A class of quasi-variable mesh methods based on off-step discretization for the solution of non-linear fourth order ordinary differential equations with Dirichlet and Neumann boundary conditions

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

We propose a class of second and third order techniques based on off-step discretizations for a general non-linear ordinary differential equation of order four, subject to the Dirichlet and Neumann boundary conditions. Our approach uses only three grid points and involves the construction of a quasi-variable mesh. This type of a mesh is framed using a mesh ratio parameter ...... whose value is chosen in accordance with the occurrence of boundary layer in the problem, and varies with the number of grid points taken. The third order technique reduces to a fourth order one when taken with ....... The stability and convergence analysis of the techniques are discussed over a model problem. Computational results obtained upon the application to seven linear as well as non-linear problems endorse the theoretically claimed accuracies. We also provide a comparison with the computational results using approaches of other authors, which shows that the proposed methods are better.

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 DOI 10.1186/s13662-016-0973-5

A class of quasi-variable mesh methods based on off-step discretization for the solution of non-linear fourth order ordinary differential equations with Dirichlet and Neumann boundary conditions

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-016-0973-5&domain=pdf

Web End = Ranjan K Mohanty1*, Md Hasan Sarwer1 and Nikita Setia2

*Correspondence: mailto:[email protected]

Web End [email protected]

1Department of Mathematics, South Asian University, New Delhi, 110021, IndiaFull list of author information is available at the end of the article

1 Introduction

Consider the following boundary value problem (BVP):

u()(x) = f [parenleftbig]x, u(x), u (x), u (x), u (x)[parenrightbig], a < x < b, (.)

subject to the prescribed natural boundary conditions:

u(a) = A, u (a) = B, u(b) = A, u (b) = B, (.)

where A, B, A, and B are real constants and < a x b < .

2016 Mohanty et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 2 of 27

The equation (.) represents general form of a fourth order non-linear ordinary differential equation (ODE), prescribed along with the Dirichlet and Neumann boundary conditions viz. (.). These conditions are also referred to as the boundary conditions of the rst kind. Fourth order BVPs represent various physical problems that are related to elastic stability theory. These appear in the modeling of viscoelastic inelastic ows [], plate deection theory [], and deformation of beams, arches, load bearing members like street lights, and robotic arms in multi-purpose engineering systems where elastic members serve as key members for shedding and transmitting loads [, ].

Another example of physical importance is the following fourth order ODE:

u()(x) u(x)u (x) = , (.)

subject to the conditions (.). This arises from the time-independent Navier-Stokes equations for the axisymmetric ow of an incompressible uid contained between innite disks which occupy planes z = d and z = d. The disks are porous and uid is injected or extracted normally with velocity A at z = d and A at z = d. Here, is the kinematic viscosity (Elcrat []).

Thus, due to the vast physical applications of fourth order BVPs, various techniques have been proposed by researchers to solve these problems. On one hand, equations of type (.) with boundary conditions of the second kind are transformable to coupled second order equations [], such type of a reduction is not possible with rst kind boundary conditions. Apart from these, a quartic non-polynomial spline approach has been proposed by researchers for the solution of the fourth [] and sixth order [] ODEs with second kind boundary conditions. In the past, several approaches have been sought for solving fourth order BVPs with rst kind boundary conditions. These include multi-derivative methods proposed by Twizell and Tirmizi [], collocation algorithms based on interpolating and approximating subdivision schemes by Ma and Silva [], sinc collocation method by Nurmuhammad et al. [], homotopy perturbation technique for a special fourth order BVP by Momani and Noor [] and nite dierence method by Usmani [], and Chen and Li []. Some of the recently proposed approaches are the quintic spline by Akram and Amin [], the septic spline by Akram and Naheed [], the Adomian decomposition by Kelesoglu [], and subdivision schemes based on collocation algorithms by Ejaz et al. []. However, all these techniques are applicable to only a linear counterpart of the problem (.)-(.). For the non-linear case, an iterative method was proposed by Agarwal and Chow [] in . In the year , Mohanty [] developed a fourth order nite difference technique for solving one-dimensional non-linear biharmonic problem of the rst kind. Variational iteration and homotopy perturbation techniques were proposed by Noor and Mohyud-Din [], Choobbasti et al. [] and Mirmoradi et al. [] in the years , and , respectively. In , Talwar and Mohanty [] framed a nite dierence method for the solution of (.)-(.) using a uniform mesh size h > .

However, a uniform grid does not always result in stable solutions when applied to the singularly perturbed boundary value problems (SPBVPs) [, ]. Formation of sharp boundary layers in numerical methods when , the coecient of highest order derivative, approaches to zero creates trouble when used in conjunction with many classical techniques. During the past decades, many approximate methods have been developed and rened, including the method of averaging, methods of matched asymptotic expansion

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 3 of 27

and multiple scales. In , Tirmizi et al. [] developed a non-polynomial spline technique for a second order self-adjoint SPBVP. In , Jiaqi [] proposed a boundary layer correction technique for the linear fourth order SPBVPs. The recently proposed spline techniques of Akram [, ] have also been successfully applied to linear problems with boundary layer. To the best of the authors knowledge, no quasi-variable mesh methods of order two and three for the solution of fourth order non-linear ODE with boundary conditions of the rst kind have been discussed in the literature so far.

In this article, with three grid points, we have derived two new methods of order two and three for the solution of the BVP (.)-(.) using a quasi-variable mesh. We use step-size hk = xk xk > , where k refers to the grid point number, with subsequent step-size being hk+ = hk, where is a positive constant whose value is chosen in accordance with the occurrence of boundary layer. This approach enables a denser grid in the boundary layer region i.e. when is very small, and hence successfully applicable to SPBVPs. We use a combination of u(x) and its derivative u (x) at each grid point, thereby obtaining the values of u (x) as a by-product. Since we ultimately need to solve the coupled non-linear system of equations at each mesh point, the iterative methods pertaining to the complicated block structure so obtained are used. We have solved the linear systems using Gauss-Seidel and Gauss-Jacobi methods, and non-linear systems by the generalized Newton method ([ ]). Our nite dierence techniques also show highly accurate results when applied to coupled non-linear fourth order BVPs with boundary conditions of the rst kind. The numerical illustrations for the same are given below in this article.

This paper is organized into ve sections: In Section , we present and derive our second and third order quasi-variable mesh techniques, which are reducible to second and fourth order techniques, respectively, upon setting the parameter = . In Section , we discuss the convergence and stability analysis of the fourth order technique applied to a model problem. Section comprises the numerical illustrations of the methods when applied to seven fourth order BVPs of the type (.)-(.). All these problems are of physical interest, as also discussed in this section. In Section , we give some concluding remarks about this article.

