MatLab solution: Absolute error values obtained when solving Example 1 using
Abstract
In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasilinearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger’sFisher equation, the FitzhughNagumo equation and the Burger’sHuxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.
Keywords
 bivariate interpolation
 spectral collocation
 quasilinearisation
 multidomain approach
 nonlinear evolution PDEs
1. Introduction
Most practical problems which model systems in nature lead to nonlinear partial differential equations (PDEs). This is evident in the fields of chemistry, physics, biology, mathematics and engineering. Many assumptions have been made to make some nonlinear PDEs solvable. It has been reported that a vast number of nonlinear PDEs that are encountered in these fields are difficult to solve analytically [1]. The investigation of solutions of such nonlinear PDEs has then been of key interest to many researchers due to their potential applications and more effort has been devoted to search for better and more efficient solution methods for these nonlinear models [2, 3].
The nonlinear PDEs that are solved in this study include the generalized Burger’sFisher equation, the generalized Burger’sHuxley equation and the FitzhughNagumo equation. The generalized Burger’sFisher equation appears in many applications such as shock wave formation, fluid mechanics, turbulence, traffic flows, gas dynamics, heat conduction and sound waves via viscous medium among other fields of applied science [4–6]. The generalized Burger’sHuxley equation models the interaction between reaction mechanisms, diffusion transports and convection effects [7–11]. The FitzhughNagumo equation arises in genetics, biology, and heat and mass transfer [12, 13].
A number of methods have been applied to solve the nonlinear PDEs such as spectral collocation method [7, 8], Adomian decomposition method [9], homotopy perturbation method [14] and the variational iteration method [4]. The spectral methods have been reported to be strikingly successful if the problem has a smooth solution and falls into various categories, namely Galerkin, Tau and collocationbased methods [15], and therefore, recent advances in the development of numerical methods for solving nonlinear PDEs has focused spectralbased approaches as they require a few grid points to give very accurate results and take less computation time. The spectral collocationbased methods are used often, chiefly because they offer the simplest treatment of boundary conditions. A newly developed spectral collocation method for solving nonlinear PDEs is the bivariate spectral quasilinearization method (QLM) [16]. This method approximates the solution of the PDE using a bivariate Lagrange interpolation polynomial [17]. It applies quasilinearization method of Bellman and Kalaba [18] to simplify the nonlinear PDE which is then discretized using spectral collocation on both time and space variables. The method has successfully been used to solve problems defined over shorter time intervals [16]. However, it has been observed that when this method is applied to solve problems defined over largetime intervals, there is no guarantee that the resulting approximate solution will be accurate [16].
In this study, we describe the multidomain bivariate spectral collocation method (MDBSCM) to solutions of nonlinear parabolic PDEs defined over largetime intervals. The MDBSCM is based on decomposing the given domain of approximation in the time variable into smaller subintervals and then solving the PDE independently in each subinterval using the bivariate spectral collocation method. The multidomain approach has been applied to solve nonlinear ordinary differential equations that model chaotic systems described as 1st order systems of equations [19–21]. In this study the same idea is extended to solutions of nonlinear parabolic PDEs. In the description of the method, the algorithm is kept as simple as possible, while retaining the heart of generality to cover many applications. The extent of the discussion of multidomain approach in this study is limited to nonoverlapping subintervals only.
2. Method of solution
In this section, we describe the algorithm to describe how the multidomain bivariate spectral collocation method can be applied to solve nonlinear parabolic PDEs. We shall consider a general secondorder nonlinear PDE,
subject to boundary conditions
and initial condition
where
2.1. The quasilinearization method
The quasilinearization method (QLM) of Bellman and Kalaba [18] is a technique that is used to simplify nonlinear ordinary and partial differential equations. The technique has been adopted and generalized in further studies presented in [22, 23]. The QLM is based on the NewtonRaphson method and is constructed from the linear terms of Taylor series expansion about an initial approximation to solution. The QLM assumes that the difference between solutions at two successive iterations denoted by
where prime denotes differentiation with respect to
where
where
The dot here denotes differentiation with respect to the time
Let
The domain
before the spectral collocation is applied. Similarly, the spatial domain
The collocation nodes are the symmetrically distributed GaussLobatto grid points defined on the interval [−1, 1] by,
To distinguish between the solutions at different subdomains we shall use,
subject to boundary and initial conditions
After the solution in the first interval
subject to boundary and initial conditions
In the solution process, the approximate solution that is searched for takes a form of a bivariate Lagrange interpolation polynomial. The solution at each subinterval is approximated as
The first and second spatial derivatives are evaluated at the collocation nodes (
where
where
and
By changing the indices, Eq. (20) can be written as
where
Eq. (21) constitutes an
where
and
These boundary conditions are imposed on the main diagonal submatrices of the matrix system (23) to obtain a new system which takes the form
where
The matrix system (26) is solved for
which denotes the solution at the boundaries of the subintervals.
3. Numerical experimentation
In this section, we illustrate the practical applicability of the multidomain approach in solving nonlinear parabolic PDEs by considering the solutions of wellknown nonlinear PDEs that have been reported in the literature.
subject to boundary conditions
and initial condition
The exact solution is given in [24] as
Eq. (29) is an example of a generalized Burger’sFisher equation that was solved in [4] using variational iteration method. Applying the QLM, we obtain the linearized system
where
In each subinterval
The matrices resulting from application of the spectral collocation in (33) are
The initial condition at different subintervals is given by
The boundary conditions at the collocation points are
Making the relevant substitution, a matrix system similar to (26) is solved to obtain the approximate solution.
subject to boundary conditions
and initial condition
The exact solution is given in [12]
Eq. (41) is an example of a generalized FitzhughNagumo equation [12, 13]. Applying the QLM, we obtain a linearized system similar to that given in Eq. (33). The coefficients in this example are given by:
In each subinterval
The application of the spectral collocation in (33) results into the following set of coefficient matrices:
The initial condition at different subintervals is given by:
The boundary conditions at the collocation points are given by:
subject to boundary conditions
and initial condition
The exact solution is given in [25] as
Eq. (52) is an example of a generalized Burger’sHuxley equation [25]. Applying the QLM, we obtain a linearized system similar to that given in Eq. (33). The coefficients in this example are given by
In each subinterval
The application of the spectral collocation in (33) results into the following set of coefficient matrices
The initial condition at different subintervals is given by
The boundary conditions at the collocation points are given by
4. Results and discussion
In this section, we present the results for the absolute error values at selected values of
where
2.0  4.0  6.0  8.0  

