Abstract
Theoretical underpinnings of differential equations have advanced greatly, particularly in the twentieth century. This expansion is due to the rapid and effective development of supporting mathematical fields (such as functional analysis, measure theory, and function spaces), as well as an ever-increasing need for applications, particularly in engineering, science, and medicine. The Lie infinitesimals method was employed to reduce the nonlinear fourth order PDE into an ordinary differential equation then the resulting ODE solved by the finite difference method. The Lie infinitesimals method is used to solve significantly more complex problems which used in manufacture.
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