A class of quasi-variable mesh methods based on off-step discretization for the numerical solution of fourth-order quasi-linear parabolic partial differential equations
- Ranjan Kumar Mohanty^{1}Email author and
- Deepti Kaur^{2}
https://doi.org/10.1186/s13662-016-1048-3
© The Author(s) 2016
Received: 14 September 2016
Accepted: 29 November 2016
Published: 13 December 2016
Abstract
Numerical schemes based on off-step discretization are developed to solve two classes of fourth-order time-dependent partial differential equations subjected to appropriate initial and boundary conditions. The difference methods reported here are second-order accurate in time and second-order accurate in space and, for a nonuniform grid, second-order accurate in time and third-order accurate in space. In case of a uniform grid, the second scheme is of order two in time and four in space. The presented methods split the original problem to a coupled system of two second-order equations and involve only three spatial grid points of a compact stencil without discretizing the boundary conditions. The linear stability of the presented methods has been examined, and it is shown that the proposed two-level finite difference method is unconditionally stable for a linear model problem. The new developed methods are directly applicable to fourth-order parabolic partial differential equations with singular coefficients, which is the main highlight of our work. The methods are successfully tested on singular problems. The proposed method is applied to find numerical solutions of the Euler-Bernoulli beam equation and complex fourth-order nonlinear equations like the good Boussinesq equation. Comparison of the obtained results with those for some earlier known methods show the superiority of the present approach.
Keywords
Euler-Bernoulli beam equation off-step nodal points quasi-variable mesh finite difference method successive tangential partial derivatives good Boussinesq equation1 Introduction
Owing to their great importance and wide range of applications, the attention of many physicists and mathematicians has been attracted to the studies of such problems. The closed-form solutions to fourth-order PDEs are necessary to know the qualitative behavior of natural processes and physical phenomena. But most fourth-order time-dependent PDEs have no closed-form solutions except for certain particular types of linear or quasi-linear equations. Therefore, construction of accurate numerical methods for finding approximate solutions to these equations are of great significance. Among the entire arsenal of numerical methods available to approximate a fourth-order PDE, such as the finite element method, spline collocation method, the finite difference method, is attractive because of its relative ease of implementation, flexibility, and accuracy in the solution values. Higher-order methods yield not only comparable accuracy but also require much coarser discretization with greater computational efficiency. Apart from this, the advantage of developing a compact scheme restricted to the patch of cells immediately surrounding any given grid point is its suitability to be used directly adjacent to the boundary without introducing any extra nodes outside the boundary of the domain. Higher-order difference approximations for one-space-dimensional nonlinear parabolic and hyperbolic differential equations were discussed in [12–20]. A meshless numerical solution of hyperbolic PDEs using an improved localized radial basis functions collocation method was proposed in [21]. Recently, a new high-order compact implicit variable mesh discretization for one-space-dimensional unsteady quasi-linear biharmonic problem was developed in [22].
Various explicit and implicit difference schemes for numerical solution of the Euler-Bernoulli equation by decomposing it into a system of second-order PDEs have been studied by Conte [23], Crandall [24], Evans [25], Fairweather and Gourley [26], and Collatz [27]. The three-level explicit method suggested by Collatz [27] is easy to implement but is very time consuming even for the most modest problems due to the stability restriction. Andrade and McKee [28] suggested high-accuracy alternating direction implicit methods for solving fourth-order parabolic equations with variable coefficients. Using a multiderivative method, Twizell and Khaliq [29] derived a stable difference scheme for fourth-order parabolic equations with constant coefficients. Evans and Yousif [30] presented an unconditionally stable second-order accurate finite difference scheme using the alternating group explicit method achieving a better accuracy level. Later, Khan et al. [31] reported a three-level difference method of accuracy \(O(k^{2}+h^{4})\) for numerical solution of the Euler-Bernoulli equation by using a sextic spline in space and finite difference discretization in time. Further, Caglar and Caglar [32] considered a family of B-spline methods to produce accurate numerical solution of the Euler-Bernoulli equation. Rashidinia and Mohammadi [33] developed an approximation for finding the numerical solution of differential equation (3) by replacing the time derivative by a finite difference approximation and the space derivative by sextic spline functions using off-step points to obtain three-level implicit methods of accuracies \(O(k^{2}+h^{4})\) and \(O(k^{4}+h^{4})\). Mittal and Jain [34] discussed two new methods for solving the Euler-Bernoulli equation using B-splines with redefined basis functions. Most recently, Mohammadi [35] proposed a sextic B-spline collocation scheme for numerical solution of fourth-order time-dependent PDEs subjected to fixed and cantilever boundary conditions. Lai and Ma [7] proposed a lattice Boltzmann model for the second-order Benjamin-Ono equation (4). Numerous numerical methods have been proposed for solving the good Boussinesq equation (5) (see [8–10]). Recently, Siddiqi and Arshed [36] developed a quintic B-spline collocation method for finding an approximate solution of the good Boussinesq equation.
