A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum
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A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum Mohammad Asadi Dalir1
Received: 14 November 2018 / Revised: 29 November 2018 / Accepted: 31 January 2020 Ó The National Academy of Sciences, India 2020
Abstract In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed-form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are introduced. Regarding the fact that the governing equation for any arbitrary system represents its inherent properties, it is shown that the nonlinear term causes that the system to have a variable identity. Hence, the original function is included as a variable in the solution to can take into account the variation of governing equation. To find the exact closed-form solution, the variation of the nonlinear differential equation tends to zero, where in this case the system with a local linear differential equation has a definite identity with a definite local answer. It is shown that the general answer is an arbitrary curve on a surface, a newly developed concept known as superfunction, and different initial conditions give different curves as particular solutions. The comparison of the results with those of finite difference shows an exact agreement for any arbitrary amplitude and initial conditions. Keywords Nonlinear differential equation Exact closed-form solution Super-function Variable identity
The new solution is developed using a number of definitions as following: & Mohammad Asadi Dalir [email protected] 1
Mechanical Engineering Department, Bu-Ali Sina University, Hamedan 65175-4161, Iran
_ x; aj in A differential equation in the form of L x€; x; which x ¼ xðtÞ and aj ðj ¼ 1; 2Þ is the constant coefficient of differential equation, is known as constant differential equation with a constant identity in all times and amplitudes of x. The governing equation of a linear mass–spring system x€ þ x2 x ¼ 0 is an example of a constant differential equation. function with a definite identity with the form A definite x t; aj in which by knowing initial condition its value and behavior is determined for all points is a function with constant identity which is called constant function here, for abbreviation. A constant function can be the answer of a constant differential equation whose most important property in this definition is that the points in an infinitesimal neighborhood satisfy the slope definition for a function below: dx t; aj x t þ Dt; aj x t; aj ¼ lim : ð1Þ Dt!0 dt Dt The function xðt; xÞ ¼ sinðxtÞ is an example of a constant function. _ yð xÞ; aj , A differential equation in the form L x€; x; where yð xÞ is a known function of x, is called a variable differential equation in which the coefficients of differential equation can vary with x. As a result, the coefficients of differential equation are constant for a determined x, but vary by amplitude. Thus, the differential
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