بهبود جواب معادله غیرخطی تعمیم یافته بنجامین بااستفاده از یک روش بدون شبکه توابع پایه ای شعاعی
Subject Areas : Numerical Analysis
مهران
نعمتی
1
(گروه ریاضی ، واحد رودبار، دانشگاه آزاد اسلامی، رودبار، ایران)
سیده فائزه
تیموری
2
(گروه کامپیوتر، واحد رودبار، دانشگاه آزاد اسلامی، رودبار، ایران)
Keywords: تفاضل متناهی, توابع پایه ای شعاعی, روش کرانک نیکلسون, معادله غیر خطی تعمیم یافته بنجامین,
Abstract :
در این مقاله براساس روش توابع پایه ای شعاعی ، معادله دیفرانسیل غیرخطی تعمیم یافته بنجامین حل شده است.برای گسسته سازی قسمت زمانی از تفاضل متناهی و کرانک نیکلسون استفاده کردیم. و قسمت فضایی با استفاده از درونیابی توابع پایه ای شعاعی تقریب زده شده است. در نتیجه یک دستگاه معادلات جبری خطی حاصل می شود که باحل این دستگاه جواب های تقریبی بدست می ایند.در ادامه باحل مثال عددی نشان داده می شود که روش پیشنهادی کارا هست واینکه خطا بهبود یافته است.
Improved solution to nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation by a meshless RBFs method
Abstract
In this paper, based on the RBF collocation method and finite differences, a numerical method is proposed to solve nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. First order finite differences and Crank-Nicolson method are applied to discretize the temporal parts. The spatial parts are approximated by MQ-RBF interpolation which results in a linear system of algebraic equations. Approximate solutions are determined by solving such a system. The proposed scheme is verified by solving some test problems and computing error norms 
and 
. Results show the efficiency of the suggested method and the error has been improved.
Key words: Radial basis functions (RBFs), Finite differences, Crank-Nicolson scheme, Nonlinear Generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation.
1. Introduction
Most phenomena in applied sciences and engineering are modeled by nonlinear partial differential equations (PDEs). Since their exact solutions cannot be found simply, studying those types of equations is often hard. Hence, numerical methods for approximating nonlinear PDEs have been regarded by many researchers and they have been successfully applied to a lot of real-world problems (e.g., [1-5]). One well-known class of numerical methods is meshless methods. These schemes have attracted great attention in recent decades and they have been applied as powerful tools especially for problems in computational mechanics. The advantage of meshless methods over the traditional numerical techniques including finite difference method (FDM), finite element method (FEM), and finite volume method (FVM), is that they do not require mesh generation and domain or surface discretization.
There exist several types of meshless methods including moving least square meshless method [6], meshless local Petrov- Galerkin method (MLPG) [7], smooth-particle hydrodynamics [8], the reproduced kernel particle method (RKPM) [9], the finite point method [10], mesh-free weak-strong form (MWS) [11], the diffuse element method (DEM) [12], and radial basis functions (RBFs) method. Each of these approaches has specific advantages for certain classes of problems. Among them, RBFs method is the simplest and the most efficient one.
RBFs method was introduced by Ronald Hardy, an Iowa State geodesist, for the first time in a paper appearing in 1971 [13]. He proposed this method for efficient interpolation of the scattered data on a topographic surface. A set of N distinct points called centers is used in RBFs interpolation. There are no constraints on the geometry of domains of the problem or on the position of the centers [14]. There are different types of RBFs such as Multiquadric (MQ), Thin Plate Spline (TPS), Gaussian, Linear, Inverse quadric (IQ), Inverse Multiquadric (IMQ). Hardy used the MQ function for his interpolation scheme. Later, Duchon proposed TPS for data interpolation in 1975 [15]. The use of radial basis functions was limited to scattered data interpolation until 1990, a physicist named Edward Kansa used RBFs to solve PDEs for the first time [16,17]. His method leads to an ill-conditioned matrix for a large number of nodes because of having asymmetrical nature of the interpolation matrix. Fasshauer modified the Kansa's method in 1996 [18] by proposing a Hermite based approach in which the collocation matrices are symmetric in nature, and their condition number is smaller. In the continuation, many efforts have done by numerous researchers to improve the method and also propose new kinds of it (e.g., [19-24]).
During recent years, RBFs method has been considered as an efficient tool for solving different kinds of problems including PDEs [25-28], integral equations [29,30], and fractional equations [31,32]. The current work is devoted to the numerical study of the following two-dimensional nonlinear PDE by the RBFs method.
(1) 
with following initial and boundary conditions.
(2) 
where 
, 
is the vector valued function, 
, 
, 
are gradient and Laplacian operators respectively. This equation is known as the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation.
