Solution of Second-Order IVP and BVP of Matrix Differential Models Using Matrix DTM

and Applied Analysis 3 Definition 2.1. If u t ∈ Rn×n is matrix analytical function in the domain T , then it will be differentiated continuously with respect to time t ∈ T , du t dtk φ t, k , 2.1 where k belongs to the set of nonnegative integer, denoted by the K domain. Therefore, for t ti, 2.1 can be written as Ui k φ ti, k [ du t dtk ]


Introduction
In this work, we study two important cases of nonlinear second-order matrix models given by the following.
1 Matrix initial-value problems are of the following form: 2 Matrix boundary-value problems are of the following form: u t f t, u t , u t , Definition 2.1. If u t ∈ R n×n is matrix analytical function in the domain T , then it will be differentiated continuously with respect to time t ∈ T , where k belongs to the set of nonnegative integer, denoted by the K domain. Therefore, for t t i , 2.1 can be written as where U i k ∈ R n×n is called the spectrum of u t at t t i , in the K domain.
Definition 2.2. If u t ∈ R n×n can be expressed by Taylor's series about fixed point t i , then u t can be represented as If u n t n k 0 u k t i /k! t − t i k is the n-partial sums of a Taylor's series 2.3 , then u t n k 0 where u n t is called the nth Taylor polynomial for u t about t i and R n t is remainder term. If U k is defined as and the n-partial sums of a Taylor's series 2.6 reduce to The U k defined in 2.5 is called the matrix differential transform of matrix function u t . For a special case, when t 0 0, then solution 2.6 reduces to From the aforementioned definitions, it can be found that the concept of the one-dimensional matrix differential transform is derived from the Taylor series expansion. With 2.5 and 2.6 , the fundamental mathematical operations performed by one-dimensional matrix differential transform can readily be obtained and listed in Table 1.

Convergence Analysis
In this section, we show that the presented matrix differential transformation method is convergence.
where · ∞ is infinity matrix norm.
Proof. By Definition 2.2, we get Abstract and Applied Analysis 5 We use the right-hand side as a "pattern" to define a function h : R n×n → R n×n . This time, we keep x fixed say, x a ∈ T and replace t 0 by a variable t. Thus we set Then h t 0 R n a and h a 0. Our assumptions imply that h is relatively continuous and finite on T and differentiable on T − Q. Differentiating 3.3 , we see that all cancels out except for one term: Then for t ∈ T we get −h t a t u n 1 t /n! a − t n dt, and a t 0 or R n a ∞ ≤ M |a − t 0 | n 1 / n 1 ! Thus 3.1 follows, because a is arbitrary value.
From Theorem 3.1, we get that if n → ∞, then R n x ∞ → 0 n×n . Then by this theorem, the convergence of this method is investigated.

Applications and Numerical Results
This section is devoted to computational results. We applied the method presented in this Example 4.1. In the first example, we consider the following non linear second-order matrix initial-value problem: Abstract and Applied Analysis e t e 2t . By applying matrix differential transform operator Table 1 on nonlinear system 4.1 , for k 0, 1, 2, . . . , n, we get where U k and C k are the differential transform of u t and c t , respectively. From 4.2 , and for k 0, 1, 2, . . ., we get we substitute the initial condition 4.2 , in recursive equation 4.3 , for k 0, 1, 2, we get U 2 −1 0 0 1/2 , U 3 0 0 0 1/6 , and U 4 0 0 0 1/24 , and then from the inverse differential transform operator 2.7 , we get The closed form of above solution is u t 2t−t 2 −1 0 e t , which is exactly the same as the exact solution.
Example 4.2. In this example, we consider the following nonlinear second-order matrix differential equations: where c t e t −e 2t −e t 2e t te t e 2t e t . The differential transform version of nonlinear system 4.5 , for k 0, 1, 2, . . . , n, is

4.6
Abstract and Applied Analysis 7 where U k and C k are the matrix differential transform of u t and c t , respectively. From 4.6 , and for k 0, 1, 2, . . ., we get following iteration equation: By utilizing the initial values U 0 and U 1 in recursive equations 4.7 , for k 0, 1, 2, the first four terms of U k are obtained as follows: U 2 1/2 −1/2 1 1/2 , U 3 1/6 −1/6 1/2 1/6 , and U 4 1/24 −1/2 1/6 1/24 , and then from 2.7 , we get The closed form of the previous solution is u t e t −e t te t e t , which is exactly the same as the exact solution. . By applying matrix differential transform operator listed in Table 1, on linear system 4.9 , for k 0, 1, 2, . . . , n, we obtain where U k , A k , B k , and C k are the matrix differential transform of u t , a t , b t , and c t , respectively. For k 0, 1, 2, . . ., 4.10 can be rewriten as in the following iterative equation: Abstract and Applied Analysis Then from 4.11 , for k 0, 1, 2, we get

4.12
In the same manner, the rest of components were obtained using the MAPLE Package. From boundary condition 4.10 we get To obtain the remain coefficient component of U k , for k ≥ 1, it is enough to find U 1 from differential transform version of boundary conditions 4.13 . Therefore, assume that Y 1 0 0 . Therefore from inverse differential transform operator 2.7 , the four-term approximation solutions is obtained as follows: which is exactly the same as the exact solution u t Example 4.4. Finally, we consider the following nonlinear second-order matrix boundaryvalue problem: Abstract and Applied Analysis 9 where c t 2 4t 2 −7t 3 3t 4 −2 6t−2t 2 5t 3 −3t 5 −2 t−3t 2 −t 3 3t 4 6t t 2 2t 4 −3t 5 . Then for k 0, 1, 2, . . . , n, we have

4.17
Therefore from 4.17 , some of the first quantities are obtained as follows:

4.18
In the same manner, the rest of components were obtained using the MAPLE Package.
In this case also from the common solution of 4.20 , we get U 1 The closed form of previous solution is u t which is exactly the same as the exact solution.

Conclusions
In this paper, we have shown that DTM can successfully be used for solving the linear and nonlinear second-order Matrix IVPs and Matrix BVPs. This method is simple and easy to use and solves the problem without any need for discretizing the variables. Also this method is useful for finding an accurate approximation of the exact solution. A symbolic calculation software package, Maple, is used in the derivations. The method gives rapidly converging series solutions. The accuracy of the obtained solution can be improved by taking more terms in the solution. In many cases, the series solutions obtained with DTM can be written in exact closed form. Further we note that DTM can also be applied to solve the nonlinear matrix differential Riccati equations first and second kinds of Riccati matrix differential equations where DTM technique provides a sequence of matrix functions which converges to the exact solution of the problem; see 19 . In fact the present work is the extension of 19 .