2 Finite difference methods and derivation

For the sake of simplicity, let us take the domain of interest to be the closed interval [, ]. We divide this interval into N + parts by introducing mesh points: = x < x < <

xN+ = , with hk+ = xk+ xk > , k = ()N, being the step-size in the (k + )th interval, and a parameter = hk+hk > , k = ()N.

Then

= xN+ x = (xN+ xN) + (xN xN) + + (x x) + (x x)

= hN+ + hN + + h + h = [parenleftbig]N + N + + + [parenrightbig]h.

This yields h = /( + + + + N), which is the rst step length, and the subsequent

step lengths can be determined using hk+ = hk for k = ()N. Let the o-step grid points be given by xk+

= xk + hk for k = ()N, and xk

= xk hk for k = ()N + .

Let uk = u(xk), u k = u (xk) for k = ()N + , and the corresponding notations hold true for higher order derivatives of u as well. Let fk = f (xk, uk, u k, u k, u k) and fk

=

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 4 of 27

f (xk

) for k = ()N. Throughout the rest of this article, we vary k = ()N, unless otherwise specied. Clearly, at each grid point xk, (.) can be written as

u()k = fk. (.)

Let us now dene

u k =

( + )hk

[bracketleftbig]uk+ ( + )uk + uk[bracketrightbig], (.a)

u k =

, uk

, u k

, u k

, u k

( + )hk

[bracketleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[bracketrightbig]

hk u k, (.b)

u k =

( + )hk [bracketleftbig]u k+ [parenleftbig]

[parenrightbig]u k u k[bracketrightbig], (.c)

u k =

( + )hk

[bracketleftbig]u k+ ( + )u k + u k[bracketrightbig]. (.d)

Expanding each of the equations (.a)-(.d) using a Taylor series expansion, we obtain the following:

u k = u k + (

)hk

u k +

( + )hk

u()k +

( + + )hk

u()k

+ (

+ + )hk

u()k

+ ( +

+ + )hk

, u()k + O[parenleftbig]hk[parenrightbig], (.a)

u k = u k + (

)hk

u()k +

( + )hk

u()k

+ ( +

+ )hk

u()k +

( + + )hk

u()k + O[parenleftbig]hk[parenrightbig], (.b)

u k = u k +

hk

u()k +

( )hk

u()k +

( + )hk

u()k

+

( + + )hk

u()k + O[parenleftbig]hk[parenrightbig], (.c)

u k = u k + (

)hk

u()k +

( + )hk

u()k +

( + + )hk

u()k

+ (

+ + )hk

u()k + O[parenleftbig]hk[parenrightbig]. (.d)

2.1 Second order technique

To discretize the left hand side of (.), let us assume

hkfk = hku()k = ahku k + ahku k + ahku k + ahku k + Tk, (.)

where a, a, a, a are parameters to be suitably determined and Tk is the truncation error.

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 5 of 27

Substituting values from (.a)-(.d) in (.), and further equating to zero the coecients of hk, hk, hk, and hk, so as to obtain Tk = O(hk), we get

a = (

+ )( + ) , a =

( )( + + ) ( + ) ,

a = (

( )

( + ) .

Since fk is a function of u k and u k, we need second order approximations for them. It can be seen from (.c) that u k of (.c) is a second order approximation to u k. Further, eliminating the coecient of hk from (.b) and (.d), we obtain the second order approximation to u k given by

u

k =

+ )( + ) , a =

( + )hk

[bracketleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[bracketrightbig]

( + )hk

[bracketleftbig]u k+ + ( + )u k + u k[bracketrightbig]. (.)

Now, dene

f k = f [parenleftbig]xk, uk, u k, u k, u

k

[parenrightbig]. (.)

Thus, we obtain the discretization

(

+ )

hk( + ) u k +

( ) hk( + ) u k

= f k + Tk, (.a)

where Tk = O(hk).

Further, eliminating u k from (.b) and (.d), and using (.b) and (.d), we obtain

[bracketleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[bracketrightbig] hk[bracketleftbig]u k+ + ( + )u k + u k[bracketrightbig] = O[parenleftbig]hk[parenrightbig]. (.b)

Varying k over internal grid points to N, equations (.a) and (.b) together form a system of N equations in N unknowns viz. u, u, . . . , uN, u , u , . . . , u N, and hence can be solved for a unique solution. We observe that for the uniform mesh case, i.e. when = , the discretization (.a)-(.b) retains its order of accuracy.

2.2 Third order technique

To obtain the third order discretization to (.), let us consider for each k:

hk[parenleftbig]fk + hkf k[parenrightbig] = hku()k + hku()k

= bhku k + bhku k + bhku k + bhku k + T()k, (.)

where , b, b, b, and b are the parameters to be suitably determined, and T()k is the truncation error. Proceeding in a similar manner to the case of the second order technique,

( )( + + )

hk( + ) u k +

( + )

hk( + ) u k +

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 6 of 27

using equations (.a)-(.d), we obtain the following values of parameters consistent with T()k = O(hk):

b = (

+ )( + ) , b =

( )( + + )( + ) , b =

( + ) ( + ) ,

(.)

b = (

)( + ) , and

= +

.

Also, simply using the Taylor series expansions, it is easy to obtain

f k =

( + )hk

[bracketleftbig]fk+

( + )fk + fk

[bracketrightbig] + O(hk). (.)

Using equations (.) and (.) in (.), we obtain

hk(

+ )( + ) u k +

hk( )( + + )

( + ) u k

+ hk(

+ )( + ) u k +

hk( )( + ) u k

= hk[bracketleftbigg](

+ )

fk +

( + )

( + ) (fk+

+ fk

)

[bracketrightbigg] + O[parenleftbig]h

k

[parenrightbig]. (.)

Now, eliminating u k from equations (.b) and (.d), we obtain

( + )hk

[bracketleftbigg]

[parenleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[parenrightbig] + hk[parenleftbig]u k+ + ( + )u k + u k[parenrightbig]

[bracketrightbigg]

( + )hk

f k + O[parenleftbig]hk[parenrightbig]. (.)

Again, with the Taylor series expansions, it is easy to obtain

f k = hk( + )[bracketleftbig]fk+

[parenleftbig] [parenrightbig]fk fk

= (

)hk

fk +

[parenrightbig]. (.)

