Laplace transform homotopy perturbation method for the approximation of variational problems

A symmetric version of the Euler equations by using Generalized Bernoulli Method

The aim of this article is to show a way to extend the usefulness of the Generalized Bernoulli Method (GBM) with the purpose to apply it for the case of variational problems with functionals that depend explicitly of all the variables. Moreover, after expressing the Euler equations in terms of this extension of GBM, we will see that the resulting equations acquire a symmetric form, which is not shared by the known Euler equations. We will see that this symmetry is useful because it allows us to recall these equations with ease. The presentation of three examples shows that by applying GBM, the Euler equations are obtained just as well as it does the known Euler formalism but with much less effort, which makes GBM ideal for practical applications. In fact, given a variational problem, GBM establishes the corresponding Euler equations by means of a systematic procedure, which is easy to recall, based in both elementary calculus and algebra without having to memorize the known formulas. Finally, in order to extend the practical applications of the proposed method, this work will employ GBM with the purpose to apply it for the case of solving isoperimetric problems.

The Novel Integral Homotopy Expansive Method

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

A practical proposal to obtain solutions of certain variational problems avoiding Euler formalism

The aim of this article is to show the way to get both, exact and analytical approximate solutions for certain variational problems with moving boundaries but without resorting to Euler formalism at all, for which we propose two methods: the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and the Moving Boundary Condition Employing Transversality Conditions (METC). It is worthwhile to mention that the first of them avoids the concept of transversality condition, which is basic for this kind of problems, from the point of view of the known Euler formalism. While it is true that the second method will utilize the above mentioned conditions, it will do through a systematic elementary procedure, easy to apply and recall; in addition, it will be seen that the Generalized Bernoulli Method (GBM) will turn out to be a fundamental tool in order to achieve these objectives.

The exact solutions and approximate analytic solutions of the (2 + 1)-dimensional KP equation based on symmetry method

In this paper, we successfully obtained the exact solutions and the approximate analytic solutions of the (2 + 1)-dimensional KP equation based on the Lie symmetry, the extended tanh method and the homotopy perturbation method. In first part, we obtained the symmetries of the (2 + 1)-dimensional KP equation based on the Wu-differential characteristic set algorithm and reduced it. In the second part, we constructed the abundant exact travelling wave solutions by using the extended tanh method. These solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. It should be noted that when the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Finally, we apply the homotopy perturbation method to obtain the approximate analytic solutions based on four kinds of initial conditions.