Non-polynomial quadratic spline method for solving fourth order singularly perturbed boundary value problems

1 Introduction

Fourth order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, Newtonian fluid mechanics, chemical reactor theory, aerodynamics, hydrodynamics, optimal control, convection diffusion processes, quantum mechanics, etc. These problems have various important applications in fluid dynamics. Ghasemi et al. (2014) gave the analysis of electrohydrodynamic flow in a circular cylindrical conduit using least square method and Hatami and Domairry (2014) investigated the transient vertically motion of a soluble particle in a Newtonian fluid media. Hatami and Ganji (2014) studied the motion of a spherical particle in a fluid forced vortex by DQM and DTM. Hatami et al. (2016) gave the optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition. Nadeem and Haq (2014) studied the effect of thermal radiation for megnetohydrodynamic boundary layer flow of a nanofluid past a stretching sheet with convective boundary conditions. Sheikholeslami and Ganji investigated the nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM in Sheikholeslami and Ganji (2015). Sheikholeslami et al. (2013) gave investigation of squeezing unsteady nanofluid flow using ADM. Sheikholeslami et al. (2012) discussed analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian Decomposition Method. Sheikholeslami et al. (2012) investigated the laminar viscous flow in a semi-porous channel in the presence of uniform magnetic field using Optimal Homotopy Asymptotic Method. Zhou et al. (2016) designed the microchannel heat sink with wavy channel and its time-efficient optimization with combined RSM and FVM methods.

Singularly perturbed problems are classified on the fact that how the order of the differential equation is affected if $∊\to 0$ , here $∊$ is a small positive parameter multiplying the highest order derivative of the differential equation. The solution of singularly perturbed boundary value problem has a multiscale character; that is, there are thin transition layers where the solution varies rapidly, while away from the layers the solution varies very slowly.

In this paper, we develop a computational method to solve fourth order singularly perturbed boundary value problems of the form:

OR

In literature, we found many numerical methods which were developed for solving second order singularly perturbed BVPs. These methods are exponentially fitted finite difference scheme (Kadalbajoo and Kumar, 2009), non-polynomial spline method (Tirmizi, 2008), cubic spline method (Kumar et al., 2007) and quintic spline method (Rashidinia et al., 2010) etc. There are few methods available for solving higher order singularly perturbed BVPs such as asymptotic finite element method (Babu and Ramanujam, 2007), reproducing kernel method (Akram and Rehman, 2012). Shanthi and Ramanujam (Shanthi and Ramanujam, 2002) solved singularly perturbed fourth-order ordinary differential equations of convection–diffusion type using asymptotic numerical methods. The authors in Akram and Amin (2012, 2013) used quintic spline and septic spline respectively for solving fourth order singularly perturbed BVPs.

However, most of these methods were used to solve fourth-order singularly perturbed boundary value problem by using a higher degree spline. Here, we use a non-polynomial quadratic spline method for solving the problem (1). In this paper, we discuss two types of boundary value problems with boundary conditions (2) and (3). Firstly a numerical system is obtained by using non-polynomial quadratic spline. Then finite difference formula of $O(h^2)$ is used for making the system consistent with the given boundary value problem. Finally the obtained scheme is used to solve fourth order singularly perturbed boundary value problems. After implementation of the problem over the method we get a system of pentadiagonal matrix which is solved by using LU decomposition method. The paper describing a non-polynomial quadratic spline method is organized into five sections. Section 2 gives a brief derivation of the method, along with boundary conditions. In Section 3 truncation error has been obtained for fourth order method. Application of the method for solving fourth order singularly perturbed BVPs is discussed in Section 4. Convergence analysis of the method is discussed in Section 5. In Section 6, numerical examples and their comparison with the existing methods are presented which demonstrate the efficiency of our method. Conclusion and the figures are presented in Section 7 also proves the accuracy.

