Pseudo-Spectral Collocation Solution of
Nonlinear Time Dependent System of Partial Differential Equations
Chandra Shekara G., T. N. Vishalakshi
Department of Mathematics, B .M.S. College of
Engineering, Bangalore, Karnataka, India
*Corresponding Author
E-mail: chandrashekarag.maths@bmsce.ac.in
ABSTRACT:
The pseudo-spectral collocation solution of
time dependent nonlinear system of partial differential equations with
Dirichlet’s boundary conditions is presented in this paper. The nonlinear systems
of partial differential equations are reduced to linear form by using
quasi-linearization along with the Taylor’s series of multi-variables. Then
the spectral collocation algorithm is developed by using Lagrange interpolating
polynomials as basis of the solution at the Chebyshev-Gauss-Labatto grid
points. This algorithm is implemented using MATLAB for test case problems and
the results are presented graphically. Error analysis is done by comparing the
numerical solution and analytical solution. Solution found by the said method
is more accurate compared to the finite difference method with uniform grid
points.
KEYWORDS: Pseudo-Spectral, Chebyshev-Collocation,
Lagrange’s-Interpolation, Quasi-linearization, Gauss-Labatto points,
Taylor’s-series.
INTRODUCTION:
We consider the system of time dependent nonlinear
partial differential equations in the form
(1)
(2)
where 
and 
are
nonlinear operators subject to the initial conditions
and 
(3)
With the boundary conditions
and 
at 
(4)
and 
at 
(5)
In order to achieve the solution of nonlinear system
of time dependent partial differential equations (1)-(2), several researchers
have applied different numerical and analytical techniques.
Like, method of lines was used by Darvishi and Javidi4
for solving system of time dependent partial differential equations. Abdou and
Soliman1 have used variational iteration method for solving Burger’s
and coupled Burger’s equations. Nowak10 applied adaptive
MOL-treatment of parabolic 1-D problems along with extrapolation. Javidi5,6
had used spectral collocation method along with Runge-Kutta method for time
stepping to solve the nonlinear system of time dependent partial differential
equation. Chandra Shekara11 have used spectral quasi-linearization
to solve the time dependent nonlinear PDE with nonlinear time derivative
boundary conditions.
In this paper, system of nonlinear time dependent
partial differential equations was solved numerically by using pseudo-spectral
collocation method along with Gauss-Labatto grid points in both time and
special direction. To demonstrate this algorithm it is considered the following
nonlinear time dependent system of partial differential equations as case study
problems posed by Javidi6 and Abdou and Soliman1. The
problem is defined as follows
(6)
(7)
Subject to initial and boundary conditions
(8)
, (9)
(10)
The exact solution of equations [6] and [7]
with the initial condition [8] and boundary conditions [9] and [10] is given by
. (11)
Quasi-Linearization:
To reduce the nonlinear system of partial differential
equation in the above section to linear it is made use of Quasi-linearization
technique along with Taylor’s series of multi-variables. The detailed
description of quasi-linearization is presented in this section
Equations (1) and (2) can be written in the general
form as sum of linear and nonlinear terms
(12)
(13)
Where 
and 
are
linear operators and 
and
are
non-linear operators.
Suppose 
and 
are
approximate solutions of equation (12) and (13) at 
iteration
with the condition that 
and
.
Expanding nonlinear terms of equation (12) and (13) as series about and by
using Taylor’s series of multi-variables, after simplification equation (12)
and (13) becomes:
(14)
(15)
where
, 
, 
, 
,
, 
and
.
and
.
Pseudo-spectral
collocation method:
In recent years, a pseudo-spectral method
for solving the differential equations is gaining attention over the
researchers. Pseudo–spectral method is one of most accurate and fast converging
method. This method is used to solve the partial differential equation (See [2-8]).
Pseudo-spectral methods have become increasingly popular for solving
differential equations and also very useful in providing highly accurate
solution to differential equations.
First the special domain and the time
domain are discretized by using Chebyshev-Gauss-Labatto collocation points
given by
and
(16)
This function gives 
grid
points in the domain 
and
the linear transformation given below is used to transform these grids into the
domain 
by
.
Further the solution of the differential
equation can be approximated by using the sum of Lagrange’s interpolation
polynomials as
(17)
and
(18)
Where 
and
are
Lagrange’s cardinal polynomials given by 
and
.
Following S.S. Motsa9 the derivatives at the collocation points are derived
and Cheb() is used to implement in MATLAB function defined as Trefethen8. The numerical experimentation of this
method is done in next section.
Numerical
Experiments
In this section the case study problem given in first
section are considered for the numerical experimentation of the method
described in previous sections. First the system of non-linear PDEs (6) and (7)
are reduced to linear form by using Quasi-linearization discussed in second
section as follows
(19)
(20)
where
,
,
,
,
,
,
,
.
and
Next, substitute the assumed solutions (18) and (19)
in to equations (19) to (20) and using Chebyshev differential matrices (S.S.
Motsa9 ) for the spatial derivatives are given by
and
The second derivatives are
and
.
Similarly, the time derivative is given by
and
Making use of these derivatives in
equations (19) and (20) becomes
These equations can be written in matrix form as
system of algebraic equations 
given
by
(21)
where
The boundary conditions becomes
This algorithm is implemented in MATLAB using the
initial conditions
as initial guess and 
is
obtained by using
.
The numerical results in terms
of residual error are presented in the Table-1 for different number of
collocation points.
Also the results are compared with exact solutions are depicted as graphs in
figure (1) and (2) at 
.
Table -1: Absolute Error in the solution for
different values of 
and 
at
|
Nz
|
30
|
30
|
40
|
40
|
50
|
|
Nt
|
10
|
20
|
10
|
20
|
10
|
|
|
4×10-11
|
3×10-12
|
2.1×10-13
|
1.1×10-13
|
0.8×10-13
|
Figure 1: Exact
solution versus Numerical solution of
Figure 2: Exact
Solution versus Numerical Solution of
Figure 3:
Residual Error in the solution at.
CONCLUSIONS:
In this paper it
is has been proposed an efficient and accurate pseudo-spectral collocation
method for solving system of time dependent partial differential equations. It
is seen that the method is fast converging with a negligibly small error with
increasing the number of collocation points as given in Table-1. Also this
method is far better compared to other method as given by several authors.
ACKNOWLEDGEMENT:
The authors Dr.
Chandra Shekara G. and T. N. Vishalakshi are thankful to the principal and
TEQIP-II, B.M.S. College of Engineering, Bangalore, India, for their support
and encouragement to carry out this research work.
CONFLICT OF INTEREST:
The authors
declare no conflict of interest.
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