Non-polynomial cubic spline discretization for system of non-linear singular boundary value problems using variable mesh

Abstract

In this paper, we propose two generalized non-polynomial cubic spline schemes using a variable mesh to solve the system of non-linear singular two point boundary value problems. Theoretical analysis proves that the proposed methods have second- and third-order convergence. Both methods are applicable to singular boundary value problems. Numerical results are also provided to show the accuracy and efficiency of the proposed methods.

1 Introduction

In this paper, we study two effective numerical techniques using a non-polynomial cubic spline based on a variable mesh to solve system of M non-linear singular boundary value problems (BVPs) of the following type:

$$\begin{aligned} y_{xx}^{(i)}=F^{(i)} \bigl(x,y^{(1)},\ldots,y^{(i)},\ldots,y^{(M)},y_{x}^{(1)}, \ldots, y_{x}^{(i)},\ldots,y_{x}^{(M)} \bigr),\quad a \le x \le b, \end{aligned}$$

(1)

subject to boundary conditions

$$ y^{(i)}(a)=A_{ i },\qquad y^{(i)}(b)=B_{ i}, \quad \text{where } y_{x}^{(i)}=\frac{d y^{(i)}}{dx},y_{xx}^{(i)}= \frac{d ^{2} y^{(i)}}{dx^{2}},i=1,2,3,\ldots,M. $$

(2)

We assume that, for $-\infty < a \le x \le b<\infty$ and $-\infty < y^{(i)}$, $y_{x}^{(i)}<\infty$, where $y^{(i)}=y^{(i)}(x)$, $y_{x}^{(i)}=y_{x}^{(i)}(x)$, we have

• (i) $F^{(i)}(x, y^{(1)},y^{(2)},\ldots,y^{(i)},\ldots,y^{(M)},y_{x}^{(1)}, y_{x}^{(2)},\ldots,y_{x}^{(i)},\ldots,y_{x}^{(M)})$ is continuous;

• (ii) $\frac{\partial F^{(i)}}{\partial y^{(j)}}$ and $\frac{\partial F^{(i)}}{\partial y_{x}^{(j)}}$ exist and are continuous;

• (iii) $\frac{\partial F^{(i)}}{\partial y^{(j)}}>0$ and $\vert \frac{\partial F ^{(i)}}{\partial {y_{x}^{(j)}}} \vert \leq C$, for some positive constant C and $i,j=1,2,3,\ldots,M$.

These conditions as proved by Keller [1] ensure us of the existence of a unique solution of the above system of boundary value problem (1)-(2).

In the present paper, we have derived generalized non-polynomial cubic spline schemes of second- and third-order using a variable mesh for solving system of two point boundary value problems (1)-(2). Such systems effectively decompose several higher-order problems into second-order boundary value problems; thus solving them efficiently. These higher-order problems are used to model various phenomena in the field of astrophysics, astronomy, hydrodynamics, beam and wave theory [2–6]. For these boundary value problems, Aftabizadeh [7], Zill [8], Regan [9] and Agarwal [10] have obtained the existence and uniqueness of the solutions.

Many authors have developed efficient numerical schemes to solve boundary value problems and splines have been rigourously used to approximate the solution of such problems. To name a few, Mohanty et al. [11–13] developed AGE, cubic spline TAGE, Newton-TAGE iteration methods using a finite difference and cubic spline method based on uniform and non-uniform mesh, respectively, to solve non-linear singular two point boundary value problems. Even singularly perturbed boundary value problems with or without first derivative terms are solved. Kadalbajoo et al. [14] developed a third-order variable-mesh cubic spline method; Mohanty et al. used a spline in compression [15, 16], a spline in tension [17, 18], and the cubic spline TAGE method [19]. Wazwaz [20] developed modified Adomian decomposition method to solve linear and non-linear fourth-order boundary value problems, Akram and Siddiqi [21] used the second-order convergent non-polynomial spline method to solve sixth-order linear special case boundary value problems. Talwar and Mohanty [22] developed a finite-difference method for the solution of a fourth-order ordinary differential equation. Twizell and Boutayeb [23] developed finite-difference methods for solving eighth-order boundary value problems. Akram and Rehman [24] solved eighth-order boundary value problems using the kernel space method. Siddiqi and Akram [25, 26] solved sixth- and eighth-order boundary value problems using a non-polynomial and septic spline. Liu and Wu [27] used a generalized differential quadrature rule to solve a special case of eighth-order boundary value problems. Also a variable mesh has been extensively used by many authors. Numerical simulations with high-order compact difference schemes depict more accurate solution values on variable meshes as compared to some high-order compact scheme on a uniform mesh network. This happens because the truncation error in a finite-difference approximation depends upon the derivative of the variable as well as mesh spacing. Therefore, to attain uniformly distributed truncation errors, it is essential to employ non-uniform meshes, i.e., finer meshes in the region for largely deviated derivatives and coarse meshes for a smooth function. In this manner, the error disperses almost uniformly over the domain of integration and renders an accurate solution to a greater extent [28]. Such ongoing work motivated us to develop an efficient non-polynomial cubic spline scheme to solve the system of non-linear singular boundary value problems using a variable mesh.

Presently, once the proposed method decomposes the higher-order boundary value problem into a system of second-order boundary value problems (1), the boundary conditions are also accordingly modified. As the boundary conditions are incorporated, we get a block tri-diagonal Jacobian from the system of discretized second-order boundary value problems. We implemented the method on non-linear as well as linear problems. In the case of a linear boundary value problem, we get a linear system of coupled difference equations, which are solved by a block Gauss elimination method and the non-linear system of coupled difference equations are solved by using the block Newton’s method.

The sections of this paper are organized as follows. In Section 2, we give details of derivation of the scheme using a second-order singular linear boundary value problem and, in Section 3, we provide a generalization of the scheme. In Section 4, we present the application of the proposed schemes to a fourth- and sixth-order singular BVP. In Section 5, we discuss a convergence analysis and in Section 6, we provide numerical illustrations to demonstrate the accuracy of the proposed schemes. Finally in Section 7, we provide concluding remarks.

2 Derivation of the schemes

We consider a scalar second-order non-linear boundary value problem of the following type:

$$\begin{aligned} y_{xx} =&F(x,y,y_{x}),\quad \text{such that } y (a)=A, y(b)=B. \end{aligned}$$

(3)

Firstly, we discretize the solution region $[a,b]$ such that $a=x_{0} < x_{1} <x_{2}<\cdots<x_{N-1}< x_{N}=b$. Let $h_{j}=x_{j}-x_{j-1}$, $j=1,2,3,\ldots,N$, be the mesh size and the mesh ratio be $\sigma = \frac{h_{j+1}}{h_{j}} >0$, $j=1,2,3,\ldots,N-1$. When $\sigma =1$, the mesh reduces to a uniform mesh, i.e., $h_{j+1}=h_{j}=h$. Also, assume $y_{j}$ and $Y_{j}$ to be the approximate and exact solution of (3) at the grid points $x_{j}$, $j=1,2,\ldots,N$. Then the interpolating non-polynomial cubic spline approximation function can be defined as

$$\begin{aligned}& {S}(x) = a_{j}+b_{j}(x-x_{j})+c_{j} \sin \bigl(k(x-x_{j})\bigr)+d_{j} \cos \bigl( k(x-x _{j})\bigr), \\& \quad k>0, x_{j-1} \le x \le x_{j},j=1,2,3,\ldots,N, \end{aligned}$$

(4)

which satisfies the following conditions:

• (i) $S(x)$ coincides with a cubic polynomial in $[x_{j-1}, x_{j}]$, $j=1,2,3,\ldots,N$.

• (ii) $S(x) \in C^{2}[a,b]$.

• (iii) $S(x_{j})=y(x_{j})$, $S_{{xx}_{j}}(x_{j})=F_{j}$, $S_{{xx}_{j}}(x_{j\pm \frac{1}{2}})=F_{j \pm \frac{1}{2}}$ for $j=1,2,3,\ldots,N$.

Using the definition of spline (4) and conditions (i) and (iii) we get the following values for $a_{j}$, $b_{j}$, $c_{j}$, $d_{j}$ and approximations:

$$\begin{aligned}& a_{j}=y_{j}+\frac{F_{j}}{k^{2}},\qquad b_{j}= \frac{y_{j+1}-y_{j}}{h_{j}}+\frac{F _{j+1}-F_{j}}{k^{2} h_{j}}, \end{aligned}$$

(5)

$$\begin{aligned}& c_{j}=-\frac{F_{j}}{k^{2}} \quad \text{and} \quad d_{j}= \frac{F_{j} \cos (k h_{j}) - F_{j+1}}{k^{2}\sin (k h_{j})}, \end{aligned}$$

(6)

$$\begin{aligned}& {S_{x}}(x_{j+1}) =\frac{y_{j+1}-y_{j}}{h_{j+1}}+h_{j+1}( \alpha F _{j}+\beta_{1} F_{j+1}), \end{aligned}$$

(7)

$$\begin{aligned}& {S_{x}}(x_{j-1}) = \frac{y_{j}-y_{j-1}}{h_{j}}-h_{j}( \gamma F_{j}+ \beta_{2} F_{j-1}), \end{aligned}$$

(8)

$$\begin{aligned}& {S_{x}}(x_{j})=\frac{y_{j+1}\beta_{3}+y_{j}(\sigma_{j}^{2} \beta _{4}-\beta_{3})-y_{j-1}\sigma_{j}^{2}\beta_{4}}{h_{j}\sigma_{j}}-h _{j+1}( \alpha \beta_{3} F_{j+1}-\gamma \beta_{4} F_{j-1}), \end{aligned}$$

(9)

where

$$\begin{aligned}& \alpha = \frac{-1}{\theta_{j+1}^{2}} \biggl(1-\frac{\theta_{j+1}}{\sin \theta_{j+1}} \biggr),\qquad \gamma =\frac{-1}{{\theta_{j}}^{2}} \biggl(1-\frac{\theta_{j}}{\sin\theta_{j}} \biggr), \end{aligned}$$

(10)

$$\begin{aligned}& \beta_{1} = \frac{1}{\theta_{j+1}^{2}} \biggl(1-\frac{\cot\theta_{j+1}}{ \theta_{j+1}} \biggr), \qquad \beta_{2}=\frac{1}{\theta_{j}^{2}} \biggl(1-\frac{\cot \theta_{j}}{\theta_{j}} \biggr), \end{aligned}$$

(11)

$$\begin{aligned}& \beta_{3} = \frac{\beta_{2}}{(\sigma_{j}\beta_{1}+\beta_{2})}, \qquad \beta _{4}= \frac{\beta_{1}}{(\sigma_{j}\beta_{1}+\beta_{2})}, \qquad \theta_{j}=k h _{j}\quad \text{and}\quad y_{j}=y(x_{j}). \end{aligned}$$

(12)

Now, by the continuity conditions of first derivative, i.e., (ii), we get the following scheme:

$$\begin{aligned} y_{j+1}-(1+\sigma_{j})y_{j}+\sigma_{j} y_{j-1} = h_{j} h_{j+1} \bigl( \sigma_{j} \alpha F_{j+1} + (\sigma_{j}\beta_{1}+ \beta_{2} )F_{j} + \gamma F_{j-1} \bigr)+T_{j}(h_{j}). \end{aligned}$$

(13)

We observe, as $\theta_{j} \rightarrow 0$, ($\alpha , \beta_{1}, \beta_{2}, \gamma )\rightarrow (\frac{1}{6},\frac{1}{3}, \frac{1}{3}, \frac{1}{6}$), scheme (13) reduces into the standard variable-mesh cubic spline scheme

$$\begin{aligned} y_{j+1}-(1+\sigma_{j})y_{j}+\sigma_{j} y_{j-1}= h_{j} h_{j+1} \biggl[ \frac{ \sigma_{j} }{6} F_{j+1}+\frac{ (\sigma_{j}+1)}{3} F_{j} +\frac{1}{6} F _{j-1} \biggr]. \end{aligned}$$