The GBBMB equation is used in different scientific fields such as the analysis of the surface waves of long wavelength in liquids, hydromagnetic waves in a cold plasma, acoustic gravity waves incompressible fluids, and acoustic waves in a harmonic crystal. The GBBMB equation and its modified or generalized form have been investigated by many authors both numerically and analytically. For further information refer to [33-44]. The nonlinear GBBMB has been solved using different analytical and numerical methods that in the following, some of them are reported. Tari and Ganji [45] approximated solutions of nonlinear GBBMB equation by He’s methods analytically. In [46], an exponential function method was presented to solve especial type of nonlinear GBBMB equation. Dehghan et al. studied nonlinear GBBMB by an RBF meshless method in [47]. Haq et al. proposed a numerical method based on Haar wavelets and finite differences to solve nonlinear GBBMB in [54]. Hajiketabi et al. introduced a new numerical method for solving the nonlinear GBBMB equation in [55]. Recently, Ali Ebrahimijahan and Mehdi Dehghan [48] have proposed a numerical technique based on the integrated radial basis functions (IRBFs) for solving the nonlinear GBBMB and regularized long-wave equations. The present study aims to apply a numerical method based on the RBF collocation method and finite differences to solve the nonlinear GBBMB equation. Finite differences and Crank-Nicolson scheme are employed for temporal part discretization, while two-dimensional RBF interpolation is implemented for approximating the spatial parts. The MQ-RBF is the focus of this paper because of its popularity in applications and its good approximation characteristic.
The manuscript is organized as following. In section 2, the RBF method is explained briefly by presenting basic concepts and definitions. In Section 3, RBF collocation method is implemented for the time discretized nonlinear GBBMB equation. Then, the proposed method is applied for some test problems, and consequently the results are reported in Section 4. Finally, a conclusion is presented in Section 5.
2. A brief review of RBFs method
In this section, some basic concepts and definitions are expressed for the radial basis functions interpolation.
Definition 1. Let 
be d-dimensional Euclidean space and 
. A radial basis function is a function which is both continuous and multivariable like 
that its value at any point 
, is dependent on the distance from a certain point . This function could be written as 
where 
and 
is the Euclidean norm on . The function 
is an univariable function in r and 
is a center of RBF .
Definition 2. Given the data 
, with 
, 
, and 
, the scattered data interpolation problem is defined as finding a smooth function s such that 
, for 
Function s is called an interpolant.
A radial basis function interpolant u at centers 
assumes the following form.
(3) 
where 
, is a radial basis function, coefficients 
are constants to be determined such that the following interpolation condition at the set of N centers, 
is hold.
(4) 
Imposing the interpolation condition (4) to (3) leads to a linear system as follows.
(5) 
where 
, 
, and B is a 
matrix called the interpolation matrix or the system matrix defined as follows.
where 
According to Definition 1, RBF is independent of the spatial dimension. This property makes it possible to transform a multivariable problem into a one-variable problem easily. This is the main advantage of the RBF interpolation scheme over the other classical methods.
Generally, RBFs are sorted into two major categories: Infinitely smooth and piecewise smooth. Some well-known RBFs are listed in Table 1. Infinitely smooth RBFs include a free parameter which is called shape parameter, denoted by 
. Although this parameter can be chosen arbitrarily, a proper choice of its value is necessary. Because in an infinitely smooth RBF interpolation, the value of a shape parameter influences the accuracy of the scheme [49]. Piecewise smooth RBFs have algebraic convergence rates while infinitely smooth RBFs achieve spectral or exponential convergence rates [50, 51].
Table 1. Some well-known RBFs
Category | Name of the function | Definition |
| Multiquadric (MQ) |
|
| Inverse Multiquadric (IMQ) |
|
Infinitely smooth RBFs | Inverse Quadric (IQ) |
|
| Gaussian |
|
Piecewise smooth RBFs | Linear | r |
Cubic |
| |
Thin Plate Spline (TPS) |
|
3. Time discretization of nonlinear GBBM equation
In this section, Let 
be an arbitrary interval in 
. by applying forward finite differences and also Crank-Nicolson scheme, the time variable is discretized for the first-order time derivative as follows.
(6) 
Now, choose 
. Then, 
Approximated non-linear terms by the following formulas
(7) 
yields to
This equation can be simplified as follows.
(8) 
Discretizing Eq. (8) in space by RBF expansion (3) results in
Considering N collocation points 
in leads to the following linear system.
Where 
, 
, 
, 
and 
. 
and 
are matrices of second derivative of the system matrix, B, respectively in 
, and 
. 
and 
are matrices of first derivative of the system matrix, B, respectively in , and .
