Abstract
The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-order nonlinear ordinary differential equations with variable and proportional delays. The main advantage of using Legendre polynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime. The method is applied to five differential equations up to sixth order, and the results are compared with the exact solutions and other numerical solutions when available. The accuracy of the method is presented in terms of absolute errors. The numerical results demonstrate that the method is accurate, effectual and simple to apply.
Highlights
Many physical phenomena are modeled by using both the present and past states of the model
There are studies on the solutions of nonlinear ordinary differential equations solved by collocation methods which are based on Bessel polynomials [39,40,41]
In order to show that the method is capable of finding the exact solution, we start with a first-order nonlinear differential equation with variable delay
Summary
Many physical phenomena are modeled by using both the present and past states of the model. Differential equations with time delays are needed in the modeling of real life situations. Applications of these equations can be seen in many areas such as human body control and multibody control systems, electric circuits, dynamical behavior of a system in fluid mechanics, chemical engineering [18], spread of bacteriophage infection [35], population dynamics, epidemiology, physiology, immunology, neural networks, and cell kinetics [9]. Several numerical techniques are introduced to find approximate solutions of nonlinear differential equations with proportional and constant delays. There are studies on the solutions of nonlinear ordinary differential equations solved by collocation methods which are based on Bessel polynomials [39,40,41]
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