Using equation (.) in (.), we obtain the following:

( + )hk

[bracketleftbigg]

[parenleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[parenrightbig] + hk[parenleftbig]u k+ + ( + )u k + u k[parenrightbig]

[bracketrightbig] + O[parenleftbig]h

k

[bracketrightbigg]

= (

)( + + )hk

fk +

( + )hk

( + ) [parenleftbig]fk+

fk

[parenrightbig]. (.)

[parenrightbig] + O[parenleftbig]h

k

Let us now dene

k+

= uk +

hk

u k +

hk

u k, (.a)

k

= uk

hk

u k +

hk

u k, (.b)

k+

= u k +

hk

u k +

hk

u k, (.c)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 7 of 27

k

= u k

u k. (.d)

Using Taylor series expansions, it can easily be observed that (.a)-(.d) are O(hk)

approximations.We now aim to nd third order approximation for u k+

. For this purpose, let us consider

u k+

= u k +

hk

u()k +

hk

u k +

hk

hk

u()k + O[parenleftbig]hk[parenrightbig]

=

u k + hkcu k + cu k + hkcu k[bracketrightbig] + T()k, (.)

where c, c, c, and c are parameters to be determined, and T()k is the truncation error. Substituting values from (.a)-(.d), the values of the parameters so obtained, such that

T()k = O(hk), are as follows:

c = ( +

+ + )( + ) , c =

hk [bracketleftbig]c

+ + ( + ) ,

c = ( +

( + )

( + ) .

Thus, we dene the third order approximation:

k+

=

+ + )( + ) , c =

( + + + )

hk( + ) u k +

( + + )( + ) u k

+ ( +

+ + )

hk( + ) u k +

( + )

( + ) u k

+ O[parenleftbig]hk[parenrightbig]. (.a)

In a similar manner to above, we nd the following third order approximations:

k

=

= u k+

( + + + ) hk( + ) u k +

( + )

( + ) u k

( +

+ + ) hk( + ) u k +

( + )

( + ) u k

= u k

+ O[parenleftbig]hk[parenrightbig], (.b)

u k = ( +

+ )

hk( + ) u k

( + )

( + ) u k

( +

+ )

hk( + ) u k +

( + )

( + ) u k

= u k + O[parenleftbig]hk[parenrightbig]. (.c)

Now, for nding third order approximation to u k+

, let us consider

u k+

= u k +

hk

u k +

hk

u()k + O[parenleftbig]hk[parenrightbig] = du k + dhku k + du k + T()k, (.)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 8 of 27

where d, d, and d are the parameters to be determined, and T()k is the truncation error. Substituting values from (.a)-(.c) in (.), and comparing the coecients of hk and hk so as to induce T()k = O(hk), we obtain

d = , d =

+

, d =

.

Thus, we dene the O(hk) approximation to u k+

as follows:

k+

=

u k +

( + )

hku k +

u k

+ O[parenleftbig]hk[parenrightbig]. (.a)

With the same approach as above, we dene the following third order approximations:

k

=

= u k+

u k

( + )

hku k +

u k

= u k

+ O[parenleftbig]hk[parenrightbig], (.b)

u k = u k + (

) hku k u k

= u k + O[parenleftbig]hk[parenrightbig]. (.c)

Now, dene

fk+

= f [parenleftbig]xk+

,k+

, k+

, k+

, k+

[parenrightbig], (.a)

fk

= f [parenleftbig]xk

,k

, k

, k

, k

[parenrightbig], (.b)

fk = f [parenleftbig]xk, uk, u k, k, k[parenrightbig]. (.c)

Then we claim that the third order discretization to equation (.), subject to the conditions (.), is given by

(

+ )

hk( + ) u k +

( )( + + )

hk( + ) u k

+ (

+ )

hk( + ) u k +

( ) hk( + ) u k

= (

+ )

fk +

( + )

( + ) (fk+

+ fk

) + T()k, (.a)

( + )hk

[bracketleftbigg]

[parenleftbig]uk+ [parenleftbig] [parenrightbig]uk uk[parenrightbig] + hk[parenleftbig]u k+ + ( + )u k + u k[parenrightbig]

[bracketrightbigg]

= (

)( + + )hk

fk +

( + )hk

( + ) [parenleftbig]fk+

fk

[parenrightbig] + T()k, (.b)

where T()k = O(hk) and T()k = O(hk).

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 9 of 27

To verify this, we observe that by the Taylor series expansions in (.a)-(.c) and using equations (.a)-(.d), (.a)-(.c), and (.a)-(.c), we get

fk+

= fk+

+ O[parenleftbig]hk[parenrightbig], (.a)

fk

= fk

+ O[parenleftbig]hk[parenrightbig], (.b)

[parenrightbig]. (.c)

Substituting values from (.a)-(.c) in (.a), and further using (.), we obtain

T()k = O(hk). Similarly, using equations (.a)-(.c) in (.b), from (.), we obtain

T()k = O(hk). It is easily observable that upon setting = , the mesh becomes uniform, and the discretization (.a)-(.b) reduces to fourth order. Note that upon varying k = ()N, equations (.a)-(.b) form a system of N equations in N unknowns viz. u, u, . . . , uN, u , u , . . . , u N.

The system of N equations so obtained in both the second and the third order methods is easily solvable by numerical techniques, as discussed in Section .

3 Convergence and stability analysis3.1 Convergence analysis

Let us consider a simple counterpart of the problem (.):

u()(x) = f (x), < x < , (.)

subject to the boundary conditions:

u() = A, u() = A,

u () = B, u () = B.

fk = fk + O[parenleftbig]h

k

(.)

On setting the parameter = , the discretization (.a)-(.b) reduces to fourth order nite dierence scheme. Applying this scheme on the model problem (.)-(.), we obtain

(uk uk + uk+) + h

[parenleftbig]u k u k+[parenrightbig]

= h

(fk+

+ fk

+ fk) + T()k, (.a)

h(uk uk+) + [parenleftbig]u k + u k + u k+[parenrightbig]

= h

(fk+

fk

) + T()k, (.b)

where k = ()N, h is the uniform step-size, and T()k and T()k are the truncation errors of

O(h).

Denote by P = [, , ], L = [, , ], and M = [, , ] the N N tridiagonal matrices.

