Efficient low-error analytical-numerical approximations for radial solutions of nonlinear Laplace equations

Abstract

We study radial solution of nonlinear elliptic partial differential equations of the form ??u=f(u) (a nonlinear Laplace equation) by means of an analytical-numerical method, namely optimal homotopy analysis. In this method, one obtain approximate analytical solutions which contain a free control parameter. This control parameter can be adjusted in order to improve the convergence or accuracy of the approximations. We outline the general technique for obtaining radial solutions of the general nonlinear elliptic partial differential equations of the form ??u=f(u), before focusing our attention on several specific equations, namely, the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind. For the general case, we outline the method by which one may control the residual errors of these analytical-numerical approximations. One benefit to this method is that one can obtain solutions with rather low residual errors after only a few terms in the analytical expansion are calculated. This makes the method rather efficient compared to a standard homotopy approach, where many terms may need to be computed to guarantee the accuracy of the solution. By studying the modified Liouville equation (with general positive nonlinearity), the Yamabe equation, and a generalized Lane-Emden equation of second kind, we demonstrate that the benefits of the method are often related to the form of the nonlinearity inherent in the problem. For certain forms of nonlinearity, the method gives very accurate solutions after relatively few terms, while for other forms of nonlinearity this is not the case.