2 Derivation of the scheme

Let $a=x_0<x_1<x_2<···<x_{n+1}=b$ , we first divide the interval [a,b] into $n+1$ equal parts by introducing $$x_i=a+\mathit{ih}\text{,}i=0\text{,}1\text{,}\dots \text{,}n+1\mathit{and}h=(b-a)/(n+1)$$

Let

To determine the coefficients $a_i\text{,}{b}_i\mathit{and}{c}_i$ , we define the following interpolatory conditions as

By using above conditions we calculated the coefficients as $$\begin{array}{cc}\hfill a_i=& \frac{y_{i+1}-y_i}{\sin \theta }+\frac{(1-e^{\theta })}{2{\tau }^2\sin \theta }(D_i+D_{i+1})\hfill \\ \hfill b_i=& \frac{1}{2{\tau }^2}(D_i+D_{i+1})\hfill \\ \hfill c_i=& y_i-\frac{1}{2{\tau }^2}(D_i+D_{i+1})\hfill \end{array}$$ where, $\theta =\tau h$

Using the continuity of first derivative, $Q_{i-1}^m\left(x_i\right)=Q_i^m\left(x_i\right)\text{,}m=1$ the following consistency relation is derived

For making the system consistent with the given boundary conditions, we use finite difference formula of $O(h^2)$

Using (6) and (8), we obtained the following relation $$\gamma y_{i-2}+(\delta -2\gamma )y_{i-1}+(1-2\delta +\gamma )y_i+(-2+\delta )y_{i+1}+y_{i+2}=h^4({\alpha }_1{y}_{i-1}^{(4)}+\beta y_i^{(4)}+{\alpha }_2{y}_{i+1}^{(4)})$$

Eq. (9) form a system of $n-2$ linear equations in n unknowns $y_i\text{,}i=1\text{,}2\text{,}\dots \text{,}n$ . Thus, we need two more equations, one at each end of the range of integration.

For case (i), the equations are obtained as

For case (ii), the equations are obtained as

3 Truncation error

Expanding (9) by using Taylor series, we obtained the following truncation error

For different values of parameters, we get the method of second order as well as fourth order. Here we discuss only fourth order method. The local truncation error for (10) and (11) is $$t_i=\left\{\begin{array}{cc}-\frac{20162/3}{8!}{h}^8{y}_i^{(8)}+O(h^9)\text{,}\hfill & \mathrm{i}=1\hfill \\ \frac{5904/133}{8!}{h}^8{y}_i^{(8)}+O(h^9)\text{,}\hfill & \mathrm{i}=\mathrm{n}\hfill \end{array}\right.$$ and the local truncation error for (12) and (13) is $$t_i=\left\{\begin{array}{cc}-\frac{6554}{8!}{h}^8{y}_i^{(8)}+O(h^9)\text{,}\hfill & \mathrm{i}=1\hfill \\ \frac{211}{8!}{h}^8{y}_i^{(8)}+O(h^9)\text{,}\hfill & \mathrm{i}=\mathrm{n}\hfill \end{array}\right.$$

For $({\alpha }_1\text{,}\beta \text{,}{\alpha }_2\text{,}\gamma \text{,}\delta )=\left(\frac{1}{6}\text{,}\frac{1}{3}\text{,}\frac{1}{6}\text{,}1\text{,}-2\right)$ the truncation error is $$t_i=\left(-\frac{1}{720}\right)h^8{y}_i^{(8)}+O(h^9)\text{,}i=2\text{,}3\text{,}\dots \text{,}n-1$$

4 Application to the fourth order singularly perturbed boundary value problems

We consider a fourth order singularly perturbed boundary value problem of the form $$-∊y^{(4)}(x)+p(x)y(x)=q(x)\text{,}a<x<b$$ subject to the boundary conditions $$\begin{array}{c}\hfill y(a)={\alpha }_1\text{,}y(b)={\beta }_1\\ \hfill y^{(2)}(a)={\alpha }_2\text{,}{y}^{(2)}(b)={\beta }_2\end{array}$$

OR $$\begin{array}{c}\hfill y(a)={\alpha }_1\text{,}y(b)={\beta }_1\\ \hfill y^{(1)}(a)={\alpha }_2\text{,}{y}^{(1)}(b)={\beta }_2\end{array}$$ where $p\text{,}q$ are sufficiently continuously differentiable functions in the interval $\left[a\text{,}b\right]\text{,}{\alpha }_i$ and ${\beta }_i$ are real constants.

