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
- Ranjan K Mohanty^{1}Email author,
- Md Hasan Sarwer^{1} and
- Nikita Setia^{2}
https://doi.org/10.1186/s13662-016-0973-5
© Mohanty et al. 2016
Received: 22 July 2016
Accepted: 13 September 2016
Published: 26 September 2016
Abstract
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 \(\eta>0\) 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 \(\eta=1\). 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.
Keywords
ordinary fourth order boundary value problems quasi-variable mesh off-step discretization Navier-Stokes equationMSC
65L10 65L12 65L20 65L50 65F051 Introduction
The equation (1.1) represents general form of a fourth order non-linear ordinary differential equation (ODE), prescribed along with the Dirichlet and Neumann boundary conditions viz. (1.2). These conditions are also referred to as the boundary conditions of the first kind. Fourth order BVPs represent various physical problems that are related to elastic stability theory. These appear in the modeling of viscoelastic inelastic flows [1], plate deflection theory [2], 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 [3, 4].
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 (1.1) with boundary conditions of the second kind are transformable to coupled second order equations [6–12], such type of a reduction is not possible with first kind boundary conditions. Apart from these, a quartic non-polynomial spline approach has been proposed by researchers for the solution of the fourth [13] and sixth order [14] ODEs with second kind boundary conditions. In the past, several approaches have been sought for solving fourth order BVPs with first kind boundary conditions. These include multi-derivative methods proposed by Twizell and Tirmizi [15], collocation algorithms based on interpolating and approximating subdivision schemes by Ma and Silva [3], sinc collocation method by Nurmuhammad et al. [16], homotopy perturbation technique for a special fourth order BVP by Momani and Noor [17] and finite difference method by Usmani [18], and Chen and Li [19]. Some of the recently proposed approaches are the quintic spline by Akram and Amin [20], the septic spline by Akram and Naheed [21], the Adomian decomposition by Kelesoglu [22], and subdivision schemes based on collocation algorithms by Ejaz et al. [23]. However, all these techniques are applicable to only a linear counterpart of the problem (1.1)-(1.2). For the non-linear case, an iterative method was proposed by Agarwal and Chow [24] in 1984. In the year 2000, Mohanty [25] developed a fourth order finite difference technique for solving one-dimensional non-linear biharmonic problem of the first kind. Variational iteration and homotopy perturbation techniques were proposed by Noor and Mohyud-Din [26], Choobbasti et al. [4] and Mirmoradi et al. [27] in the years 2007, 2008 and 2009, respectively. In 2012, Talwar and Mohanty [28] framed a finite difference method for the solution of (1.1)-(1.2) using a uniform mesh size \(h>0\).
However, a uniform grid does not always result in stable solutions when applied to the singularly perturbed boundary value problems (SPBVPs) [21, 29]. Formation of sharp boundary layers in numerical methods when ϵ, the coefficient 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 refined, including the method of averaging, methods of matched asymptotic expansion and multiple scales. In 2008, Tirmizi et al. [30] developed a non-polynomial spline technique for a second order self-adjoint SPBVP. In 2010, Jiaqi [31] proposed a boundary layer correction technique for the linear fourth order SPBVPs. The recently proposed spline techniques of Akram [20, 21] 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 first 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 (1.1)-(1.2) using a quasi-variable mesh. We use step-size \(h_{k}=x_{k}-x_{k-1}>0\), where k refers to the grid point number, with subsequent step-size being \(h_{k+1}=\eta h_{k}\), where η 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 ϵ 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 ([32–34]). Our finite difference techniques also show highly accurate results when applied to coupled non-linear fourth order BVPs with boundary conditions of the first kind. The numerical illustrations for the same are given below in this article.
This paper is organized into five sections: In Section 2, 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 \(\eta=1\). In Section 3, we discuss the convergence and stability analysis of the fourth order technique applied to a model problem. Section 4 comprises the numerical illustrations of the methods when applied to seven fourth order BVPs of the type (1.1)-(1.2). All these problems are of physical interest, as also discussed in this section. In Section 5, 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 \([0,1]\). We divide this interval into \(N+1 \) parts by introducing mesh points: \(0 = x_{0} < x_{1} < \cdots< x_{N+1} = 1 \), with \(h_{k+1} = x_{k+1} - x_{k} > 0, k=0(1)N\), being the step-size in the \((k+1)\)th interval, and a parameter \(\eta= \frac{h_{k+1}}{h_{k}}> 0, k = 1(1)N \).