Then the system of equations (.a)-(.b) can be reformulated in matrix form:

[bracketleftBigg](L I) h

M

h M (L + I)

[bracketrightBigg][bracketleftBigg]u u

[bracketrightBigg]

=

[bracketleftBigg]d1 d2

[bracketrightBigg]

+

[bracketleftBigg]T(1)

T(2)

[bracketrightBigg]

,

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 10 of 27

where u = [u, u, . . . , uN]t and u = [u , u , . . . , u N]t are N-dimensional solution vectors, d1 and d2 are vectors with right side functions along with boundary conditions as components, T(1) and T(2) are the truncation error vectors and I is the identity matrix. Assuming

U and U to be the approximate solution vectors corresponding to u and u , respectively, the modied block successive over relaxation (BSOR) method for the scheme (.a)-(.b) is given by (see [])

U(n+) =

h

(

L I)MU (n) + ( )U(n) + (L I)d1, (.a)

U (n+) =

L + I)MU(n) + ( )U (n) + (L + I)d2, (.b)

where n = , , , , . . . , refers to the iteration number, and and are the relaxation parameters.

The associated SOR iteration matrix of (.a)-(.b) is given by

S =

[bracketleftBigg] ( )I

h (

(L I)M

h (L + I)M ( )I

[bracketrightBigg]

.

h

The associated Jacobi iteration matrix is given by

J =

[bracketleftBigg] h

[bracketrightBigg]

.

(L I)M

h (L + I)M

From the SOR theory [], we know that if is an eigenvalue of J, then is an eigenvalue of S, where they are related by the following equation:

( + )( + ) = .

To evaluate the value of , we let [bracketleftbig]

vv [bracketrightbig] be a partitioned eigenvector of J. Then we have

h

(

L I)Mv = v,

h (

L + I)Mv = v.

Eliminating v from the above two equations, we get

(

L I)M(L + I)Mv = v.

The rate of convergence of the BSOR method is dependent on the eigenvalues of the Jacobi matrix J, which in turn are given by

=

,

denoting the eigenvalues of (L I)M(L + I)M.

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Hence, we determine the optimal parameter = = as

= + [radicalBig] ()

,

where = S((L I)M(L + I)M), S being the spectral radius.The convergence factor is given by

= =

[radicalBig] ()

+ [radicalBig] ()

.

For convergence, we must have |

| < , which gives the range <

< . Thus, we establish

the following result.

Theorem The iterative method of the form (.a)-(.b) for the solution of u()(x) = f (x)

converges if <

< , where

= S((L I)M(L + I)M), S being the spectral radius, L = [, , ], and M = [, , ] being the N N tridiagonal matrices, and I being the N N

identity matrix.

3.2 Stability analysis

An iterative method for (.a)-(.b) can be written as

U(n+) =

PU(n) + h

MU (n) + r, (.a)

U (n+) = h

PU (n) + r, (.b)

where U(n) and U (n) are approximate solution vectors at the nth iteration and r and r are right hand side vectors consisting of the boundary conditions.

The above iterative method can be written in matrix form:

[bracketleftBigg]U(n+)

U (n+)

[bracketrightBigg]

MU(n)

= G

[bracketleftBigg]U(n)

U (n)

[bracketrightBigg]

+ R,

where

G =

[bracketleftBigg]

P

h M

h M

P

[bracketrightBigg]

and

R =

[bracketleftBigg]r r

[bracketrightBigg]

.

The eigenvalues of P and M are cos( n

(N+) ) and i cos(

n(N+) ), respectively, where n =

, , . . . , N. The characteristic equation of matrix G is given by

det [bracketleftBigg]

P I

h M

h M

P I

[bracketrightBigg]

= ,

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 12 of 27

where are the eigenvalues of G given by

[parenleftbigg]det[bracketleftbigg]

P I[bracketrightbigg][parenrightbigg]

P I[parenrightbigg][bracketrightbigg][parenrightbigg] = . (.)

The proposed iterative method (.a)-(.b) is stable, if maximum absolute eigenvalues of the iteration matrix are less than or equal to . It has been veried computationally that all the eigenvalues are less than . Hence, the scheme is stable.

4 Numerical illustrations

For the uniform mesh case, we know that the step size h is equal to

(N+) , thereby giving O(h) = O(N). However, for the quasi-variable mesh case, we need to appropriately chose the parameter so as to retain the claimed order of convergence as the number of intervals are varied, taking also into account the region of boundary layer, if any. As discussed in Section , if h, h, . . . , hN+ are the step-sizes over the N + sub-intervals of the domain [, ], then let

Hmax = max

kN+ hk. (.)

Let us choose, without loss of generality, > . Then it is easy to observe the following:

Hmax = hN+

= Nh

[parenleftbigg]det[bracketleftbigg][parenleftbigg]

P I[parenrightbigg] +

M[parenleftbigg]

N (N + )

N

< N . (.)

Thus, if we x C = N to be a constant, then we obtain Hmax < C/N. Further, let h = {hk}N+k=.

Then

h = Hmax < C/N. (.)

Table 1 Problem 1: Absolute errors with C = 1

x K = 1 K = 10

Second order (2.7a)-(2.7b)

Fourth order (2.21a)-(2.21b)

Absolute error [17]

0.0 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+000.1 5.1e07 2.1e10 1.7e09 2.2e06 1.7e09 1.2e090.2 1.6e06 6.4e10 5.8e09 7.2e06 5.2e09 4.3e090.3 2.9e06 1.1e09 1.1e08 1.3e05 9.0e09 8.2e090.4 3.9e06 1.5e09 1.6e08 1.7e05 1.2e08 1.2e080.5 4.4e06 1.7e09 2.1e08 2.0e05 1.3e08 1.5e080.6 4.2e06 1.6e09 2.2e08 1.9e05 1.3e08 1.6e080.7 3.3e06 1.3e09 2.0e08 1.6e05 1.0e08 1.5e080.8 2.0e06 7.5e10 1.4e08 9.5e06 6.2e09 1.1e080.9 6.2e07 2.3e10 5.6e08 3.2e06 1.9e09 4.5e091.0 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00 0.0e+00

Absolute error [17]

Second order (2.7a)-(2.7b)

Fourth order (2.21a)-(2.21b)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 13 of 27

