MODIFIED RESIDUAL POWER SERIES METHOD FOR SOLVING SYSTEM OF DIFFERENTIAL ALGEBRAIC EQUATIONS

1Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, 2600 Irbid, Jordan 2Department of Mathematics, College of arts and Sciences Tabarjal, Jouf University, KSA 3School of Quantitative Sciences, Universiti Utara Malaysia (UUM), Kedah, Sintok, 06010 Malaysia 4Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 211-63, Jordan 5Department of Computer Science, Faculty of Science and Information Technology, Irbid National University, 2600 Irbid, Jordan


INTRODUCTION
Linear and nonlinear system of differential equations appears in various fields of applied science and engineering. Obtaining exact solutions of these systems are not easy to find. Different numerical or approximated methods have been applied such as homotopy analysis method, optimal homotopy analysis method, A domian decomposition method, Differential transformation method, cubic splin, and so on [1,2,3,4,5,6,7,8,9,10,11,12,13]. The existing of a powerful and perfect method with high performance results is much more difficulties in related to the size of computational work, especially when the system is strongly nonlinear. In this regard, the differential algebraic equation is kind of differential equations, but the unknown functions in these equations are satisfying additional algebraic equations, such tha the derivatives is not in general expressed explicitly and typically derivatives of some of the dependent variables may not arise in the equations at all. In the last decades, several studies about the differential algebraic equations have been appeared. In this point, it is commonly difficult to solve these types of equations analytically. Hence, there are many powerful numerical methods in literature that can be employed to find approximate solutions, for example, the backward differentiation formula is first numerical method employed to find the solutions of class of algebraic differential equations. The implicit Runge-Kutta method also has been employed to solve these type of equations numerically. Furthermore, the variational iteration method and the homotopy perturbation method have been applied successfully for solving different types of equations and their applications in engineering [14,15,16,17,18,19].
The fundamental motivation of this article is to utilized the RPSM for developing a technique to obtain the exact solutions of strongly linear and nonlinear system of differential algebraic equations. This technique is simple, in addition it can be applied directly to the given problems and does't require big effort to achieve accurate approximate solutions. The RPSM is an effective, easy and powerful technique that was employed in extensive scale in the last years for different types of differential equations without any restrictions such as lineariazation, descretization and perturbation [20,21,22,23,24,25,26,27]. However, the accuracy of the   [28,29] In this research article, we explain the solution procedure for system of linear and nonlinear algebraic differential equations which has the form of a power series expansion about the initial point t = t 0 for the given problem where f : [0, a] × R− > R are nonlinear continuous function, u(t) are unknown functions of independent variable t to be determined, and a > 0. To reach our goal, we assume the solution in the following form where u m (t) are terms of approximations, note that, when m = 0, we have u 0 (t) = u(t 0 ) = c 0 , which is the initial guess approximation, then we evaluate u m (t), ∀m = 1, 2, ... and approximate the solution u(t) of the given problem by k'th truncated series To apply the RPSM, we write the given problem (1) in the following form: Now, the kth residual function will be obtained by substituting the k th truncated series (3) into Eq. (4), as given below , and the following ∞ 'th residual function: Clearly, it is easy to see that Res ∞ (t) = 0 for each t ∈ (t 0 , T ), are infinitely differentiable func- .., k, this relation is considered a basic rule in the RPSM and its applications. Now, in order to obtain the first order-approximate solutions, we put k = 1, and substituting t = 0 into Eq. (5), and using the fact that Res (0)). Thus, using first-truncated series the first approximation for the given problem can be written as Similarly, the second-order approximation will be will be obtained by substituting k = 2 into Eq. (2) to be ∑ k=2 m=0 u m (t) and by differentiate both sides of Eq. (5) with respect to t which yields to d dt Res . Therefore, by consider the values of c 1 and c 2 into Eq.(3) when k = 2, the second-order approximate solution for the given problem becomes: The same process will be repeated to compute more components of the solution-order to obtain higher accuracy. The next theorem shows convergence of the RPS method.
2.2. Padé approximation. [29,30] The [L/M] Padé approximants of a function u(x) is given where P L (t) and Q M (t) are polynomials of degrees at most L and M, respectively. We know the formal power series The coefficients of the polynomials P L (t) and Q M (t) are obtained from the equation When the fraction of the numerator and denominator P L (t) Q M (t) is multiplying by a nonzero constant the fractional values remain unchanged, then we can define the normalization condition as (10) Q M (0) = 1.
Hence, we note that P L (t) and Q M (t) have no public factors. If we express the coefficient of then, by Eq.s (10) and (11), we may multiply (9) by Q M (x), which linearizes the coefficient equations. We can write out Eq. (9) in more detail as . a L+M + a L+M−1 q 1 + · · · + a L q M = 0 . a L + a L−1 q 1 + · · · + a 0 q L = p L To solve these equations, we start with Eq.
Now, we can obtain Padé approximants diagonal matrix of different order using software such as Mathematica, Matlab and son.

NUMERICAL RESULTS AND DISSECTIONS
This section is devoted to present some numerical examples to check the validity and performances of our procedure.
which is the exact solution.
The obtained results, show that the proposed technique give us high accurate solutions identical to the exact solution. This obtained throughout using a few number's of truncated series solution of the standard RPSM, this advantage overcomes the difficulties and efforts of evaluated more terms of the solutions order. Figs. 1 and 2, are displayed to represent the absolute errors of the standard RPSM solutions, from these figures we observed that high accuracy will be obtained by evaluating more and more terms of the approximation.

CONCLUSIONS
In this research paper, an accurate and efficient modified procedure is presented and employed to handle linear and nonlinear system of algebraic differential equations based on the RPSM. The obtained results confirmed that the presented procedure is precise, convenient and straightforward for the solution of such systems of algebraic equations.