2.1 Second order technique
2.2 Third order technique
The system of 2N equations so obtained in both the second and the third order methods is easily solvable by numerical techniques, as discussed in Section 4.
3 Convergence and stability analysis
3.1 Convergence analysis
Theorem 1
The iterative method of the form (3.4a)-(3.4b) for the solution of \(u^{(4)}(x) = f(x) \) converges if \(0 < \bar {\boldsymbol {\tau}} < \frac{2}{3} \), where \(\bar{\boldsymbol {\tau}} = S((\boldsymbol {L}-3\boldsymbol {I})^{-1}\boldsymbol {M}(\boldsymbol {L}+3\boldsymbol {I})^{-1}\boldsymbol {M})\), ‘S’ being the spectral radius, \(\boldsymbol {L} = [1,1,1] \), and \(\boldsymbol {M} = [1,0,-1] \) being the \(N \times N \) tridiagonal matrices, and I being the \(N \times N\) identity matrix.
3.2 Stability analysis
4 Numerical illustrations
In a similar manner to above, it can be verified that if \(\eta<1\), then \(\|\mathbf{h}\|_{\infty} \leq1/CN\) and \(\|\mathbf{h}\|_{2} \leq 1/CN\), where \(C=\eta^{N}\) is taken as a constant. Thus, upon defining η 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 η, which in turn needs to be chosen so as to have a finer grid in the region of boundary layer. For \(\eta>1\), the mesh will be finer near \(x=0\), and coarser on the other side, while for \(\eta<1\), the mesh will be finer near \(x=1\), and coarser on the other side. If the boundary layer appears on both sides, the domain can be decomposed into two equal parts, and η be chosen less than 1 on first half, and greater than 1 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, \(C=\eta=1\).
We have tested our numerical methods on five 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 iterations were stopped once the error tolerance ≤10^{−12} was achieved. The numerical results support the theoretical order of accuracy of our methods.
Problem 1
Problem 1 : Absolute errors with \(\pmb{C=1}\)
x | K = 1 | K = 10 | ||||
---|---|---|---|---|---|---|
Absolute error [ 17 ] | Absolute error [ 17 ] | |||||
0.0 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 |
0.1 | 5.1e−07 | 2.1e−10 | 1.7e−09 | 2.2e−06 | 1.7e−09 | 1.2e−09 |
0.2 | 1.6e−06 | 6.4e−10 | 5.8e−09 | 7.2e−06 | 5.2e−09 | 4.3e−09 |
0.3 | 2.9e−06 | 1.1e−09 | 1.1e−08 | 1.3e−05 | 9.0e−09 | 8.2e−09 |
0.4 | 3.9e−06 | 1.5e−09 | 1.6e−08 | 1.7e−05 | 1.2e−08 | 1.2e−08 |
0.5 | 4.4e−06 | 1.7e−09 | 2.1e−08 | 2.0e−05 | 1.3e−08 | 1.5e−08 |
0.6 | 4.2e−06 | 1.6e−09 | 2.2e−08 | 1.9e−05 | 1.3e−08 | 1.6e−08 |
0.7 | 3.3e−06 | 1.3e−09 | 2.0e−08 | 1.6e−05 | 1.0e−08 | 1.5e−08 |
0.8 | 2.0e−06 | 7.5e−10 | 1.4e−08 | 9.5e−06 | 6.2e−09 | 1.1e−08 |
