Newton-homotopy analysis method for nonlinear equations

The homotopy analysis method and the Liénard equation

“…Liao [15,16] has developed this purely analytic technique to solve nonlinear problems in science and engineering. The HAM has been applied successfully to many nonlinear problems such as free oscillations of self-excited systems [17], the generalized Hirota-Satsuma coupled KdV equation [18], heat radiation [19], finding the root of nonlinear equations [20], finding solitary-wave solutions for the fifth-order KdV equation [21], finding solitary wave solutions for the Kuramoto-Sivashinsky equation [22], finding the solitary solutions for the Fitzhugh-Nagumo equation [23], boundary-layer flows over an impermeable stretched plate [24], unsteady boundarylayer flows over a stretching flat plate [25], exponentially decaying boundary layers [26], a nonlinear model of combined convective and radiative cooling of a spherical body [27], and many other problems (see [28,29,30,31,32,33,34,35,36], for example).…”

The Analysis Approach of Boundary Layer Equations of Power-Law Fluids of Second Grade

“…Using the above definitions, we construct the zeroorder deformation equations (17) subject to the boundary conditions…”

“…It is shown that HAM solutions agree well with the numerical solutions. The HAM has been applied successfully to many nonlinear problems in engineering and science, such the generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation [13], heat radiation [14], finding solitarywave solutions for the fifth-order KdV equation [15], finding the solutions of the generalized BenjaminBona-Mahony equation [16], finding the root of nonlinear equations [17], finding the solitary-wave solutions for the Fitzhugh-Nagumo equation [18], unsteady boundary-layer flows over a stretching flat plate [19], exponentially decaying boundary layers [20], a nonlinear model of combined convective and radiative cooling of a spherical body [21], and many other problems (see [22 -31], for example).…”

Travelling wave solutions: A new approach to the analysis of nonlinear physical phenomena

“…There are many papers that deal with HAM. Abbasbandy et al [13] applied the Newton-homotopy analysis method to solve nonlinear algebraic equations, Allan [14] constructed the analytical solutions to Lorenz system by the HAM, Bataineh et al [15,16] proposed a new reliable modification of the HAM, M. Ganjiani et al [17] constructed the analytical solutions to coupled nonlinear diffusion reaction equations by the HAM, Alomari et al [18] applied the HAM to study delay differential equations, Chen and Liu. [19] applied the HAM to increase the convergent region of the harmonic balance method.…”