Abstract
In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.
Highlights
Since the emergence of humanity, people have taken a great interest in understanding natural phenomena, beginning from fire, lightning, thunder, earthquakes, and volcanoes, ranging all to way to the nano-particle
Partial differential equations have been applied to the study of many phenomena in different fields, such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, solid state physics, fluid mechanics, hydrodynamics, optics, plasma physics, chemical kinetics, biological phenomena, and more
It is known that there exist various analytic solution methods for NPDEs [7–21]
Summary
Since the emergence of humanity, people have taken a great interest in understanding natural phenomena, beginning from fire, lightning, thunder, earthquakes, and volcanoes, ranging all to way to the nano-particle. The most important results in determining explicit solutions of nonlinear partial differential equations (NLPDEs) were derived in [1]. These types of methods are best known as continuous symmetry transformation groups [2–6]. Many researchers tried in conformity with finding out extra properties expecting that form of derivatives They discovered partially about methods in accordance with changing the nonlinear fractional half differential equations among everyday differential equation together with integer order. Fractional modified equal width equation: This model refers to the replica of one-dimensional wave propagation in nonlinear form with dispersion processes, and has the following formula: Dtθu + h u2 Dxθu − r Dx3xθtu, 0 < θ < 1,.
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