(14)

Now, we consider the following approximations evaluated at the grid points $x_{j}$, $j=1,2,3,\ldots,N-1$:

$$\begin{aligned}& s_{j}=\sigma_{j}(\sigma_{j}+1), \end{aligned}$$

(15)

$$\begin{aligned}& \bar{y}_{x_{j+1}} =\frac{(1+2 \sigma_{j})y_{j+1}-(1+\sigma_{j})^{2} y_{j} +\sigma_{j}^{2} y_{j-1} }{h_{j} s_{j}}, \end{aligned}$$

(16)

$$\begin{aligned}& \bar{y}_{x_{j-1}} =\frac{- y_{j+1} +(1+\sigma_{j})^{2} y_{j} -\sigma _{j}(2+\sigma_{j}) y_{j-1} }{h_{j} s_{j}}, \end{aligned}$$

(17)

$$\begin{aligned}& \bar{y}_{x_{j}} =\frac{y_{j+1} +(\sigma_{j}^{2}-1)y_{j} -\sigma_{j} ^{2} y_{j-1} }{h_{j} s_{j}}, \end{aligned}$$

(18)

$$\begin{aligned}& \bar{{F}}_{j+1}=F (x_{j+1},{y_{j+1}}, \bar{y}_{x_{j+1}}), \end{aligned}$$

(19)

$$\begin{aligned}& \bar{{F}}_{j-1}=F(x_{j-1}, {y_{j-1}}, \bar{y}_{x_{j-1}}), \end{aligned}$$

(20)

$$\begin{aligned}& \bar{{F}}_{j}=F(x_{j},y_{j}, \bar{y}_{x_{j}}), \end{aligned}$$

(21)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j+1}}=\frac{y_{j+1}-y_{j}}{h_{j+1}}+h_{j+1}( \alpha \bar{{F}}_{j}+\beta_{1} \bar{{F}}_{j+1}), \end{aligned}$$

(22)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j-1}}=\frac{y_{j}-y_{j-1}}{h_{j}}-h_{j}(\gamma \bar{ {F}}_{j}+\beta_{2} \bar{{F}}_{j-1}), \end{aligned}$$

(23)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j}}=\frac{y_{j+1}\beta_{3}+y_{j}(\sigma_{j}^{2} \beta_{4}-\beta_{3})-y_{j-1}\sigma_{j}^{2}\beta_{4}}{h_{j}\sigma_{j}}-h _{j+1}(\alpha \beta_{3} \bar{{F}}_{j+1}-\gamma \beta_{4} \bar{{F}} _{j-1}), \end{aligned}$$

(24)

$$\begin{aligned}& \bar{\bar{F}}_{j+1}=F(x_{j+1}, {y_{j+1}},\bar{ \bar{y}}_{x_{j+1}}), \end{aligned}$$

(25)

$$\begin{aligned}& \bar{\bar{F}}_{j-1}=F(x_{j-1}, {y_{j-1}},\bar{ \bar{y}}_{x_{j-1}}), \end{aligned}$$

(26)

$$\begin{aligned}& \bar{\bar{F}}_{j}=F(x_{j},y_{j},\bar{ \bar{y}}_{x_{j}}). \end{aligned}$$

(27)

Simplifying (10)-(12) and the approximations (16)-(27), for $j=1,2,3,\ldots,N$ we get

$$\begin{aligned}& \alpha =\frac{1}{6}+O\bigl(h_{j}^{2}\bigr)=\gamma, \end{aligned}$$

(28)

$$\begin{aligned}& \beta_{1}=\frac{1}{3}+O\bigl(h_{j}^{2} \bigr)=\beta_{2}, \end{aligned}$$

(29)

$$\begin{aligned}& \bar{y}_{x_{j}} =y_{x_{j}} +\frac{1}{6} \sigma_{j} h_{j}^{2} y_{xxx _{j}} +O \bigl(h_{j}^{3}\bigr), \end{aligned}$$

(30)

$$\begin{aligned}& \bar{y}_{x_{j+1}} =y_{x_{j+1}} -\frac{1}{6} \sigma_{j}(1+\sigma_{j})h _{j}^{2} y_{xxx_{j}} +O\bigl(h_{j}^{3}\bigr), \end{aligned}$$

(31)

$$\begin{aligned}& \bar{y}_{x_{j-1}} =y_{x_{j-1}} -\frac{1}{6}(1+ \sigma_{j}) h_{j}^{2} y_{xxx_{j}} +O \bigl(h_{j}^{3}\bigr), \end{aligned}$$

(32)

$$\begin{aligned}& \bar{F}_{j+1} =F_{j+1} -\frac{1}{6} \sigma_{j}(1+\sigma_{j}) h_{j} ^{2} y_{xxx_{j}} G +O\bigl(h_{j}^{3}\bigr), \end{aligned}$$

(33)

$$\begin{aligned}& \bar{F}_{j-1} =F_{j-1} -\frac{1}{6}(1+ \sigma_{j}) h_{j}^{2} y_{xxx _{j}} G +O \bigl(h_{j}^{3}\bigr), \end{aligned}$$

(34)

$$\begin{aligned}& \bar{F}_{j} =F_{j} +\frac{1}{6} \sigma_{j} h_{j}^{2} y_{xxx} G +O\bigl(h _{j}^{3}\bigr), \end{aligned}$$

(35)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j}} ={y}_{x_{j}} + O\bigl(h_{j}^{3} \bigr), \end{aligned}$$

(36)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j \pm 1}} ={{{y_{x}} }_{j\pm 1}}+O \bigl(h_{j}^{3}\bigr), \end{aligned}$$

(37)

$$\begin{aligned}& \bar{\bar{F}}_{j} =F_{j} +O\bigl(h_{j}^{3}\bigr), \end{aligned}$$

(38)

$$\begin{aligned}& \bar{\bar{F}}_{j \pm 1} =F_{j\pm 1} +O\bigl(h_{j}^{3} \bigr), \end{aligned}$$

(39)

$$\begin{aligned}& \quad \text{where } G =\frac{\partial F }{ \partial y_{x_{j}} }. \end{aligned}$$

(40)

Now, using (30)-(39) we generate a family of variable-mesh non-polynomial cubic spline schemes of second- and third-order for different values of $P_{j}$, $R_{j}$ and $Q_{j}$ in the following scheme:

$$ y_{j+1}-(1+\sigma_{j})y_{j}+\sigma_{j} y_{j-1} = h_{j} h_{j+1} (P _{j} F_{j+1} + Q_{j} F_{j} +R_{j} F_{j-1} )+T_{j}(h_{j}). $$

• (I) Second-order scheme. For $P_{j}=\sigma_{j} \alpha$, $Q_{j}= ( \sigma_{j}\beta_{1}+\beta_{2} )$, $R_{j}=\gamma$, the local truncation error $T_{j}$ is $O(h_{j}^{4})$, thus leading to a second-order method. $\theta_{j}$ satisfies the consistency condition $\tan (\frac{k h _{j}}{2})+(\frac{k h_{j}}{2})=(\frac{1}{2})$ and this equation has an infinite number of roots. We can use the smallest positive non-zero root of the equation as the value of $\theta_{j}$ i.e., $k h_{j}=8.986818916$.

• (II) Third-order scheme. Again, for the following values of $P_{j}$, $R_{j}$ and $Q_{j}$, we get the second scheme:

$$ \begin{aligned} &P_{j}=\frac{(\sigma_{j}^{2}+\sigma_{j}-1)}{12},\qquad Q_{j}=\frac{(\sigma_{j}+1) (\sigma_{j}^{2}+3\sigma_{j}+1)}{12},\qquad R_{j}= \frac{\sigma_{j}(1+ \sigma_{j}-\sigma_{j}^{2})}{12}. \end{aligned} $$

(41)

• Also, using (38)-(39) we can easily prove that $T_{j}=O(h_{j}^{5})$ and the local truncation error in the case of the second discretization scheme with uniform mesh is $O(h^{6})$. Note that the coefficients $P_{j}$, $Q_{j}$, $R_{j}$ are positive if $\frac{(\sqrt{5}-1)}{2}<\sigma _{j}<\frac{(\sqrt{5}+1)}{2}$, thus satisfying condition of convergence of the scheme [29].

3 Generalization of the schemes

We generalize our method for the solution of the system of M non-linear boundary value problems (1). At the grid point $x_{j}$, $j=1,2,\ldots,N-1$, we use the following approximations and schemes:

$$\begin{aligned}& {S}^{(i)}(x) = a_{j}+b_{j}(x-x_{j})+c_{j} \sin \bigl( k(x-x_{j})\bigr)+d_{j} \cos\bigl(k(x-x_{j})\bigr), \end{aligned}$$

(42)

$$\begin{aligned}& \quad k>0, x_{j-1} \le x \le x_{j},j=1,2,3,\ldots,N. \\& s_{j} = \sigma_{j}(\sigma_{j}+1), \end{aligned}$$

(43)

$$\begin{aligned}& \bar{y}_{x_{j+1}}^{(i)} = \frac{(1+2 \sigma_{j})y_{j+1}^{(i)}-(1+ \sigma_{j})^{2} y_{j}^{(i)}+\sigma_{j}^{2} y_{j-1}^{(i)}}{h_{j} s_{j}}, \end{aligned}$$

(44)

$$\begin{aligned}& \bar{y}_{x_{j-1}}^{(i)} = \frac{- y_{j+1}^{(i)}+(1+\sigma_{j})^{2} y _{j}^{(i)}-\sigma_{j}(2+\sigma_{j}) y_{j-1}^{(i)}}{h_{j} s_{j}}, \end{aligned}$$

(45)

$$\begin{aligned}& \bar{y}_{x_{j}}^{(i)} = \frac{y_{j+1}^{(i)}+(\sigma_{j}^{2}-1)y_{j} ^{(i)}-\sigma_{j}^{2} y_{j-1}^{(i)}}{h_{j} s_{j}}, \end{aligned}$$

(46)

$$\begin{aligned}& \bar{F}_{r}^{(i)} = F^{(i)}\bigl(x_{r},y_{r}^{(1)},y_{r}^{(2)},\ldots,y_{r}^{(i)},\ldots,y_{r}^{(M)}, \bar{y}_{x_{r}}^{(1)},\bar{y}_{x_{r}} ^{(2)},\ldots, \bar{y}_{x_{r}}^{(i)},\ldots,\bar{y}_{x_{r}}^{(M)} \bigr), \end{aligned}$$

(47)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j+1}}^{(i)} = \frac{{y^{(i)}_{j+1}} -{y^{(i)}_{j}} }{h_{j+1}}+h_{j+1} \bigl(\alpha \bar{F}^{(i)}_{j}+\beta_{1} \bar{F}^{(i)} _{j+1}\bigr), \end{aligned}$$

(48)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j-1}}^{(i)} = \frac{{y^{(i)}_{j}} -{y^{(i)}_{j-1}} }{h_{j}}-h_{j} \bigl(\gamma \bar{F}^{(i)}_{j}+\beta_{2} \bar{F}^{(i)}_{j-1}\bigr), \end{aligned}$$

(49)

$$\begin{aligned}& \bar{\bar{y}}_{x_{j}}^{(i)} = \frac{{y^{(i)}_{j+1}} \beta_{3}+{y ^{(i)}_{j}} (\sigma_{j}^{2} \beta_{4}-\beta_{3})-{y^{(i)}_{j-1}} \sigma_{j}^{2}\beta_{4}}{h_{j}\sigma_{j}}-h_{j+1} \bigl(\alpha \beta_{3} \bar{F}^{(i)}_{j+1}-\gamma \beta_{4} \bar{F}^{(i)}_{j-1}\bigr), \end{aligned}$$