The consideration of using off-step nodal points for discretization is motivated by the polar form of one space Laplacian operator \(\nabla ^{2}\equiv\partial^{2}/\partial r^{2}+(\alpha/r)(\partial/\partial r)\), which has a singular coefficient associated with the first-order derivative term. Using only three grid points at each time level, three-level compact difference methods of order two in time and four in space for the solution of differential equation (1) for uniform mesh were reported by Mohanty and Evans [37], but these methods fail at singular points, and a special technique was needed to solve singular problems. To this concern, in the present article, using the same number of grid points (\(3+3+3\)) of a single compact cell, we have proposed two new off-step discretizations for the solution of the fourth-order quasi-linear PDE (1) having the foremost advantage that these are directly applicable to the singular problems without requiring any fictitious points. Recently, Mohanty and Kaur [11] proposed an implicit high-order two-level finite difference scheme for the solution of particular type of fourth-order equation (6). However, that scheme featured a major shortcoming that it is not directly applicable to the singular problems and requires a special treatment to handle singular points. In this paper, we have developed two new two-level unconditionally stable implicit methods using off-step nodal points for the solution of the differential equation (6). The proposed new methods are convenient to implement at singular points without requiring any modification, and we do not need to discretize the boundary conditions, which is a main attraction.
An outline of the paper is as follows: In Section 2, we formulate and derive three-level quasi-variable mesh difference methods using off-step points for the solution of quasi-linear fourth-order PDE (1). In Section 3, we present and derive new quasi-variable mesh two-level off-step discretizations to solve the particular type of fourth-order PDE (6). Further, in Section 4, the stability analysis of the derived methods for linear model problems have been discussed. In Section 5, we apply the proposed methods to a linear fourth-order PDE in polar coordinates. In Section 6, the performance of the proposed methods is illustrated by numerical experiments done on a collection of test problems having physical significance including the highly nonlinear good Boussinesq equation. Some concluding remarks about this paper are given in Section 7.
2 Three-level quasi-variable mesh off-step discretization and derivation
Since the value of u and \(u_{t}\) is prescribed at \(t=0\), this implies that the values of all successive tangential partial derivatives \(u_{x}, u_{xx},\ldots\) of u are known at \(t=0\). Since \(v(x,0)=u_{xx}(x,0)\), the value of v is also known at \(t=0\). Also, note that the values of u and v are given at \(x=a\) and \(x=b\).