0.00.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+00

0.19.2e067.9e091.2e112.4e051.8e081.5e146.9e042.1e081.5e10

0.22.7e052.0e084.0e114.9e053.4e082.9e133.8e053.6e083.7e08

0.34.4e053.3e088.5e117.0e054.8e083.1e127.2e044.8e089.0e07

0.45.8e054.2e081.6e108.7e055.8e081.9e117.0e052.1e108.5e06

0.56.6e054.7e082.9e109.8e056.4e087.4e117.5e045.7e084.8e05

0.66.7e054.6e085.1e101.0e046.4e082.0e108.5e056.3e081.9e04

0.75.8e053.9e087.4e109.5e055.7e083.9e107.7e045.7e086.4e04

0.84.0e052.5e088.1e107.8e054.1e085.1e106.7e054.3e081.7e03

0.91.5e058.5e094.7e104.5e051.5e083.4e107.7e042.0e084.2e03

1.00.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+000.0e+00

Absolute

error[17]

Fourthorder

(2.21a)-(2.21b)

Secondorder

(2.7a)-(2.7b)

Absolute

error[17]

Fourthorder

(2.21a)-(2.21b)

Secondorder

(2.7a)-(2.7b)

Absolute

error[17]

Table2Problem1:AbsoluteerrorswithC=1

xK=102 K=103 K=106

Fourthorder

(2.21a)-(2.21b)

Secondorder

(2.7a)-(2.7b)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 14 of 27

Hence, O(h) = O(N) in the maximum absolute norm. Similarly, in the sense of the root mean square norm, we have

h =

[radicaltp]

[radicalvertex]

[radicalvertex]

[radicalbt]

N+

[summationdisplay]

k=

hk

(N + )

N +

[radicalBig](N + )Hmax

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 15 of 27

Table 3 Problem 2: MAEs using (2.7a)-(2.7b) and (2.21a)-(2.21b) with C = 1

N Second order Fourth order Method given by [20]

= 1/16

16 5.7082e07 4.94e09 1.7094e04 1.666e06 32 4.0505e08 7.72e11 4.7425e05 1.31e07 64 2.6061e09 1.06e12 1.2094e05 2.614e09 128 1.6406e10 1.20e14 3.0303e06 6.716e11 Order 3.9896 6.47

= 1/32

16 2.9413e07 2.51e09 4.4022e5 8.537e07 32 2.0827e08 3.78e11 1.2203e5 6.736e08 64 1.3400e09 4.66e13 3.1220e06 1.344e09 128 8.4357e11 5.10e15 7.7974e07 3.452e11 Order 3.9896 6.51

= 1/64

16 1.5656e07 1.29e09 1.1706e05 4.520e07 32 1.1037e08 1.81e11 3.2459e06 3.569e08 64 7.1086e10 1.99e13 8.2662e07 7.128e10 128 4.4747e11 2.0714e07 1.829e11 Order 3.9897 6.51

= 1/128

16 9.0137e08 7.06e10 2.60e07 32 6.3500e09 9.09e12 2.049e08 64 4.0842e10 8.19e14 4.092e10 128 2.5709e11 1.05e11 Order 3.9897 6.79

= hN+

= Nh

N/(N + ) = C/(N + )

< C/N, (.)

where C = N is a constant.

In a similar manner to above, it can be veried that if < , then h /CN and h /CN, where C = N is taken as a constant. Thus, upon dening as a function

of N, we are able to retain the order of accuracy upon varying N. It is to be noted that the choice of constant C needs to be compatible with the range of , which in turn needs to be chosen so as to have a ner grid in the region of boundary layer. For > , the mesh will be ner near x = , and coarser on the other side, while for < , the mesh will be ner near x = , and coarser on the other side. If the boundary layer appears on both sides, the domain can be decomposed into two equal parts, and be chosen less than on rst half, and greater than on the second half of the domain. Then the method vice versa should be followed in the case an interior layer appears in the middle. In the case of a uniform mesh, C = = .

We have tested our numerical methods on ve linear and two non-linear problems. The right hand side functions and the boundary conditions can be determined from the exact solution. All the numerical computations are performed using double arithmetic. The iter-

Method given by [21]

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 16 of 27

Table 4 Problem 3: MAEs using (2.7a)-(2.7b) with C = 1

1 10 102 103 104

N u u u u u u u u u u

8 7.11e07 2.36e05 1.61e04 1.21e03 5.25e03 3.95e02 3.48e02 2.57e01 1.29e01 7.07e01 16 4.61e07 3.84e06 4.46e05 3.03e04 1.31e03 9.53e03 7.96e03 1.19e01 3.89e02 4.31e01 32 1.36e07 8.95e07 1.15e05 7.85e05 3.27e04 2.27e03 1.94e03 3.21e02 7.36e03 2.48e01 64 3.51e08 2.42e07 2.90e06 1.98e05 8.21e05 5.69e04 4.84e04 7.28e03 1.83e03 8.49e02 128 8.86e09 6.16e08 7.27e07 4.96e06 2.05e05 1.42e04 1.21e04 1.78e03 4.58e04 1.94e02 256 2.29e09 1.50e08 1.82e07 1.24e06 5.13e06 3.55e05 3.03e05 4.43e04 1.14e04 4.60e03 Order 1.95 2.04 2.00 2.00 2.00 2.00 2.00 2.01 2.00 2.08

ations were stopped once the error tolerance was achieved. The numerical results

support the theoretical order of accuracy of our methods.

Problem Solve (see [])

u()(x) = ( + K)u (x) Ku(x) +

x , < x < . (.)

Here, K is a constant. The exact solution for this problem is given by

u(x) = +

x +

sinh(x).

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 17 of 27

1 for

Table 5 Problem 3: MAEs using (2.7a)-(2.7b) with C = for 0 < x < 12 and C =

12 x < 1 102 104 106 108

N u u u u u u u u

32 2.50e04 2.15e03 9.24e04 2.71e02 1.74e02 1.36e01 1.97e01 1.48e+00 64 6.12e05 5.72e04 2.84e04 7.84e03 2.43e04 1.74e02 7.29e02 7.06e01 128 1.50e05 1.45e04 7.31e05 1.96e03 1.85e05 4.65e03 3.26e06 8.23e03 256 3.74e06 3.64e05 1.84e05 4.91e04 4.72e06 1.16e03 8.49e07 2.07e03 Order 2.01 1.99 1.99 2.00 1.97 2.00 1.94 1.99

Table 6 Problem 3: MAEs using (2.21a)-(2.21b) with C = 1

102 103 104

N u u u u u u

8 1.17e05 9.57e05 3.89e04 4.67e03 3.03e03 2.43e02 16 8.21e07 6.06e06 4.25e05 6.63e04 6.87e04 1.37e02 32 5.27e08 3.67e07 3.02e06 4.94e05 8.75e05 2.98e03 64 3.33e09 2.31e08 1.95e07 2.93e06 6.84e06 3.08e04 128 2.14e10 1.46e09 1.23e08 1.81e07 4.55e07 1.90e05 Order 3.96 3.98 3.99 4.02 3.91 4.01

