Robert A. Van Gorder, University of Central Florida


A perturbation scheme which preserves the structure of nonlinear differential equations

Abstract: In the late 1980's, Carl M. Bender and colleagues introduced a type of perterbation technique, the $\delta$ - expansion method, in which one expands in powers of a nonlinearity present in a nonlinear differential equation. At first applied to problems in quantum field theory [1], the method found additional application to nonlinear differential equations in many areas of science (see, for instance, [2] and the references therein). The beauty of the method was that it appeared to give a faster convergence to the solution of a nonlinear differential equation than, say, a standard ``small-parameter" perturbation expansion. Furthermore, the method did not even require the existence of any small parameters in the original model, making it far more versatile than some other perturbation techniques. The downside of the method is that finding each term in the $\delta$ - expansion usually required more computation than many other methods. In the present talk, we shall discuss the $\delta$ - expansion method from a slightly more general framework; in particular, we view the auxiliary parameter $\delta$ as a mapping from a nonlinear differential operator to a linear differential operator. Successive inversion of the linear operator then yields the higher order terms in this perturbation expansion. Indeed, standard perturbation methods may be viewed in a similar light; however, we show that the $\delta$ - expansion technique preserves more of the structure of of the nonlinear differential equation (at least in the cases where it is useful) . This is why the convergence to the solution often appears more rapid: the very first term in the $\delta$ expansion is a much more accurate representation of the true solution (and the higher order corrections are much more accurate) in those cases where the method has been applied successfully. Unfortunately, the greater accuracy of the initial approximations results in more challenging computations which must be performed in order to obtain the higher order terms; as such, the method is often best coupled with numerical methods, when there is a need for higher accuracy (i.e., a need for the higher order terms). After discussing the general idea of the method, we proceed to consider some applications to nonlinear differential equations fairly quickly, as the solutions will more or less fall into place once we understand the general method. First, we consider the Lane-Emden equation of the second kind, a second-order nonlinear ODE describing what is now commonly known as Bonnor-Ebert gas spheres: isothermal gas spheres embedded in a pressurized medium at the maximum possible mass allowing for hydrostatic equilibrium [3]. For our second application, we consider the flow of a laminar power-law non-Newtonian fluid past a semi-infinite flat plate (the x-y plane); the fluid is assumed to be incompressible. The laminar boundary-layer approximation, at constant temperature, yields a nonlinear ordinary boundary value problem (after an appropriate similarity transformation); in the case in which the power-law index is equal to one, the equation reduces to the famous Blasius equation [4]. Interestingly, we have been able to demonstrate certain behaviors of the solutions which have been predicted analytically, but never demonstrated computationally, as the previously applied purely numerical methods break down for large values of the power-law index. Time permitting, we may discuss an application to stochastic nonlinear differential equations [5] or the Painleve transcendents [6].

References:
[1] C. M. Bender, K. A. Milton, M. Moshe, S. S. Pinsky, and L. M. Simmons, Jr., Novel Perturbative Scheme in Quantum Field Theory, Physical Review D 37 (1988) 1472.

[2] C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, Jr., A New Perturbative Approach to Nonlinear Problems, Journal of Mathematical Physics 30 (1989) 1447.

[3] R. A. Van Gorder, An elegant perturbation solution for the Lane-Emden equation of the second kind, New Astronomy (2010), in press.

[4] R. A. Van Gorder, Two-dimensional Blasius viscous flow of a power-law fluid over a semi-infinite flat plane, under review (2010).

[5] R. A. Van Gorder, $\delta$ - expansion method for nonlinear stochastic differential equations describing wave propagation in a random medium, Physical Review E (2010), revisions requested.

[6] R. A. Van Gorder, Perturbation method for the Painleve transcendents, under review (2010).