(50)

$$\begin{aligned}& \bar{\bar{F}}_{r}^{(i)} = F^{(i)} \bigl(x_{r}, y_{r}^{(1)},y_{r}^{(2)}, \ldots, y_{r}^{(i)},\ldots,y_{r}^{(M)}, \bar{\bar{y}}_{x_{r}}^{(1)},\bar{ \bar{y}}_{x_{r}}^{(2)}, \ldots,\bar{\bar{y}}_{x_{r}}^{(i)},\ldots,\bar{ \bar{y}}_{x_{r}}^{(M)}\bigr), \end{aligned}$$

(51)

$$\begin{aligned}& y^{(i)}_{j+1}-(1+\sigma_{j})y^{(i)}_{j}+ \sigma_{j} y^{(i)}_{j-1} = h _{j} h_{j+1} \bigl(\sigma_{j} \alpha \bar{F}^{(i)}_{j+1} + (\sigma_{j} \beta_{1}+\beta_{2} ) \bar{F}^{(i)}_{j} + \gamma \bar{F}^{(i)}_{j-1} \bigr)+T_{j}(h_{j}), \end{aligned}$$

(52)

and also

$$\begin{aligned} y^{(i)}_{j+1}-(1+\sigma_{j})y^{(i)}_{j}+ \sigma_{j} y^{(i)}_{j-1} = h _{j} h_{j+1} \bigl(P_{j} \bar{\bar{F}}^{(i)}_{j+1} + Q_{j} \bar{ \bar{F}}^{(i)}_{j} + R_{j} \bar{\bar{F}}^{(i)}_{j-1} \bigr)+T_{j}(h_{j}), \end{aligned}$$

(53)

where

$$\begin{aligned}& a_{j} = {y^{(i)}_{j}}+\frac{F_{j}^{(i)}}{k^{2}},\qquad b_{j}= \frac{y^{(i)} _{j+1}-y^{(i)}_{j}}{h_{j}}+\frac{F^{(i)}_{j+1}-F^{(i)}_{j}}{k^{2} h _{j}}, \end{aligned}$$

(54)

$$\begin{aligned}& c_{j} = -\frac{F^{(i)}_{j}}{k^{2}} \quad \text{and} \quad d_{j}= \frac{F^{(i)}_{j}\cos (k h_{j}) - {F^{(i)}_{j+1}}}{k^{2}\sin (k h_{j})}, \end{aligned}$$

(55)

$$\begin{aligned}& \alpha = \frac{-1}{\theta_{j+1}^{2}} \biggl(1-\frac{\theta_{j+1}}{\sin\theta_{j+1}} \biggr),\qquad \gamma =\frac{-1}{\theta_{j}^{2}} \biggl(1-\frac{ \theta_{j}}{\sin \theta_{j}} \biggr), \end{aligned}$$

(56)

$$\begin{aligned}& \beta_{1} = \frac{1}{\theta_{j+1}^{2}} \biggl(1-\frac{\cot \theta_{j+1}}{\theta_{j+1}} \biggr),\qquad \beta_{2}=\frac{1}{\theta_{j}^{2}} \biggl(1-\frac{\cot\theta_{j}}{\theta_{j}} \biggr), \end{aligned}$$

(57)

$$\begin{aligned}& \beta_{3} = \frac{\beta_{2}}{(\sigma_{j}\beta_{1}+\beta_{2})}, \qquad \beta _{4}= \frac{\beta_{1}}{(\sigma_{j}\beta_{1}+\beta_{2})}, \qquad \theta_{j}=k h _{j}\quad \text{and}\quad r=j,j \pm 1, \end{aligned}$$

(58)

$$\begin{aligned}& P_{j} = \frac{(\sigma_{j}^{2}+\sigma_{j}-1)}{12},\qquad Q_{j}=\frac{(\sigma _{j}+1) (\sigma_{j}^{2}+3\sigma_{j}+1)}{12},\qquad R_{j}= \frac{\sigma_{j}(1+ \sigma_{j}-\sigma_{j}^{2})}{12}. \end{aligned}$$

(59)

4 Application to fourth-order singular boundary value problem

We consider a linear fourth-order singular boundary value problem:

$$\begin{aligned}& \frac{d^{4}y}{dx^{4}}= F\biggl(x,y,\frac{dy}{dx},\frac{d^{2}y}{dx^{2}}, \frac{d ^{3}y}{dx^{3}}\biggr),\quad 0 < x \le b, \end{aligned}$$

(60)

$$\begin{aligned}& \text{subject to }\quad y(0)=A_{1}, \qquad y(b)=B_{1},\qquad \frac{d^{2} }{dx^{2}}y(0)=A _{2},\qquad \frac{d^{2} }{dx^{2}}y(b)=B_{2}, \end{aligned}$$

(61)

where $F(x,y,\frac{dy}{dx},\frac{d^{2}y}{dx^{2}}, \frac{d^{3}y}{dx^{3}})=a(x)\frac{d^{3}y}{dx^{3}}+b(x)\frac{d^{2}y}{dx ^{2}}+c(x) \frac{dy}{dx}+d(x) y(x)+ g(x)$; $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$ are real constants and at least one of the coefficients $a(x)$, $b(x)$, $c(x)$, $d(x)$ or $g(x)$ may be singular at $x=0$. We may rewrite the problem (60)-(61) as a system of second-order boundary value problems:

$$\begin{aligned}& \frac{d^{2}y}{dx^{2}} = z(x), \end{aligned}$$

(62)

$$\begin{aligned}& \frac{d^{2}z}{dx^{2}} = a(x) \frac{dz}{dx}+b(x)z(x)+c(x)\frac{dy}{dx}+d(x) y(x)+ g(x), \end{aligned}$$

(63)

subject to

$$\begin{aligned} y(0)=A_{1}, \qquad z(0)=A_{2},\qquad y(b)=B_{1},\qquad z(b)=B_{2}. \end{aligned}$$

(64)

Applying the difference scheme (52) to the coupled second-order boundary value problem (62)-(63), we obtain the following difference scheme:

$$\begin{aligned}& \sigma_{j} y_{j-1} - (1+\sigma_{j})y_{j}+ y_{j+1} = h_{j} h_{j+1}(P _{j} {z_{j+1}}+ Q_{j}{z_{j }}+R_{j} {z_{j-1}}), \end{aligned}$$

(65)

$$\begin{aligned}& \sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} = h_{j} h_{j+1} \bigl[P_{j}( a_{j+1}\bar{z}_{x_{j+1}}+b_{j+1} {z_{j+1}}+c_{j+1} \bar{y} _{x_{j+1}}+d_{j+1} y_{j+1}+g_{j+1}) \\& \hphantom{\sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} =}{}+ Q_{j}( a_{j}\bar{z}_{x_{j}}+b_{j}{z_{j}}+c_{j} \bar{y}_{x_{j}}+d _{j}y_{j}+g_{j}) \\& \hphantom{\sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} =}{}+ R_{j}( a_{j-1}\bar{z}_{x_{j-1}}+b_{j-1}{z_{j-1}}+c_{j-1} \bar{y} _{x_{j-1}}+d_{j-1}y_{j-1}+g_{j-1}) \bigr], \end{aligned}$$

(66)

where $P_{j}=\sigma_{j} \alpha$, $Q_{j}= (\sigma_{j}\beta_{1}+\beta _{2} )$, $R_{j}=\gamma$ and $j=1,2,\ldots,N-1$. The boundary value problem has a singularity at some $x=0$ and hence, the scheme fails at $j=1$. Therefore, we define the following approximations around the jth node to evade the singularity:

$$\begin{aligned}& a_{j-1}^{*} = a_{j}- h_{j}{a_{{x}_{j}}} + O\bigl(h_{j}^{2}\bigr), \end{aligned}$$

(67)

$$\begin{aligned}& a_{j+1}^{*} = a_{j}+\sigma_{j} h_{j} a_{x_{j}} + O\bigl(h_{j}^{2}\bigr), \end{aligned}$$

(68)

$$\begin{aligned}& a_{j-1}^{**} = a_{j}-h_{j}a_{x_{j}}+ \frac{( h_{j} )^{2}}{2}a_{{xx} _{j}} + O\bigl(h_{j}^{3}\bigr), \end{aligned}$$

(69)

$$\begin{aligned}& a_{j+1}^{**} = a_{j}+\sigma_{j} h_{j}a_{{x}_{j}}+ \frac{(\sigma_{j} h _{j})^{2}}{2}a_{{xx}_{j}} + O \bigl(h_{j}^{3}\bigr). \end{aligned}$$

(70)

Similar relations for $b_{j\pm 1}$, $c_{j\pm 1}$, $d_{j\pm 1}$, $g_{j\pm 1}$ can also be defined. Now, using equations (67)-(68) in (65)-(66) we get

$$\begin{aligned}& \sigma_{j} y_{j-1} - (1+\sigma_{j})y_{j}+ y_{j+1} = h_{j} h_{j+1} (P _{j} {z_{j+1}}+Q_{j} {z_{j }}+R_{j} {z_{j-1}}), \end{aligned}$$

(71)

$$\begin{aligned}& \sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} = h_{j} h_{j+1} \bigl[P_{j} \bigl( a_{j+1}^{*}\bar{z}_{x_{j+1}}+b_{j+1}^{*} {z_{j+1}}+c_{j+1} ^{*}\bar{y}_{x_{j+1}}+d_{j+1}^{*} y_{j+1}+g_{j+1}^{*}\bigr) \\& \hphantom{\sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} =}{}+ Q_{j}( a_{j}\bar{z}_{x_{j}}+b_{j}{z_{j}}+c_{j} \bar{y}_{x_{j}}+d _{j}y_{j}+g_{j}) \\& \hphantom{\sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} =}{}+ R_{j}\bigl(a_{j-1}^{*}\bar{z}_{x_{j-1}}+b_{j-1}^{*}{z_{j-1}}+c_{j-1} ^{*}\bar{y}_{x_{j-1}}+d_{j-1}^{*}y_{j-1}+g_{j-1}^{*} \bigr) \bigr], \\& \quad \text{for } j=1,\ldots,N-1. \end{aligned}$$

(72)

Finally, substituting (44)-(46) in (71)-(72), we obtain the vector difference equation of boundary value problem (60)-(61) as follows:

$$\begin{aligned} \begin{bmatrix} \operatorname{sub}_{j}^{11}&\operatorname{sub}_{j}^{12} \\ \operatorname{sub}_{j}^{21}&\operatorname{sub}_{j}^{22} \end{bmatrix} \begin{bmatrix} y_{j-1} \\ z_{j-1} \end{bmatrix} + \begin{bmatrix} \operatorname{diag}_{j}^{11}&\operatorname{diag}_{j}^{12} \\ \operatorname{diag}_{j}^{21}&\operatorname{diag}_{j}^{22} \end{bmatrix} \begin{bmatrix} y_{j} \\ z_{j} \end{bmatrix} + \begin{bmatrix} \operatorname{sup}_{j}^{11}&\operatorname{sup}_{j}^{12} \\ \operatorname{sup}_{j}^{21}&\operatorname{sup}_{j}^{22} \end{bmatrix} \begin{bmatrix} y_{j+1} \\ z_{j+1} \end{bmatrix} = \begin{bmatrix} \phi_{j}^{1} \\ \phi_{j}^{2} \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned}& \operatorname{sub}_{j}^{11} = -\sigma_{j},\qquad \operatorname{sub}_{j}^{12}= h_{j}^{2} \sigma_{j} P_{j}, \\& \operatorname{sub}_{j}^{21} = \frac{h_{j} c_{j} }{ (\sigma_{j}+1)} \bigl[(P_{j}-Q_{j}) \sigma_{j}^{2}-R_{j}\sigma_{j}(2+ \sigma_{j}) \bigr]+h_{j}^{2} \biggl[ \frac{c _{x_{j}}(P_{j}\sigma_{j}^{2}+R_{j}\sigma_{j}(2+\sigma_{j}))}{(1+\sigma _{j})}+R_{j}\sigma_{j}{d}_{j} \biggr] \\& \hphantom{\operatorname{sup}_{j}^{22} = }{} +h_{j}^{3}(-\sigma_{j} d_{x_{j}}R_{j}), \\& \operatorname{sub}_{j}^{22} = -\sigma_{j}+\frac{h_{j} a_{j} }{ (\sigma_{j}+1)} \bigl[(P _{j}-Q_{j})\sigma_{j}^{2}-R_{j} \sigma_{j}(2+\sigma_{j}) \bigr] \\& \hphantom{\operatorname{sup}_{j}^{22} = }{}+h_{j} ^{2} \biggl[\frac{a_{x_{j}}(P_{j}\sigma_{j}^{2}+R_{j}\sigma_{j}(2+\sigma _{j}))}{(1+\sigma_{j})}+R_{j}\sigma_{j}{b}_{j} \biggr] \\& \hphantom{\operatorname{sup}_{j}^{22} = }{} +h_{j}^{3}(-\sigma_{j} b_{x_{j}}R_{j}), \\& \operatorname{diag}_{j}^{11} = (1+\sigma_{j}),\qquad \operatorname{diag}_{j}^{12}= \frac{h_{j}^{2}\sigma_{j}(\sigma_{j}+1)}{6}, \\& \operatorname{diag}_{j}^{21} = \frac{h_{j} c_{j} }{ (\sigma_{j}+1)} \bigl[(-P_{j}+R _{j}) (1+\sigma_{j})^{2}+Q_{j}\bigl( \sigma_{j}^{2}-1\bigr) \bigr]+h_{j}^{2} \bigl[ c _{x_{j}}(-P_{j}\sigma_{j}-R_{j}) (1+\sigma_{j}) +Q_{j}\sigma_{j}{d} _{j} \bigr], \\& \operatorname{diag}_{j}^{22} = (1+\sigma_{j})+ \frac{h_{j} a_{j}}{(\sigma_{j}+1)}+ \bigl[(-P_{j}+R_{j}) (1+ \sigma_{j})^{2}+Q_{j}\bigl(\sigma_{j}^{2}-1 \bigr) \bigr] \\& \hphantom{\operatorname{diag}_{j}^{22}=}{}+h_{j}^{2} \bigl[a_{x_{j}}(-P_{j} \sigma_{j}-R_{j}) (1+\sigma_{j})+Q_{j} \sigma_{j}{b}_{j} \bigr], \\& \operatorname{sup}_{j}^{11} = -1,\qquad \operatorname{sup}_{j}^{12}=h_{j}^{2} R_{j}, \\& \operatorname{sup}_{j}^{21} = \frac{h_{j} c_{j} }{ (\sigma_{j}+1)} \bigl[P_{j}(2 \sigma _{j}+1)-R_{j}+Q_{j} \bigr] \\& \hphantom{\operatorname{sup}_{j}^{22} = }{}+h_{j}^{2} \biggl[\frac{c_{x_{j}}(P_{j}(1+2\sigma _{j})\sigma_{j}+R_{j})}{(1+\sigma_{j})}+P_{j}\sigma_{j}{d}_{j}\biggr]+h _{j}^{3}\bigl(\sigma_{j}^{2}d_{x_{j}}R_{j}\bigr), \\& \operatorname{sup}_{j}^{22} = -1+\frac{h_{j} a_{j} }{ (\sigma_{j}+1)} \bigl[P_{j}(2\sigma_{j}+1)-R_{j}+Q_{j} \bigr] \\& \hphantom{\operatorname{sup}_{j}^{22} = }{} +h_{j}^{2} \biggl[\frac{a_{x_{j}}(P_{j}(1+2\sigma_{j})\sigma_{j}+R_{j})}{(1+\sigma_{j})}+P_{j} \sigma_{j}{b}_{j} \biggr]+h_{j}^{3} \bigl(\sigma_{j}^{2} b _{x_{j}}P_{j}\bigr), \\& \phi_{j}^{1 } = 0,\qquad \phi_{j}^{2}=-h_{j}^{2} \bigl[ \sigma_{j}(P_{j}+Q_{j}+R_{j}) g_{j}+ h _{j} \sigma_{j}(P_{j} \sigma_{j}-R_{j}) g_{x_{j}} \bigr]. \end{aligned}$$

(73)

Similarly, using (43)-(51) and (69)-(70) up to $O(h_{j}^{5})$ terms in scheme (53) we get the second difference scheme of higher order.

4.1 Application to sixth-order singular boundary value problem

Let us consider a linear singular sixth-order boundary value problem of the following form:

$$\begin{aligned} \frac{d^{6}y}{dx^{6}} &= a(x)\frac{d^{5}y}{dx^{5}}+b(x)\frac{d^{4}y}{dx^{4}} \\ &\quad {}+c(x)\frac{d^{3} y}{dx^{3}}+d(x)\frac{d^{2} y}{dx^{2}}+e(x) \frac{dy}{dx}+f(x)y(x)+g(x),\quad 0 < x \le b, \end{aligned}$$

(74)

subject to boundary conditions:

$$\begin{aligned}& \begin{aligned} &y(0)=A_{1},\qquad \frac{d^{2} }{dx^{2}}y(0)=A_{2},\qquad \frac{d^{4}}{dx^{4}}y(0)=A _{3}, \\ &y(b)=B_{1},\qquad \frac{d^{2}}{dx^{2}}y(b)=B_{2}, \qquad \frac{d^{4}}{dx^{4}}y(b)=B _{3}, \end{aligned} \end{aligned}$$

(75)

where $A_{1}$, $A_{2}$, $A_{3}$, $B_{1}$, $B_{2}$, $B_{3}$ are real constants and any one of the coefficients $a(x)$, $b(x)$, $c(x)$, $d(x)$, $e(x)$, $f(x)$ or $g(x)$ may be singular at $x=0$. We may rewrite the problem (74)-(75) as a system of second-order boundary value problems:

$$\begin{aligned}& \frac{d^{2} y}{dx^{2}} = z(x), \end{aligned}$$

(76)

$$\begin{aligned}& \frac{d^{2} z}{dx^{2}} = v(x), \end{aligned}$$

(77)

$$\begin{aligned}& \frac{d^{2} v}{dx^{2}} = a(x)\frac{d v}{dx}+b(x)v(x)+c(x) \frac{d z}{dx}+d(x) z(x)+e(x)\frac{dy}{dx} +f(x)y(x)+g(x), \end{aligned}$$

(78)

subject to

$$ \begin{aligned} &y(0)=A_{1}, \qquad z(0)=A_{2},\qquad v(0)=A_{3}, \\ &y(b)=B_{1},\qquad z(b)=B_{2},\qquad v(b)=B_{3} . \end{aligned} $$

(79)

Applying the difference scheme (52) to the coupled second-order boundary value problem (76)-(78), we obtain the following difference scheme:

$$\begin{aligned}& \sigma_{j} y_{j-1} - (1+\sigma_{j})y_{j}+ y_{j+1} = h_{j} h_{j+1}(P _{j} {z_{j+1}}+ Q_{j}{z_{j }}+R_{j} {z_{j-1}}), \end{aligned}$$

(80)

$$\begin{aligned}& \sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} = h_{j} h_{j+1}(P _{j} {v_{j+1}}+ Q_{j}{v_{j }}+R_{j} {v_{j-1}}), \end{aligned}$$

(81)

$$\begin{aligned}& \sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = h_{j} h_{j+1} \bigl[P_{j}(a_{j+1} \bar{v}_{x_{j+1}}+b_{j+1}v_{j+1}+c_{j+1} \bar{z} _{x_{j+1}} \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{} +d_{j+1} z_{j+1}+e_{j+1} \bar{y}_{x_{j+1}} +f_{j+1}y_{j+1}+g_{j+1}) \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{}+ Q_{j}(a_{j} \bar{v}_{x_{j}}+b_{j}v_{j}+c_{j} \bar{z}_{x_{j}}+d_{j} z _{j}+e_{j} \bar{y} _{x_{j}} +f_{j}y_{j}+g_{j}) \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{}+ R_{j}(a_{j-1} \bar{v}_{x_{j-1}}+b_{j-1}v_{j-1}+c_{j-1} \bar{z}_{x _{j-1}} \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{}+d_{j-1} z_{j-1}+e_{j-1} \bar{y}_{x_{j-1}} +f_{j-1}y_{j-1}+g_{j-1}) \bigr], \end{aligned}$$

(82)

where $P_{j}$, $Q_{j}$, $R_{j}$ are defined in Section 2. The boundary value problem has a singularity at $x=0$. Therefore, as in Section 4 we use the approximation (67)-(68) in (80)-(82) and we get

$$\begin{aligned}& \sigma_{j} y_{j-1} - (1+\sigma_{j})y_{j}+ y_{j+1} = h_{j} h_{j+1}(P _{j} {z_{j+1}}+ Q_{j}{z_{j }}+R_{j} {z_{j-1}}), \end{aligned}$$

(83)

$$\begin{aligned}& \sigma_{j} z_{j-1} - (1+\sigma_{j})z_{j}+ z_{j+1} = h_{j} h_{j+1}(P _{j} {v_{j+1}}+ Q_{j}{v_{j }}+R_{j} {v_{j-1}}), \end{aligned}$$

(84)

$$\begin{aligned}& \sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = h_{j} h_{j+1} \bigl[P_{j} \bigl(a_{j+1}^{*} \bar{v}_{x_{j+1}}+b_{j+1}^{*}v_{j+1}+c_{j+1} ^{*} \bar{z}_{x_{j+1}} \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{}+d_{j+1}^{*} z_{j+1}+e_{j+1}^{*} \bar{y}_{x_{j+1}} +f_{j+1}^{*}y_{j+1}+g _{j+1}^{*}\bigr) \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{} + Q_{j}(a_{j} \bar{v}_{x_{j}}+b_{j}v_{j}+c_{j} \bar{z}_{x_{j}}+d_{j} z _{j}+e_{j} \bar{ y}_{x_{j}} +f_{j}y_{j}+g_{j}) \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{} + R_{j}\bigl(a_{j-1}^{*} \bar{v}_{x_{j-1}}+b_{j-1}^{*} v_{j-1}+c_{j-1} ^{*} \bar{z}_{x_{j-1}} \\& \hphantom{\sigma_{j} v_{j-1} - (1+\sigma_{j})v_{j}+ v_{j+1} = }{} +d_{j-1}^{*} z_{j-1}+e_{j-1}^{*} \bar{y}_{x_{j-1}} +f_{j-1}^{*}y_{j-1}+g _{j-1}^{*}\bigr) \bigr]. \end{aligned}$$

(85)

Finally, substituting (43)-(47) in (83)-(85), we obtain the vector difference equation of boundary value problem (74) as follows:

$$\begin{aligned}& \begin{bmatrix} \operatorname{sub}_{j}^{11}&\operatorname{sub}_{j}^{12}&\operatorname{sub}_{j}^{13} \\ \operatorname{sub}_{j}^{21}&\operatorname{sub}_{j}^{22} &\operatorname{sub}_{j}^{23} \\ \operatorname{sub}_{j}^{31}&\operatorname{sub}_{j}^{32}&\operatorname{sub}_{j}^{33} \end{bmatrix} \begin{bmatrix} y_{j-1} \\ z_{j-1} \\ v_{j-1} \end{bmatrix}+ \begin{bmatrix} \operatorname{diag}_{j}^{11}&\operatorname{diag}_{j}^{12}&\operatorname{diag}_{j}^{13} \\ \operatorname{diag}_{j}^{21}&\operatorname{diag}_{j}^{22}&\operatorname{diag}_{j}^{23} \\ \operatorname{diag}_{j}^{31}&\operatorname{diag}_{j}^{32}&\operatorname{diag}_{j}^{33} \end{bmatrix} \begin{bmatrix} y_{j} \\ z_{j} \\ v_{j} \end{bmatrix} \\& \quad {}+ \begin{bmatrix} \operatorname{sup}_{j}^{11}&\operatorname{sup}_{j}^{12}&\operatorname{sup}_{j}^{13} \\ \operatorname{sup}_{j}^{21}&\operatorname{sup}_{j}^{22}&\operatorname{sup}_{j}^{23} \\ \operatorname{sup}_{j}^{31}&\operatorname{sup}_{j}^{32}&\operatorname{sup}_{j}^{33} \end{bmatrix} \begin{bmatrix} y_{j+1} \\ z_{j+1} \\ v_{j+1} \end{bmatrix} = \begin{bmatrix} \phi_{j}^{1} \\ \phi_{j}^{2} \\ \phi_{j}^{3} \end{bmatrix}, \end{aligned}$$

