Abstract
In this study, by means of the matrix relations between the Laguerre polynomials, and their derivatives, a novel matrix method based on collocation points is modified and developed for solving a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms, via mixed conditions. The method reduces the solution of the nonlinear equation to the solution of a matrix equation corresponding to system of nonlinear algebraic equations with the unknown Laguerre coefficients. Also, some illustrative examples along with an error analysis based on residual function are included to demonstrate the validity and applicability of the proposed method.
Highlights
Nonlinear differential equations and the related initial and boundary value problems play an important role in astrophysics, physics and engineering
These type of mathematical models can be described by particular names such as Riccati equation, nonlinear equations of motion, Duffing’s equation, Van Der Pol’s equation, the equation of motion with quadratic damping, Emden’s equation, Liouville’s equation [1,2,3,4,5]
We introduce a matrix method depending on Laguerre polynomials in order to solve a class of second-order nonlinear ordinary differential equations having quadratic and cubic terms numerically
Summary
Nonlinear differential equations and the related initial and boundary value problems play an important role in astrophysics, physics and engineering. It may not be possible to find the analytical solutions of such problems for all coefficient functions. These type of mathematical models can be described by particular names such as Riccati equation, nonlinear equations of motion, Duffing’s equation, Van Der Pol’s equation, the equation of motion with quadratic damping, Emden’s equation, Liouville’s equation [1,2,3,4,5]. We consider the second-order nonlinear ordinary differential equations with quadratic and cubic terms: 2p. Modified operational matrix method for second-order nonlinear ordinary differential equations .
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