0.9 | 6.2e−07 | 2.3e−10 | 5.6e−08 | 3.2e−06 | 1.9e−09 | 4.5e−09 |
1.0 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 |
Problem 1 : Absolute errors with \(\pmb{C=1}\)
x | \(\boldsymbol {K=10^{2}}\) | \(\boldsymbol {K=10^{3}} \) | \(\boldsymbol {K=10^{6}} \) | ||||||
---|---|---|---|---|---|---|---|---|---|
Absolute error [ 17 ] | Absolute error [ 17 ] | Absolute error [ 17 ] | |||||||
0.0 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 |
0.1 | 9.2e−06 | 7.9e−09 | 1.2e−11 | 2.4e−05 | 1.8e−08 | 1.5e−14 | 6.9e−04 | 2.1e−08 | 1.5e−10 |
0.2 | 2.7e−05 | 2.0e−08 | 4.0e−11 | 4.9e−05 | 3.4e−08 | 2.9e−13 | 3.8e−05 | 3.6e−08 | 3.7e−08 |
0.3 | 4.4e−05 | 3.3e−08 | 8.5e−11 | 7.0e−05 | 4.8e−08 | 3.1e−12 | 7.2e−04 | 4.8e−08 | 9.0e−07 |
0.4 | 5.8e−05 | 4.2e−08 | 1.6e−10 | 8.7e−05 | 5.8e−08 | 1.9e−11 | 7.0e−05 | 2.1e−10 | 8.5e−06 |
0.5 | 6.6e−05 | 4.7e−08 | 2.9e−10 | 9.8e−05 | 6.4e−08 | 7.4e−11 | 7.5e−04 | 5.7e−08 | 4.8e−05 |
0.6 | 6.7e−05 | 4.6e−08 | 5.1e−10 | 1.0e−04 | 6.4e−08 | 2.0e−10 | 8.5e−05 | 6.3e−08 | 1.9e−04 |
0.7 | 5.8e−05 | 3.9e−08 | 7.4e−10 | 9.5e−05 | 5.7e−08 | 3.9e−10 | 7.7e−04 | 5.7e−08 | 6.4e−04 |
0.8 | 4.0e−05 | 2.5e−08 | 8.1e−10 | 7.8e−05 | 4.1e−08 | 5.1e−10 | 6.7e−05 | 4.3e−08 | 1.7e−03 |
0.9 | 1.5e−05 | 8.5e−09 | 4.7e−10 | 4.5e−05 | 1.5e−08 | 3.4e−10 | 7.7e−04 | 2.0e−08 | 4.2e−03 |
1.0 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 | 0.0e+00 |
Problem 2
N | Second order | Fourth order | Method given by [ 20 ] | Method given by [ 21 ] |
---|---|---|---|---|
ϵ = 1/16 | ||||
16 | 5.7082e−07 | 4.94e−09 | 1.7094e−04 | 1.666e−06 |
32 | 4.0505e−08 | 7.72e−11 | 4.7425e−05 | 1.31e−07 |
64 | 2.6061e−09 | 1.06e−12 | 1.2094e−05 | 2.614e−09 |
128 | 1.6406e−10 | 1.20e−14 | 3.0303e−06 | 6.716e−11 |
Order | 3.9896 | 6.47 | ||
ϵ = 1/32 | ||||
16 | 2.9413e−07 | 2.51e−09 | 4.4022e−5 | 8.537e−07 |
32 | 2.0827e−08 | 3.78e−11 | 1.2203e−5 | 6.736e−08 |
64 | 1.3400e−09 | 4.66e−13 | 3.1220e−06 | 1.344e−09 |
128 | 8.4357e−11 | 5.10e−15 | 7.7974e−07 | 3.452e−11 |
Order | 3.9896 | 6.51 | ||
ϵ = 1/64 | ||||
16 | 1.5656e−07 | 1.29e−09 | 1.1706e−05 | 4.520e−07 |
32 | 1.1037e−08 | 1.81e−11 | 3.2459e−06 | 3.569e−08 |
64 | 7.1086e−10 | 1.99e−13 | 8.2662e−07 | 7.128e−10 |
128 | 4.4747e−11 | – | 2.0714e−07 | 1.829e−11 |
Order | 3.9897 | 6.51 | ||
ϵ = 1/128 | ||||
16 | 9.0137e−08 | 7.06e−10 | – | 2.60e−07 |
32 | 6.3500e−09 | 9.09e−12 | – | 2.049e−08 |
64 | 4.0842e−10 | 8.19e−14 | – | 4.092e−10 |
128 | 2.5709e−11 | – | – | 1.05e−11 |
Order | 3.9897 | 6.79 |
Problem 3
λ | 1 | 10 | \(\boldsymbol {10^{2}} \) | \(\boldsymbol {10^{3}} \) | \(\boldsymbol {10^{4}} \) | |||||