(86)

where

$$\begin{aligned}& \textstyle\begin{cases} \operatorname{sub}_{j}^{11}=-\sigma_{j}, \qquad \operatorname{sub}_{j}^{12}= h_{j}^{2}\sigma_{j} R_{j}, \qquad \operatorname{sub}_{j}^{13}=0, \\ \operatorname{sub}_{j}^{21}=0, \qquad \operatorname{sub}_{j}^{22}= -\sigma_{j}, \qquad \operatorname{sub}_{j}^{23}= h_{j}^{2} \sigma_{j} R_{j}, \\ \operatorname{sub}_{j}^{31}= C_{3}, \qquad \operatorname{sub}_{j}^{32}=B_{3}, \qquad \operatorname{sub}_{j}^{33}=-\sigma_{j}+A_{3}, \end{cases}\displaystyle \\& \textstyle\begin{cases} \operatorname{diag}_{j}^{11}=(1+\sigma_{j}), \qquad \operatorname{diag}_{j}^{12}= \frac{h_{j}^{2}\sigma_{j}(\sigma_{j}+1)}{6}, \qquad \operatorname{diag}_{j}^{13}=0, \\ \operatorname{diag}_{j}^{21}=0, \qquad \operatorname{diag}_{j}^{22}=(1+\sigma_{j}), \qquad \operatorname{diag}_{j}^{23}= \frac{h_{j}^{2}\sigma_{j}(\sigma_{j}+1)}{6}, \\ \operatorname{diag}_{j}^{31}=C_{2}, \qquad \operatorname{diag}_{j}^{32}=B_{2}, \qquad \operatorname{diag}_{j}^{33}=1+\sigma_{j}+A_{2}, \end{cases}\displaystyle \\& \textstyle\begin{cases} \operatorname{sup}_{j}^{11}=-1, \qquad \operatorname{sup}_{j}^{12}= h_{j}^{2}\sigma_{j}^{2} P_{j}, \qquad \operatorname{sup}_{j}^{13}=0, \\ \operatorname{sup}_{j}^{21}=0, \qquad \operatorname{sup}_{j}^{22}= -1, \qquad \operatorname{sup}_{j}^{23}= h_{j}^{2} \sigma_{j}^{2} P_{j}, \\ \operatorname{sup}_{j}^{31}= C_{1}, \qquad \operatorname{sup}_{j}^{32}=B_{1}, \qquad \operatorname{sup}_{j}^{33}=-1+A_{1}, \end{cases}\displaystyle \\& A_{1}=\frac{h_{j} a_{j}}{S_{j}}\bigl(\sigma_{j}^{2} \alpha (1+2\sigma _{j})+\sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}\gamma \bigr)+h _{j}^{2}\biggl( \frac{ a_{x_{j}}}{S_{j}}\bigl( \sigma_{j}^{3} \alpha (1+2\sigma _{j})+ \sigma_{j} \gamma \bigr)+\sigma_{j}^{2} \alpha b_{j}\biggr) \\& \hphantom{ C_{3}=}{} +h_{j}^{3}\biggl( \frac{ a_{{xx}_{j}}}{2 S_{j}}\bigl( \sigma_{j}^{4}\alpha (1+2 \sigma_{j})- \sigma_{j} \gamma \bigr)+ \sigma_{j}^{2} \alpha b_{x_{j}}\biggr)+h _{j}^{4}\biggl( \sigma_{j}^{4} \alpha \frac{ b_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(87)

$$\begin{aligned}& A_{2}=\frac{h_{j} a_{j}}{S_{j}}\bigl(-\sigma_{j}^{2} \alpha (1+\sigma _{j})^{2}+\sigma_{j}( \sigma_{j} \beta_{1}+\beta_{2}) \bigl( \sigma_{j}^{2}-1\bigr)+ \sigma_{j}\gamma (1+ \sigma_{j})^{2} \bigr) \\& \hphantom{A_{2}=}{}+h_{j}^{2} \biggl( \frac{ a_{x_{j}}(1+ \sigma_{j})^{2}}{S_{j}}\bigl(-\sigma_{j}^{3} \alpha - \sigma_{j} \gamma \bigr)+ \sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2}) b_{j}\biggr) \\& \hphantom{A_{2}=}{}+h_{j}^{3}\biggl( \frac{ a_{{xx}_{j}}(1+\sigma_{j})^{2}}{2 S_{j}}\bigl(- \sigma _{j}^{4}\alpha +\gamma \sigma_{j} \bigr) \biggr), \end{aligned}$$

(88)

$$\begin{aligned}& A_{3}=\frac{h_{j} a_{j}}{S_{j}}\bigl(\sigma_{j}^{4} \alpha -\sigma_{j} ^{3}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}^{2} \gamma (2+\sigma _{j}) \bigr)+h_{j}^{2}\biggl( \frac{ a_{x_{j}}}{S_{j}}\bigl(\sigma_{j}^{5} \alpha + \sigma_{j}^{2} \gamma (2+\sigma_{j}) \bigr)- \sigma_{j} \gamma b_{j}\biggr) \\& \hphantom{ C_{3}=}{} +h_{j}^{3}\biggl( \frac{ a_{{xx}_{j}}}{2 S_{j}}\bigl( \sigma_{j}^{6}\alpha -\sigma _{j}^{2} \gamma (2+\sigma_{j}) \bigr) -\sigma_{j} \gamma b_{x_{j}}\biggr)+h_{j} ^{4}\biggl( \sigma_{j}^{2} \alpha \frac{ b_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(89)

$$\begin{aligned}& B_{1}=\frac{h_{j} c_{j}}{S_{j}}\bigl(\sigma_{j}^{2} \alpha (1+2\sigma _{j})+\sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}\gamma \bigr)+h _{j}^{2}\biggl( \frac{ c_{x_{j}}}{S_{j}}\bigl( \sigma_{j}^{3} \alpha (1+2\sigma _{j})+ \sigma_{j} \gamma \bigr)+\sigma_{j}^{2} \alpha d_{j}\biggr) \\& \hphantom{ C_{3}=}{} +h_{j}^{3}\biggl( \frac{ c_{{xx}_{j}}}{2 S_{j}}\bigl( \sigma_{j}^{4}\alpha (1+2 \sigma_{j})- \sigma_{j} \gamma \bigr)+ \sigma_{j}^{2} \alpha d_{x_{j}}\biggr)+h _{j}^{4}\biggl( \sigma_{j}^{4} \alpha \frac{ d_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(90)

$$\begin{aligned}& B_{2}=\frac{h_{j} c_{j}}{S_{j}}\bigl(-\sigma_{j}^{2} \alpha (1+\sigma _{j})^{2}+\sigma_{j}( \sigma_{j} \beta_{1}+\beta_{2}) \bigl( \sigma_{j}^{2}-1\bigr)+ \sigma_{j}\gamma (1+ \sigma_{j})^{2} \bigr) \\& \hphantom{B_{2}=}{}+h_{j}^{2} \biggl( \frac{ c_{x_{j}}(1+ \sigma_{j})^{2}}{S_{j}}\bigl(-\sigma_{j}^{3} \alpha - \sigma_{j} \gamma \bigr)+ \sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2}) d_{j}\biggr) \\& \hphantom{B_{2}=}{}+h_{j}^{3}\biggl( \frac{ c_{{xx}_{j}}(1+\sigma_{j})^{2}}{2 S_{j}}\bigl(- \sigma _{j}^{4}\alpha +\gamma \sigma_{j} \bigr) \biggr), \end{aligned}$$

(91)

$$\begin{aligned}& B_{3}=\frac{h_{j} c_{j}}{S_{j}}\bigl(\sigma_{j}^{4} \alpha -\sigma_{j} ^{3}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}^{2} \gamma (2+\sigma _{j}) \bigr)+h_{j}^{2}\biggl( \frac{ a_{x_{j}}}{S_{j}}\bigl(\sigma_{j}^{5} \alpha + \sigma_{j}^{2} \gamma (2+\sigma_{j}) \bigr)- \sigma_{j} \gamma d_{j}\biggr) \\& \hphantom{ C_{3}=}{}+h_{j}^{3}\biggl( \frac{ a_{{xx}_{j}}}{2 S_{j}}\bigl( \sigma_{j}^{6}\alpha -\sigma _{j}^{2} \gamma (2+\sigma_{j}) \bigr) -\sigma_{j} \gamma d_{x_{j}}\biggr)+h_{j} ^{4}\biggl( \sigma_{j}^{2} \alpha \frac{ b_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(92)

$$\begin{aligned}& C_{1}=\frac{h_{j} e_{j}}{S_{j}}\bigl(\sigma_{j}^{2} \alpha (1+2\sigma _{j})+\sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}\gamma \bigr)+h _{j}^{2}\biggl( \frac{ e_{x_{j}}}{S_{j}}\bigl( \sigma_{j}^{3} \alpha (1+2\sigma _{j})+ \sigma_{j} \gamma \bigr)+\sigma_{j}^{2} \alpha f_{j}\biggr) \\& \hphantom{ C_{3}=}{} +h_{j}^{3}\biggl( \frac{ e_{{xx}_{j}}}{2 S_{j}}\bigl( \sigma_{j}^{4}\alpha (1+2 \sigma_{j})- \sigma_{j} \gamma \bigr)+ \sigma_{j}^{2} \alpha f_{x_{j}}\biggr)+h _{j}^{4}\biggl( \sigma_{j}^{4} \alpha \frac{ f_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(93)

$$\begin{aligned}& C_{2}=\frac{h_{j} e_{j}}{S_{j}}\bigl(-\sigma_{j}^{2} \alpha (1+\sigma _{j})^{2}+\sigma_{j}( \sigma_{j} \beta_{1}+\beta_{2}) \bigl( \sigma_{j}^{2}-1\bigr)+ \sigma_{j}\gamma (1+ \sigma_{j})^{2} \bigr) \\& \hphantom{ C_{2}=}{}+h_{j}^{2} \biggl( \frac{ e_{x_{j}}(1+ \sigma_{j})^{2}}{S_{j}}\bigl(-\sigma_{j}^{3} \alpha - \sigma_{j} \gamma \bigr)+ \sigma_{j}(\sigma_{j} \beta_{1}+\beta_{2}) f_{j}\biggr) \\& \hphantom{ C_{2}=}{} +h_{j}^{3}\biggl( \frac{ e_{{xx}_{j}}(1+\sigma_{j})^{2}}{2 S_{j}}\bigl(- \sigma _{j}^{4}\alpha +\gamma \sigma_{j} \bigr) \biggr), \end{aligned}$$

(94)

$$\begin{aligned}& C_{3}=\frac{h_{j} e_{j}}{S_{j}}\bigl(\sigma_{j}^{4} \alpha -\sigma_{j} ^{3}(\sigma_{j} \beta_{1}+\beta_{2})-\sigma_{j}^{2} \gamma (2+\sigma _{j}) \bigr)+h_{j}^{2}\biggl( \frac{ e_{x_{j}}}{S_{j}}\bigl(\sigma_{j}^{5} \alpha + \sigma_{j}^{2} \gamma (2+\sigma_{j}) \bigr)- \sigma_{j} \gamma f_{j}\biggr) \\& \hphantom{ C_{3}=}{} +h_{j}^{3}\biggl( \frac{ e_{{xx}_{j}}}{2 S_{j}}\bigl(\sigma_{j}^{6}\alpha -\sigma _{j}^{2} \gamma (2+\sigma_{j}) \bigr) -\sigma_{j} \gamma f_{x_{j}}\biggr)+h_{j} ^{4}\biggl( \sigma_{j}^{2} \alpha \frac{ f_{{xx}_{j}}}{2 } \biggr), \end{aligned}$$