In order to obtain a numerical solution of above initial boundary value problem, we superimpose on the solution domain Ω a rectangular grid with spacing \(h_{l}=x_{l}-x_{l-1}\), \(l=1(1)N+1\), in the x-direction such that \(a=x_{0}< x_{1}<\cdots<x_{N}<x_{N+1}=b\), N being a positive integer, and \(k=t_{j+1}-t_{j} >0\) in time direction. Spatial grid points are defined by \(x_{l}=x_{0}+\sum _{i=1}^{l}h_{i}, l=1(1)N+1\), and time steps are given by \(t_{j}=jk\), \(j=0,1,2,\ldots,J\), where J is a positive integer. The mesh ratio is denoted by \(\eta_{l}=(h_{l+1}/h_{l}) >0\), \(l=1(1)N\). The neighboring off-step points are defined as \(x_{l+1/2}=x_{l}+\frac{\eta _{l}h_{l}}{2}\) and \(x_{l-1/2}=x_{l}-\frac{h_{l}}{2}\), \(l=1(1)N\). For \(\eta_{l}=1\), it reduces to the uniform mesh case. Let \(u^{j}_{l}\), \(v^{j}_{l}\) denote approximate solution values of \(u(x,t)\), \(v(x,t)\) at the grid point \((x_{l},t_{j})\), and \(U^{j}_{l}\), \(V^{j}_{l}\) be their exact solution values at the the grid point \((x_{l},t_{j})\), respectively. For \(E=A, A_{x}\), and \(A_{xx}\), let the values \(E(x_{l},t_{j})\) be denoted by \(E^{j}_{l}\). For simplicity, we consider \(\eta_{l}=\eta\) (a constant ≠ 1), \(l=1(1)N\). Such a mesh is called a quasi-variable mesh.
The derivation of the numerical methods (15a)-(15b) is straightforward. So, we discuss in detail the derivation of the novel off-step discretization technique given by (16a)-(16b).
Note that for the constant mesh case, the difference method (28a)-(28b) is fourth-order accurate in space for a fixed value of the mesh ratio parameter \(\lambda=k/h^{2}\).
3 Two-level off-step discretization strategy and truncation error analysis
In this section, we develop new quasi-variable mesh off-step finite difference methods for the differential equation (6) with initial and boundary conditions given by (2a)-(2c).
4 Stability analysis using characteristic equation
The eigenvalues of \(S_{11}, S_{12}, S_{21}\), and \(S_{22}\) are given by \(-48 \theta\sin^{2}\phi\), \(-h^{2}\theta(12-4\sin^{2}\phi)\), \(12-4\sin^{2}\phi\), and \(-48 \lambda^{2}h^{2}\epsilon\theta\sin ^{2}\phi\), respectively. Further, the eigenvalues of \(T_{11}, T_{12}, T_{21}\), and \(T_{22}\) are given by \(48 \sin^{2}\phi\), \(h^{2}(12-4\sin ^{2}\phi)\), 0, and \(48 \lambda^{2}h^{2}\epsilon\sin^{2}\phi\), respectively.
5 Application of the proposed difference methods to a linear singular equation
6 Computational results
In order to test the accuracy of the proposed methods, we have solved a large variety of linear and nonlinear fourth-order parabolic problems. In each case, the exact solution is prescribed and the right-hand side functions, the initial and boundary conditions, are obtained using the exact solution as a test procedure. We have chosen \(\theta=0.5\) in each case for computing the solution of PDE (1), and all the computations were performed using MATLAB. The matrices represented by the new formulas are block tridiagonal. The Gauss-Seidel iteration method has been used for solving linear coupled system of equations, whereas the Newton nonlinear iteration method has been applied to determine the solution of nonlinear equations (see [38, 39]), and in each case, the iterations are terminated once the absolute error tolerance 10^{−12} is reached.