0.4775  4.91994e007  3.31285e007  1.99894e008  6.67958e008 
1.3650  4.36608e007  7.17567e007  1.76673e008  1.70637e007 
2.5000  3.26724e006  1.43695e006  1.42849e007  2.19591e007 
3.6350  2.35521e007  2.54426e006  1.48557e007  6.72760e007 
4.5225  7.14695e006  6.81188e006  2.08164e006  2.15409e007 
CPU time (sec)  0.103606 




2.0  4.0  6.0  8.0 
0.4775  4.77396e014  7.99361e015  1.16573e014  2.66454e014 
1.3650  7.41074e013  1.29896e014  1.13243e014  2.81997e014 
2.5000  3.31513e013  3.83027e014  1.25455e014  2.44249e015 
3.6350  3.37175e012  7.40519e014  8.88178e015  4.21885e015 
4.5225  2.55729e012  1.52323e013  2.68674e014  2.10942e014 
CPU time (sec)  0.019602 
The results obtained from approximating the solution of (41) are given in Tables 3 and 4. Table 3 shows the results generated when the bivariate spectral collocation method (single domain) is used whereas Table 4 presents the results obtained when using the multidomain bivariate spectral collocation method. The results are similar to those of Example 1, thus the multidomain approach is more efficient than singledomain approach when it is applied in solving nonlinear parabolic PDEs defined over largetime domain.