(95)

$$\begin{aligned}& \begin{aligned} &\phi_{j}^{1 }=0, \qquad \phi_{j}^{2 }=0, \\ &\phi_{j}^{3}=-h_{j}^{2}\sigma_{j} \biggl[ \bigl(\sigma_{j} \alpha + (\sigma_{j} \beta_{1}+\beta_{2})+\gamma \bigr) g_{j}+ h_{j} \bigl(\sigma_{j}^{2} \alpha -\gamma \bigr) g_{x_{j}}+h_{j}^{2}\bigl( \sigma_{j}^{3}\alpha +\gamma \bigr)\frac{g _{{xx}_{j}}}{2} \biggr]. \end{aligned} \end{aligned}$$

(96)

5 Convergence analysis

We provide the vector convergence analysis for $M=2$, i.e., a fourth-order non-linear singular boundary value problem (60)-(61). We apply the difference scheme (53) to the boundary value problem and obtain the following difference scheme:

$$\begin{aligned}& \sigma y_{j-1} - (1+\sigma )y_{j}+ y_{j+1} = h_{j}^{2} [P_{j} \bar{ \bar{z}}_{j+1}+Q_{j} \bar{\bar{z}}_{j}+R_{j} \bar{\bar{z}}_{j-1} ] +T_{1}(h_{j}), \end{aligned}$$

(97)

$$\begin{aligned}& \sigma z_{j-1} - (1+\sigma )z_{j}+ z_{j+1} = h_{j}^{2} [P_{j} \bar{ \bar{F}}_{j+1}+Q_{j} \bar{\bar{F}}_{j}+R_{j} \bar{\bar{F}}_{j-1} ] +T_{2}(h_{j}), \end{aligned}$$

(98)

where

$$ P_{j}=\frac{(\sigma_{j}^{2}+\sigma_{j}-1)}{12},\qquad Q_{j}=\frac{(\sigma _{j}+1) (\sigma_{j}^{2}+3\sigma_{j}+1)}{12},\qquad R_{j}= \frac{\sigma_{j}(1+ \sigma_{j}-\sigma_{j}^{2})}{12}. $$

Now, let $\hat{\mathbf{y}}=( \hat{y}_{1},\hat{y}_{2},\ldots,\hat{y}_{N-1} )^{T}$, $\hat{\mathbf{z}}=( \hat{z}_{1},\hat{z}_{2},\ldots,\hat{z}_{N-1} )^{T}$ represent the exact solutions and $\mathbf{y}=( {y}_{1},{y}_{2},\ldots, {y}_{N-1} )^{T}$ and $\mathbf{z}=( {z}_{1},{z}_{2},\ldots,{z}_{N-1} )^{T}$ be the approximate solutions. Then we define the error as $\hat{\mathbf{y}}- \mathbf{y}=(e_{1,1},e_{2,1},\ldots,e_{N-1,1})^{T}$ and $\hat{\mathbf{z}}- \mathbf{z}=(e_{1,2},e_{2,2},\ldots,e_{N-1,2})^{T}$. Next, we define the following approximation:

$$\begin{aligned}& \hat{\bar{F}}_{j \pm 1}=\bar{F}_{j \pm 1}+e_{j \pm 1,1}G_{j \pm 1}^{1}+(\hat{\bar{y}}_{x_{j \pm 1}}-\bar{y}_{x_{j \pm 1}})H_{j \pm 1}^{1}+e_{j \pm 1,2}V_{j \pm 1}^{1}+(\hat{ \bar{z}}_{x_{j \pm 1}}- \bar{z}_{x_{j \pm 1}})W_{j \pm 1}^{1}, \end{aligned}$$

(99)

$$\begin{aligned}& \hat{\bar{\bar{F}}}_{j \pm 1}=\bar{\bar{F}}_{j \pm 1}+e_{j \pm 1,1} G_{j \pm 1}^{2}+(\hat{\bar{\bar{y}}}_{x_{j \pm 1}}-\bar{ \bar{y}} _{x_{j \pm 1}})H_{j \pm 1}^{2}+e_{j \pm 1,2}V_{j \pm 1}^{2}+( \hat{\bar{\bar{z}}}_{x_{j \pm 1}}-\bar{\bar{z}}_{x_{j \pm 1}})W_{j\pm 1}^{2}, \end{aligned}$$

(100)

$$\begin{aligned}& \hat{\bar{{F}}}_{j }={{\bar{F}}_{j}}+e_{j,1 }G_{j}^{1}+( \hat{\bar{{y}}}_{x_{j }}-\bar{{y}}_{x_{j}})H_{j }^{1}+e_{j,2 }V_{j }^{2}+(\hat{\bar{{z}}}_{x_{j }}- \bar{{z}}_{x_{j}})W_{j }^{2}, \end{aligned}$$

(101)

$$\begin{aligned}& \hat{\bar{\bar{F}}}_{j }={\bar{\bar{F}}_{j \pm 1}}+e_{j,1 }G_{j }^{1}+(\hat{\bar{\bar{y}}}_{x_{j }}-\bar{\bar{y}}_{x_{j}})H_{j }^{1}+e _{j,2 }V_{j }^{2}+(\hat{\bar{\bar{z}}}_{x_{j }}- \bar{\bar{z}}_{x _{j}})W_{j }^{2}, \end{aligned}$$

(102)

where

$$\begin{aligned}& G_{r}^{1}=\frac{\partial \bar{F}}{\partial {y_{r}}},\qquad H_{r}^{1}=\frac{\partial \bar{F}}{\partial {y_{x_{r}}}},\qquad V_{r}^{1}= \frac{ \partial \bar{F}}{\partial {z_{r}}},\qquad W_{r}^{1}=\frac{\partial \bar{F}}{\partial {z_{x_{r}}}}, \end{aligned}$$

(103)

$$\begin{aligned}& G_{r}^{2}=\frac{\partial \bar{\bar{F}}}{\partial {y_{r}}},\qquad {H_{r}} ^{2}=\frac{\partial \bar{\bar{F}}}{\partial {y_{x_{r}}}},\qquad V_{r}^{2}= \frac{ \partial \bar{\bar{F}}}{\partial {z_{r}}},\qquad W_{r}^{2}=\frac{\partial \bar{\bar{F}}}{\partial {z_{x_{r}}}},\quad r=j,j\pm 1. \end{aligned}$$

(104)

Further, a singularity is at $x=0$. Therefore, we may define $G_{j \pm 1}^{k},k=1,2$, as the following approximation. Moreover, similar approximations can be also defined for $H_{j \pm 1}^{k}$, $V_{j\pm 1}^{k}$ and $W_{j \pm 1}^{k}$, $k=1,2$. We have

$$\begin{aligned}& G_{j + 1}^{k} =G_{j }^{k}+h_{j}\sigma {{G_{x}}_{j }^{k}} +\frac{(h_{j}\sigma )^{2}}{2}{{G_{xx}}_{j }^{k}}, \end{aligned}$$

(105)

$$\begin{aligned}& G_{j- 1}^{k} =G_{j }^{k}-h_{j} {{G_{x}}_{j }^{k}}+ \frac{h_{j} ^{2}}{2}{{G_{xx}}_{j }^{k}}. \end{aligned}$$

(106)

Thus, as we use the approximations (44)-(47), (105)-(106) in equations (97)-(98), we get the error equation in matrix form as follows:

$$\begin{aligned} L {E}+T(h_{j} )=0, \end{aligned}$$

(107)

where ${E}= ((e_{1,1},e_{1,2}),(e_{2,1},e_{2,2}),\ldots,(e_{N-1,1},e _{N-1,2}) )^{T}$, $T(h_{j} )= ( (T_{1}(h_{1}),T_{2}(h_{1})), (T_{1}(h_{2}), T_{2}(h _{2})),\ldots, (T_{1}(h_{N-1}),T_{2}(h_{N-1}) ) )^{T}$, and $L=(L_{k,j})_{k,j=1}^{N-1}$ denote the block tri-diagonal matrix. The block elements of L are as follows:

$$ l_{j,j}= \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} 1+\sigma &h_{j}^{2} Q_{j} \\ B_{1} & 1+\sigma +B_{2} \end{array}\displaystyle \right ] , \qquad l_{j,j+1}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} -1 & h_{j}^{2} P_{j} \\ C_{1} & -1+C_{2} \end{array}\displaystyle \right ] , \qquad l_{j,j-1}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} -\sigma & h_{j}^{2} R_{j} \\ A_{1} & -\sigma +A_{2} \end{array}\displaystyle \right ], $$

where

$$\begin{aligned}& A_{1}=h_{j} \bigl(-R_{j} H_{j}^{2}-Q_{j} \sigma \beta_{4} H_{j}^{2} \bigr)+ \bigl(R_{j} H_{x_{j}}^{2} +R_{j} G_{j}^{2} \bigr) \\& \hphantom{ A_{2}=}{}+h_{j}^{3} \biggl( \frac{R_{j}}{2}H_{{xx}_{j}}^{2}-R_{j} G_{{x}_{j}}^{2} \biggr) +h _{j}^{4} \biggl( \frac{R_{j}}{2}G_{{xx}_{j}}^{2} \biggr), \end{aligned}$$

(108)

$$\begin{aligned}& A_{2}=h_{j} \bigl(-R_{j} W_{j}^{2} -Q_{j} \sigma \beta_{4} W_{j}^{2} \bigr)+ \bigl(R_{j} W_{x_{j}}^{2} +R_{j} V_{j}^{2} \bigr) \\& \hphantom{ A_{2}=}{}+h_{j}^{3} \biggl( \frac{R_{j}}{2}W_{{xx}_{j}}^{2}-R_{j} V_{{x}_{j}}^{2}-R_{j} \beta_{2}H_{j}^{2}+Q_{j} \sigma \beta_{4}\gamma H_{j}^{2} \biggr) \\& \hphantom{ A_{2}=}{}+h_{j}^{4} \biggl( \frac{R_{j}}{2}V_{{xx}_{j}}^{2}+R_{j} \beta_{2} H_{j} ^{2} \biggr), \end{aligned}$$

(109)

$$\begin{aligned}& B_{1}=h_{j}\biggl(\frac{-P_{j} H_{j}^{2}}{\sigma }+R_{j} H_{j}^{2}+Q_{j}\frac{ \sigma^{2}\beta_{4}-\beta_{3}}{\sigma }H_{j}^{2} \biggr) \\& \hphantom{ A_{2}=}{}+h_{j}^{2}\bigl(-P_{j} H _{x_{j}}^{2}-R_{j} H_{x_{j}}^{2}+Q_{j} G_{j}^{2}\bigr)+h_{j}^{3}\biggl( \frac{-P _{j}\sigma +R_{j}}{2}H_{{xx}_{j}}^{2}\biggr), \end{aligned}$$

(110)

$$\begin{aligned}& B_{2}=h_{j}\biggl(\frac{-P_{j} W_{j}^{2}}{\sigma }+R_{j} W_{j}^{2}+Q_{j}\frac{ \sigma^{2}\beta_{4}-\beta_{3}}{\sigma }W_{j}^{2} \biggr)+h_{j}^{2}\bigl(-P_{j} W _{x_{j}}^{2}-R_{j} W_{x_{j}}^{2}+Q_{j} V_{j}^{2}\bigr) \\& \hphantom{B_{2}=}{}+h_{j}^{3}\biggl( \frac{-P_{j}\sigma +R_{j}}{2}W_{{xx}_{j}}^{2}\biggr)+h_{j}^{2}( \sigma \alpha P_{j}-R_{j} \gamma )+h_{j}^{4}\bigl(H_{j}^{2} \bigl(\sigma^{2} \alpha P_{j}-R_{j} \gamma \bigr)\bigr), \end{aligned}$$