---|---|---|---|---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) |
8 | 7.11e−07 | 2.36e−05 | 1.61e−04 | 1.21e−03 | 5.25e−03 | 3.95e−02 | 3.48e−02 | 2.57e−01 | 1.29e−01 | 7.07e−01 |
16 | 4.61e−07 | 3.84e−06 | 4.46e−05 | 3.03e−04 | 1.31e−03 | 9.53e−03 | 7.96e−03 | 1.19e−01 | 3.89e−02 | 4.31e−01 |
32 | 1.36e−07 | 8.95e−07 | 1.15e−05 | 7.85e−05 | 3.27e−04 | 2.27e−03 | 1.94e−03 | 3.21e−02 | 7.36e−03 | 2.48e−01 |
64 | 3.51e−08 | 2.42e−07 | 2.90e−06 | 1.98e−05 | 8.21e−05 | 5.69e−04 | 4.84e−04 | 7.28e−03 | 1.83e−03 | 8.49e−02 |
128 | 8.86e−09 | 6.16e−08 | 7.27e−07 | 4.96e−06 | 2.05e−05 | 1.42e−04 | 1.21e−04 | 1.78e−03 | 4.58e−04 | 1.94e−02 |
256 | 2.29e−09 | 1.50e−08 | 1.82e−07 | 1.24e−06 | 5.13e−06 | 3.55e−05 | 3.03e−05 | 4.43e−04 | 1.14e−04 | 4.60e−03 |
Order | 1.95 | 2.04 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.01 | 2.00 | 2.08 |
λ | \(\boldsymbol {10^{2}} \) | \(\boldsymbol {10^{4}} \) | \(\boldsymbol {10^{6}} \) | \(\boldsymbol {10^{8}} \) | ||||
---|---|---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) |
32 | 2.50e−04 | 2.15e−03 | 9.24e−04 | 2.71e−02 | 1.74e−02 | 1.36e−01 | 1.97e−01 | 1.48e+00 |
64 | 6.12e−05 | 5.72e−04 | 2.84e−04 | 7.84e−03 | 2.43e−04 | 1.74e−02 | 7.29e−02 | 7.06e−01 |
128 | 1.50e−05 | 1.45e−04 | 7.31e−05 | 1.96e−03 | 1.85e−05 | 4.65e−03 | 3.26e−06 | 8.23e−03 |
256 | 3.74e−06 | 3.64e−05 | 1.84e−05 | 4.91e−04 | 4.72e−06 | 1.16e−03 | 8.49e−07 | 2.07e−03 |
Order | 2.01 | 1.99 | 1.99 | 2.00 | 1.97 | 2.00 | 1.94 | 1.99 |
λ | \(\boldsymbol {10^{2}} \) | \(\boldsymbol {10^{3}} \) | \(\boldsymbol {10^{4}} \) | |||
---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) |
8 | 1.17e−05 | 9.57e−05 | 3.89e−04 | 4.67e−03 | 3.03e−03 | 2.43e−02 |
16 | 8.21e−07 | 6.06e−06 | 4.25e−05 | 6.63e−04 | 6.87e−04 | 1.37e−02 |
32 | 5.27e−08 | 3.67e−07 | 3.02e−06 | 4.94e−05 | 8.75e−05 | 2.98e−03 |
64 | 3.33e−09 | 2.31e−08 | 1.95e−07 | 2.93e−06 | 6.84e−06 | 3.08e−04 |
128 | 2.14e−10 | 1.46e−09 | 1.23e−08 | 1.81e−07 | 4.55e−07 | 1.90e−05 |
Order | 3.96 | 3.98 | 3.99 | 4.02 | 3.91 | 4.01 |
λ | \(\boldsymbol {10^{2}} \) | \(\boldsymbol {10^{4}} \) | \(\boldsymbol {10^{6}} \) | \(\boldsymbol {10^{8}} \) | ||||
---|---|---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) |
32 | 3.08e−05 | 1.71e−04 | 1.11e−04 | 2.84e−03 | 5.24e−05 | 1.24e−02 | 1.25e−05 | 3.01e−02 |
64 | 3.96e−06 | 2.10e−05 | 8.40e−06 | 2.29e−04 | 4.06e−06 | 9.84e−04 | 1.22e−06 | 2.89e−03 |
128 | 5.28e−07 | 2.69e−06 | 7.38e−07 | 2.15e−05 | 3.18e−07 | 8.45e−05 | 9.06e−08 | 2.32e−04 |
256 | 6.91e−08 | 3.43e−07 | 7.49e−08 | 2.31e−06 | 2.91e−08 | 8.39e−06 | 7.76e−09 | 2.16e−05 |