Table 7 Problem 3: MAEs using (2.21a)-(2.21b) with C = for 0 < x < 12 and C =

12 x < 1 102 104 106 108

N u u u u u u u u

32 3.08e05 1.71e04 1.11e04 2.84e03 5.24e05 1.24e02 1.25e05 3.01e02 64 3.96e06 2.10e05 8.40e06 2.29e04 4.06e06 9.84e04 1.22e06 2.89e03 128 5.28e07 2.69e06 7.38e07 2.15e05 3.18e07 8.45e05 9.06e08 2.32e04 256 6.91e08 3.43e07 7.49e08 2.31e06 2.91e08 8.39e06 7.76e09 2.16e05 Order 2.93 2.97 3.30 3.22 3.45 3.33 3.55 3.43

Tables and illustrate the absolute errors so obtained using our second and fourth order methods, respectively, over a uniform mesh. We obtain successful results for value of K as large as . The tables also draw a comparison between the proposed results and the results of []. It is observed that on one hand, for large value of K, the accuracy of the numerical results of [] deteriorates as the value of x increases from to , our methods are unaected by the same. The proposed results are clearly better than that of []. Figure (a) and (b) provide the plots of absolute error vs. x and Figure (c) depicts a comparison of the exact and numerical solutions so obtained with the fourth order technique.

Problem Solve

u(x) + u(x) = [parenleftbig]sin( x)[parenleftbig],x ,x + ,x ,x

+ ,x x(x )

x(x ) x(x ) + x(x )[parenrightbig]

+ cos( x)[parenleftbig] x(x )

+ , x(x ) + , x(x ) + , x(x )

1 for

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 18 of 27

x(x ) x(x )[parenrightbig][parenrightbig]

+ x sin( x)(x ), < x < . (.)

The exact solution is given by

u(x) = x(x ) sin( x).

The maximum absolute errors (MAEs) corresponding to dierent values of with a uniform mesh are tabulated in Table , along with a comparison drawn with the results of

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 19 of 27

Table 8 Problem 4: MAEs using (2.7a)-(2.7b) with C = 1

1 10 102

N u u u u u u

16 9.83e06 3.37e05 3.25e03 2.03e02 7.81e02 8.71e01 32 2.49e06 8.22e06 8.19e04 4.93e0332 1.48e02 4.98e01 64 6.23e07 2.04e06 2.05e04 1.24e03 3.68e03 1.70e01 128 1.56e07 5.10e07 5.12e05 3.08e04 9.20e04 3.89e02 256 3.90e08 1.27e07 1.28e05 7.70e05 2.30e04 9.21e03 Order 2.00 2.00 2.00 2.00 2.00 2.08

Table 9 Problem 4: MAEs using (2.7a)-(2.7b) with C =

10 102 103 104 105

N u u u u u u u u u u

32 3.10e04 2.13e03 1.54e04 8.59e03 3.56e05 1.91e02 3.44e05 3.37e02 1.43e02 6.35e02 64 7.77e05 5.34e04 3.87e05 2.16e03 8.95e06 4.83e03 1.59e06 8.60e03 2.47e07 1.33e02 128 1.94e05 1.33e04 9.68e06 5.41e04 2.24e06 1.22e03 3.99e07 2.16e03 6.24e08 3.37e03 256 4.86e06 3.33e05 2.42e06 1.35e04 5.60e07 3.05e04 9.98e08 5.41e04 1.56e08 8.46e04 Order 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 1.99

Table 10 Problem 4: MAEs using (2.21a)-(2.21b) with C = 1

10 102

u u u u

8 3.06e04 2.11e03 9.80e02 5.45e01 16 2.08e05 1.29e04 1.99e02 2.66e01 32 1.33e06 7.98e06 2.00e03 5.80e02 64 8.32e08 5.02e07 1.43e04 6.10e03 128 5.23e09 3.14e08 9.23e06 3.80e04 Order 3.99 4.00 3.95 4.01

Table 11 Problem 4: MAEs using (2.21a)-(2.21b) with C =

102 103 104

u u u u u u

32 2.49e05 7.22e04 2.38e02 1.02e01 4.70e01 2.00e+00 64 5.35e06 1.50e04 1.58e06 4.13e04 1.00e02 4.45e02 128 8.32e07 2.33e05 2.83e07 7.34e05 1.60e05 9.37e05 256 1.15e07 3.22e06 4.15e08 1.07e05 1.42e08 2.41e05 512 1.49e08 4.23e07 5.52e09 1.43e06 1.32e09 3.35e06 Order 2.95 2.93 2.91 2.90 3.43 2.85

[] and []. The tables clearly depict a better result with the proposed techniques. A plot of MAE vs. N and exact and numerical solutions vs. x, obtained with the fourth order method, are presented in Figure (a) and (b), respectively.

Problem Solve

u()(x) + u (x) = , < x < . (.)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 20 of 27

This is a reaction type equation, which arises in beam theory. The exact solution is given by

u(x) =

exp(x) + (exp() )x + x

+ ( + ) exp() .

exp(x) exp(( x)) + (exp() + )x + exp() + ( + ) exp() .

The MAEs obtained for a range of values of , using the second order technique with a uniform mesh are given in Table and that with the fourth order technique are given in Table . Using the second and third order quasi-variable mesh methods, the MAEs so obtained are depicted in Tables and , respectively. In the quasi-variable mesh case, we have chosen C = for the rst half of the domain and C =

for the rest half. It is observed that while uniform mesh methods fail for high values of , the quasi-variable mesh methods are successful. Figure provides the plots using the third order technique.

Problem Solve

u()(x) + u (x) = , < x < . (.)

