝜀¹¹+𝑎39916800𝜀¹⁰−𝑎3628800𝜀⁹+𝑎362880𝜀⁸−𝑎40320𝜀⁷+𝑎5040𝜀⁶−𝑎720𝜀⁵+𝑎120𝜀⁴−𝑎24𝜀³+𝑎6𝜀²−𝑎2𝜀=𝑒^−1/𝜀. (3.22)
From (3.22), 𝑎 is evaluated as 𝑎=−130767436800𝑒^−1/𝜀−1+𝑒^1/𝜀𝜀¹⁴/1−15𝜀+210𝜀²−2730𝜀³+32760𝜀⁴−360360𝜀⁵+3603600𝜀⁶−32432400𝜀⁷[phone]𝜀⁸−1816214400𝜀⁹[phone]𝜀¹⁰−54486432000𝜀¹¹[phone]𝜀¹²−653837184000𝜀¹³[phone]𝜀¹⁴. (3.23) By using this value of the missing boundary condition, the approximate solution can be obtained easily.
Comparison of the approximate solution with the exact solution (3.18) for 𝜀=2⁻⁵ is sketched in Figure 3.

Figure 3 

The approximate solution (dotted curve) versus the analytic solution (solid curve) for 𝜀=2−5.

Example 3.4. Finally, we consider the following problem [23, 24] −𝜀𝑦^′′+𝑦^′=𝑒^𝑥;𝑥∈[0,1] (3.24) subject to the boundary conditions 𝑦(0)=0,𝑦(1)=0. (3.25) Its exact solution is given by 𝑦(𝑥)=1𝑒1−𝜀^𝑥−1−𝑒^1−(1/𝜀)+(𝑒−1)𝑒^{𝑥−(1/𝜀)}1−𝑒^−1/𝜀. (3.26) By applying the fundamental mathematical operations performed by differential transform, the differential transform of (3.24) is obtained as 𝑌()=()𝑌()𝑘+2−1/𝑘!+𝑘+1𝑘+1𝜀(𝑘+1)(𝑘+2). (3.27) The boundary conditions in (3.25) can be transformed at 𝑥₀=0 as 𝑌(0)=0,^𝑁_𝑘=0𝑌(𝑘)=0. (3.28) By using the inverse transformation rule in (2.2), the approximate solution is evaluated up to 𝑁=20. The first few terms of the series solution are given by 𝑦(𝑥)−1=𝑎𝑥+6𝜀²+𝑎6𝜀²−1𝑥6𝜀³+−124𝜀³+𝑎24𝜀³−124𝜀²−1𝑥24𝜀⁴+⋯, (3.29) where 𝑎=𝑦^′(0).The solution obtained from (2.3) has yet to satisfy the second boundary condition in (3.25), which has not been manipulated in obtaining this approximate solution. Applying this boundary condition and then solving the resulting equation for 𝑎 will determine the unknown constant 𝑎 and eventually the numerical solution.
Graphical result for 𝜀=1/1000 with comparison to the exact solution (3.26) is shown in Figure 4.

Figure 4 

The approximate solution (dotted curve) versus the analytic solution (solid curve) for 𝜀=1/1000.

4. Conclusion

In this study, the differential transformation method (DTM) has been employed, for the first time, successfully for solving linear singularly perturbed two-point boundary value problems. Four examples with boundary layers have been treated. This new method accelerated the convergence to the solutions. As it can be seen, this method leads to tremendously accurate results. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, discretization, or restrictive assumptions. The Mathematica software system has been used for all the symbolic and numerical computations in this paper.