Order | 2.93 | 2.97 | 3.30 | 3.22 | 3.45 | 3.33 | 3.55 | 3.43 |
Problem 4
λ | 1 | 10 | \(\boldsymbol {10^{2}}\) | |||
---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) |
16 | 9.83e−06 | 3.37e−05 | 3.25e−03 | 2.03e−02 | 7.81e−02 | 8.71e−01 |
32 | 2.49e−06 | 8.22e−06 | 8.19e−04 | 4.93e−0332 | 1.48e−02 | 4.98e−01 |
64 | 6.23e−07 | 2.04e−06 | 2.05e−04 | 1.24e−03 | 3.68e−03 | 1.70e−01 |
128 | 1.56e−07 | 5.10e−07 | 5.12e−05 | 3.08e−04 | 9.20e−04 | 3.89e−02 |
256 | 3.90e−08 | 1.27e−07 | 1.28e−05 | 7.70e−05 | 2.30e−04 | 9.21e−03 |
Order | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.08 |
λ | 10 | \(\boldsymbol {10^{2}}\) | \(\boldsymbol {10^{3}} \) | \(\boldsymbol {10^{4}} \) | \(\boldsymbol {10^{5}} \) | |||||
---|---|---|---|---|---|---|---|---|---|---|
N | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) | u | \(\boldsymbol {u'}\) |
32 | 3.10e−04 | 2.13e−03 | 1.54e−04 | 8.59e−03 | 3.56e−05 | 1.91e−02 | 3.44e−05 | 3.37e−02 | 1.43e−02 | 6.35e−02 |
64 | 7.77e−05 | 5.34e−04 | 3.87e−05 | 2.16e−03 | 8.95e−06 | 4.83e−03 | 1.59e−06 | 8.60e−03 | 2.47e−07 | 1.33e−02 |
128 | 1.94e−05 | 1.33e−04 | 9.68e−06 | 5.41e−04 | 2.24e−06 | 1.22e−03 | 3.99e−07 | 2.16e−03 | 6.24e−08 | 3.37e−03 |
256 | 4.86e−06 | 3.33e−05 | 2.42e−06 | 1.35e−04 | 5.60e−07 | 3.05e−04 | 9.98e−08 | 5.41e−04 | 1.56e−08 | 8.46e−04 |
Order | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 1.99 |
λ | 10 | \(\boldsymbol {10^{2}}\) | ||
---|---|---|---|---|
u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | |
8 | 3.06e−04 | 2.11e−03 | 9.80e−02 | 5.45e−01 |
16 | 2.08e−05 | 1.29e−04 | 1.99e−02 | 2.66e−01 |
32 | 1.33e−06 | 7.98e−06 | 2.00e−03 | 5.80e−02 |
64 | 8.32e−08 | 5.02e−07 | 1.43e−04 | 6.10e−03 |
128 | 5.23e−09 | 3.14e−08 | 9.23e−06 | 3.80e−04 |
Order | 3.99 | 4.00 | 3.95 | 4.01 |
λ | \(\boldsymbol {10^{2}}\) | \(\boldsymbol {10^{3}} \) | \(\boldsymbol {10^{4}} \) | |||
---|---|---|---|---|---|---|
u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | |
32 | 2.49e−05 | 7.22e−04 | 2.38e−02 | 1.02e−01 | 4.70e−01 | 2.00e+00 |
64 | 5.35e−06 | 1.50e−04 | 1.58e−06 | 4.13e−04 | 1.00e−02 | 4.45e−02 |
128 | 8.32e−07 | 2.33e−05 | 2.83e−07 | 7.34e−05 | 1.60e−05 | 9.37e−05 |
256 | 1.15e−07 | 3.22e−06 | 4.15e−08 | 1.07e−05 | 1.42e−08 | 2.41e−05 |
512 | 1.49e−08 | 4.23e−07 | 5.52e−09 | 1.43e−06 | 1.32e−09 | 3.35e−06 |
Order | 2.95 | 2.93 | 2.91 | 2.90 | 3.43 | 2.85 |
Problem 5
λ | N | Uniform Mesh ( C = 1) | Quasi-Variable Mesh ( C = 0.6) | |||
---|---|---|---|---|---|---|
MAE | RMSE | MAE | RMSE | |||
1 | 8 | u | 1.79e−04 | 1.13e−04 | 4.34e−05 | 2.60e−05 |
