Abstract
This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.
Highlights
The calculus of variations is a powerful branch of the analysis with many applications in both pure and practical mathematics (Lanczos 1986)
As a matter of fact, we proposed a combination between LT-homotopy perturbation method (HPM) and NDLT-HPM method, in order to obtain better results
This work introduced LT-HPM and some of its modifications in order to find analytical approximate solutions for some Euler’s ordinary differential equations, whose solutions extremizing the value of integrals of the form (1)
Summary
The calculus of variations is a powerful branch of the analysis with many applications in both pure and practical mathematics (Lanczos 1986). It has been found that even, the laws of physics can be expressed in a compact and elegant way through variational principles, as occurs with the Lagrange equations of mechanics, which can be deduced from the variational principle of Hamilton. Unlike elementary calculus problems, which seeks to find the points at which a function reaches its maximum and minimum values, the variational calculus considers the problem of some magnitude, whose values depend on a entire curve, throughout an integral (for instance: surface area or descent time). The aim is to find the curve that extremizes the aforementioned quantity in question. The procedure that we will follow consist in finding the differential equation for a function, that leads an integral to take an extreme value. As will be seen the procedure that we will follow consist in finding the differential equation for a function, that leads an integral to take an extreme value.
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