Example 1
Methods | Time | N + 1 | k | x = 0.1 | x = 0.2 | x = 0.3 | x = 0.4 | x = 0.5 | |
---|---|---|---|---|---|---|---|---|---|
0.02 | 20 | 0.00125 | u | 1.53 (−08) | 2.91 (−08) | 4.00 (−08) | 4.70 (−08) | 4.95 (−08) | |
\(u_{xx}\) | 1.52 (−07) | 2.90 (−07) | 3.99 (−07) | 4.69 (−07) | 4.93 (−07) | ||||
0.02 | 40 | 0.00125 | u | 5.07 (−09) | 9.64 (−09) | 1.33 (−08) | 1.56 (−08) | 1.64 (−08) | |
\(u_{xx}\) | 5.01 (−08) | 9.53 (−08) | 1.31 (−07) | 1.54 (−07) | 1.62 (−07) | ||||
0.05 | 20 | 0.005 | u | 4.79 (−07) | 9.11 (−07) | 1.25 (−06) | 1.47 (−06) | 1.55 (−06) | |
\(u_{xx}\) | 1.08 (−05) | 2.05 (−05) | 2.82 (−05) | 3.31 (−05) | 3.48 (−05) | ||||
0.05 | 40 | 0.005 | u | 4.19 (−07) | 7.96 (−07) | 1.10 (−06) | 1.29 (−06) | 1.35 (−06) | |
\(u_{xx}\) | 3.16 (−06) | 6.02 (−06) | 8.28 (−06) | 9.74 (−06) | 1.02 (−05) | ||||
Mohammadi [35] | 0.02 | 20 | 0.00125 | u | 4.29 (−07) | 2.51 (−07) | 1.24 (−07) | 1.38 (−07) | 1.40 (−07) |
0.02 | 40 | 0.00125 | u | 8.54 (−08) | 6.23 (−08) | 4.91 (−08) | 5.07 (−08) | 5.12 (−08) | |
0.05 | 20 | 0.005 | u | 2.96 (−06) | 1.77 (−06) | 1.64 (−06) | 2.28 (−06) | 2.65 (−07) | |
0.05 | 40 | 0.005 | u | 9.07 (−07) | 7.84 (−07) | 7.69 (−07) | 8.27 (−07) | 8.61 (−08) | |
Mittal and Jain [34] | 0.02 | 181 | 0.005 | u | 1.50 (−07) | 2.90 (−07) | 3.90 (−07) | 4.60 (−07) | 4.90 (−07) |
0.05 | 181 | 0.005 | u | 1.10 (−06) | 2.09 (−06) | 2.88 (−06) | 3.38 (−06) | 3.56 (−06) | |
Rashidinia and Mohammadi [33] | 0.02 | 20 | 0.00125 | u | 4.47 (−07) | 2.66 (−07) | 1.39 (−07) | 1.55 (−07) | 1.57 (−07) |
0.05 | 20 | 0.005 | u | 2.91 (−06) | 1.73 (−06) | 1.60 (−06) | 2.23 (−06) | 2.60 (−07) | |
Caglar and Caglar [32] | 0.02 | 121 | 0.005 | u | 4.80 (−06) | 9.70 (−06) | 1.40 (−05) | 1.90 (−05) | 2.40 (−05) |
0.02 | 191 | 0.005 | u | 5.20 (−06) | 2.10 (−06) | 3.10 (−06) | 4.20 (−06) | 5.20 (−06) | |
0.02 | 521 | 0.005 | u | 4.90 (−07) | 9.90 (−07) | 1.40 (−06) | 1.90 (−06) | 2.40 (−06) |
Example 2
Methods | λ | Time steps | x = 0.1 | x = 0.2 | x = 0.3 | x = 0.4 | x = 0.5 | |
---|---|---|---|---|---|---|---|---|
Proposed \(O(k^{2}+h^{4})\)- | 0.5 | 16 | u | 2.09 (−11) | 2.63 (−11) | 1.23 (−10) | 1.93 (−10) | 2.85 (−10) |
\(u_{xx}\) | 5.06 (−09) | 1.54 (−08) | 6.00 (−08) | 3.40 (−08) | 3.87 (−08) | |||
\(O(k^{2}+h^{4})\)-method in [33] | 0.5 | 16 | u | 7.46 (−10) | 2.91 (−10) | 8.65 (−10) | 6.87 (−10) | 6.98 (−10) |
\(O(k^{4}+h^{4})\)-method in [33] | 0.5 | 16 | u | 6.25 (−10) | 2.22 (−10) | 4.53 (−10) | 4.41 (−10) | 5.03 (−10) |
Example 3
The Maximum absolute relative errors for Example 3 at \(\pmb{t=0.01}\) for various values of λ (uniform mesh)
λ | Proposed \(\boldsymbol {O(k^{2}+ h^{4})}\) -method ( 28a )-( 28b ) | \(\boldsymbol {O(k^{2}+ h^{4})}\) -method discussed in [ 33 ] | \(\boldsymbol {O(k^{4}+ h^{4})}\) -method discussed in [ 33 ] | Method discussed in [ 29 ] | Method discussed in [ 28 ] | |
---|---|---|---|---|---|---|