\(u' \) | 6.45e−04 | 4.21e−04 | 1.35e−04 | 8.96e−05 | ||
16 | u | 4.65e−05 | 2.95e−05 | 9.98e−06 | 5.92e−06 | |
\(u' \) | 1.52e−04 | 1.04e−04 | 3.33e−05 | 2.18e−05 | ||
32 | u | 1.17e−05 | 7.46e−06 | 2.44e−06 | 1.45e−06 | |
\(u' \) | 3.75e−05 | 2.60e−05 | 8.57e−06 | 5.41e−06 | ||
64 | u | 2.94e−06 | 1.87e−06 | 6.07e−07 | 3.59e−07 | |
\(u' \) | 9.35e−06 | 6.51e−06 | 2.15e−06 | 1.35e−06 | ||
128 | u | 7.36e−07 | 4.68e−07 | 1.52e−07 | 8.97e−08 | |
\(u' \) | 2.33e−06 | 1.63e−06 | 5.40e−07 | 3.38e−07 | ||
256 | u | 1.84e−07 | 1.17e−07 | 2.69e−08 | 1.51e−08 | |
order | 2.00 | 2.00 | 2.49 | 2.57 | ||
\(u' \) | 5.83e−07 | 4.06e−07 | 1.04e−07 | 6.32e−08 | ||
order | 2.00 | 2.00 | 2.38 | 2.42 | ||
10 | 8 | u | 1.40e−03 | 8.98e−04 | 1.29e−03 | 8.40e−04 |
\(u' \) | 4.36e−03 | 3.12e−03 | 4.05e−03 | 2.91e−03 | ||
16 | u | 3.47e−04 | 2.22e−04 | 3.26e−04 | 2.06e−04 | |
\(u' \) | 1.11e−03 | 7.68e−04 | 1.00e−03 | 7.18e−04 | ||
32 | u | 8.64e−05 | 5.53e−05 | 8.11e−05 | 5.14e−05 | |
\(u' \) | 2.75e−04 | 1.91e−04 | 2.51e−04 | 1.79e−04 | ||
64 | u | 2.16e−05 | 1.38e−05 | 2.03e−05 | 1.28e−05 | |
\(u' \) | 6.89e−05 | 4.78e−05 | 6.29e−05 | 4.47e−05 | ||
128 | u | 5.40e−06 | 3.46e−06 | 5.06e−06 | 3.21e−06 | |
\(u' \) | 1.72e−05 | 1.19e−05 | 1.57e−05 | 1.12e−05 | ||
256 | u | 1.35e−06 | 8.64e−07 | 1.26e−06 | 7.95e−07 | |
order | 2.00 | 2.00 | 2.01 | 2.01 | ||
\(u' \) | 4.30e−06 | 2.99e−06 | 3.90e−06 | 2.77e−06 | ||
order | 2.00 | 2.00 | 2.01 | 2.01 | ||
100 | 8 | u | 4.03e−03 | 2.70e−03 | 4.04e−03 | 2.72e−03 |
\(u' \) | 1.34e−02 | 8.99e−03 | 1.41e−02 | 9.05e−03 | ||
16 | u | 9.68e−04 | 6.46e−04 | 9.80e−04 | 6.46e−04 | |
\(u' \) | 3.21e−03 | 2.16e−03 | 3.20e−03 | 2.18e−03 | ||
32 | u | 2.40e−04 | 1.60e−04 | 2.42e−04 | 1.60e−04 | |
\(u' \) | 8.06e−04 | 5.35e−04 | 7.87e−04 | 5.38e−04 | ||
64 | u | 5.99e−05 | 3.98e−05 | 6.04e−05 | 3.98e−05 | |
\(u' \) | 2.01e−04 | 1.33e−04 | 1.96e−04 | 1.34e−04 | ||
128 | u | 1.50e−05 | 9.95e−06 | 1.51e−05 | 9.93e−06 | |
\(u' \) | 5.02e−05 | 3.33e−05 | 4.89e−05 | 3.35e−05 | ||
256 | u | 3.74e−06 | 2.49e−06 | 3.77e−06 | 2.48e−06 | |
order | 2.00 | 2.00 | 2.00 | 2.00 | ||
\(u' \) | 1.25e−05 | 8.33e−06 | 1.22e−05 | 8.37e−06 | ||
order | 2.00 | 2.00 | 2.00 | 2.00 |
λ | N | Uniform mesh ( C = 1) | Quasi-variable mesh ( C = 0.7) | |||
---|---|---|---|---|---|---|
MAE | RMSE | MAE | RMSE | |||
1 | 16 | u | 1.59e−07 | 1.20e−08 | 1.01e−06 | 4.22e−08 |
\(u' \) | 5.68e−07 | 3.72e−07 | 3.29e−06 | 1.59e−06 | ||
32 | u | 1.00e−08 | 2.13e−10 | 1.11e−07 | 1.20e−09 | |
\(u' \) | 3.55e−08 | 1.34e−08 | 3.52e−07 | 9.15e−08 | ||
64 | u | 6.29e−10 | 3.54e−12 | 1.29e−08 | 3.57e−11 | |