0.2  0.4  0.6  0.8 

1.0039  1.35033e006  1.14349e005  1.42724e005  2.50691e006 
1.9283  8.19001e008  1.97335e008  4.17650e008  4.41645e009 
3.0000  6.03945e009  1.22066e008  9.50467e009  8.55427e009 
4.0717  8.95142e011  1.11732e009  2.53453e009  3.16956e009 
4.9372  2.23332e012  2.90505e011  1.05300e010  1.94311e010 
CPU time (sec)  0.135906 




0.2  0.4  0.6  0.8 
1.0039  7.57705e012  3.20854e012  6.38556e012  2.23954e011 
1.9283  8.88178e015  2.10942e014  3.37508e014  2.22045e015 
3.0000  1.55431e015  7.32747e015  1.77636e014  1.95399e014 
4.0717  6.83897e014  5.66214e014  8.79297e014  3.50830e014 
4.9372  1.11511e012  2.31637e012  7.83817e014  4.10783e014 
CPU time (sec)  0.026619 
The results obtained from approximating the solution of (52) are given below. Table 5 shows the results generated when the bivariate spectral collocation method (single domain) is used whereas Table 6 presents the results obtained when using the multidomain bivariate spectral collocation method. The results indicate that the multidomain approach is very accurate and computationally faster when it is applied to solve nonlinear PDEs defined over largetime intervals.




2.0  4.0  6.0  8.0 
0.0010  1.61398e005  9.61285e005  7.30576e005  7.70011e006 
0.2321  2.99891e005  1.09535e004  9.97674e005  2.21837e006 
0.5000  2.15105e00  9.64207e005  1.07763e004  7.47171e006 
0.7679  1.07315e005  1.90612e005  6.33297e005  5.44788e006 
0.9843  3.75286e005  1.10078e004  2.79722e005  6.65481e007 
CPU time (sec)  0.026619 




2.0  4.0  6.0  8.0 
0.0010  5.21284e010  4.77699e010  3.01078e010  3.88833e010 
0.2321  2.05589e011  4.15302e011  4.59769e011  2.75941e011 
0.5000  2.27406e011  1.40028e011  1.33067e011  1.88674e011 
0.7679  5.37154e011  3.15495e011  1.85438e011  2.47591e012 
0.9843  2.02505e010  6.77300e011  2.10308e010  1.70352e010 
CPU time (sec)  0.028104 
The lesser computational time that is evident in the case when the multidomain approach is applied to solve the nonlinear PDE is attributed to the fact that the multidomain approach uses very few number of collocation points in each subinterval for the time variable than in the singledomain approach. This reduction in the number of collocation points significantly reduces the size of the resulting coefficient matrices. The small sized coefficient matrices are less dense and take less CPU time to produce results. The high accuracy and less computational time substantiate our claim that the multidomain bivariate spectral collocation method is a powerful numerical method for solving nonlinear parabolic PDEs that are defined over largetime intervals. The QLM is a powerful technique for simplifying nonlinear PDEs as very accurate results are obtained after 10 iterations only. The spectral collocationbased methods yield very accurate results with a few number of grid points as the approximate solution that is searched for is a higher degree polynomial. In the numerical experimentation, the symmetrically distributed GaussLobatto (GL) collocation points have been used instead of equispaced grid points as the GL nodes have a feature that tends to uniformly distribute the approximation errors across the entire interval of approximation [26]. The equispaced nodes, on the other hand, produce oscillations near the end of interval of approximation, a behavior referred to as Runge phenomena [27].
5. Conclusion
The multidomain bivariate spectral collocation method has been used successfully to solve nonlinear parabolic PDEs that arise in a wide range of applications like genetics, biology, heat and mass transfer and wave processes. The approximate results confirm that the multidomain bivariate spectral collocation method is very accurate and computationally faster when it is used to solve nonlinear parabolic PDEs that are defined over largetime domains. This approach is an alternative to other numerical methods that can be used to solve nonlinear parabolic partial differential equations. The multidomain bivariate spectral collocation method being more accurate and computationally faster can therefore be adopted and extended to solve similar problems that model reallife phenomenon.
Acknowledgments
This work is based on the research supported in part by the National Research Foundation of South Africa (Grant No. 85596).
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