Abstract
In this article, a new analytical method has been devised to solve higher-order initial value problems for ordinary differential equations. This method was implemented to construct a series solution for higher-order initial value problems in the form of a rapidly convergent series with easily computable components using symbolic computation software. The proposed method is based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial and reproduces the exact solution when the solution is polynomial. This technique is applied to a few test examples to illustrate the accuracy, efficiency, and applicability of the method. The results reveal that the method is very effective, straightforward, and simple.
Highlights
The study of nonlinear problems is of crucial importance in mathematics and in physics, engineering, economic, and other disciplines, since most phenomena in our world are essentially nonlinear and are described by nonlinear equations
We focus on finding approximate solution to higher-order initial value problems (IVPs), which are a combination of higher-order ordinary differential equations subject to given initial conditions
There are a number of differential equations which we studied in calculus to get closed form solutions
Summary
The study of nonlinear problems is of crucial importance in mathematics and in physics, engineering, economic, and other disciplines, since most phenomena in our world are essentially nonlinear and are described by nonlinear equations. In the fields of engineering and science, we come across physical and natural phenomena which, when represented by mathematical models, happen to be differential equations. There are a number of differential equations which we studied in calculus to get closed form solutions. All differential equations do not possess closed form of finite form solutions. Even if they possess closed form solutions, we do not know the method of getting it. In such situations, depending on the need of the hour, we go in for numerical solutions of differential equations. Especially after the advent of computer, the numerical solutions of differential equations have become easy for manipulations
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