order | 4.00 | 5.91 | 3.10 | 5.07 | ||
\(u' \) | 2.23e−09 | 4.49e−10 | 4.10e−08 | 5.47e−09 | ||
order | 4.00 | 4.90 | 3.10 | 4.06 | ||
10 | 16 | u | 9.90e−07 | 1.24e−07 | 2.08e−06 | 1.58e−07 |
\(u' \) | 4.63e−06 | 3.69e−06 | 8.43e−06 | 5.61e−06 | ||
32 | u | 6.20e−08 | 2.25e−09 | 1.82e−07 | 3.64e−09 | |
\(u' \) | 2.85e−07 | 1.40e−07 | 7.32e−07 | 2.70e−07 | ||
64 | u | 3.88e−09 | 3.79e−11 | 1.82e−08 | 9.09e−11 | |
order | 4.00 | 5.89 | 3.33 | 5.32 | ||
\(u' \) | 1.77e−08 | 4.78e−09 | 7.18e−08 | 1.38e−08 | ||
order | 4.01 | 4.87 | 3.35 | 4.30 | ||
100 | 16 | u | 5.57e−06 | 1.55e−06 | 7.17e−06 | 1.33e−06 |
\(u' \) | 3.70e−05 | 3.70e−05 | 3.99e−05 | 3.99e−05 | ||
32 | u | 3.30e−07 | 3.14e−08 | 4.53e−07 | 2.65e−08 | |
\(u' \) | 2.18e−06 | 1.77e−06 | 2.55e−06 | 1.82e−06 | ||
64 | u | 2.03e−08 | 5.59e−10 | 3.36e−08 | 5.33e−10 | |
order | 4.02 | 5.81 | 3.75 | 5.64 | ||
\(u' \) | 1.34e−07 | 6.76e−08 | 1.89e−07 | 7.79e−08 | ||
order | 4.02 | 4.71 | 3.75 | 4.55 |
Problem 6
N | Second order method | Fourth order method | ||||||
---|---|---|---|---|---|---|---|---|
u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | |||||
MAE | RMSE | MAE | RMSE | MAE | RMSE | MAE | RMSE | |
8 | 9.512e−04 | 6.737e−04 | 3.162e−03 | 2.137e−03 | 5.80e−05 | 3.97e−05 | 1.55e−04 | 9.83e−05 |
16 | 2.212e−04 | 7.021e−04 | 1.501e−04 | 4.995e−04 | 3.68e−06 | 2.44e−06 | 1.94e−05 | 7.78e−06 |
32 | 5.410e−05 | 1.729e−04 | 3.576e−05 | 1.206e−04 | 2.36e−07 | 1.54e−07 | 1.67e−06 | 5.65e−07 |
64 | 1.339e−05 | 4.298e−05 | 8.769e−06 | 2.972e−05 | 1.51e−08 | 9.75e−09 | 1.23e−07 | 3.86e−08 |
128 | 3.337e−06 | 1.073e−05 | 2.175e−06 | 7.386e−06 | 9.64e−10 | 6.19e−10 | 8.43e−09 | 2.54e−09 |
256 | 8.335e−07 | 2.682e−06 | 5.421e−07 | 1.841e−06 | 3.66e−11 | 2.45e−11 | 5.37e−10 | 1.68e−10 |
Order | 2.0015 | 2.0001 | 2.0046 | 2.0038 | 4.72 | 4.66 | 3.97 | 3.92 |
N | Second order scheme ( C = 0.5) | Third order scheme ( C = 0.6) | ||||||
---|---|---|---|---|---|---|---|---|
u | \(\boldsymbol {u'} \) | u | \(\boldsymbol {u'} \) | |||||
MAE | RMSE | MAE | RMSE | MAE | RMSE | MAE | RMSE | |
8 | 2.254e−03 | 8.560e−03 | 1.521e−03 | 5.047e−03 | 2.40e−04 | 1.67e−04 | 7.48e−04 | 4.76e−04 |
16 | 5.225e−04 | 1.969e−03 | 3.401e−04 | 1.164e−03 | 2.17e−05 | 1.45e−05 | 8.88e−05 | 4.60e−05 |
32 | 1.259e−04 | 4.542e−04 | 7.999e−05 | 2.754e−04 | 2.19e−06 | 1.43e−06 | 8.30e−06 | 4.76e−06 |
64 | 3.091e−05 | 1.112e−04 | 1.945e−05 | 6.718e−05 | 2.43e−07 | 1.57e−07 | 8.73e−07 | 5.29e−07 |
Order | 2.0266 | 2.0301 | 2.0395 | 2.0356 | 3.17 | 3.19 | 3.25 | 3.17 |
Problem 7
N | Second order | Fourth order | |||
