Shijun Liao

On the homotopy analysis method for nonlinear problems

A powerful, easy-to-use analytic tool for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e. the algebraically decaying viscous boundary layer flow due to a moving sheet. Two rules, the rule of solution expression and the rule of coefficient ergodicity, are proposed, which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering. An explicit analytic solution is given for the first time, with recursive formulas for coefficients. This analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem.

Homotopy Analysis Method in Nonlinear Differential Equations

Basic Ideas.- Systematic Descriptions.- Advanced Approaches.- Convergent Series For Divergent Taylor Series.- Nonlinear Initial Value Problems.- Nonlinear Eigenvalue Problems.- Nonlinear Problems In Heat Transfer.- Nonlinear Problems With Free Or Moving Boundary.- Steady-State Similarity Boundary-Layer Flows.- Unsteady Similarity Boundary-Layer Flows.- Non-Similarity Boundary-Layer Flows.- Applications In Numerical Methods.

An approximate solution technique not depending on small parameters: A special example

One simple, typical non-linear equation is used in this paper to describe a kind of analytical technique for non-linear problems. This technique is based on both homotopy in topology and the Maclaurin series. In contrast to perturbation techniques, the proposed method does not require small or large parameters. The example shows that the proposed method can give much better approximations than those given by perturbation techniques. In addition the proposed method can be used to obtain formulae uniformly valid for both small and large parameters in non-linear problems.

On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet

A powerful, easy-to-use analytic technique for nonlinear problems, the homotopy analysis method, is employed to give analytic solutions of magnetohydrodynamic viscous flows of non-Newtonian fluids over a stretching sheet. For the so-called second-order and third-order power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for real power-law index and magnetic field parameter in a quite large range, an analytic approach is proposed. All of our analytic results agree well with numerical ones. In particular, a simple analytic formula of the dimensionless velocity gradient at the wall is found, which is accurate for all real power-law indices and magnetic field parameters. This analytic formula can give sufficiently accurate results for the skin friction on the moving sheet that it would find wide application in industries. Physically, they indicate that the magnetic field tends to increase the skin friction, and that this effect is more pronounced for shear-thinning than for shear-thickening fluids.

An explicit, totally analytic approximate solution for Blasius' viscous flow problems

Abstract By means of using an operator A to denote non-linear differential equations in general, we first give a systematic description of a new kind of analytic technique for non-linear problems, namely the homotopy analysis method (HAM). Secondly, we generally discuss the convergence of the related approximate solution sequences and show that, as long as the approximate solution sequence given by the HAM is convergent, it must converge to one solution of the non-linear problem under consideration. Besides, we illustrate that even though a non-linear problem has one and only one solution, the sole solution might have an infinite number of expressions. Finally, to show the validity of the HAM, we apply it to give an explicit, purely analytic solution of the 2D laminar viscous flow over a semi-infinite flat plate. This explicit analytic solution is valid in the whole region η=[0, +∞) and can give, the first time in history (to our knowledge), an analytic value f ″(0)=0.33206 , which agrees very well with Howarth’s numerical result. This verifies the validity and great potential of the proposed homotopy analysis method as a new kind of powerful analytic tool.

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

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 (2o)th-order linear differential equations, where 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 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.

Analytic solutions of the temperature distribution in Blasius viscous flow problems

We apply a new analytic technique, namely the homotopy analysis method, to give an analytic approximation of temperature distributions for a laminar viscous flow over a semi-infinite plate. An explicit analytic solution of the temperature distributions is obtained in general cases and recurrence formulae of the corresponding constant coefficients are given. In the cases of constant plate temperature distribution and constant plate heat flux, the first-order derivative of the temperature on the plate at the 30th order of approximation is given. The convergence regions of these two formulae are greatly enlarged by the Pade technique. They agree well with numerical results in a very large region of Prandtl number 1[les ] Pr [les ]50 and therefore can be applied without interpolations.

Homotopy analysis of nonlinear progressive waves in deep water

This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive a solution for the classical problem of nonlinear progressive waves in deep water. The method is based on a continuous variation from an initial trial to the exact solution. A Maclaurin series expansion provides a successive approximation of the solution through repeated application of a differential operator with the initial trial as the first term. This approach does not require the use of perturbation parameters and the solution series converges rapidly with the number of terms. In the framework of this approach, a new technique to apply the Padé expansion is implemented to further improve the convergence. As a result, the calculated phase speed at the 20th-order approximation of the solution agrees well with previous perturbation solutions of much higher orders and reproduces the well-known characteristics of being a non-monotonic function of wave steepness near the limiting condition.

Explicit analytic solution for similarity boundary layer equations

In this paper the homotopy analysis method for strongly non-linear problems is employed to give two kinds of explicit analytic solutions of similarity boundary-layer equations. The analytic solutions are explicitly expressed by recurrence formulas for constant coefficients and can give accurate results in the whole regions of physical parameters.

Application of Homotopy Analysis Method in Nonlinear Oscillations

In this paper, we apply a new analytical technique for nonlinear problems, namely the Homotopy Analysis Method Liao 1992a), to give two-period formulas fo oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. These two formulas are uniformly valid for any possible amplitudes of oscillation. Four examples are given to illustrate the validity of the two formulas. This paper also demonstrates the general validity and the great potential of the Homotopy Analysis Method.