Abstract
In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.
Highlights
We assume that f(x) ∈ C∞(I); for any point Journal of Mathematics x0 ∈ I, the Taylor series of f(x) about x0 can be represented by f(x)
K 0 which implies that the concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symbolically
Let U (k) and V (k) be the differential transformations of the functions u (x) and v (x), respectively; we have for the constants α and β the following properties: (1) If f(x) αu(x) ± βv(x), F(k) αU(k) ±
Summary
In the first part of this section, we introduce a general algorithm depending on the differential transform method to solve systems of n linear differential equations with constant coefficients. e systems are assumed be autonomous, which means that the independent variable t is not present explicitly. E systems are assumed be autonomous, which means that the independent variable t is not present explicitly. Let X(k) be the kth differential transform of X (t) and Ak. be the kth power of the matrix A. en, the solution of the system given by (5) can be expressed as. Where Aj is the power the matrix A; the solution of the system is given by (12). Let X(k) and W (k) be the kth differential transforms of X (t) and W (t), respectively, and Ak be the kth power of the matrix A. en, the solution of the system given by (13) can be expressed as. En, the general solution of the system is given by k∞ 0k1!AkX(0)tk ((kj))!!Ak−1−jW(j)⎫⎪⎬⎪⎭tk
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