Let
and
By the assumption 
is non-singular, 
is obtained as follows.
Recalling that 
, the approximate PDE solution at 
is obtained as follows.
(9) 
Where 
and 
.
4. Numerical experiments
In this section, the following some test problems are numerically solved for the purpose of verifying the ability of the proposed method with regards to the nonlinear GBBMB equations. Among all of the RBFs, MQ, the most popular RBF, is used in computations due to the rapid convergent rate. Here, following MQ radial basis function is used.
Where is the shape parameter.
Meshless methods which are based on radial basis functions (RBFs) contain a free shape parameter that plays an important role for the accuracy and condition number of the coefficient matrix of the method. Most authors use the trial and error method for obtaining a good shape parameter that results in best accuracy. Here, the shape parameter is chosen by trial and error method.
The domain 
is chosen as the unit region, i.e. 
. In order to test the accuracy, two error norms, 
and 
defined as follows are computed.
where 
and 
denote the approximate and exact solutions, respectively.
4.1. Test Problem 1
Consider the nonlinear GBBMB equation as follows.
with conditions
and the source term
The exact solution of the problem is 
.
Numerical solutions are calculated at various values of time variable 
with 
and time step 
and shape parameter= 1.4. Consequently, the results are provided in Table 2, showing that the proposed method is accurate sufficiently. The numerical results at 
, 
and different time steps 
and the numerical results presented in [54], are all provided in Table 3. Values of shape parameter are derived by trial and error method. Comparison the results show that better approximations are obtained by the proposed scheme.
Approximate and exact solutions and also absolute error are illustrated in Fig.1,. According to this figure, it can be seen that approximate solutions are very close to the exact ones.
Table 2. Error norms of test problem 1 at 
and 
T |
|
|
0.1 | 6.7949e-06 | 3.4022e-05 |
0.2 | 1.2265e-05 | 6.1129e-05 |
0.5 | 2.1897e-05 | 1.0869e-04 |
1 | 2.9094e-05 | 1.2306e-04 |
Fig 1. Graphs of approximate and exact solutions and absolute error of test problem 1 at 
and 
Table 3. Error norms of test problem 1 at 
|
|
|
|
| CPU time | Shape parameter |
1/10 | 1.9000e-3 | 3.4000e-3 | 4.1369e - 03 | 1.0243e - 02 | 0.1550 | 1.5 |
1/20 | 9.9446e-04 | 1.8000e-3 | 2.4115e - 03 | 5.8698e - 03 | 0.1718 | 1.1 |
1/40 | 5.7353e-04 | 9.8716e-04 | 1.5207e - 03 | 3.6346e - 03 | 0.1747 | 0.8 |
1/80 | 3.1700e-04 | 5.5014e-04 | 1.0682e - 03 | 2.5188 - 03 | 0.1958 | 0.6 |
1/160 | 1.2022e-04 | 1.7277e-04 | 8.4022e - 04 | 1.9708 - 03 | 0.2301 | 0.6 |
1/320 | 5.0058e-05 | 8.2924e-05 | 7.2574e - 04 | 1.7034 - 03 | 0.3083 | 0.5 |
1/640 | 2.0853e-05 | 3.9993e-05 | 6.7980e - 04 | 1.5724 - 03 | 0.47338 | 0.4 |
1/1280 | 3.2936e-05 | 5.0085e-05 | 6.6576e - 04 | 1.5079 - 03 | 0.7473 | 0.3 |
1/2560 | 2.4328e-05 | 4.8605e-05 | 6.5874e - 04 | 1.4759 - 03 | 1.3567 | 0.3 |
4.2. Test Problem 2
Consider the nonlinear GBBMB equation as follows.
With initial and boundary conditions
and the source term
with the exact solution 
.
In table 4, Numerical solutions are calculated at various values of time variable 
with and time step and shape parameter= 1.4. The results are provided in Table 2 show that the proposed method is accurate sufficiently.
Table 4. Error norms of test problem 2 at and
T |
|
|
0.1 | 3.7096e-06 | 1.5701e-05 |
0.2 | 7.4830e-05 | 2.8961e-05 |
0.5 | 1.9129e-05 | 6.5406e-05 |
1 | 3.9390e-05 | 1.5772e-04 |
5. Conclusion
In this study, an PDE called nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation was studied numerically. The finite difference formula and Crank Nicolson technique were implemented to discretized the temporal parts. As a result, a time semi-discrete formula was obtained. After that, a fully discrete formula was achieved by approximating the spatial terms using RBF interpolation. Numerical results show that the suggested method has better accuracy and the error has been improved.
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