After applying the scheme (9) to the problem (1) with boundary conditions (2) and (3), we get the following relation

Eqs. (10) and (11) takes the form $$A_1{y}_1+A_2{y}_2+A_3{y}_3=∊A_0{y}_0-h^2∊y_0^{(2)}-h^4\left(\frac{7}{90}{q}_0+\frac{49}{72}{q}_1+\frac{7}{45}{q}_2+\frac{1}{360}{q}_3\right)$$ $$C_{n-2}{y}_{n-2}+C_{n-1}{y}_{n-1}+C_n{y}_n=∊C_{n+1}{y}_{n+1}-h^2∊y_{n+1}^{(2)}-h^4\left(\frac{7}{90}{q}_{n+1}+\frac{49}{72}{q}_n+\frac{7}{45}{q}_{n-1}+\frac{1}{360}{q}_{n-2}\right)$$ where, $$\begin{array}{c}\hfill A_0=2∊-\frac{7}{9}{h}^4{p}_0\text{,}{A}_1=5∊-\frac{49}{72}{h}^4{p}_1\text{,}{A}_2=-4∊-\frac{7}{45}{h}^4{p}_2\text{,}{A}_3=∊-\frac{1}{360}{h}^4{p}_3\\ \hfill C_{n-2}=5∊-\frac{49}{72}{h}^4{p}_{n-2}\text{,}{C}_{n-1}=-4∊-\frac{7}{45}{h}^4{p}_{n-1}\text{,}{C}_n=∊-\frac{1}{360}{h}^4{p}_n\text{,}{C}_{n+1}=2∊-\frac{7}{90}{h}^4{p}_{n+1}\end{array}$$

Eqs. (12) and (13) takes the form $$A_1{y}_1+A_2{y}_2+A_3{y}_3=∊A_0{y}_0-3∊{\mathit{hy}}_0^{(1)}-h^4\left(\frac{8}{280}{q}_0+\frac{151}{280}{q}_1+\frac{52}{280}{q}_2-\frac{1}{280}{q}_3\right)$$ $$C_{n-2}{y}_{n-2}+C_{n-1}{y}_{n-1}+C_n{y}_n=∊C_{n+1}{y}_{n+1}-3∊{\mathit{hy}}_{n+1}^{(1)}-h^4\left(\frac{8}{280}{q}_{n+1}+\frac{151}{280}{q}_n+\frac{52}{280}{q}_{n-1}-\frac{1}{280}{q}_{n-2}\right)$$ where, $$\begin{array}{cc}\hfill & A_0=\frac{11}{2}∊-\frac{8}{280}{h}^4{p}_0\text{,}{A}_1=9∊-\frac{151}{280}{h}^4{p}_1\text{,}{A}_2=-\frac{9}{2}∊-\frac{52}{280}{h}^4{p}_2\text{,}{A}_3=∊+\frac{1}{280}{h}^4{p}_3\hfill \\ \hfill & C_{n-2}=9∊-\frac{151}{280}{h}^4{p}_{n-2}\text{,}{C}_{n-1}=-\frac{9}{2}∊-\frac{52}{280}{h}^4{p}_{n-1}\text{,}{C}_n=∊+\frac{1}{280}{h}^4{p}_n\text{,}\hfill \\ \hfill & C_{n+1}=\frac{11}{2}∊-\frac{8}{280}{h}^4{p}_{n+1}\hfill \end{array}$$

5 Convergence analysis

The developed method leads to the following matrix form

$Y={[y_1\text{,}{y}_2\text{,}\dots \text{,}{y}_{n-1}]}^T$ and the right hand side vector is $C={[c_1\text{,}{c}_2\text{,}\dots \text{,}{c}_{n-1}]}^T$ .

