This paper develops a modified variational iteration method coupled with the Legendre wavelets, which can be used for the efficient numerical solution of nonlinear partial differential equations (PDEs). The approximate solutions of PDEs are calculated in the form of a series whose components are computed by applying a recursive relation. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The main advantage of the new method is that it can avoid solving the nonlinear algebraic system and symbolic computation. Furthermore, the developed vector-matrix form makes it computationally efficient. The results show that the proposed method is very effective and easy to implement.
Nonlinear phenomena are of fundamental importance in applied mathematics and physics and thus have attracted much attention. It is well known that most engineering problems are nonlinear, and it is very difficult to achieve the solution analytically or numerically. The analytical methods commonly used to solve them are very restricted, while the numerical techniques involving discretization of the variables on the other hand give rise to rounding off errors. Considerable attention has been paid to developing an efficient and fast convergent method. Recently, several approximate methods are introduced to find the numerical solutions of nonlinear PDEs, such as Adomian’s decomposition method (ADM) [
The variational iteration method (VIM) proposed by He [
In recent years, wavelets have found their way into many different fields of science and engineering. Various wavelets [
Recently, some efficient modifications of ADM (using [
This section introduces the basic ideas of variational iteration method (VIM). Here a description of the method [
Legendre wavelets
A function
If the infinite series in (
A two-dimensional function
The integration and derivative operation matrices of the Legendre wavelets have been derived in [
The integration of the vector
The derivative of the vector
The integration of
Similarly, the integration of
The derivative of
Similarly, the derivative of
The block pulse functions (BPFs) form a complete set of orthogonal functions that are defined on the interval Disjointness: the BPFs are disjoined with each other in the interval
for Orthogonality: the BPFs are orthogonal with each other in the interval
for Completeness: the BPFs set is complete when
where
Let
Assuming that
According to the disjointness property of BPFS in (
Let
The Legendre wavelets can be expanded into
The operational matrix of product of Legendre wavelets can be obtained by using the properties of BPFs. Let
From (
By employing Lemma
In this section, we present a new modification of variational iteration method using Legendre wavelets (called VIMLW). This algorithm can be implemented for solving nonlinear PDEs effectively.
To deduce the basic relations of our proposed algorithm, consider the following forms of initial value problems:
According to the traditional VIM, we can construct the correction functional for (
In order to improve the performance of VIM, we introduce Legendre wavelets to approximate
Now for the nonlinear part, by nonlinear term approximation described in Section
For the linear part, we have
Then the iteration formula (
If
When
Substituting (
Finally, we get the iteration formula as follows:
To demonstrate the effectiveness and good accuracy of the VIMLW, four different examples will be examined.
Consider the regularized long-wave (RLW) equation [
By assuming
We utilize the methods presented in this paper to solve (
Numerical values when
|
0.25 | 0.50 | ||||||
---|---|---|---|---|---|---|---|---|
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
VIM | 0.20002 | 0.40004 | 0.60006 | 0.80007 | 0.16702 | 0.33405 | 0.50107 | 0.66810 |
VIMLW | 0.20002 | 0.40004 | 0.60006 | 0.80006 | 0.16702 | 0.33405 | 0.50107 | 0.66809 |
Exact | 0.20000 | 0.40000 | 0.60000 | 0.80000 | 0.16667 | 0.33333 | 0.50000 | 0.66667 |
| ||||||||
|
0.75 | 1.00 | ||||||
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
| ||||||||
VIM | 0.14461 | 0.28921 | 0.43382 | 0.57842 | 0.12989 | 0.25977 | 0.38966 | 0.51955 |
