A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

Abstract

Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k, where the order k is even unnecessary to be equal to the order n. In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve and 9600 Garsington Road, Oxford, OX4 2DQ, UK. 298S. Liao and Y. Tan nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study.

On the existence of steady-state resonant waves in experiments

“…Besides, the HAM provides great freedom to choose the equation type of related linear equations for high-order approximations. With these advantages, the HAM has been successfully applied to solve many nonlinear problems (Liao & Tan 2007;Liao 2012).…”

On the homotopy analysis method for backward/forward-backward stochastic differential equations

“…It should be emphasized that, it is the HAM that provides us great freedom to choose such a simple auxiliary linear operator, as mentioned by Liao [36,37]. For example, using such kind of freedom of the HAM, one can solve the 2nd-order Gelfand differential equation even by means of a 2d-th order linear differential operator in the frame of the HAM, where d = 1, 2, 3 is the dimensionality [49]. In addition, a nonlinear differential equation can be solved in the frame of the HAM even by means of directly defining an inverse mapping, i.e.…”

Optimal homotopy asymptotic method with application to thin film flow

“…(2) when H( ) = − , and the Homotopy Analysis Method (HAM) proposed by Liao in [17][18][19][20][21][22] is also another special case of Eq. (2) when H( ) = ¯ , where the parameter¯ is determined from so-called "¯ -curves".…”

“…Some modifications of this method were also reported [16]. Liao [17][18][19][20][21][22] employed the basic ideas of the homotopy in topology to propose a general analytical method for non-linear problems, namely homotopy analysis method (HAM). This method has been successfully applied to solve many types of nonlinear problems [23][24][25][26][27][28].…”