For Case (i), we have $$\begin{array}{cc}\hfill c_1=& A_0∊y_0-h^2∊y_0^{(2)}-h^4\left(\frac{7}{90}{q}_0+\frac{49}{72}{q}_1+\frac{7}{45}{q}_2+\frac{1}{360}{q}_3\right)\hfill \\ \hfill c_2=& -A∊y_0-h^4({\alpha }_1{q}_1+\beta q_2+{\alpha }_2{q}_3)\hfill \\ \hfill c_i=& -h^4({\alpha }_1{q}_{i-1}+\beta q_i+{\alpha }_2{q}_{i+1})\text{,}i=3\text{,}4\text{,}\dots \text{,}n-1\hfill \\ \hfill c_n=& ∊C_{n+1}{y}_{n+1}-h^2∊y_{n+1}^{(2)}-h^4\left(\frac{7}{90}{q}_{n+1}+\frac{49}{72}{q}_n+\frac{7}{45}{q}_{n-1}+\frac{1}{360}{q}_{n-2}\right)\hfill \end{array}$$

For Case (ii),we have $$\begin{array}{cc}\hfill c_1=& A_0∊y_0-3h∊y_0^{(1)}-h^4\left(\frac{8}{280}{q}_0+\frac{151}{280}{q}_1+\frac{52}{280}{q}_2-\frac{1}{280}{q}_3\right)\hfill \\ \hfill c_2=& -A∊y_0-h^4({\alpha }_1{q}_1+\beta q_2+{\alpha }_2{q}_3)\hfill \\ \hfill c_i=& -h^4({\alpha }_1{q}_{i-1}+\beta q_i+{\alpha }_2{q}_{i+1})\text{,}i=3\text{,}4\text{,}\dots \text{,}n-1\hfill \\ \hfill c_n=& ∊C_{n+1}{y}_{n+1}-3h∊y_{n+1}^{(1)}-h^4\left(\frac{8}{280}{q}_{n+1}+\frac{151}{280}{q}_n+\frac{52}{280}{q}_{n-1}-\frac{1}{280}{q}_{n-2}\right)\hfill \end{array}$$

Also we have,

From (16) and (17) we have $$\begin{array}{cc}\hfill P(\stackrel{\sim }{Y}-Y)=& T\hfill \\ \hfill \mathit{PE}=& T\hfill \\ \hfill E=& \stackrel{\sim }{Y}-Y={[e_1\text{,}{e}_2\text{,}\dots \text{,}{e}_{n-1}]}^T\hfill \end{array}$$

Now we have to calculate sum of each row of matrix P:

For case (i), $$\begin{array}{cc}\hfill S_1=& 2∊-\frac{49}{72}{h}^4{p}_1-\frac{7}{45}{h}^4{p}_2-\frac{1}{360}{h}^4{p}_3\hfill \\ \hfill S_2=& -\gamma ∊-h^4{\alpha }_1{p}_1-h^4\beta p_2-h^4{\alpha }_2{p}_3\hfill \\ \hfill S_i=& -h^4{\alpha }_1{p}_{i-1}-h^4\beta p_i-h^4{\alpha }_2{p}_{i+1}\text{,}i=3\text{,}\dots \text{,}n-1\hfill \\ \hfill S_n=& 2∊-\frac{49}{72}{h}^4{p}_{n-2}-\frac{7}{45}{h}^4{p}_{n-1}-\frac{1}{360}{h}^4{p}_n\hfill \end{array}$$

For case (ii), $$\begin{array}{cc}\hfill S_1=& \frac{11}{2}∊-\frac{151}{280}{h}^4{p}_1-\frac{52}{280}{h}^4{p}_2+\frac{1}{280}{h}^4{p}_3\hfill \\ \hfill S_2=& -\gamma ∊-h^4{\alpha }_1{p}_1-h^4\beta p_2-h^4{\alpha }_2{p}_3\hfill \\ \hfill S_i=& -h^4{\alpha }_1{p}_{i-1}-h^4\beta p_i-h^4{\alpha }_2{p}_{i+1}\text{,}i=3\text{,}\dots \text{,}n-1\hfill \\ \hfill S_n=& \frac{11}{2}∊-\frac{151}{280}{h}^4{p}_{n-2}-\frac{52}{280}{h}^4{p}_{n-1}+\frac{1}{280}{h}^4{p}_n\hfill \end{array}$$

Let $0<M\in Z^{+}$ is the minimum of $|p_{i-1}|\text{,}|p_i|\text{,}|p_{i+1}|$ . For sufficiently small h we can say that:

For case (i), $$S_i⩾\left\{\begin{array}{cc}\frac{151}{180}{h}^{4}M\text{,}\hfill & \mathrm{i}=1\hfill \\ ({\alpha }_1+\beta +{\alpha }_2)h^{4}M\text{,}\hfill & \mathrm{i}=2\hfill \\ ({\alpha }_1+\beta +{\alpha }_2)h^{4}M\text{,}\hfill & \mathrm{i}=3\text{,}\dots \text{,}\mathrm{n}-1\hfill \\ \frac{151}{180}{h}^{4}M\text{,}\hfill & \mathrm{i}=\mathrm{n}\hfill \end{array}\right.$$

For case (ii), $$S_i⩾\left\{\begin{array}{cc}\frac{51}{70}{h}^{4}M\text{,}\hfill & \mathrm{i}=1\hfill \\ ({\alpha }_1+\beta +{\alpha }_2)h^{4}M\text{,}\hfill & \mathrm{i}=2\hfill \\ ({\alpha }_1+\beta +{\alpha }_2)h^{4}M\text{,}\hfill & \mathrm{i}=3\text{,}\dots \text{,}\mathrm{n}-1\hfill \\ \frac{51}{70}{h}^{4}M\text{,}\hfill & \mathrm{i}=\mathrm{n}\hfill \end{array}\right.$$

Further,we get for case (i) $$\frac{1}{S_i}⩽\left\{\begin{array}{cc}\frac{180}{151h^{4}M}\text{,}\hfill & \mathrm{i}=1\text{,}\mathrm{n}\hfill \\ \frac{1}{({\alpha }_1+\beta +{\alpha }_2)h^{4}M}\text{,}\hfill & \mathrm{i}=2\text{,}3\text{,}\dots \text{,}\mathrm{n}-1\hfill \end{array}\right.$$

For sufficiently small h, we can easily show that the matrix P is irreducible and monotone. Therefore, $P^{-1}$ exist and $P^{-1}⩾0$ . Hence,

Therefore, $$\begin{array}{cc}\hfill p_{i\text{,}j}^{\ast }⩽& \frac{1}{S_i}\hfill \\ \hfill \Vert P^{-1}\Vert =& {\displaystyle \underset{1⩽i⩽n}{\max }}{\displaystyle \sum _{j=1}^n}|p_{i\text{,}j}^{\ast }|⩽{\displaystyle \sum _{i=1}^n}\frac{1}{S_i}=\frac{1}{h^{4}M}\left(\frac{360}{151}+\frac{2}{{\alpha }_1+\beta +{\alpha }_2}\right)\text{,}i=1\text{,}\dots \text{,}n\mathit{and}\hfill \\ \hfill \Vert T_i\Vert =& {\displaystyle \underset{1⩽i⩽n}{\max }}{\displaystyle \sum _{i=1}^n}|T_i|\hfill \end{array}$$ and error is given by:

For case (i), $$\Vert E\Vert =\Vert P^{-1}\Vert \Vert T\Vert ⩽\frac{1}{h^{4}M}\left(\frac{360}{151}+\frac{2}{{\alpha }_1+\beta +{\alpha }_2}\right)\Vert T\Vert$$

Similarly for Case (ii), the error is given by $$\Vert E\Vert =\Vert P^{-1}\Vert \Vert T\Vert ⩽\frac{1}{h^{4}M}\left(\frac{140}{51}+\frac{2}{{\alpha }_1+\beta +{\alpha }_2}\right)\Vert T\Vert$$

By using (14), we have truncation error $\Vert T\Vert =O(h^8)$ , then we get $$\Vert E\Vert ⩽\frac{1}{h^{4}M}\left(\frac{360}{151}+\frac{2}{{\alpha }_1+\beta +{\alpha }_2}\right)O(h^8)=O(h^4)$$

Hence, the scheme is fourth order convergent. $$\Vert E\Vert =O(h^4)$$

6 Numerical illustrations

In the present paper, we consider two linear fourth order singularly perturbed boundary value problems whose exact solutions are known. The maximum absolute errors for h = 1/16, 1/32, 1/64 and 1/128 are tabulated in Tables 1 and 2 and comparison are also shown in graphs 1–2. The obtained results are compared with the results of quintic spline method (Akram and Amin, 2012) and septic spline method (Akram and Naheed, 2013).