(111)

$$\begin{aligned}& C_{1}=h_{j}\biggl(\frac{P_{j}+Q_{j}\beta_{3}}{\sigma }H_{j}^{2}\biggr)+h_{j}^{2} \bigl(P _{j} H_{x_{j}}+P_{j} G_{j}^{2}\bigr) \\& \hphantom{ C_{1}=}{}+h_{j}^{3} P_{j}\sigma \biggl(\frac{H_{ {xx}_{j}}^{2}}{2}+G_{x_{j}}^{2}\biggr)+h_{j}^{4}\biggl(\frac{P_{j}\sigma^{2} G_{{xx}_{j}}^{2}}{2}\biggr), \end{aligned}$$

(112)

$$\begin{aligned}& C_{2}=h_{j}\biggl(\frac{P_{j}+Q_{j}\beta_{3}}{\sigma }W_{j}^{2}\biggr)+h_{j}^{2} \bigl(P _{j} W_{x_{j}}+P_{j} V_{j}^{2} \bigr) \\& \hphantom{ C_{2}=}{}+h_{j}^{3} \biggl(P_{j}\sigma \biggl(\frac{W_{{xx}_{j}}^{2}}{2}+V_{x_{j}}^{2} -Q_{j}\sigma \alpha \beta_{3} H_{j} ^{2}+ \sigma \beta_{1} P_{j} H_{x_{j}}^{2}\biggr)\biggr) \\& \hphantom{ C_{2}=}{} +h_{j}^{4}\biggl( \frac{P_{j}\sigma^{2} V_{{xx}_{j}}^{2}}{2}+P_{j}\sigma^{2} \beta_{1} H_{x_{j}}^{2}\biggr). \end{aligned}$$

(113)

Further, for some $K_{1},K_{2} >0$, we assume

$$\begin{aligned}& K_{1}=\max \bigl(\bigl\vert G_{j}^{2} \bigr\vert ,\bigl\vert H_{j}^{2} \bigr\vert ,\bigl\vert G_{x_{j}}^{2} \bigr\vert ,\bigl\vert H_{x_{j}}^{2} \bigr\vert ,\bigl\vert G_{{xx}_{j}}^{2}\vert , \vert H_{{xx}_{j}}^{2} \bigr\vert \bigr), \end{aligned}$$

(114)

$$\begin{aligned}& K_{2}=\max \bigl(\bigl\vert V_{j}^{2} \bigr\vert ,\bigl\vert W_{j}^{2} \bigr\vert ,\bigl\vert V_{x_{j}}^{2} \bigr\vert ,\bigl\vert W_{x_{j}}^{2} \bigr\vert ,\bigl\vert V_{{xx}_{j}}^{2}\vert , \vert W_{{xx}_{j}}^{2} \bigr\vert \bigr). \end{aligned}$$

(115)

Then, using $l_{j,j+1}\ l_{j,j-1}$ and (114)-(115), we get

$$\begin{aligned}& \Vert \operatorname{sup}_{j} \Vert _{\infty }\leq \max_{\substack{1\leq j\leq N-2}} \textstyle\begin{cases} 1+ h_{j}^{2} P_{j}, \\ 1+h_{j} ( (\frac{P_{j}+Q_{j}\beta_{3}}{\sigma })+ 2 h_{j} P_{j} + h_{j}^{2} \frac{3P_{j}\sigma }{2}+h_{j}^{3} \frac{P_{j}\sigma_{j}^{2}}{2} )(\vert K_{1} \vert +\vert K_{2} \vert ) \\ \quad {}+h_{j}^{3} ( \vert P_{j} \beta_{1}-Q\sigma_{j}\alpha \beta_{4} \vert +P_{j} \sigma_{j}^{2} \beta_{1} )\vert K_{1} \vert , \end{cases}\displaystyle \\& \Vert \operatorname{sub}_{j} \Vert _{\infty }\leq \max_{\substack{1\leq j\leq N-2}} \textstyle\begin{cases} \sigma + h_{j}^{2} R_{j}, \\ \sigma +h_{j} ( (R_{j}+Q_{j}\sigma \beta_{4})+2 h_{j} R_{j} + h _{j}^{2} \frac{3R_{j}}{2}+h_{j}^{3} \frac{R_{j}}{2} )(\vert K_{1} \vert +\vert K_{2} \vert ) \\ \quad {}+h_{j}^{3} ( \vert R_{j} \beta_{2}-Q\sigma_{j}\gamma \beta_{4} \vert +R_{j} \beta_{2} )\vert K_{1} \vert . \end{cases}\displaystyle \end{aligned}$$

Thus for sufficiently small $h_{j}$, we get $\Vert \operatorname{sub}_{j} \Vert _{\infty } \leq \sigma$ and $\Vert \operatorname{sup}_{j} \Vert _{\infty }\leq 1$. Hence, L is irreducible. Next we prove that L is monotone. We let the sum of the elements of the kth row of L be $\operatorname{SUM}_{k}$. Then

$$\begin{aligned}& \operatorname{SUM}_{k}=\sigma +\frac{h_{j}^{2}}{12}(P_{j}+Q_{j}),\quad k=1, \end{aligned}$$

(116)

$$\begin{aligned}& \operatorname{SUM}_{k}=\sigma +h_{j} r_{1,j} +h_{j}^{2} \bigl((P_{j}+Q_{j}) \bigl(G_{j} ^{2}+V_{j} ^{2}\bigr)-R \bigl(H_{x_{j}}^{2}+W_{x_{j}}^{2}\bigr) \bigr) +O\bigl(h_{j}^{3}\bigr), \quad k=2, \end{aligned}$$

(117)

$$\begin{aligned}& \operatorname{SUM}_{k}= h_{j}^{2}(P_{j}+Q_{j}+R_{j}),\quad k=3,5, \ldots,N-4, \end{aligned}$$

(118)

$$\begin{aligned}& \operatorname{SUM}_{k}= h_{j}^{2}(P_{j}+Q_{j}+R_{j}) \bigl(G_{j}^{2}+V_{j} ^{2}\bigr),\quad k=4,6, \ldots,N-3, \end{aligned}$$

(119)

$$\begin{aligned}& \operatorname{SUM}_{k}=1+ h_{j}^{2}(R_{j}+Q_{j} ),\quad k=N-2, \end{aligned}$$

(120)

$$\begin{aligned}& \operatorname{SUM}_{k}=1+h_{j} \bigl((r_{2,j} )+h_{j}\bigl((R_{j}+Q_{j}) \bigl(G_{j}^{2}+V_{j} ^{2}\bigr) \\& \hphantom{\operatorname{SUM}_{k}=}{}-P_{j}\bigl(H_{x_{j}}^{2}+W_{x_{j}}^{2} \bigr)\bigr) \bigr) +O\bigl(h_{j}^{3}\bigr), \quad k=N-1, \end{aligned}$$

(121)

where

$$ \begin{aligned} r_{1,j}= (R_{j}+Q_{j}\sigma\beta_{4}) \bigl(H_{j}^{2}+W _{j}^{2} \bigr), \\ r_{2,j}=\frac{-P_{j}-Q_{j}\beta_{3}}{\sigma }\bigl(H_{j}^{2}+W _{j}^{2}\bigr). \end{aligned} $$

(122)

Now, using (114)-(115), we have

$$\begin{aligned}& \vert r_{1,j} \vert \le \bigl\vert (R_{j}+Q_{j} \sigma \beta_{4}) \bigr\vert \bigl(\vert K_{1} \vert + \vert K_{2} \vert \bigr), \qquad \vert r_{2,j}\vert =\biggl\vert \frac{P_{j}+Q_{j}\beta_{3}}{\sigma }\biggr\vert \bigl( \vert K_{1}\vert + \vert K_{2} \vert \bigr). \end{aligned}$$

(123)

Finally, for sufficiently small $h_{j}$ and (116)-(121), we can easily prove that L is monotone. Therefore, $L^{-1}$ exists and $L^{-1}\geq 0$. Hence by (107) we have

$$\begin{aligned}& \Vert E \Vert = \bigl\Vert L^{-1} \bigr\Vert \Vert T \Vert . \end{aligned}$$

(124)

Now for sufficiently small $h_{j}$ and by (116)-(121) we can say that

$$\begin{aligned}& \operatorname{SUM}_{k} > \textstyle\begin{cases} h_{j}^{2} (P+Q),&k=1, \\ h_{j}^{2} ((P+Q)(\vert K_{1} \vert +\vert K_{2} \vert ) ),& k=2, \end{cases}\displaystyle \end{aligned}$$

(125)

$$\begin{aligned}& \operatorname{SUM}_{k} \ge \textstyle\begin{cases} h_{j}^{2}(P+Q+R),& k=3,5,\ldots,N-4, \\ h_{j}^{2}(P+Q+R)(\vert K_{1} \vert +\vert K_{2} \vert ),& k=4,6,\ldots,N-3, \end{cases}\displaystyle \end{aligned}$$

(126)

$$\begin{aligned}& \operatorname{SUM}_{k} > \textstyle\begin{cases} h_{j}^{2} (R+Q),&k=N-2, \\ h_{j}^{2} ((R+Q)(\vert K_{1} \vert +\vert K_{2} \vert ) ),& k=N-1. \end{cases}\displaystyle \end{aligned}$$

(127)

Since $\sigma \ne 0$ we can say that

$$\begin{aligned} \operatorname{SUM}_{k} &> \max \bigl[h_{j}^{2} (P+Q), h_{j}^{2} \bigl((P+Q) \bigl(\vert K_{1} \vert +\vert K_{2} \vert \bigr) \bigr)\bigr] \\ &=h_{j}^{2} \bigl((P+Q) \bigl(\vert K_{1} \vert +\vert K_{2} \vert \bigr) \bigr),\quad \text{for }k=1,2, \end{aligned}$$

(128)

$$\begin{aligned} \operatorname{SUM}_{k} &\geq \max \bigl[h_{j}^{2}(P+Q+R), h_{j}^{2}(P+Q+R) \bigl(\vert K_{1} \vert +\vert K_{2} \vert \bigr) \bigr] \\ &=h_{j}^{2}(P+Q+R) \bigl(\vert K_{1} \vert + \vert K_{2} \vert \bigr),\quad \text{for } k =3,4, \ldots,N-3, \end{aligned}$$

(129)

$$\begin{aligned} \operatorname{SUM}_{k} &>\max \bigl[h_{j}^{2}(R+Q), h_{j}^{2} (R+Q) \bigl(\vert K_{1} \vert +\vert K_{2} \vert \bigr)\bigr] \\ &=h_{j}^{2} \bigl((R+Q) \bigl(\vert K_{1} \vert +\vert K_{2} \vert \bigr) \bigr),\quad \text{for } k=N-2,N-1 . \end{aligned}$$

(130)

Let $L_{i,k}^{-1}$ be the $(i,k)$th element of $L^{-1}$, then by the theory of matrices [30] for $i=1,2,\ldots,N-1$

$$\begin{aligned} L_{i,k}^{-1}\leq \frac{1}{\operatorname{SUM}_{k} }. \end{aligned}$$

(131)

By using (128)-(130), we have

$$\begin{aligned} \frac{1}{\operatorname{SUM}_{k} } \leq \textstyle\begin{cases} \frac{1}{h_{j}^{2}(P+Q)(\vert K_{1} \vert +\vert K_{2} \vert )},&k=1,2, \\ \frac{1}{h_{j}^{2}(P+Q+R)(\vert K_{1} \vert +\vert K_{2} \vert )},&k=3,4,5,\ldots,N-3, \\ \frac{1}{h_{j}^{2}(R+Q)(\vert K_{1} \vert +\vert K_{2} \vert )},&k=N-2,N-1. \end{cases}\displaystyle \end{aligned}$$

(132)

Now let us define

$$\begin{aligned} \bigl\Vert L_{i,k}^{-1} \bigr\Vert =\max _{\substack{1\leq i \leq N-1}} \sum_{k=1}^{N-1} \bigl\vert L_{i,k}^{-1} \bigr\vert \quad \mbox{and}\quad \Vert T_{j} \Vert =\max_{\substack{1\leq j\leq N-1}} \sum_{j =1}^{N-1}\big\vert T_{j}(h_{j})\big\vert . \end{aligned}$$

(133)

Therefore, as discussed in Section 2 in scheme (53), $T_{j}(h_{j})=O(h _{j}^{5})$ and using (124), (131)-(133) we get

$$\begin{aligned} \Vert E \Vert \leq & \frac{1}{h_{j}^{2}(\vert K_{1} \vert +\vert K_{2} \vert )} \biggl[ \frac{1}{ (P+Q) } +\frac{1}{ (P+Q+R)}+\frac{1}{ (R+Q)} \biggr]O \bigl(h_{j}^{5}\bigr)=O\bigl(h_{j}^{3} \bigr). \end{aligned}$$

(134)

Hence, the third-order vector convergence of scheme (53) follows. Along similar lines, we can prove the second-order vector convergence of scheme (52) for a system of second-order boundary value problems (1).