This is a convection type equation. The exact solution is given by

u(x) =

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 21 of 27

Table 12 Problem 5: MAEs and RMSEs using (2.7a)-(2.7b) with different values of C

N Uniform Mesh (C = 1) Quasi-Variable Mesh (C = 0.6)

MAE RMSE MAE RMSE

1 8 u 1.79e04 1.13e04 4.34e05 2.60e05 u 6.45e04 4.21e04 1.35e04 8.96e05

16 u 4.65e05 2.95e05 9.98e06 5.92e06 u 1.52e04 1.04e04 3.33e05 2.18e05

32 u 1.17e05 7.46e06 2.44e06 1.45e06 u 3.75e05 2.60e05 8.57e06 5.41e06

64 u 2.94e06 1.87e06 6.07e07 3.59e07 u 9.35e06 6.51e06 2.15e06 1.35e06

128 u 7.36e07 4.68e07 1.52e07 8.97e08 u 2.33e06 1.63e06 5.40e07 3.38e07

256 u 1.84e07 1.17e07 2.69e08 1.51e08 order 2.00 2.00 2.49 2.57u 5.83e07 4.06e07 1.04e07 6.32e08 order 2.00 2.00 2.38 2.42

10 8 u 1.40e03 8.98e04 1.29e03 8.40e04 u 4.36e03 3.12e03 4.05e03 2.91e03

16 u 3.47e04 2.22e04 3.26e04 2.06e04 u 1.11e03 7.68e04 1.00e03 7.18e04

32 u 8.64e05 5.53e05 8.11e05 5.14e05 u 2.75e04 1.91e04 2.51e04 1.79e04

64 u 2.16e05 1.38e05 2.03e05 1.28e05 u 6.89e05 4.78e05 6.29e05 4.47e05

128 u 5.40e06 3.46e06 5.06e06 3.21e06 u 1.72e05 1.19e05 1.57e05 1.12e05

256 u 1.35e06 8.64e07 1.26e06 7.95e07 order 2.00 2.00 2.01 2.01u 4.30e06 2.99e06 3.90e06 2.77e06 order 2.00 2.00 2.01 2.01

100 8 u 4.03e03 2.70e03 4.04e03 2.72e03 u 1.34e02 8.99e03 1.41e02 9.05e03

16 u 9.68e04 6.46e04 9.80e04 6.46e04 u 3.21e03 2.16e03 3.20e03 2.18e03

32 u 2.40e04 1.60e04 2.42e04 1.60e04 u 8.06e04 5.35e04 7.87e04 5.38e04

64 u 5.99e05 3.98e05 6.04e05 3.98e05 u 2.01e04 1.33e04 1.96e04 1.34e04

128 u 1.50e05 9.95e06 1.51e05 9.93e06 u 5.02e05 3.33e05 4.89e05 3.35e05

256 u 3.74e06 2.49e06 3.77e06 2.48e06 order 2.00 2.00 2.00 2.00u 1.25e05 8.33e06 1.22e05 8.37e06 order 2.00 2.00 2.00 2.00

The MAEs obtained with a uniform mesh are given in Table using the proposed second order method and in Table using the fourth order method. The MAEs obtained using second order quasi-variable mesh method are shown in Table , and that using third order method in Table . Here, we have chosen C = . It is observed that as increases, quasi-variable mesh methods produce successful results while the uniform mesh methods fail. The plots of MAE vs. N and the exact and numerical solutions vs. x with the third order technique are presented in Figure (a) and (b), respectively.

Problem Solve (see [])

u()(x) u(x)u (x) = f (x), < x < . (.)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 22 of 27

Table 13 Problem 5: MAEs and RMSEs using (2.21a)-(2.21b) with different values of C

N Uniform mesh (C = 1) Quasi-variable mesh (C = 0.7)

MAE RMSE MAE RMSE

1 16 u 1.59e07 1.20e08 1.01e06 4.22e08 u 5.68e07 3.72e07 3.29e06 1.59e06

32 u 1.00e08 2.13e10 1.11e07 1.20e09 u 3.55e08 1.34e08 3.52e07 9.15e08

64 u 6.29e10 3.54e12 1.29e08 3.57e11 order 4.00 5.91 3.10 5.07u 2.23e09 4.49e10 4.10e08 5.47e09 order 4.00 4.90 3.10 4.06

10 16 u 9.90e07 1.24e07 2.08e06 1.58e07 u 4.63e06 3.69e06 8.43e06 5.61e06

32 u 6.20e08 2.25e09 1.82e07 3.64e09 u 2.85e07 1.40e07 7.32e07 2.70e07

64 u 3.88e09 3.79e11 1.82e08 9.09e11 order 4.00 5.89 3.33 5.32u 1.77e08 4.78e09 7.18e08 1.38e08 order 4.01 4.87 3.35 4.30

100 16 u 5.57e06 1.55e06 7.17e06 1.33e06 u 3.70e05 3.70e05 3.99e05 3.99e05

32 u 3.30e07 3.14e08 4.53e07 2.65e08 u 2.18e06 1.77e06 2.55e06 1.82e06

64 u 2.03e08 5.59e10 3.36e08 5.33e10 order 4.02 5.81 3.75 5.64u 1.34e07 6.76e08 1.89e07 7.79e08 order 4.02 4.71 3.75 4.55

The exact solution is given by u(x) = ( x) exp(x). The physical signicance of this nonlinear problem has been discussed in Section . The MAEs and the root mean square errors (RMSEs) are tabulated in Tables and using second, and third and fourth order techniques, respectively. When used with a quasi-variable mesh, we have xed C = . for the second, and C = . for the third order discretization. The tables clearly illustrate the accuracy of our methods. Figure provides a comparative plot of the exact and numerical solutions.

Problem Solve

u(r)

or, equivalently,

u()(r) = r u (r) +

[parenleftbigg] d dr +

r

d dr

u(r) = f (r), < r < , (.)

r u (r) + f (r), < r < .

This is a fourth order singular problem in cylindrical polar coordinates. The exact solution is given by u(r) = r sin(r). The MAEs and RMSEs so obtained are tabulated in Table using a uniform mesh and in Table using a quasi-variable mesh. In the case of quasi-variable mesh, we have taken C = . for the second and C = . for the third order techniques, respectively. The plots of the exact and numerical solutions and MAE vs. N with the third order method are presented in Figure .

r u (r)

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 23 of 27

Table 14 Problem 6: MAEs and RMSEs using (2.7a)-(2.7b) and (2.21a)-(2.21b) with C = 1

N Second order method Fourth order method

u u u u

MAE RMSE MAE RMSE MAE RMSE MAE RMSE

8 9.512e04 6.737e04 3.162e03 2.137e03 5.80e05 3.97e05 1.55e04 9.83e05 16 2.212e04 7.021e04 1.501e04 4.995e04 3.68e06 2.44e06 1.94e05 7.78e06 32 5.410e05 1.729e04 3.576e05 1.206e04 2.36e07 1.54e07 1.67e06 5.65e07 64 1.339e05 4.298e05 8.769e06 2.972e05 1.51e08 9.75e09 1.23e07 3.86e08 128 3.337e06 1.073e05 2.175e06 7.386e06 9.64e10 6.19e10 8.43e09 2.54e09 256 8.335e07 2.682e06 5.421e07 1.841e06 3.66e11 2.45e11 5.37e10 1.68e10 Order 2.0015 2.0001 2.0046 2.0038 4.72 4.66 3.97 3.92