VIMLW | 0.14460 | 0.28921 | 0.43382 | 0.57841 | 0.12989 | 0.25977 | 0.38966 | 0.51953 |
Exact | 0.14286 | 0.28571 | 0.42857 | 0.57143 | 0.12500 | 0.25000 | 0.37500 | 0.50000 |
Exact solution and VIMLW approximate solution of Example
Consider the following equation [
By assuming
We employ the methods presented in this paper to solve (
Numerical values when
|
0.25 | 0.50 | ||||||
---|---|---|---|---|---|---|---|---|
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
VIM | 0.05001 | 0.20002 | 0.45004 | 0.80007 | 0.04176 | 0.16702 | 0.37581 | 0.66810 |
VIMLW | 0.05000 | 0.20002 | 0.45004 | 0.80021 | 0.04168 | 0.16699 | 0.37572 | 0.66831 |
Exact | 0.05000 | 0.20000 | 0.45000 | 0.80000 | 0.04167 | 0.16667 | 0.37500 | 0.66667 |
| ||||||||
|
0.75 | 1.00 | ||||||
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
| ||||||||
VIM | 0.03615 | 0.14461 | 0.32536 | 0.57842 | 0.03247 | 0.12989 | 0.29225 | 0.51955 |
VIMLW | 0.03577 | 0.14443 | 0.32491 | 0.57751 | 0.03128 | 0.12932 | 0.29083 | 0.51421 |
Exact | 0.03571 | 0.14286 | 0.32143 | 0.57143 | 0.03125 | 0.12500 | 0.28125 | 0.50000 |
Exact solution and VIMLW approximate solution of Example
We consider the following equation [
By assuming
Table
Numerical values when
|
0.25 | 0.50 | ||||||
---|---|---|---|---|---|---|---|---|
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
VIM | 0.06250 | 0.12500 | 0.18750 | 0.25000 | 0.12508 | 0.25015 | 0.37523 | 0.50030 |
VIMLW | 0.06250 | 0.12500 | 0.18750 | 0.25000 | 0.12508 | 0.25015 | 0.37523 | 0.50030 |
Exact | 0.06250 | 0.12500 | 0.18750 | 0.25000 | 0.12500 | 0.25000 | 0.37500 | 0.50000 |
| ||||||||
|
0.75 | 1.00 | ||||||
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
| ||||||||
VIM | 0.18882 | 0.37764 | 0.56646 | 0.75527 | 0.25992 | 0.51983 | 0.77975 | 1.03970 |
VIMLW | 0.18882 | 0.37764 | 0.56646 | 0.75527 | 0.25992 | 0.51983 | 0.77975 | 1.03970 |
Exact | 0.18750 | 0.37500 | 0.56250 | 0.75000 | 0.25000 | 0.50000 | 0.75000 | 1.00000 |
Exact solution and VIMLW approximate solution of Example
Consider the following Burgers-Poisson (BP) equation of the form [
By assuming
And
Table
Numerical values when
|
0.25 | 0.50 | ||||||
---|---|---|---|---|---|---|---|---|
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
VIM | 0.00009 | 0.20011 | 0.40013 | 0.60015 | −0.16488 | 0.00215 | 0.16917 | 0.33620 |
VIMLW | 0.00009 | 0.20011 | 0.40013 | 0.59864 | −0.16489 | 0.00215 | 0.16920 | 0.33283 |
Exact | 0.00000 | 0.20000 | 0.40000 | 0.60000 | −0.16667 | 0.00000 | 0.16667 | 0.33333 |
| ||||||||
|
0.75 | 1.00 | ||||||
|
0.25 | 0.50 | 0.75 | 1.00 | 0.25 | 0.50 | 0.75 | 1.00 |
| ||||||||
VIM | −0.27697 | −0.13237 | 0.01224 | 0.15694 | −0.35057 | −0.22068 | −0.09079 | 0.03909 |
VIMLW | −0.27701 | −0.13237 | 0.01228 | 0.15213 | −0.35070 | −0.22068 | −0.09078 | 0.03422 |
Exact | −0.28571 | −0.14286 | 0.00000 | 0.14286 | −0.37500 | −0.25000 | −0.12500 | 0.00000 |
Exact solution and VIMLW approximate solution of Example
A new modification of variational iteration method using Legendre wavelets is proposed and employed to solve a number of nonlinear partial differential equations. The proposed method can give approximations of higher accuracy and closed form solutions if existed. There are four important points to make here. First, unlike the VIM, the VIMLW can easily overcome the difficulty arising in the evaluation integration and the derivative of nonlinear terms and does not need symbolic computation. Second, by using the properties of BPFs, operational matrices of product of Legendre wavelets are derived and utilized to deal with nonlinear terms. Third, compared with Legendre wavelets method, the VIMLW only needs a few iterations instead of solving a system of nonlinear algebraic equations. Fourth and most important, VIMLW is computer oriented and can use existing fast algorithms to reduce the computation cost.
This work is supported by the National Natural Science Foundation of China (Grant no. 41105063). The authors are very grateful to the reviewers for carefully reading the paper and for thier comments and suggestions which have improved the paper.