0.05 | u | 6.0176 (−12) | 5.21 (−08) | 5.33 (−08) | 9.90 (−08) | 1.90 (−06) |
\(u_{xx}\) | 2.4614 (−12) | |||||
0.1 | u | 4.4223 (−12) | 1.03 (−07) | 9.97 (−08) | 8.10 (−08) | 7.20 (−07) |
\(u_{xx}\) | 3.1911 (−12) | |||||
0.25 | u | 5.2459 (−12) | 3.74 (−08) | 3.51 (−08) | 6.90 (−08) | 4.10 (−07) |
\(u_{xx}\) | 4.9774 (−12) |
Example 4
The MAEs for Example 4 at \(\pmb{t=1.0}\) for a fixed \(\pmb{\lambda =(k/h^{2})=1.6}\) (uniform mesh)
h | |||||||
---|---|---|---|---|---|---|---|
ϵ = 0.1 | ϵ = 0.01 | ϵ = 0.001 | ϵ = 0.1 | ϵ = 0.01 | ϵ = 0.001 | ||
1/8 | u | 3.0988 (−04) | 2.9825 (−05) | 3.5010 (−06) | 3.2430 (−02) | 1.0781 (−02) | 1.1932 (−03) |
\(u_{xx}\) | 2.4409 (−03) | 3.8680 (−04) | 4.4212 (−05) | 3.9514 (−01) | 1.1690 (−01) | 1.2865 (−02) | |
1/16 | u | 1.9387 (−05) | 1.8905 (−06) | 2.2211 (−07) | 8.0703 (−03) | 2.7648 (−03) | 3.0660 (−04) |
\(u_{xx}\) | 1.5294 (−04) | 2.4412 (−05) | 2.7933 (−06) | 9.9261 (−02) | 3.0176 (−02) | 3.3273 (−03) | |
1/32 | u | 1.2121 (−06) | 1.1856 (−07) | 1.3934 (−08) | 2.0148 (−03) | 6.9554 (−04) | 7.7174 (−05) |
\(u_{xx}\) | 9.5655 (−06) | 1.5293 (−06) | 1.7506 (−07) | 2.4840 (−02) | 7.6039 (−03) | 8.3890 (−04) |
Example 5
The MAEs for Example 5 at \(\pmb{t=1.0, \eta=0.94}\) (quasi-variable mesh)
N + 1 | \(\boldsymbol {O(k^{2}+k^{2}h_{l}+ h_{l}^{3})}\) -method ( 16a )-( 16b ) | \(\boldsymbol {O(k^{2}+ h_{l}^{2})}\) -method ( 15a )-( 15b ) | |||
---|---|---|---|---|---|
α = 1 | α = 2 | α = 1 | α = 2 | ||
8 | u | 1.8743 (−04) | 5.4079 (−04) | 3.6061 (−03) | 8.7376 (−03) |
\(u_{rr}\) | 1.4353 (−03) | 8.2498 (−03) | 8.6067 (−02) | 7.9741 (−02) | |
16 | u | 1.7017 (−05) | 4.5019 (−05) | 6.8621 (−04) | 1.6912 (−03) |
\(u_{rr}\) | 1.5498 (−04) | 1.3071 (−03) | 2.6282 (−02) | 4.7895 (−02) | |
32 | u | 3.1833 (−06) | 7.4946 (−06) | 1.2889 (−04) | 3.5410 (−04) |
\(u_{rr}\) | 5.1980 (−05) | 4.3430 (−04) | 1.1033 (−02) | 3.0090 (−02) |
Example 6
Example 7
t | Parameters | Collocation method discussed in [ 36 ] | |||
---|---|---|---|---|---|
u | \(\boldsymbol {u_{xx}}\) | u | |||
0.5 | h = 1/40 | \(x_{0}=30\) | 8.1007 (−07) | 4.4585 (−06) | 8.2943 (−07) |
1.0 | h = 1/60 | \(x_{0}=40\) | 7.6030 (−09) | 3.1598 (−08) | 7.3326 (−09) |
1.5 | h = 1/80 | \(x_{0}=50\) | 5.8974 (−11) | 2.0970 (−09) | 6.4525 (−11) |
2.0 | h = 1/100 | \(x_{0}=60\) | 2.9068 (−13) | 3.2836 (−13) | 5.2066 (−13) |
Example 8
The MAEs for Example 8 at \(\pmb{t=1.0, \eta=0.92}\) (quasi-variable mesh)
N + 1 | \(\boldsymbol {O(k^{2}+ k^{2}h_{l}+h_{l}^{3})}\) -method ( 16a )-( 16b ) | \(\boldsymbol {O(k^{2}+ h_{l}^{2})}\) -method ( 15a )-( 15b ) | |||||
---|---|---|---|---|---|---|---|
α = 10 | α = 20 | α = 40 | α = 10 | α = 20 | α = 40 | ||
8 | u | 3.3667 (−05) | 7.7683 (−05) | 1.7212 (−02) | 2.6072 (−04) | 3.0874 (−04) | 2.6133 (−02) |
\(u_{xx}\) | 3.3298 (−04) | 7.6915 (−04) | 1.7290 (−01) | 2.3571 (−04) | 7.0446 (−04) | 2.5645 (−01) | |