Table 15 Problem 6: MAEs and RMSEs using (2.7a)-(2.7b) and (2.21a)-(2.21b)

N Second order scheme (C = 0.5) Third order scheme (C = 0.6)

u u u u

MAE RMSE MAE RMSE MAE RMSE MAE RMSE

8 2.254e03 8.560e03 1.521e03 5.047e03 2.40e04 1.67e04 7.48e04 4.76e04 16 5.225e04 1.969e03 3.401e04 1.164e03 2.17e05 1.45e05 8.88e05 4.60e05 32 1.259e04 4.542e04 7.999e05 2.754e04 2.19e06 1.43e06 8.30e06 4.76e06 64 3.091e05 1.112e04 1.945e05 6.718e05 2.43e07 1.57e07 8.73e07 5.29e07 Order 2.0266 2.0301 2.0395 2.0356 3.17 3.19 3.25 3.17

Problem Solve

u()(r) = r [bracketleftbig]u (r)v (r) + v (r)u (r)[bracketrightbig] + f (r), < r < , (.a)

v()(r) = r u (r)u (r) + g(r), < r < . (.b)

The exact solution is given by u(r) = cos(r), v(r) = exp(r). These coupled non-linear equations represent a model of equilibrium for a load symmetric about the center (see []). With a quasi-variable mesh, we have used C = . for the second and C = . for the third order method. The MAEs and RMSEs obtained with the uniform mesh methods are tabulated in Table and that obtained with quasi-variable mesh methods in Table . Comparative plots of the exact and numerical solutions obtained with the third order technique are presented in Figure .

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 24 of 27

Table 16 Problem 7: MAEs and RMSEs using (2.7a)-(2.7b) and (2.21a)-(2.21b) with C = 1

N Second order Fourth order

MAE RMSE MAE RMSE

10 u 7.1411e06 4.8598e06 1.5183e08 1.0322e08 u 2.4886e05 1.6832e05 4.8261e08 3.5393e08 v 3.3399e06 2.2408e06 2.0788e08 1.4229e08 v 1.1264e05 8.1372e06 7.3344e08 4.6608e08

20 u 1.8060e06 1.1822e06 9.4738e10 6.2681e10 u 6.4868e06 4.1284e06 3.0174e09 2.1522e09 v 8.5494e07 5.5949e07 1.2768e09 8.4967e10 v 2.6592e06 1.9574e06 4.7506e09 2.9017e09

40 u 4.5147e07 2.9169e07 5.9265e11 3.8611e11 u 1.6208e06 1.0204e06 1.8898e10 1.3259e10 v 2.1498e07 1.3896e07 8.0073e11 5.2627e11 v 6.7508e07 4.8234e07 3.0237e10 1.8155e10

Order u 2.0046 2.0092 4.00 4.02 Order u 2.0020 2.0087 4.00 4.02

Order v 2.0253 2.0288 4.00 4.01 Order v 2.0268 2.0305 3.97 4.00

5 Concluding remarks

In this article, we derived nite dierence techniques (.a)-(.b) of second and (.a)-(.b) of third order accuracies for the fourth order BVPs of the type (.)-(.), using a quasi-variable mesh. While the second order method retained its accuracy, the third or-

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Table 17 Problem 7: MAEs and RMSEs using (2.7a)-(2.7b) and (2.21a)-(2.21b)

N Second order (C = 0.45) Third order (C = 1.1)

MAE RMSE MAE RMSE

10 u 2.3524e06 1.2294e06 3.3425e09 2.1271e09 u 1.6070e05 6.8720e06 1.5594e08 1.0533e08 v 1.2026e05 8.0087e06 1.9955e08 1.2704e08 v 3.9650e05 2.8558e05 1.0046e07 5.9011e08

20 u 7.4405e07 3.8633e07 1.5475e09 1.0102e09 u 4.0111e06 1.9062e06 4.8890e09 3.5842e09 v 3.0216e06 1.9569e06 3.8218e09 2.4468e09 v 9.9398e06 6.9784e06 1.3573e08 8.9719e09

40 u 1.9760e07 1.0079e07 2.7490e10 1.7786e10 u 1.0611e06 4.8541e07 8.6380e10 6.1758e10 v 7.5983e07 4.8342e07 5.2919e10 3.3868e10 v 2.4813e06 1.7237e06 1.7523e09 1.1953e09

Order u 1.9128 1.9385 2.49 2.51 u 1.9185 1.9734 2.50 2.54 v 1.9916 2.0172 2.85 2.85 v 2.0021 2.0173 2.95 2.91

der method transformed into a fourth order technique, upon setting the parameter = . Further, we conducted the convergence and stability analysis of the fourth order technique applied to a model problem. We solved seven physical problems, including a singular and a coupled non-linear BVP. The developed methods were directly applicable to problems

Mohanty et al. Advances in Dierence Equations (2016) 2016:248 Page 26 of 27

in polar coordinates. As a by-product of our methods, we obtained the high order approximations to the values of u as well, at each grid point. The numerical results conrmed that the proposed quasi-variable mesh schemes yield results of desired accuracies, as theoretically claimed. Also, we observed that while in some cases, for higher values of the perturbation parameter , the uniform mesh techniques failed, the quasi-variable mesh techniques still yielded good results. A comparison of the proposed techniques with that of previously developed techniques clearly depicted the superiority of our methods.

Competing interests

The authors declare that they have no competing interest.

Authors contributions

RKM discussed the quasi-variable mesh methods and the convergence analysis. HS partly discussed the derivation of the methods and partly carried out the computational work. HS also discussed the stability analysis. NS partly carried out the derivation of the methods and the computational work. All the authors read and approved the nal manuscript.

Author details

1Department of Mathematics, South Asian University, New Delhi, 110021, India. 2Department of Mathematics, Shaheed Bhagat Singh College, University of Delhi, New Delhi, 110017, India.

Acknowledgements

The second author is supported by SAARC Silver Jubilee Scholarship under the scholarship grant no. SAU/(S/ship)/003/2013-14. The authors are thankful to the reviewers for their valuable suggestions, which greatly improved the standard of this paper.

Received: 22 July 2016 Accepted: 13 September 2016

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