Hermite Matrix Polynomial Collocation Method for Linear Complex Differential Equations and Some Comparisons

In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given.


Introduction
Complex differential equations and their solutions play a major role in science and engineering.A physical event can be modeled by complex differential equations.Since a few of these equations cannot be solved explicitly, it is often necessary to resort to approximation and numerical techniques.In recent years, the studies on complex differential equations were developed very rapidly [1][2][3][4][5][6].
Since 1994, matrix polynomial collocation approaches such as Taylor and Bessel matrix collocation methods have been used by Sezer and colleagues [7][8][9][10][11] to solve the complex linear differential equations.
The present work contains two main parts, in the first part, we use Hermite matrix collocation method to find the approximate solution of higher-order linear complex differential equations of the following form.
which is a generalized case of the complex differential equations given in [5,6], with themixed conditions in the following rectangular domain or elliptic domain In the second part, we will study the effect of using different polynomial classes on the matrix polynomial methods.
The outline of this paper is as follows.In Section 2, we briefly introduce Hermite polynomial and describe details of using these polynomials in matrix polynomial collocation method.Section 3 focuses on the comparison of matrix collocation methods when different polynomials are used.We present the results of numerical experiments in Section 4. Finally, conclusions are drawn in Section 5.
proximate solution of (1) defined by a truncated Hermite series form where are the Hermite polynomials defined by if n is odd and .It is well known [12] that the relation between the powers By using the expression ( 6) and ( 7) and taking 0,1, , n N   we find the corresponding matrix relation , and where   , for odd N and for even N   Then, by taking into account (5), we obtain and we can replace series (6) in the matrix form Furthermore, the relation between the matrix   Z z and its derivative From the matrix Equation ( 10), we get the following relations: By using relations ( 9) and ( 11), we have a recurrence relation in what follows For the collocation points , the matrix relation (12) becomes For one can write the relation ( 13) in the following form where Moreover, substituting the collocation points into Equation (3), we have By means of the expressions ( 13) and ( 14), we acquire the fundamental matrix equation In which . With the aid of relation (12), we can obtain the corresponding matrix form due to the condition (4) as follows where Briefly, the system of the matrix Equation ( 17) can be written in the matrix form where We can write Equation (16) in the form where G q is defined in (16).The augmented matrix of Equation (19) becomes
Consequently, to find the unknown Hermite coefficients an, 0,1, , n N   related to approximate solution of the problem consisting of Equation ( 3 ; If det (W * ) ≠ 0 then we can write . The unknown Hermite coefficients matrix A, is determined by solving this linear system and 0 1 N a a a    n are substituted in Equation ( 3).Thus, we obtain the Hermite polynomial solution

Comparison of Matrix Polynomial Collocation Methods
Theorem3.1.Let 0 i be a base for vector space S, then every member s  S has a unique representation in the form of linear combination of these vectors.
Proof.[13].Based on the above theorem, if the bases of approximate space in collocation methods are chosen from complex polynomials up to degree N, using different bases or choosing of different complex polynomial classes as the base has no effect on the approximate solution, theoreticcally.This means that if and two different bases according to the uniqueness representation, then approximate solution of (1) can be written For this reason when we use different polynomials (such as Taylor, Bessel, Hermite, etc.) in polynomial Collocation methods one expects the equal results obtained.
In the numerical implementation, to determine coefficients an, in (5), we should solve a system of equations in the form of WF = G and properties of matrix W is directly depended on choosing the base.So different bases result different matrix W with different properties.Some of these properties such as condition number has the direct influence on solution's accuracy.In addition CPU time for solving these systems differs for different bases.Hence, different polynomial bases can cause solutions with different accuracy.
Our experiences show that when we use different polynomial classes in matrix polynomial collocation methods, there is negligible difference among approximated solutions.In Section 4, we compare this matter for several examples by using Taylor, Bessel and Hermite polynomials.

Numerical Examples
Several numerical examples are studied in this section to illustrate the accuracy and efficiently properties of Taylor, Bessel and Hermite collocation method.In this paper, collocation points in the rectangular domain (3) are defined by pq p z x iy q   , such that , , ; , 0,1, , and in the elliptic domain ( 4) are defined by Examples show that the difference among collocation methods based on these polynomials is negligible.All of them are performed on a computer using programs written in MATLAB 2011a.In this regard, we have reported in the Tables the value of absolute error function .
at the selected points of the domain.

Example 1
As the first example, [10], we consider the linear second order complex differential equation and    .Absolute errors are listed in able 1. T

Example 2
In this example, [10], we consider the third order linear complex differential equation        b  and    .Absolute errors of the obtain solutions are given in Table 2.

Example 3
The last example, [11] is the second order complex differential equation  3 and 4.

Conclusion
In this article, approximate solutions which can be obtained by different polynomial collocation methods have been compared.Our experiments show that using different polynomials cannot significantly affect the numerical solutions and the results are similar to each other.
) and condition (4), we replace the matrices (20) by the last m rows of the augmented matrix (19).Hence, we have a new aug- exact solution is   e z f z  on rectangular domain with a = −1, b = 1, c = −1, d = 1.Absolute errors are listed in Tables