---|---|---|---|---|---|
MAE | RMSE | MAE | RMSE | ||
10 | u | 7.1411e−06 | 4.8598e−06 | 1.5183e−08 | 1.0322e−08 |
\(u' \) | 2.4886e−05 | 1.6832e−05 | 4.8261e−08 | 3.5393e−08 | |
v | 3.3399e−06 | 2.2408e−06 | 2.0788e−08 | 1.4229e−08 | |
\(v' \) | 1.1264e−05 | 8.1372e−06 | 7.3344e−08 | 4.6608e−08 | |
20 | u | 1.8060e−06 | 1.1822e−06 | 9.4738e−10 | 6.2681e−10 |
\(u'\) | 6.4868e−06 | 4.1284e−06 | 3.0174e−09 | 2.1522e−09 | |
v | 8.5494e−07 | 5.5949e−07 | 1.2768e−09 | 8.4967e−10 | |
\(v' \) | 2.6592e−06 | 1.9574e−06 | 4.7506e−09 | 2.9017e−09 | |
40 | u | 4.5147e−07 | 2.9169e−07 | 5.9265e−11 | 3.8611e−11 |
\(u'\) | 1.6208e−06 | 1.0204e−06 | 1.8898e−10 | 1.3259e−10 | |
v | 2.1498e−07 | 1.3896e−07 | 8.0073e−11 | 5.2627e−11 | |
\(v' \) | 6.7508e−07 | 4.8234e−07 | 3.0237e−10 | 1.8155e−10 | |
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 |
N | Second order ( C = 0.45) | Third order ( C = 1.1) | |||
---|---|---|---|---|---|
MAE | RMSE | MAE | RMSE | ||
10 | u | 2.3524e−06 | 1.2294e−06 | 3.3425e−09 | 2.1271e−09 |
\(u' \) | 1.6070e−05 | 6.8720e−06 | 1.5594e−08 | 1.0533e−08 | |
v | 1.2026e−05 | 8.0087e−06 | 1.9955e−08 | 1.2704e−08 | |
\(v' \) | 3.9650e−05 | 2.8558e−05 | 1.0046e−07 | 5.9011e−08 | |
20 | u | 7.4405e−07 | 3.8633e−07 | 1.5475e−09 | 1.0102e−09 |
\(u'\) | 4.0111e−06 | 1.9062e−06 | 4.8890e−09 | 3.5842e−09 | |
v | 3.0216e−06 | 1.9569e−06 | 3.8218e−09 | 2.4468e−09 | |
\(v' \) | 9.9398e−06 | 6.9784e−06 | 1.3573e−08 | 8.9719e−09 | |
40 | u | 1.9760e−07 | 1.0079e−07 | 2.7490e−10 | 1.7786e−10 |
\(u'\) | 1.0611e−06 | 4.8541e−07 | 8.6380e−10 | 6.1758e−10 | |
v | 7.5983e−07 | 4.8342e−07 | 5.2919e−10 | 3.3868e−10 | |
\(v' \) | 2.4813e−06 | 1.7237e−06 | 1.7523e−09 | 1.1953e−09 | |
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 |
5 Concluding remarks
In this article, we derived finite difference techniques (2.7a)-(2.7b) of second and (2.21a)-(2.21b) of third order accuracies for the fourth order BVPs of the type (1.1)-(1.2), using a quasi-variable mesh. While the second order method retained its accuracy, the third order method transformed into a fourth order technique, upon setting the parameter \(\eta=1\). 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 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 confirmed 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 λ, 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.
Declarations
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.
Open Access 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.
Authors’ Affiliations
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