Theorem 1

The scheme (53) for the numerical solution of system of non-linear singular boundary value problem (1) with sufficiently small $h_{j}$ and $\sigma \neq 1$ has third-order convergence under appropriate conditions.

6 Numerical illustration

To illustrate the comparative performance of our method with existing methods, we solved the following eight problems. The root mean square errors (RMSs) in the case of a variable mesh, the maximum absolute error (MA) and the relative error (RE) for a uniform mesh are tabulated in Tables 1-8. Let $h_{1}=\frac{(\sigma -1)}{(\sigma^{N}-1)}$, $\sigma \neq 1$. Therefore, the rest of the $h_{j}$ can be obtained: $h_{j+1}=\sigma h_{j}$, $j=1,\ldots,N-1$. In the case of the presence of a boundary layer near the left or right end of the domain, take

$$h_{1}= \textstyle\begin{cases} \frac{\sigma -1}{\sigma^{N}-1},&\sigma >1, \\ \frac{1- \sigma }{1-\sigma^{N}},& \sigma < 1. \end{cases} $$

This ensures the presence of mesh points in the boundary layer region near the left or right end of the interval. The linear system of difference equations have been solved by the block Gauss elimination method and the non-linear system of difference equations by the block Newton’s method in which we have considered $y_{0}=0$ as the initial approximation. Also, without loss of generality, throughout we will use $\theta_{j+1}= \theta_{j}=\theta$. This does not affect the accuracy of the scheme. All calculations have been done in Matlab 07.

Table 1 Example 6.1

Example 6.1

Consider the fourth-order linear boundary value problem of the form [31]

$$\begin{aligned} &\frac{d^{4}y}{dx^{4}}(x)- y(x)=-8 x \cos (x)- 12 \sin (x),\quad 0 \le x \le 1, \\ &y(0)=y(1)=0,\qquad \frac{d^{2}}{dx^{2}}y(0)=0,\qquad \frac{d^{2}}{dx^{2}}y(1)=2 \sin (1)+4\cos(1). \end{aligned}$$

The exact solution is given by $y(x)=( x^{2} -1)\sin (x)$. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 1. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=32$ is given by Figure 1.

Figure 1

Graph of the exact solution $\pmb{y(x)=( x^{2} -1)\sin (x) }$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 32}$ and $\pmb{\sigma =1}$ for Example  6.1 .

Example 6.2

We consider the sixth-order linear boundary value problem [21, 32]

$$\begin{aligned}& \biggl(\frac{d^{6}}{dx^{6}}+1\biggr)y(x)=6\bigl(2 x \cos (x) + 5 \sin (x) \bigr),\quad 0\le x \le 1, \\& y(0)= y(1)=0,\qquad \frac{d^{2}}{dx^{2}}y(0)=0,\qquad \frac{d^{2}}{dx^{2}}y(1)=2 \sin (1)+4\cos (1), \\& \frac{d^{4}}{dx^{4}}y(0)=0,\qquad \frac{d^{4}}{dx^{4}}y(1)=-12 \sin (1)- 8 \cos (1). \end{aligned}$$

The exact solution is given by $y(x)= (x^{2}-1)\sin (x)$. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 2. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=32$ is given by Figure 2.

Figure 2

Graph of the exact solution $\pmb{y(x)= (x^{2}-1)\sin (x)}$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 32}$ and $\pmb{\sigma =1}$ for Example  6.2 .

Table 2 Example 6.2

Example 6.3

Consider the fourth-order non-linear boundary value problem [20]:

$$\begin{aligned}& \frac{d^{4}y}{dx^{4}}(x)=-6 \exp \bigl(-4y(x)\bigr), \quad 0 \le x \le 4-e, \\& y(0)=1, \qquad y(4-e)=\ln (4),\qquad \frac{d^{2}}{dx^{2}}y(0)=-\frac{1}{e^{2}},\qquad \frac{d ^{2}}{dx^{2}}y(4-e)=-\frac{1}{16}. \end{aligned}$$

The exact solution is given by $y(x)=\ln (e+x)$. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 3. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=64$ is given by Figure 3.

Figure 3

Graph of the exact solution $\pmb{y(x)=\ln (e+x)}$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 64}$ and $\pmb{\sigma =1}$ for Example  6.3 .

Table 3 Example 6.3

Example 6.4

Consider the sixth-order non-linear boundary value problem [33]:

$$\begin{aligned}& \frac{d^{6}y}{dx^{6}}=\exp (-x)y^{2},\quad 0 \le x \le 1, \\& y(0)=\frac{d^{2}}{dx^{2}}y(0)=\frac{d^{4}}{dx^{4}}y(0)=1, \qquad y(1)= \frac{d ^{2}}{dx^{2}}y(1)=\frac{d^{4}}{dx^{4}}y(1)=e. \end{aligned}$$

The exact solution is given by $y(x)=\exp (x)$. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 4. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=32$ is given by Figure 4.

Figure 4

Graph of the exact solution $\pmb{y(x)=\exp (x)}$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 32}$ and $\pmb{\sigma =1}$ for Example  6.4 .

Table 4 Example 6.4

Example 6.5

Consider a sixth-order singular boundary value problem of the form [34]:

$$\begin{aligned}& 15x(1-x)\frac{d^{6}y}{dx^{6}}+ x \frac{d^{5}y}{dx^{5}} +\bigl(25+\exp (2x)\bigr) \frac{d ^{4}y}{dx^{4}}+\bigl(5+\exp (x)\bigr)\frac{d^{2}y}{dx^{2}}+x y=f(x),\quad 0< x < 1, \\& y(0)=y(1)=0, \qquad \frac{d^{2}}{dx^{2}}y(0)=\frac{d^{2}}{dx^{2}}y(1)=0,\qquad \frac{d ^{4}}{dx^{4}}y(0)=\frac{d^{4}}{dx^{4}}y(1)=0, \end{aligned}$$

where

$$\begin{aligned}& f(x)=\pi^{2} x\cos (\pi x)+\bigl(-\bigl(5+\exp (x)\bigr) \pi^{2}+\bigl(25+\exp (2 x)\bigr)\pi ^{4}+x+15 \pi^{6}(x-1)x\bigr)\sin (\pi x). \end{aligned}$$

The exact solution is given by $y(x)= \sin ( \pi x)$. The RMS errors for a fixed value $\sigma =0.9$ and RE error for $\sigma =1$ are tabulated in Table 5. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=160$ is given by Figure 5.

Figure 5

Graph of the exact solution $\pmb{y(x)= \sin ( \pi x) }$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 160}$ and $\pmb{\sigma =1}$ for Example  6.5 .

Table 5 Example 6.5

Example 6.6

Consider a fourth-order non-linear singular boundary value problem of the form

$$\begin{aligned}& \biggl(\frac{d^{4}}{dx^{4}}+\frac{4}{x}\frac{d^{3}}{dx^{3}} \biggr)y=y^{2} +\frac{ \cos (x)}{x}\bigl(x \sin (2x)-4\bigr),\quad 0 < x \le 1. \end{aligned}$$

The exact solution is given by $y(x)= \sin (x)$. The boundary conditions are obtained from the exact solution by a test procedure. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 6. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=64$ is given by Figure 6.

Figure 6

Graph of the exact solution $\pmb{y(x)= \sin (x) }$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 64}$ and $\pmb{\sigma =1}$ for Example  6.6 .

Table 6 Example 6.6

Example 6.7

We consider a sixth-order non-linear singular boundary value problem of the form

$$\begin{aligned} &\biggl(\frac{d^{6}}{dx^{6}}+\frac{6}{x}\frac{d^{5}}{dx^{5}}+2 \biggr)y=e^{y} +6e ^{x}\biggl(\frac{1+x}{x}\biggr),\quad 0< x \le 1. \end{aligned}$$

The exact solution is given by $y(x)= \exp (x)$. The boundary conditions are obtained from the exact solution by a test procedure. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 7. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=64$ is given by Figure 7.

Figure 7

Graph of the exact solution $\pmb{y(x)= \exp (x) }$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 64}$ and $\pmb{\sigma =1}$ for Example  6.7 .

Table 7 Example 6.7

Example 6.8

We consider a system of second-order boundary value problem of the form:

$$\begin{aligned} &\frac{d^{2} y}{dx^{2}}= y \frac{dz}{dx}+z\frac{dy}{dx}+f(x),\quad 0\le x \le 1, \\ &\frac{d^{2} z}{dx^{2}}= z \frac{dz}{dx}+y \frac{dy}{dx}+g(x). \end{aligned}$$

The exact solution is given by $y(x)= \sinh (x)$, $z(x)= \cosh (x)$. The boundary conditions are obtained from the exact solution by a test procedure. The RMS errors for a fixed value $\sigma =0.9$ and MA error for $\sigma =1$ are tabulated in Table 8. The graph of the exact solution versus the approximate solution using the fourth-order method for $N=64$ is given by Figure 8.

Figure 8

Graph of the exact solution $\pmb{y(x)= \sinh (x)}$ versus the approximate solution in fourth-order uniform mesh method for $\pmb{{N} = 64}$ and $\pmb{\sigma =1}$ for Example  6.8 .

Table 8 Example 6.8

7 Conclusion

We derived second- as well as third-order variable-mesh schemes for solving linear, non-linear even-order cases and systems of second-order boundary value problems. Although, in this paper, only fourth-order and sixth-order non-linear and linear singular boundary value problems are considered, the method is general enough to implement in the case of higher even-order linear and non-linear singular boundary value problems.

Tables 1-4 shows the presence of refinement in results when compared with other linear and non-linear boundary value problems solved by computational methods using the quintic non-polynomial spline, a septic non-polynomial and a parametric spline, a sextic spline, and a modified Adomian decomposition method. In Table 5 results have been compared with solutions derived analytically by reproducing kernel method [34]. Computationally our methods seem to be more viable due to the usage of only three grid points at a time which leads to solving of a tri-diagonal matrix. However, in the case of the quintic, the sextic and septic splines a higher number of grid points are required, which results in pentadiagonal or septadiagonal matrices. Also, in the end we have solved fourth- and sixth-order non-linear singular and a system of second-order boundary value problems. As per the literature available, such a class of boundary value problems has not been solved so far. Therefore, due to the unavailability of any prior results we were unable to present a comparative study. Hence we have compared our own results in Table 6, 7, 8 and have also provided the computational order of convergence (COC^∗) for the uniform mesh method. Our methods are also applicable to problems in cartesian as well as polar coordinates with minor modifications and even higher-order singularly perturbed boundary value problems can be solved easily due to the use of a variable mesh.