16 | u | 2.1139 (−06) | 4.7259 (−06) | 1.0805 (−03) | 8.6976 (−05) | 1.0909 (−04) | 9.5763 (−03) |
\(u_{xx}\) | 2.1334 (−05) | 4.7049 (−05) | 1.0817 (−02) | 1.2171 (−04) | 3.4424 (−04) | 9.4655 (−02) | |
32 | u | 1.3319 (−07) | 2.7901 (−07) | 7.8958 (−05) | 4.1496 (−05) | 5.1293 (−05) | 3.9491 (−03) |
\(u_{xx}\) | 1.5345 (−06) | 2.9686 (−06) | 7.9178 (−04) | 5.4398 (−05) | 1.5469 (−04) | 3.9160 (−02) |
Example 9
The MAEs for Example 9 at \(\pmb{t=4.0, \eta=0.92}\) (quasi-variable mesh)
N + 1 | \(\boldsymbol {O(k^{2}+ k h_{l}+h_{l}^{3})}\) -method ( 37a )-( 37b ) | \(\boldsymbol {O(k^{2}+ h_{l}^{2})}\) -method ( 36a )-( 36b ) | |||||
---|---|---|---|---|---|---|---|
α = 1 | α = 5 | α = 10 | α = 1 | α = 5 | α = 10 | ||
8 | u | 2.5627 (−05) | 3.5373 (−05) | 7.1052 (−05) | 1.1611 (−04) | 1.1059 (−04) | 8.9291 (−05) |
\(u_{xx}-u_{t}\) | 2.8854 (−05) | 8.8904 (−05) | 4.3557 (−04) | 1.6636 (−05) | 4.6297 (−05) | 2.4532 (−04) | |
16 | u | 1.6009 (−06) | 2.2252 (−06) | 4.5252 (−06) | 4.3184 (−05) | 4.3903 (−05) | 4.6318 (−05) |
\(u_{xx}-u_{t}\) | 1.8900 (−06) | 5.7189 (−06) | 2.8222 (−05) | 2.0745 (−06) | 8.1349 (−06) | 3.5144 (−05) | |
32 | u | 9.1931 (−08) | 1.3617 (−07) | 2.9842 (−07) | 2.1699 (−05) | 2.2259 (−05) | 2.4254 (−05) |
\(u_{xx}-u_{t}\) | 1.4804 (−07) | 4.2072 (−07) | 2.0094 (−06) | 1.7714 (−06) | 6.1484 (−06) | 2.7953 (−05) |
Example 10
The MAEs for Example 10 at \(\pmb{t=1.0}\) for a fixed \(\pmb{\lambda =(k/h^{2})=1.6}\) (uniform mesh)
h | |||||||
---|---|---|---|---|---|---|---|
α = 1 | α = 10 | α = 20 | α = 1 | α = 10 | α = 20 | ||
1/8 | u | 1.0745 (−05) | 1.3706 (−05) | 1.9813 (−05) | 9.7462 (−03) | 1.2495 (−02) | 1.7658 (−02) |
\(u_{xx}-u_{t}\) | 2.4924 (−04) | 2.2351 (−04) | 1.7245 (−04) | 3.9007 (−02) | 6.3033 (−02) | 1.0975 (−01) | |
1/16 | u | 6.6313 (−07) | 8.4307 (−07) | 1.2147 (−06) | 2.4146 (−03) | 3.0834 (−03) | 4.3337 (−03) |
\(u_{xx}-u_{t}\) | 1.5550 (−05) | 1.3987 (−05) | 1.1042 (−05) | 9.7392 (−03) | 1.5641 (−02) | 2.7121 (−02) | |
1/32 | u | 4.1496 (−08) | 5.2605 (−08) | 7.5414 (−08) | 6.0229 (−04) | 7.6836 (−04) | 1.0785 (−03) |
\(u_{xx}-u_{t}\) | 9.6855 (−07) | 8.7423 (−07) | 6.8805 (−07) | 2.4340 (−03) | 3.9032 (−03) | 6.7769 (−03) |
7 Conclusions
In this paper, we propose finite difference approximations for the fourth-order time-dependent parabolic PDEs (1) and (6). The methods were tested on several examples taken from the literature to observe the accuracy and efficiency of the new methods. The results illustrate that the errors in the numerical solution obtained by the current approach are smaller than those obtained by earlier research studies. The main conclusions are:
(i) High-order accuracy: In the case of the uniform mesh, for a fixed value of the mesh ratio parameter \(\lambda=\frac {k}{h^{2}}\), the proposed three-level method (28a)-(28b) and two-level method (49a)-(49b) are fourth-order accurate in space. The numerical results for Examples 1, 2, and 3 indicate that the methods produce better results in comparison to the existing methods [28, 29, 32–35] for the Euler-Bernoulli beam equation. Also, it is seen from Table 6 that the proposed algorithm performs significantly better than the scheme in [7] for the second-order Benjamin-Ono equation and is in good agreement with [36] for the nonlinear good Boussinesq equation.
(ii) Compact stencil: The finite difference methods discussed here are based only on three spatial grid points. In each time step, every iteration involves solving a tridiagonal system.
(iii) No Ghost points: The boundary conditions are incorporated in a natural way without the use of any extra nodes or special schemes adjacent to the boundary, thereby eliminating the usual complexity encountered with the difference methods.
(iv) Directly applicable to singular problems: The existing fourth-order implicit difference method of [37] for solving the fourth-order quasi-linear parabolic equation (1) is not directly applicable to problems in polar coordinates and requires a special technique to handle singular points because of the presence of the terms of the form \(1/r_{l-1}\), which give rise to singularity at \(l=1\) as \(r_{0}=0\). In the present paper, by using off-step nodal points the singularity at \(r=0\) is avoided, which enables a direct application of the proposed stable methods for finding the numerical solution of fourth-order parabolic equations with singular coefficients.
(v) Unconditional stability of the two-level method: The two-level implicit methods for the particular type of the fourth-order parabolic PDE (6) are unconditionally stable. Thus, the time step can be considerably large, which is extremely useful when the problem is solved on a long time interval. In Example 6, the maximum absolute errors has been calculated at large time levels \(t=5,10,15,20\), and in Example 9, the errors are computed at \(t=4\). The accuracy of the schemes is not degraded at large time intervals.
Also, the numerical solution of \(u_{xx}\), in case of solution of (1) and the one-dimensional time-dependent Laplacian \(u_{xx}-u_{t}\) and in case of solution of (6), which are quite often of interest in various applied problems, are computed as a byproduct of the proposed methods. We are currently working on extension of these methods to solve 2D and 3D fourth-order nonlinear parabolic PDEs. Application of these new methods to some more physical problems in science and engineering will be the content of our further research.
Declarations
Acknowledgements
The authors thank the reviewers for their valuable comments and suggestions, which substantially improved the standard of the 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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